Philosophy A/ Int. Level: Logic Notes and Readings · PDF filePhilosophy A/ Int. Level: Logic Notes and Readings Colette Sciberras, PhD (Dunelm) 2016 Please do not distribute without

Philosophy A/ Int. Level: Logic Notes and Readings

Colette Sciberras, PhD (Dunelm)

2016

Please do not distribute without acknowledging the author!

Colette Sciberras, PhD (Dunelm) Logic, 2016 1

PART 1: Readings from Copi: Introduction to Logic.

Deduction and Induction

Truth and Validity

Fallacies

PART 2: Propositional Logic

1: Introduction to Logic ................................................................................................................ 2

2: Elementary Propositions .......................................................................................................... 5

3: The Use of Elementary Assertions ............................................................................................ 8

4: Complex Propositions I – Negations – and the Negator (NOT) .................................................. 10

5: Working out truth-tables I (Negator) ....................................................................................... 13

6: Complex Propositions II – Conjunctions – and the Conjunctor (AND) ........................................ 15

7: Translations ........................................................................................................................... 18

8: Working out Truth- Tables II (Negators and Conjunctors) ........................................................ 21

9: The Adjunctor (OR) ................................................................................................................. 26

10: The Disjunctor ...................................................................................................................... 29

11: The Subjunctor and Bi-Subjunctor ......................................................................................... 31

12: Arguments and Implications ................................................................................................. 36

PART 3: Formal Logic

13: Concepts from Formal Logic .................................................................................................. 37

14: Logical Truth ........................................................................................................................ 40

15: Implications ......................................................................................................................... 41

16: Standard Implications (Modus Ponens and Modus Tollens) ................................................... 44

17: Equivalence .......................................................................................................................... 48

18: Properties of Implications and Equivalences: (Reflexivity, Symmetry and Transitivity) ............ 49

19: Generalisation, Instantiation and Partial Replacement Rules ................................................. 55

From the old syllabus

20: Deriving Equivalent Formulae: De Morgan’s and Double Negation Equivalence...................... 56

21: Deriving Valid Implications I: The Duality Principle ................................................................ 59

22: Deriving Valid Implications II: Contraposition ........................................................................ 61

23: Deriving Valid Implications III: Transportation Rule ............................................................... 64

24: A Complete System of Junctors ............................................................................................. 66


1: Introduction to Logic

What is logic? Why do we need to study it in phi losophy?

Logic is a tool for checking whether our reasoning and arguments are valid, and for this reason it is

essential for philosophy. Philosophers usually want to show that their view or theory (whether it’s

about physical reality, religion, ethics, language and so on) is the correct one - and they do this by

providing arguments, which they and other philosophers can check, using logic. Having a good grasp

of logic will help us to determine whether our reasoning, and that of others, is correct or not.

Hurley (2008)1 puts it this way; “Logic may be defined as the science that evaluates arguments…The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own…” (p. 1). However, before we can start to check whether arguments are valid or not, we need to build up our knowledge of a few more basic features of logic and of language in general. Content vs. Form An important distinction, which we should try to understand from the start, is that between content and form. Let’s take a few examples: (1) Philosophy is an easy subject. (2) Socrates was a Greek philosopher. (3) I am really bored right now. (4) Giovanni Curmi Higher Secondary is the best school in Malta! If you study these sentences, you will see that they each share a similar form; in each case we are

saying that something IS (or was) something else. Let us use a variable to represent these

‘somethings’ – for instance, in sentence (1), let A stand for “Philosophy” and let B stand for “an easy

subject.” We can then represent sentence (1) in the following way:

A is B

In this way, we have brought out the form of sentence (1) without saying anything about the content; that is, I have shown the structure of my sentence, without specifying what it is about. The content of a sentence, therefore, refers to the ideas, or concepts, the people, things and so on

that we are talking about. In our examples, these are philosophy, easy subjects, Socrates, Greek

philosophers, me, being bored, our school, etc.

The form of a sentence is its structure, in other words, it shows only how those concepts are related

to each other, without specifying what the concepts are. We can denote the form of a sentence

using symbols and variables, e.g. A is B.


The logic we study here is sometimes called formal logic, and this is because it is concerned ONLY

with the form of arguments, and not with their content. That is, we are not interested in whether

the argument is about Socrates, schools, or whatever; the content is irrelevant.

Defining Variables In fact, sentences 1 – 4 can all be represented by ‘A is b’ – providing we define A and B each time to

stand for the particular concepts we are talking about. For instance,

Let A stand for ‘Socrates’ Let b stand for ‘a Greek Philosopher’ Sentence (2) can be written as +Ab Let A stand for ‘I’ Let b stand for ‘really bored right now’ Sentence (3) can be written as +Ab

Importantly, for each exercise in logic, when we define a variable, for instance, by writing down, “Let

A stand for ‘Socrates,’” that variable must always be used in the same way. That is to say, A must

always represent ‘Socrates’ and nothing else, and conversely, whenever the word ‘Socrates’ appears

in that exercise, we must always use the variable A to represent it. We can use the same variable to

represent another word or phrase only if we have started a separate exercise.

It is important to understand that a variable can stand for a single word, a phrase, or even an entire

sentence too (later, we will refer to these sentences as propositions). For example, I can represent

the whole sentence “Socrates was a Greek philosopher” by a single variable, say, a.

The first step in logic, therefore, is to decide what variables we are going to use, and what they are

going to stand for. We call this ‘defining our variables.’

Using symbols for logical particles Our objective in logic, what we are trying to do, is to build a new sort of language, by removing the

content from arguments and representing only their form. There are some words which are

represented by variables, as we saw above, and the others, which are known as logical particles are

represented by symbols. An useful analogy here is mathematics. Suppose I have two apples in my

bag and another two in my fridge. I can represent this as:

2 + 2 = 4 Of course, this can also represent two people in class and two outside, or two days at the weekend

and two mid-term holidays. Therefore, ‘2’ and ‘4’ can be compared to our variables, in that they can

represent anything at all.

The ‘+’ and ‘=’ signs, on the other hand, represent a sort of function, for instance ‘+’ means that we

add up whatever is to the left and to the right of the ‘+’ sign. These signs are similar to our logical

particles, and in fact logic too has such symbols, which represent certain functions. In sentences 1 –

4, the word IS relates the concept A to the concept B (by identifying them with each other.) The


word ‘is,’ therefore, is a logical particle (known as a copula), and next week, we shall see that in

logic, ‘is’ can be represented by the ‘+’ sign.

Logic is a system of rules So, in logic, we first define variables to stand for certain words or phrases, and then relate these

variables to each other by using symbols.

Just as in maths, we need to know how these symbols work. If we say that “2 + 2 = 0” we have

clearly not understood what ‘+’ means, that is, we haven’t understood the rules of addition.

Similarly, in logic we need to know the rules for our logical particles. Just as in Mathematics, 2+2 is

always equal to 4, whether we are talking about apples, people, holidays or whatever, in logic, if we

apply our rules correctly, we will always obtain the correct result. This is why Riolo refers to logic as

an “ortholanguage” (which, roughly means, a good or correct language) (Riolo 2001, 4).

Exercises: 1. Identify the form and logical particles in these sentences.

Logic is not difficult. The day before yesterday was not a very good day for me. He is not the person I thought he was.

Make sure you have understood the following:

- The content versus the form of a sentence

- How to define variables

- What a logical particle is


2: Elementary Propositions

Sentences, assert ions, and proposit ions We all know what a sentence is; in written language, it is a group of words (the first of which has a

capital letter) which has a full-stop, a question mark or an exclamation mark at the end. Here are

some sample sentences:

(1) Socrates was Greek. (assertion) (2) Do you study philosophy? (question) (3) Open the door please. (request/ command) Logic is only concerned with sentences of type (1) that is, with assertions. How can we tell whether a

sentence is an assertion or not? It’s very easy – ask yourself ‘can this sentence be true or false?’ If

the answer is yes, then the sentence is an assertion.

Only sentence (1) is an assertion here – the others cannot be true or false. In fact, sentence (2) is a

question, whereas (3) is an imperative (a request or a command). To repeat then, in logic, we are

only interested in those sentences which can be either true or false, and we call these assertions.

Another commonly used term in logic is ‘proposition.’ Propositions and assertions are more or less

the same, and many authors use the words interchangeably. A proposition can be thought of as the

meaning of an assertion. For example, the two assertions “Socrates was Greek” and “Sokrate kien

Grieg” both express the same proposition.

Riolo says that a proposition is the “taking of a position” or the “commitment” behind an assertion

(2001, p7). In other words, we usually make an assertion because we believe that the proposition it

expresses is true. However, at this stage you needn’t worry too much about the difference between

propositions and assertions – the main thing to understand at this point is that propositions and

assertions are those types of sentences which must be either true or false.

Represent ing proposit ions symbol ically As we have seen, we can represent the form of sentence symbolically, by using variables to

represent the content, and symbols to represent the logical particles. It is important to remember

that there are several ways of representing the same proposition.

For example, the proposition expressed as (1) above, might be represented as follows:

(i) Let A stand for “Socrates” Let B stand for “Greek” +Ab

Alternatively, we could use a single variable to represent the entire proposition:

(ii) Let a stand for “Socrates was Greek” a


Elementary and complex proposit ions The first thing we must understand is how to distinguish elementary propositions from complex

ones. An elementary proposition is one which contains a subject, an object (sometimes called a

complement) and some form of the verb ‘to be’ (i.e. past, present or future, singular or plural, first,

second or third person). An elementary proposition must contain nothing else. Most of our

examples, so far, have been elementary propositions. Consider the following:

(1) Philosophy is an easy subject. (2) Socrates was a Greek philosopher. (3) I am really bored right now. (4) Giovanni Curmi Higher Secondary is the best school in Malta. All of these contain a subject, a predicate, the verb ‘to be,’ and nothing else. Even though sentence

(4) seems a bit longer and more complex than the others, careful examination will show that it is

actually an elementary proposition. The subject is “Giovanni Curmi Higher Secondary,” the predicate

is “the best school in Malta” and of course there is the verb ‘to be’ (“is”).

In logic, we call the subject a nominator. The nominator is whatever it is we are talking about, or

referring to. We can usually point to something and say “This” or we can use its name such as

“Philosophy” as in example (1).

The predicator is what we want to say about our nominator, in this case, that it is “an easy subject”.

The verb ‘to be’ is called the copula (plural copulae). We can have either a positive copula (“is,”

“are,” or “was”) or a negative one (“isn’t”, “aren’t”, “weren’t”). With our copula we either affirm or

negate the predicator of the nominator.

Compare elementary propositions with the following:

(1) He is either studying or else he is on the phone. (2) Socrates was not Indian. (3) I am really bored and I am also tired now. (4) If GCHSS is the best school in Malta, then I want to go there!

Some of these sentences contain more than one copula and more than one predicator, i.e. they

contain more than one elementary proposition. They also contain other logical particles – the word

‘or’ in (1), ‘not’ in (2), ‘and’ in (3), and the words ‘if…then’ in (4). These are logical particles called

junctors which we shall be meeting soon. Any proposition which contains a junctor is a complex

proposition.


Exercises: 1. Which of these sentences are assertions?

1. I like logic. 2. Did you learn anything new today? 3. She didn’t study and that’s why she failed. 4. Go home and do some work now! 5. Hamlet asked the question “to be or not to be?”

2. Are these assertions complex or

elementary?

1. That girl is my cousin. 2. The girl I saw yesterday is my cousin’s Polish babysitter. 3. The girl I saw yesterday was wearing a blue top and she had short, spiky hair.


- Assertions/ propositions are sentences that must be true or false

- Elementary propositions vs. complex ones

- Nominators, predicators and copulae


3: The Use of Elementary Assertions How can an elementary assertion be used? In other words, what sort of action might we take, having received and accepted a simple proposition? Riolo discusses this at some length and introduces the terms virtual bi-location, and virtual bi-temporation (2001, 11-13). We all understand what ‘virtual’ means these days; a virtual reality, say, is a kind of experience in which it feels, or seems as if you’re in different circumstances, even though you’re not. Second Life, a virtual reality game, is apparently so realistic that some people become more involved in it than in their actual lives. You can spend virtual money there, which, again, although not real money, functions as though it were. You can even send or receive virtual gifts via Facebook, and again, even though they are not really gifts, getting one makes you feel as though you really received a present (or so I’m told!) The prefix “bi-” means two; location has to do with place and temporation to do with time. Therefore, ‘virtual bi-location’ means something like “it seems like I am in two places at once,” while virtual bi-temporation would be the impression of being present at two different times. Virtual bi - location Suppose you are at school, sitting in class, and the lesson has just started. Your friend is not there and you are worried that she may not know the classroom number. So you text her a message: “The logic lesson is in room 7.” (Please remember that the use of mobile phones is prohibited in class!) Here you are the speaker (let’s call you person A) and you make the assertion “the logic lesson is in room 7.” You are witnessing the event, i.e. the lesson in room 7, and communicate it to your friend (call her person B). She receives your assertion when she opens and reads your text. If she understands what you say, and believes you, your friend accepts the assertion. Importantly, in the case of virtual bi-location, the event and the assertion occur at the same time. Person A tells person B about something that is happening now. Person A is witnessing the event, whereas person B is not. How can person A’s assertion be useful to person B? Your friend, who was waiting in room 9 all this time, now has information about what is happening in another place, i.e. in room 7. Even though she is not in room 7, it’s as if she were there. That is, thanks to your message, she is virtually present in the right classroom, witnessing the start of the lesson, but she is actually in another place. Once she has received and accepted your assertion, that is, it is as if she were in two places at once. Moreover, your friend can now act upon this information. In this case, she can leave room 9 and go to room 7. Therefore, one way in which an assertion can be useful to someone who receives and accepts it, is through virtual bi-location. For that person, it is as if she were in two places at once; she is actually present in one place and virtually present in another, witnessing (virtually) the event taking place there, about which she has received the assertion. Moreover, that person can then decide to act upon the information she has received. Virtual bi -temporat ion Virtual bi-temporation is similar to virtual bi-location, but concerns time, rather than place. For the person who accepts and receives the assertion, it is as though they were present in two different


times. Of course, that person is actually in the present, but he or she can also be virtually in the past or in the future. Therefore, there are two types of virtual bi-temporation: Present and Past: In this case, A tells B about an event which happened before, and so for B it is as though he was there at the time of the event, as well as being in the present time. For example, you tell your friend that “yesterday’s logic lesson was really interesting” and as a result, for your friend it is as if he is both in the present, listening to you speak, and also in the past, witnessing yesterday’s lesson. He can then decide to act on the information (assuming that he accepts your assertion, i.e. he believes you) and he decides to go for the next lesson, because he believes it will be interesting. Another example of this occurs when I recall something I witnessed earlier, in which case the maker and receiver of the assertion are the same person - me. Otherwise, I might read about an event that occurred in the past, say in a newspaper or history book. In this case, the author is A, who makes the assertion, while I, who receive the assertion, am person B. Present and Future: As above, but this time, A makes an assertion about the future. The most common cases of virtual bi-temporation involving present and future are those of promising, and forecasting. For example, I make the assertion “there will be a test next week.” This is an example of a forecast. You receive that assertion if you are listening and hear what I say, and you accept it if you believe me, and you don’t think that I am bluffing. Again, for you, it is as though you are both in the present, listening to my assertion, and in the future, sitting for your test. Based on the information you receive, you will (hopefully) go home and do some revision. It is important, when discussing the use of assertions, that the receiver takes some action with respect to it. Thus, if someone tells me, for instance, that share prices are going down, I will decide to sell mine quickly (or else to buy some). If I hear on the weather forecast that it is going to rain tomorrow, I will decide to take an umbrella! Exercise: In not more than 10 lines, explain how an elementary assertion might be useful to someone who receives and accepts it.


- Virtual bi-location

- Virtual bi-temporation

- The use of elementary assertions


4: Complex Propositions I – Negations – and the Negator (NOT) You will remember that a simple or elementary proposition is one which is made up of a nominator,

a predicator, and a copula. Now we are going to turn to complex propositions. Therefore, we shall

be using a different type of notation instead of +Np or –Np, as this is only useful for elementary

propositions.

Another important point is that from now on, wherever the word ‘not’ occurs, we shall be treating

that proposition as a complex one. Previously, we interpreted a sentence such as “Socrates was not

Italian” as an elementary proposition, symbolized as –Np. Now, we shall think of it as a complex

proposition, and in fact, the word ‘not’ is the first junctor we shall be using to make our first type of

complex propositions – the negation.

A complex proposition can be defined as any proposition that is made up of at least one

elementary proposition and at least one junctor. Therefore the sentence “Socrates was not Italian”

is made up of the elementary proposition “Socrates was Italian” plus the junctor ‘not.’ Here are

some more examples:

(1) If George Washington was beheaded, then George Washington is dead. This is made up of two elementary propositions, “George Washington was beheaded” and “George

Washington is dead” which are combined with the junctor “if… then”.

(2) London is North of Paris and South of Edinburgh. Again, this contains two elementary propositions, “London is north of Paris” and “London is south of

Edinburgh” joined together with the junctor “and.”

There are several other junctors, which we will be encountering later on. This week, we shall only be

dealing with our first junctor – the negator (NOT). The negator can appear as the word “not,” or as

the phrases “it is not the case that…”, “it is not true that…” and so on.

Expressing negat ions symbol ically Important! We use single letters as variables for elementary propositions when these form part of a

complex proposition. That is, instead of writing +Np for “Socrates was Italian” we will express it as

follows:

Let a stand for “Socrates was Italian” a

Riolo uses German letters, e.g. a. Do not let this worry you – they are simply different systems.

We represent the junctor NOT through the symbol Therefore, we can represent the complex proposition “Socrates was not Italian” as follows:

Let a stand for ‘Socrates was Italian’

a


Notice that when we define a variable it must stand for an elementary proposition. Therefore it is

wrong to say Let a stand for “Socrates was not Italian.” Instead, we use the symbol to express the

word ‘not’.

The negat ion of a proposit ion is its counterpart We obtain the counterpart of a single elementary proposition by changing the copula from a positive

to a negative or vice-versa. For example, the counterpart of “Socrates is Greek” is “Socrates is not

Greek.”

Therefore, to add the word ‘not’ to a proposition, that is, to negate it, is to obtain its counterpart.

This is true also of complex propositions. Whatever complex proposition the variable [a] stands for,

if we negate it we are saying something like “it is not the case that *a+” or “it is not true that *a+”

(symbolized as a). This will give us the counterpart of [a].

With elementary propositions, we added ‘not’ to the copula. With complex ones, we put the negator

in front of the entire proposition. For example:

Complex proposition 1: If George Washington was beheaded, then George Washington is dead Counterpart: It is not true that [if George Washington was beheaded then George

Washington is dead] Complex proposition 2: London is North of Paris and South of Edinburgh. Counterpart: It is not the case that [London is North of Paris and South of Edinburgh.] It should be obvious that if a proposition is true then its counterpart will be false, and that if a

proposition is false, its counterpart will be true. From this fact, we can obtain the rules for the use

of the negator, as we shall see below.

Rules for the negator

We use the symbol to show that we are permitted to move from that which is on the left, to that

which is on the right. We can read it out as “leads to” or “results in.”

E.g. a b stands for proposition [a] leads to proposition [b] The symbol shows that we are permitted to move in both directions. That which is on the right

leads to that which is on the left and vice-versa.

E.g. a b stands for proposition a leads to proposition b and proposition b leads to proposition a

Rule 1 (R1): If a proposition [a] is true then its counterpart [a] will be false

If a proposition [a] is false, then its counterpart [a] will be true We can express this symbolically as follows:

(i) a is true a is false

(ii) a is false a is true RI


Truth-table for Negator (NOT) Every junctor we introduce will have a truth-table as a summary of its rules. It is important to know

how to write these rules, both symbolically (as above) and as a truth-table.

The truth-table has two sets of double lines; one vertical and one horizontal. To the left of the

vertical double-lines we write down the variables for all the elementary propositions which occur in

our rules. With the negator, there is only one variable – a – and therefore we only need one column

before the vertical double line. We draw the horizontal double-line underneath this variable:

a

Underneath our variable, we write down all the possible truth-values that the variable can have.

Clearly, a single proposition [a] can only be true or false, and therefore, we will have two rows

underneath our variable. We write T to represent true, and F for false:

a T F

On the right of our vertical double-line, we write down the complex proposition which occurs in our

rules (here a)

a a T F

Below that, we put down the truth-values which are stated in our rules. We know from our rules

that when a is true, a is false and that when a is false, a is true. Therefore, in the first row

(where a is T) we put F under a, and in the second (where a is F) we put T.

a a T F F T

This is the truth-table for our first junctor, the negator (NOT). It is simply a summary of R1.

Exercise: Study the rules and the truth-table for the negator until you know them by heart!


- A complex proposition is any proposition that contains at least one elementary proposition and at least one junctor

- The negator is the junctor ‘NOT’ and its symbol is

- The negation of any proposition is its counterpart

- Rules and truth- tables for the negator


5: Working out truth-tables I (Negator) Truth-tables do not only summarize rules; we can also use them to discover whether a complex

proposition is true or false, given the truth or falsity of its constituent elementary propositions. For

example:

Let a stand for “Today is Thursday”

It is quite clear that if [a] is true then [ a] is false. We can also work out mentally that [ a] is true.

([ a] will mean something like “It is not the case that today is not Thursday – which simply means

“Today is Thursday”)

But what about more complex propositions, like [a]?

Fortunately, we do not have to work this out mentally; there is a very easy, systematic way of doing

this.

Example 1: Work out the truth-table for a We start, as before, by putting the variables for all the elementary propositions which our complex

proposition contains, to the left of our double-line, and writing the complex proposition to the right.

As before, there is only one variable, namely, [a]:

a a

Under [a] we put all the possible truth-values for it (i.e. T or F):

a a T F

This time we are going to work out the values of [a] rather than just copying them from the

rules. We start by copying out the values of [a] under the variable [a] to the right of the double-line

(i.e. under column 1):

1 a a T T F

F

Using the rules for the negator () we can work out the values of the column to the near left of the

last one we have worked out. That is, we first work out the values for [a] under column 2. When [a]

is T [a] is F and when [a] is F, [a] is T.


2 1 a a T F T F T

F

Once we have worked out the negation of a column, we do not use that column any more. This is

why I have now put the values under column 1 in grey. We now have the values for [ a], so we do

not need the values of [a] any more, and therefore, we can ignore them from now on.

The next step is to work out the values of [ a], i.e. again, the column to the near left of the last

one we worked out (line 3). We do this by using the same rules for the negator:

3 2 1 a a T T F T F F T

F

Where we have F, its negation is T, and where we have T, its negation is F. Again, once I have this

new set of values, I ignore the last one.

We work out the final column in the same way:

4 3 2 1 a a T F T F T F T F T F *

We put an asterisk below the last column we worked out to show that this is our final answer. What

we have discovered in short is the following:

When [a] is true, [ a] is false.

When [a] is false, [a] is true. Exercise: Riolo, p. 20, no. 1* (p. 19 of the first edition)

Rules for

a is T a is F

a is F a is T


- Step-by step method for working out truth-tables


Complex Propositions II – Conjunctions – and the Conjunctor (AND) So far we have introduced our first junctor, the negator. We have set out the rules for its use, and

summarized them as a truth-table. We have also seen how to work out the truth-tables for further

complex propositions involving negation, e.g. [a], [a] etc. (see Riolo, p.20-21).

This week, we shall introduce our second junctor, the conjunctor, which, in spoken English we read

as “and”, and in logic we represent with the symbol . A complex proposition that involves a

conjunctor is called a conjunction.

Expressing conjunctions symbolically

The conjunctor is a “two-place junctor” which means that it requires at least two elementary

propositions. This is easy to understand if you know what the word ‘and’ means. Consider the

following proposition:

(1) I like ice-cream and chocolate

This is a complex proposition, although it might seem like an elementary one. It is made up, in

fact, of two elementary propositions:

(i) I like ice-cream (ii) I like chocolate When we combine these two, using the conjunctor, we get the above sentence (1).

I like ice-cream AND (I like) chocolate

All the junctors we will encounter from now on are two-place junctors. Only the negator is not.

As we saw in the previous lesson, the negator can be added to a single elementary proposition

to form a complex one, represented by, e.g. [ a]. The conjunctor, on the other hand, is used to

join at least two elementary propositions, and therefore, we will be requiring more than one

variable to symbolize a conjunction.

This is how we represent sentence (1) symbolically:

Let a stand for “I like chocolate” Let b stand for “I like ice-cream

a b

Rules for the Conjunctor Let us think about proposition (1) above. Under which conditions is it true? Clearly, if it is true that I

like both chocolate and ice-cream, then I have not told a lie, and my proposition is true. If, on the

other hand, I don’t like ice-cream, then my sentence “I like ice-cream and chocolate” is false. In the

same way, the sentence is false if I don’t happen to like chocolate. Finally, the sentence is also false

if I like neither chocolate nor ice-cream.

In other words, the conjunctor has the following rules:


(i) a is true, b is true a b is true

(ii) a is false a b is false R2

(iii) b is false a b is false

Truth-table for Conjunctor (AND) Recall that we begin our truth-table by writing down, to the left of the vertical double-line, all the

variables that occur in our rules (representing the elementary propositions). With the negator, there

was only one variable and therefore we only needed one column before the vertical double line.

The conjunctor, as we said, is a two-place junctor, and that means that our rules are going to involve

two elementary propositions (as seen above). Therefore, this time we need to put two columns

before the vertical double-line; one for each variable that occurs in the rules (that is, [a] and [b]). We

put our complex proposition, the conjunction [a b], to the right.

We draw a single line between every individual proposition, whether elementary or complex.

Therefore, I put a line between [a] and [b] because these are representing single elementary

propositions. On the other hand, I left [a b] as a single column because this represents a single

complex proposition.

The next step, remember, is to write down all the truth-values for [a] and [b] to the left of the

double-line. This was very easy in the case of the negator, as there was only one variable [a], which

could only be either true or false.

When there is more than one variable, it gets a little more complicated – however, there is a

systematic way of doing it which we will look at here. First of all, the important thing to remember is

that we want to write down all the different combinations of truth-values that there can be for all

the variables we have. When there are two variables, as in this case, we get four combinations:

(1) [a] and [b] are both true, (2) [a] and [b] are both false, (3) [a] is true and [b] is false (4) [a] is false and [b] is true.

Here is the conventional way of doing it, step-by-step:

To calculate how many combinations there are, that is, how many rows you are going to need under

your variables, take the number of variables (in this case 2) and calculate 2 to the power of that

number, i.e. 22 = 4

This is how we know that there are four combinations (as seen above).

a b a b


In the first column furthest to the left, that is, under [a], we write half that number as T and the

other half as F. Half of four is two, and therefore, this gives us two T’s and two F’s:

For the next column, we divide that number into two again. Half of two is one, and so this time, we

are going to write one T and one F. We repeat this pattern until we have filled our four rows:

Now we have all the different combinations of truth-values for [a] and [b], we can start writing in the

values for our complex proposition [a b] to the right of the double-line. Our rules say that [a b] is

only true when both [a] and [b] are true. If either one of [a] or [b] is false, then [a b] is false. From

this, we can see that [a b] is clearly also false when both [a] and [b] are false. Therefore, our

conjunctor rules can be summarized in the following truth-table:

This is the truth-table for our second junctor, the conjunctor (AND). It is simply a summary of R2.

Exercises: 1. Study the rules and truth-table for the conjunctor until you know them by heart!

2. What if there were three elementary propositions in a conjunction? How many different

combinations of truth-values would there be, and how would we write them out?

a b a B T T F F

a b a B T T T F F T F F

a b a B T T T T F F F T F F F F


- The conjunctor is the junctor ‘AND’ and its symbol is

- Writing conjunctions symbolically

- Rules and truth-table for the conjunctor

- How to work out the different combinations of truth-values for any number of variables


7: Translations Now that we have two junctors (AND, NOT), we are in a position to start doing some translating. It is

important to understand from the start that in logic, translations can mean both:

(i) From ordinary English sentences into logical symbols (ii) From logical symbols into ordinary English

Translating from English into logical notat ion Let us see how to translate the sentence:

(1) It is not the case that Anne is both reading and watching T.V. (from Riolo, p. 29)

First of all, remember that since we are dealing with complex propositions, we use single letters as

variables to represent their constituent elementary propositions. It is important that for every

translation, or every single task, once we have defined a variable, it must always be used in exactly

the same way.

The first thing to do is to identify the elementary propositions in the sentence (or sentences) to be

translated. Our example contains two, namely:

Anne is reading Anne is watching T.V How do we know this? Remember an elementary proposition has a nominator, a predicator, a

copula and nothing else. ‘Anne’ is the nominator in both our elementary propositions, they both

contain the copula ‘is,’ and they each have a predicator - ‘reading’ in the first and ‘watching T.V.’ in

the second.

What about the other words or phrases in our sentence? As you will recall, ‘it is not the case’ is one

way of expressing the negator, the junctor NOT, whereas ‘and’ is the conjunctor. We will see what

the word ‘both’ signifies below.

If we leave out the negator for a while, we have the sentence

(2) Anne is (both) reading and watching T.V.

We can translate this as follows:

Let a stand for ‘Anne is reading’ Let b stand for ‘Anne is watching T.V.’

a b (2) Anne is reading and watching T.V.

The phrase ‘it is not the case that’ negates everything that follows it, in other words, that Anne is

both reading and watching T.V. Therefore, the translation of our complete sentence is as follows:



(a b) (1) It is not the case that Anne is both reading and watching T.V.

Notice the use of brackets around [a b+. This is where the word ‘both’ comes in. If I had left out the

brackets in my translation, the meaning would have changed altogether:


a b (3) Anne is not reading and Anne is watching T.V.

Sentence (3) means something very different from sentence (1). Sentence (1) means it is not true

that she is doing both; that is to say, she may either be reading or watching T.V. but not both. It does

not tell us which of the two activities she is doing; only that she is not doing both, and in fact, she

may be doing neither. Sentence (3) on the other hand, does tell us which she is doing; it specifies

that she is watching T.V., and not reading.

The word ‘both,’ therefore, specifies that the negator, the phrase “it is not the case that,” applies to

the conjunction of our elementary propositions. It is usually an indicator that we are going to need

to use brackets somewhere. Compare with the following:

(4) It is not true that it rained, but the streets still got wet

Again, there are two elementary propositions.

Let a stand for ‘it rained’ Let b stand for ‘the streets still got wet’

a b (4) It is not true that it rained, but the streets still got wet

NOTE: the word ‘but’ must be interpreted as a conjunctor, i.e. as another form of the word ‘and.’

A final example:

(5) It is not the case that it both rained and the streets did not get wet.

Remember, when working with complex propositions we define our variables to stand for

affirmative elementary propositions (i.e. with an affirmative copula). We will add the ‘not’ in ‘did

not get wet’ later on, as the negator.

Let a stand for ‘it rained’ Let b stand for ‘the streets got wet’

(a b) (5) It is not the case that it both rained and the streets did not get wet.


Translating from logical notat ion to Engl ish sentences This is just the reverse of what we have been doing above. This time we are provided with a

proposition in logical symbols, and with the definition of the variables used, and our task is to

translate these into ordinary English.

For example:

Translate the following proposition into English:

(a b)

Where a stands for ‘today is Monday’ b stands for ‘we have a logic lesson’ We do this step by step, starting from inside the brackets, and translating any negations of elementary propositions first:

b We do not have a logic lesson We can then move onto the conjunction:

a b Today is Monday and we do not have a logic lesson

Finally, we add the negation outside the brackets:

(a b) It is not both the case that today is Monday and (the case that) we do not have a logic lesson

Exercises:

1. Riolo p. 29 ex. 5

2. Translate the following into English:

i) a b

Where a stands for ‘I like ice-cream’ b stands for ‘I like chocolate’

ii) (a b) Where a stands for ‘she studies English’ b stands for ‘she studies philosophy’


- How to identify the elementary propositions in a complex one.

- Variables are always defined to stand for affirmative elementary propositions.

- How to translate sentences into logical symbols

- How to translate from logical notation to English

- How to use brackets


8: Working out Truth- Tables II (Negators and Conjunctors) Once we have translated a complex proposition, we are often asked to write down its truth-table, or

to work out its truth-value. Let us look again at an example from lesson 7.

Example 1:

It is not the case that Anne is both reading and watching T.V.

We translated this as follows:


(a b)

How do we go about working out the truth-table for such a proposition? Here, I shall set out the

steps that we must follow.

As usual, we put all the variables for the elementary propositions to the left of our double-line, and

the complex proposition itself on the right. We know that there are two variables ([a] and [b]) and

therefore there will be four rows of values (22). We enter these values under [a] and [b]

respectively.

The next step is to work out the values for the complex proposition itself. Importantly, when there

are brackets we must always begin with whatever is inside them.

In this case, we have a conjunction [a b]. Using the rules for the conjunctor, we work out what the

values will be, and write them down, under the symbol for the conjunctor (i.e. )

The rules for AND tell us that [a b] is only true when both [a] and [b] are true, i.e. only in the first

row. All the other combinations of values result in false:

NB: It is not necessary to write down numbers corresponding to your steps – I have only done this here to make it easier for you to follow what is going on.

a b (a b) T T T F F T F F

1 a b (a b) T T T T F F F T F F F F


Finally, we work out the values for the negator outside the brackets, using the last set of values we

have obtained (under column 1). We do this by applying the rules for the negator to the values in

line 1, and write our answers under the negator sign. After we work out line 2, the values under 1

can be ignored:

We put an asterisk under the last column we have worked to show that this is our final answer.

What we have found out is the following:

When [a] is true and [b] is true, our proposition, (a b) is false

When [a] is true and [b] is false, (a b) is true

When [a] is false and [b] is true, (a b) is true

When [a] is false and [b] is false, (a b) is true In other words, the proposition it is not the case that Anne is both reading and watching T.V. is

FALSE when ‘Anne is reading’ is true and ‘Anne is watching T.V.’ is also true. In every other case, the

sentence is true. This makes sense, of course, since our sentence says, precisely, that it cannot be

true that Anne is both reading and watching T.V.

Example 2: Work out the truth-table for the following proposition:

(a b) c

Since there are three variables here, this will give us eight rows (23). Notice the system of distributing

T’s and F’s is the same; half T’s and half F’s in the first column (i.e. 4 and 4), half of that in the second

(2 and 2), and half again in the last (alternating T’s and F’s). With such long propositions, it is helpful

to write down the values of [a], [b] and [c] wherever they appear in the complex proposition:

Once again, we start from the brackets. This

time, note that one of the elementary propositions inside the brackets (i.e. a) has a negator next to

2 1 a b (a b) T T F T T F T F F T T F F F T F *

1 2 3 a b c ( A b) c

T T T T T T T T F T T F T F T T F T T F F T F F F T T F T T F T F F T F F F T F F T F F F F F F


it. It is important, in such cases, to start from that negator. We apply the rules of to the values in

line 1, giving us the following:

Next we work out the conjunctor inside the brackets. Note that we apply the rules of to the values

under 4 and 2 (not those under 1). This is because we have already worked out the values of [ a],

and therefore we don’t need those of [a] any more. Our conjunction is [a b], whereas, if we used

the values under 1 and 2, we would be working out [a b] instead:

We now have the final values for everything which is contained inside the brackets (under line 5),

and therefore the values under lines 1, 2, and 4 are no longer needed.

There is a negator outside the brackets which belongs to those brackets alone. Therefore, we work

that out next, applying the rules of to the values under 5.

(If there had been another set of brackets, i.e. ((a b) b) we would have left that negator for

the very end, as it would belong to the outer set of brackets)

the values for (a b) under Now we have

column 6. There are a couple of steps left.

4 1 2 3 a b c ( A b) c

T T T F T T T T T F F T T F T F T F T F T T F F F T F F F T T T F T T F T F T F T F F F T T F F T F F F T F F F

4 1 5 2 3 a b c ( a b) c

T T T F T F T T T T F F T F T F T F T F T F F T T F F F T F F F F T T T F T T T F T F T F T T F F F T T F F F T F F F T F F F F

6 4 1 5 2 3 a b c ( a b) c

T T T T F T F T T T T F T F T F T F T F T T F T F F T T F F T F T F F F F T T F T F T T T F T F F T F T T F F F T T T F F F T F F F T T F F F F


First, note the negator next to [c] in line 7. We need to work that out next, applying the rules of to

the values under line 3. Then, line 3 is no longer needed.

Finally, we work out the conjunctor in the middle, that joining (a b) with c. We do this, by

applying the rules for to the values in lines 6 and 7. Since this is our final answer, we mark it with

an asterisk:

6 4 1 5 2 7 3 a b c ( a b) c

T T T T F T F T F T T T F T F T F T T F T F T T F T F F F T T F F T F T F F T F F T T F T F T T F T F T F F T F T T T F F F T T T F F F F T F F F T T F F F T F

6 4 1 5 2 8 7 3 a b c ( a b) c

T T T T F T F T F F T T T F T F T F T T T F T F T T F T F F F F T T F F T F T F F T T F F T T F T F T T F F T F T F F T F T T F T F F F T T T F F F F F T F F F T T F F F T T F *


Working out the truth -value of a proposit ion: Sometimes, we are asked to work out a truth-value instead of a truth-table. In this case, we are

always given the truth-values of each elementary proposition.

For example: Work out the truth-value of

a (b a)

Given that: a is True b is False In this case, we do not have to write down all the different combinations; instead, we know this

time, that a is T and b is F. For this reason, we will only have one line under our variables:

We follow the usual steps to get the following answer:

Answer: when a is True and b is False, the proposition a (b a) is FALSE

Exercises: Riolo pp. 23 -24, ex. 2, 3, 4.

a b a ( b a)

T F

7 1 8 6 4 3 2 5 1 a b a ( b a)

T F F T F T F T F F T *


- How to work out truth-tables with two types of junctors (the negator and conjunctor)

- How to work out truth-tables with three variables.

- How to work out truth-values


9: The Adjunctor (OR) The adjunctor as one way of expressing the word ‘or’ in logic The word ‘or’ can be interpreted in two different ways, as we shall see, and each of these is

represented by a different junctor. This lesson and the next will examine these in detail.

The first junctor we shall be looking at is the most common way of interpreting ‘or,’ sometimes

written as ‘either…or.’ It is called the adjunctor or the inclusive or and it is symbolized by .

To take an example, suppose I am expecting two guests for dinner, Bob and Tom. I hear the door bell

and think “It’s either Bob or Tom.” Clearly, if I find Bob standing at the doorway, the sentence is

true. If Tom is standing there, the sentence is still true. What if Bob and Tom arrived together? If we

interpret ‘either … or’ as an adjunctor, then the sentence “it’s either Bob or Tom” is still true, even

though it’s actually both Bob and Tom standing at my door.

Another example: I am at a restaurant and ask for the desert menu. The waiter says, “you can have

ice-cream or cake.” Clearly here, what he means is I can have ice-cream alone, cake alone, or I can

have both ice-cream and cake.

What these examples show is that with an adjunction, our rules and truth-table are going to show

three cases where the complex proposition is true (see below).

Translating Adjunctions To translate sentences with an adjunction, we use the usual method. First we need to identify the elementary propositions (making sure they are affirmative) and define variables to represent them.

Do not confuse the symbol for the inclusive ‘or’ () with that for ‘and’ (). Example 1: It’s either Bob or Tom Let a stand for “It’s Bob” Let b stand for “It’s Tom”

a b Example 2: You can have ice-cream or cake Let a stand for “you can have ice-cream” Let b stand for “you can have cake”

a b Rules for the Adjunctor As we saw above, the adjunction suggests three possibilities. In each of these, the sentences in the

examples above will be true:

1. Only the first elementary proposition is true (“It’s Bob”; “You can have ice-cream”) 2. Only the second elementary proposition is true (“It’s Tom”; “You can have cake”),


3. Both elementary propositions are true (i.e. both Bob and Tom are standing behind my door; The waiter means I can have both ice-cream and cake if I like).

The only option which is ruled out, i.e. which will make the sentences false, is where neither of the

elementary propositions is true (i.e. there is someone else at my door, not Bob or Tom; there is

neither ice-cream nor cake available after all).

This will give us the following rules:

a is true a b is true

b is true a b is true R3

a is false, b is false a b is false Truth-Table for Adjunctor

We draw the truth-table for the adjunctor rules in the usual way.

a b a b T T T T F T F T T F F F

N.B. Many students confuse the truth-table for the conjunctor (and) with that of the adjunctor

(inclusive or.) Note that the adjunctor has three T’s and one F; the conjunctor has one T and three F’s

(see lesson 6).

Working complex truth-tables with adjunctors

The adjunctor follows the same rules as the conjunctor – that is, we work any negators which are

exactly next to the variables before an adjunctor. After working out negators, we work any junctor,

i.e. an adjunctor or conjunctor, within brackets. Then we work out negators outside brackets, and

finally any junctors left (conjunctors or adjunctors) outside brackets.

It is important that, just like we did with conjunctors, after working out an adjunctor, we must ignore

the values of those elementary propositions that it combines into a complex proposition.

These rules remain the same for all the junctors we will introduce, and will not be repeated. Example 1: “It is not the case that either Bob or Tom is at the door”

Let a stand for “Bob is at the door” Let b stand for “Tom is at the door”

(a b)

a b (a b) T T F T T F F T F T F T F F T F *


Notice that our result corresponds to the sentence, which is only true when neither Bob nor Tom is at the door. Example 2: “It is not the case that either Bob or Tom is at the door, but Jack is here” Let a stand for “Bob is at the door” Let b stand for “Tom is at the door” Let c stand for “Jack is here:

(a b) c Again, the results show that the sentence is only true when both Bob and Tom are not at the door, and Jack is here. Exercises: 1. Translate the following into logical symbols: i) She is either watching T.V. or doing her H.W. ii) He must take philosophy or English at A’ level. iii) It is not the case that he must take philosophy or English at A’ level, but he must take at least one

A’ level. 2. Work out the following truth-tables:

i) a b

ii) (a b) c

iii) ( a b)

a b C (a b) c T T T F T F T T T F F T F F T F T F T F T T F F F T F F F T T F T F T F T F F T F F F F T T F T T F F F T F F F *


- There are two junctors corresponding to “or”

- The most common interpretation is the inclusive or, or adjunctor

- The symbol, rules and table for the adjunctor


10: The Disjunctor The Disjunctor compared to the Adjunctor The second way or interpreting the word “or” is known as the exclusive ‘or’ and corresponds to the disjunctor, which is symbolized as ⨆. It is also sometimes written as “either…or” and the only way to determine whether we should translate ‘or’ as an adjunctor or disjunctor is to think about the meaning of the sentence and its context. Suppose this time Bob and Tom are identical twins. I can see one of them, but because it is dark, or I don’t know them very well, I cannot tell which. Although I might utter exactly the same proposition as I did in last week’s example, “It’s either Bob or Tom,” here the context suggests that this time it is only one of them that I can see. In other words, there will be only two situations in which my sentence is true – when it’s Bob (on his own) or Tom (on his own). Again, suppose I am discussing a menu with my dietician. She says to me “you can have either ice-cream or cake.” What she probably means is that I must choose one or the other, but not both. In fact, the disjunctor often contains the phrase “but not both” to make it very clear that it is the disjunctor, the exclusive ‘or,’ which is meant. In general, choose the disjunctor for your translations when it is extremely obvious that only one of the options can be true, and that they can’t both be true at the same time. If it’s not obviously an exclusive ‘or,’ then use the adjunctor. Here are some examples of sentences which are clearly disjunctions: Today is either Monday or Tuesday. The teacher is either in the staff room or in a lesson. She will either pass or fail her exam. It cannot be both Monday and Tuesday; the teacher cannot literally be in two places at once; if she passes then she hasn’t failed and vice-versa. Therefore, for each of these sentences we should interpret “or” as the disjunctor. Compare with these sentences which are somewhat vague and should be interpreted as an adjunctor: You can attend logic lessons on Mondays or on Tuesdays. The teacher is either in the staff room or on the phone. Here there is the possibility that you might attend logic lessons on Mondays and on Tuesdays; the teacher could be on the phone in the staff room. That is, both of the alternatives could be true at the same time, and so we interpret “or” as an adjunctor. Translating Disjunctions Example 1: “The person I saw was either Bob or Tom” Let a stand for “The person I saw was Bob” Let b stand for “The person I saw was Tom” a ⨆ b Example 2: “You can have ice-cream or cake (but not both)”


Let a stand for “you can have ice-cream” Let b stand for “you can have cake” a ⨆ b Note that we do not have to translate the phrase “but not both” – we include this in our translation by using the disjunctor. Rules for the Dis junctor As we saw above, there are only two cases when a disjunction is true (as opposed to the adjunction which is true in three possible cases; see lesson 9). 1. The first elementary proposition is true and the second is false 2. The first elementary proposition is false and the second is true In the other two situations, i.e. when both elementary propositions are true, and when they are both false, the disjunction is false. Thus we have the following rules:

a is true, b is true a ⨆ b is false

a is true, b is false a ⨆ b is true R4

a is false, b is true a ⨆ b is true

a is false, b is false a ⨆ b is false Truth-Table for the Disjunctor As usual the rules for the disjunctor can be summarized as a truth-table:

A B a ⨆ b T T F T F T F T T F F F

Exercises: 1. Translate the following:

i) She is either in bed or at school ii) She is either in bed or reading a book iii) a ⨆ b where a stands for “we go to the cinema”, b stands for “we go to the party” 2. Work out the following truth-values

i) a ⨆ b where a is True, b is True

ii) (a b) ⨆ (c a) where a is False, b is True and c is True

Make sure you have understood the following: - The symbol, rules and table for the disjunctor

- A disjunction is true when only one of its elementary propositions is true; it is false when they are both true, and when they are both false.


11: The Subjunctor and Bi-Subjunctor Antecedent and Consequent

One of the most important logical particles is expressed by the words “if…then” and by other

words which have the same meaning, such as “when,” “suppose that…”, “assuming that…”, or

“provided that…” or “unless” (which means “if not”). Almost every argument that is formed in

philosophy makes use of some word which has the sense of “if.”

All sentences containing the word “if” (or similar words) are made up of two parts, called the

antecedent and the consequent. Consider the following examples

i) The streets get wet when it rains. ii) If I pass my exams, I will be so happy. iii) My parents will buy me a car, if and only if I pass my exams. iv) Provided that I pass my exams, I will go to university.

Each of these sentences contains two parts.

1. The first contains the word “if” (or some other word with the same meaning) and expresses a

condition which will make the rest of the sentence true. This is called the antecedent.

2. The second expresses the result that will follow, if the antecedent is true. For this reason, it is

called the consequent.

N.B. The antecedent and consequent do not always appear in that order – in fact sentences i)

and iii) have the consequent first, followed by the antecedent.

To summarize:

Antecedent Consequent

It rains The streets get wet

I pass my exams I will be so happy

I pass my exams My parents will buy me a car

I pass my exams I will go to university The Subjunctor and Bi -Subjunctor compared

Just like the word “or,” “if” can be interpreted in two ways, and these correspond to the

subjunctor and the bi-subjunctor.

To look at the first two examples: i) The streets get wet when it rains. ii) If I pass my exams, I will be so happy. What sentence (i) is telling us is that when the antecedent (“it rains”) is true, then the consequent (“the streets get wet”) will also be true. This should be easy to see from the sense of the sentence.


However, does the antecedent have to be true for the consequent to be true? In other words, do the

streets get wet only when it rains? If we give it a little thought, we will easily see that there could be

other cases when the streets get wet, e.g. there’s a burst water pipe, or some overzealous

housewife has decided to wash the road. In other words, even when the antecedent is false (it didn’t

actually rain) the consequent might still be true (the streets got wet for another reason).

Again, sentence (ii) says that if the antecedent is true (“I pass my exams”) the consequence is true (“I

will be so happy”). But it does not suggest that the consequent is true only if the antecedent is; I

could be happy for lots of other reasons besides having passed my exams. I could be happy because

it’s a beautiful day, because somebody special has called etc. I could even have failed my exams, but

I’m still happy because at least they’re over now! So again, the antecedent could be false, while the

consequent is true.

This first interpretation of “if” is the subjunctor and is symbolized by →. The antecedent (let’s call it

A) goes before the arrow and the consequent (C) goes after. A subjunction will look like this:

A → C

Notice the arrow points in one direction only. The subjunction is only saying that if A is true, then C is

true – it is saying nothing more. It is not saying that C is true only when A is, and it is not saying that

if A is false, C must also be false.

We will understand this better, when we look at the other two examples.

iii) My parents will buy me a car, if and only if I pass my exams. iv) Provided that I pass my exams, I will go to university. Sentence (iii) contains the clearest case of bi-subjunction; the term “if and only if.” As in the previous

examples, the sentence is saying that if the antecedent is true (“I pass my exams”) then the

consequent will also be true (“my parents will buy me a car”.)

Unlike the first two examples, however, sentence (iii) also suggests that if the consequent is true, i.e.

if my parents buy me a car, then the antecedent must also be true, and I pass my exams. My parents

will buy me a car only if I pass my exams, there is no way they will buy it for me otherwise.

Therefore, in a bi-subjunction, for the antecedent to be true, the consequent must be true, and for

the consequent to be true, the antecedent must be true.

Compare this with sentence ii) where I could be happy for other reasons besides passing my exams.

Sentence ii) does not suggest that I will not be happy if I don’t pass – it only says that I will be happy

if I do pass. Sentence iii) on the other hand says that if I do not pass, I don’t get a car – my parents

will buy me a car, if and only if I pass.

The term “provided that” in sentence (iv) is another way of say “if and only if” and therefore

indicates another bi-subjunctor. The obvious meaning of this sentence is that if I pass my exams, I

will go to university. But it also means that if I go to university, I have passed my exams. Again, if the

antecedent is true, then so is the consequent, and if the consequent becomes true, the antecedent

must have been true.

The bi-subjunctor is symbolized by ↔ and a bi-subjunction will look like this:


A ↔ C

Where A stands for the antecedent and C stands for the consequent.

The symbol for the bi-subjunction is intuitive. The arrow is pointing in both directions to show that A

leads to C (if A is true then so is C) and C leads to A (If C is true, then so is A).

We only interpret “if” as a bi-subjunctor when it is absolutely clear that what is meant is “if and only

if.” If in doubt as to whether an instance of the word “if” refers to a subjunctor or to a bi-subjunctor,

then choose the subjunctor, as this is the most common interpretation.

Translating Subjunct ions and Bi -subjunctions The first thing to do is to determine whether the word “if” means “if and only if” or not. In sentences

(i) and (ii) as we saw, “if” and “when” do not mean “only if” or “only when.” Therefore we use the

subjunctor.

With a subjunctor, it is important that you put your antecedent and consequent in the right place, i.e. make sure the arrows points to the consequent. i) The streets get wet when it rains. ii) If I pass my exam, I will be so happy Let a stand for “the streets get wet” Let a stand for “I pass my exam” Let b stand for “it rains” Let b stand for “I will be so happy” b → a a → b Notice that in sentence (i) we have defined b to stand for the antecedent, and therefore the arrow must point to a. An exception is “only if” for which we invert the arrow. The streets get wet only if it rains. Let a stand for “the streets get wet” Let b stand for “it rains” b ← a Sentences (iii) and (iv) contain the words “if and only if” and “provided that” and therefore are

interpreted as bi-subjunctors.

iii) My parents will buy me a car, if and only if I pass my exams. Let a stand for “my parents will buy me a car” Let b stand for “I pass my exams” b ↔ a

iv)Provided that I pass my exams, I will go to university


Let a stand for “I pass my exams” Let b stand for “I will go to university” a ↔ b

Rules for the Subjunctor We should be able to derive the rules for the subjunctor by considering an example.

If I pass my exam, I will be so happy Let a stand for “I pass my exam” Let b stand for “I will be so happy” a → b The sentence says that if I pass, I will be happy. We have not said what will happen if I don’t pass; I

might be happy (for another reason) or I might not be happy if I don’t pass. The only case in which

my sentence could be false is when I pass and I am still not happy. This gives us the following rules:

a is false a → b is true

b is true a → b is true R5

a is true, b is false a → b is false

Rules for the Bi -subjunctor Provided I pass my exams, I will go to university Let a stand for “I pass my exams” Let b stand for “I will go to university” a ↔ b This sentence says if I pass I will go to university, and if I go to university I have passed my exams. It

also suggests that if I don’t pass then I don’t go to university and if I don’t go to university, I haven’t

passed. What is ruled out is that I pass and don’t go to university, or that I go to university without

having passed. This gives us the following rules:

a is true, b is true a ↔ b is true

a is false, b is false a↔ b is true R6

a is true, b is false a ↔ b is false

a is false, b is true a ↔ b is false


Truth-Tables for Subjunctor and Bi -subjunctor The rules can be summarized as follows:

Subjunctor:

Bi-subjunctor

N.B. Do not confuse the bi-subjunctor table with that of the disjunctor (lesson 10).

Exercises: 1. Translate the following: i) We’ll have a great time, as long as we find the place. ii) Unless she comes soon, I am leaving. iii) The lesson is cancelled if and only if the teacher does not come. 2. Work out the following truth-tables:

i) (a b) → (a c)

ii) (a ↔ b)

a b a → b T T T T F F F T T F F T

a b a ↔ b T T T T F F F T F F F T

Make sure you have understood the following: - The difference between the subjunctor and bi-subjunctor

- The symbol, rules and table for the subjunctor and bi-subjunctor


12: Arguments and Implications What is an argument? Recall that in the very first lesson, it was said that logic is a tool for constructing valid arguments, and

for checking the validity of other philosophers’ arguments. So far, though, we have not seen any

arguments; instead we have been focusing on propositions. Propositions, as we shall see, are the

building blocks of arguments; when we put a number of (related) propositions together, we get an

argument.

It might help to keep this scheme in mind:

Argument

(Complex) Proposition 1 Proposition 2 *Proposition 3, 4…+

Elementary Proposition 1 *Elementary proposition 2, 3…+

What this diagram shows is that an argument is made up of at least two propositions (there may be

more) which can be complex or elementary. A complex proposition, as we know, is made up of at

least one elementary proposition and one junctor.

So, as we can see, we have got all the parts needed to understand how arguments are made up. We

have seen what an elementary proposition is, we have all the different types of complex

propositions (because we have learnt all the main junctors) and so now we can go up to the next

level, which is the argument.

An argument, is a group of propositions, one or more of which (the premises) provide support for,

or reasons to believe, one of the others (the conclusion) (from Hurley 2008, 1).

That is, the shortest type of argument contains two propositions; one of these is the premise and the

other is the conclusion. The conclusion is normally marked by the word “therefore.” If the argument

is valid, then the truth of the conclusion will follow from the truth of the premise. In other words, if

the premise is true and the argument is valid, then the conclusion will also be true. This is why the

premise “provides support for,” or “reasons to believe” the conclusion.

Here are some examples of arguments:

i) Provided I did well in my exams, my parents will buy me a car. Premise Therefore, if my parents buy me a car, I did well in my exams. Conclusion

ii) Today is either Monday or Tuesday Premise Therefore, if today is Monday then today is not Tuesday Conclusion iii) It is not the case that both Mary and Jane went out Premise 1 Mary went out Premise 2 Therefore, Jane did not go out Conclusion


Val id vs . Sound Arguments If we think a little about these arguments, we will see that if we assume the premises are true, then

the conclusion will also have to be true. In other words, it is impossible for the premises to be true

and the conclusion to be false. This is what we call a valid argument. Later, we will learn to check

for validity, using truth-tables.

Notice that for an argument to be valid the premises (and conclusion) do not have to be true. For

an argument to be valid, all that is required is that if the premises are true, then so is the conclusion.

Thus, in examples (i)-(iii), it is quite possible that the premises (and therefore the conclusion) are

false. Still, we say the argument is valid.

Let’s take a more obviously false example. This argument is valid (but not sound): iv) The logic teacher is either a man or a woman Premise 1 (True) The logic teacher is not a woman Premise 2 (False) Therefore, the logic teacher is a man Conclusion (False) Notice that the second premise and the conclusion are both false. However, the argument itself is

valid. This is because if the premises had been true, then the conclusion would also have to be true.

That your logic teacher happens to be a woman does not alter the fact that this argument is

perfectly valid.

In other words we have to distinguish between two concepts which we use to evaluate arguments –

an argument is a good one if it is valid, however it is better if it is both valid and sound. We have

seen that to say an argument is valid does not tell us anything about whether its propositions are

true or not; ‘validity’ simply means that the conclusion is supported by the premise(s). That means

that even though we might know that an argument is valid, we cannot be certain that the conclusion

is true. To repeat, all we know about a valid argument is that if the premises are true, then the

conclusion is also true.

A sound argument, on the other hand, is one which is valid, and which also has true premises. Once

we determine that an argument is valid and has true premises then we can be absolutely certain

that the conclusion is also true. To check for soundness, that is, we need to check the facts, we need

to know for sure whether certain propositions are true or not.

Clearly example (iv) can easily be re-written to make it both valid and sound: v) The logic teacher is either a man or a woman Premise 1 (True) The logic teacher is not a man Premise 2 (True) Therefore, the logic teacher is a woman Conclusion (True)

13: Concepts from Formal Logic

Make sure you have understood the following: - An argument is made up of premises and a conclusion

- A valid argument is one where it is impossible for the premises to be true and the conclusion false


What is Formal Logic? We have already been working with formal logic; in fact, we do formal logic whenever we work with

truth-tables. Formal logic is opposed to what Riolo calls “material logic.” In material logic, our

variables together with the symbols for junctors stand for actual propositions, i.e. actual sentences

of which we know the content.

In formal logic we use the same type of notation (except that we use capital letters), but this time

we are not representing actual sentences or actual arguments, but rather we are representing the

form of these sentences or arguments. Let’s look at some examples:

Material Logic: i) Let a stand for “Mary went out”

Let b stand for “Jane went out”

(a b)

stands for the sentence “It is not the case that both Mary and Jane went out.”

ii) Let a stand for “I’m eating ice-cream” Let b stand for “I’m eating cake”

(a b) Stands for “It is not the case that I’m eating both ice-cream and cake” Formal Logic:

(A B)

This does not stand for any particular sentence, but represents instead the form of sentences (i) and (ii) and any other similar sentences.

Material Logic: Formal Logic:

Variables (small letters) stand for particular Variables (capital letters) stand for elementary propositions “primary formulae” which represent any elementary proposition Variables and junctors represent particular Variables and junctors make up complex complex propositions formulae which represent the form of complex propositions

Interpretations and Models of Formulae


In formal logic, our variables stand for formulae, not propositions, and therefore, we talk about

interpretations, rather than possibilities or “scenarios.” For example, in material logic, we might

have the proposition a b where a stands for “I like ice-cream” and b stands for “I like chocolate.”

When drawing a truth-table, we saw that there are four possibilities or scenarios, namely where “I

like ice-cream” and “I like chocolate” are both true, where one is true and the other false etc.

Because in formal logic, there is no actual sentence that the variable corresponds to we have to give

each variable an interpretation. The truth-table remains the same, only the terminology is different.

Each row under the formula is called an ‘interpretation.’

A B A B T T T Interpretation 1; A and B are interpreted as true and the complex formula is true T F F Interpretation 2; A is interpreted as true, B as false. The complex formula is false F T F Interpretation 3… F F F Interpretation 4…

When an interpretation of a complex formula gives the result “True” we call that interpretation a

model of the formula.

Exercise: How many interpretations do the following formulae have? Which of these are models of the

formula?

i. (A B)

ii. A

iii. (A B) → C

Make sure you have understood the following: - Primary and complex formulae

- What an interpretation of a formula is

- What a model of a formula is


14: Logical Truth So far, the truth-tables we have worked out have only told us what the result is in a number of cases,

which we are now calling interpretations. That is, we have only found out that, for instance A → B is

true for all interpretations except when A is true and B false (see rules for the subjunctor, lesson 11).

We can also use truth-tables to check for various outcomes; we can check whether a formula is

logically true, whether an implication is valid (lesson 15), and whether an equivalence (introduced in

lesson 16) is valid.

The concept of logical truth needs to be distinguished from ordinary truth. Whether a proposition

happens to be true or not is a contingent matter – we need to check the facts to find out. For

example “Today is Saturday” happens to be true while I am writing this, but may not be true when

you are reading it. If in doubt, we look at a calendar. Again, “Today is Saturday and I am in Dubai” is

true for me at the moment, but won’t be true in a week’s time.

Logical truth is different. A proposition or formula which is logically true is necessarily true; it has to

be true for everybody at all times. The most common examples of logically true propositions are

negations of contradictions.

A contradiction is a proposition or formula which is always false. It usually contains a conjunction

of counterparts; eg. A A. If we work out this table, we will see that there are no interpretations

that are models.

A A A T T F F F F F T *

If we negate this, i.e. (A A), we get a logically true proposition or formula, which is always

true.

A (A A) T T T F F F T F F T * A logically true formula can be defined as one in which every interpretation of that formula is a

model of that formula (i.e. every interpretation results as true).

Exercise: Which of the following are logically true?

i. (A → B) → (A B)

ii. (A ⨆ B) ↔ ( A ⨆ B)

iii. ((A B) (A B))

Make sure you have understood the following: - The concept of logical truth

- A logically true formula is one in which every interpretation is a model of that formula


15: Implications Implicat ions (arguments) in Formal Logic We need to introduce two new symbols in order to represent arguments in formal logic. Recall that

arguments are made up of premises and conclusions. We have already seen that formulae can be

separated through the double comma (,,) and we use this to separate one premise from another.

The conclusion is marked through the implication sign () and thus an example of an implication

(the word for ‘argument’ in formal logic) is the following:

A,, B C

Here A and B are the premises and C is the conclusion. We can read it as “A, B, therefore C.”

Of course our premises and conclusion can be made up of complex formulae. For example:

(A B),, A B

“It is not the case that both A and B. A. Therefore, not B.”

As we will see, this is a valid implication. Recall that a valid argument is one where it is impossible for

the premises to be true and the conclusion to be false. Of course, it is hard to work out mentally

whether this is valid when A and B do not stand for actual propositions. Through working out a

truth-table, however, we can check whether there are any interpretations which result in true

premises and a false conclusion. If there are, then our implication is not valid.

Notice that the second interpretation is the only one

where both premises are true, and in it the conclusion

is also true.

However, it is NOT simply the fact that there is one interpretation where all premises and the

conclusion are true, which makes the implication valid. The implication is valid because there are no

interpretations where the premises are all true and the conclusion is false. This is an important

point to understand which many students have difficulties grasping.

Let’s look at an implication which is not valid:

(A B),, B A

Here we have two interpretations where both

premises are true. In the second interpretation, the

conclusion is true, but in the fourth, it is false.

A B (A B) A B T T F T T F T F T F T T F T T F F F F F T F F T

A B (A B) B A T T F T F T T F T F T T F T T F F F F F T F T F


Because we have one interpretation where all premises are true, but the conclusion is false, the

implication is not valid. An interpretation where all premises are true, but the conclusion is false is

called a counterinterpretation.

Therefore, a valid implication can be defined as one in which every interpretation which is a model

of all the premises, is also a model of the conclusion. In other words, every interpretation which has

true premises also has a true conclusion.

Checking for val idity with truth -tables It should be evident that we can check for validity using truth-tables. In the previous lesson, we

learnt how to check for logical truth, and this week we will see how to check for validity.

It is important to understand what exactly we can check for validity and for logical truth. Only a

single proposition or formula can be logically true. It can be a relatively simple one, e.g. (A

A), or a more complex one, e.g. (A → B) → (A B). Importantly, though, it is a single formula and

this means there will be only one column which we mark as our answer. If the results are all true,

then the formula is logically true. For example:

A (A A) T T T F F F T F F T * When checking for validity, on the other hand, we need either an implication or an equivalence (next

week) and this means there have to be at least two formula. In the case of an implication, we need

at least one premise and one conclusion. This means we are going to have more than one column as

‘answers’ and by comparing these, we determine whether the implication (or equivalence) is valid.

Example: Us ing truth-tables , f ind out whether the fol lowing implications are valid.

i. (A ⨆ B) B

This implication has one premise and one conclusion. The third interpretation is a model of the premise but not of the conclusion, i.e. it is a counterinterpretation (T F). Thus, the implication is not valid.

ii. (A A) (A B)

This implication also has one premise, and a conclusion. There are no interpretations which are models of the premise (because the premise is a contradiction) and therefore, there are no counterinterpretations, i.e. no interpretations which are models of the premise but not of the conclusion

A B (A ⨆ B) B

T T F F T F T T F T T F

F F F T

A B (A A) (A B) T T T F F T T F T F F F F T F F T F F F F F T F * *


(no T F). Thus, the implication is valid.

iii. (A → B),, B A

Here we have two premises and a conclusion. The fourth

interpretation is a counterinterpretation (T T F) and the

implication is therefore not valid.

Exercise: Are the following implications valid? Find out using truth-tables.

i. (A → B),, B A

ii. (A ⨆ B) (A ↔ B)

iii. (A B),, (B A) A

A B (A → B) B A

T T T F T T F F T T F T T F F F F T T F

Make sure you have understood the following: - An implication is valid when there are no counterinterpretations

- A counterinterpretation is an interpretation which is a model of all the premises but not of the conclusion; i.e. the premises are all true but the conclusion is false.

- How to check for validity using truth-tables.


16: Standard Implications (Modus Ponens and Modus Tollens)

Standard Implicat ions and Fallacies There are some argument-forms, or implications, which are so commonly used, that it is worth

knowing their Latin names so that we can immediately recognize that they are valid arguments. At

the same time, there are some common mistakes in reasoning, which we call fallacies which are very

similar to these standard arguments. Again, it is worth knowing their names, so that we can

immediately recognize that these arguments are invalid.

All of these implications involve a subjunction (→) as the first premise. They then proceed by

affirming or denying part of that subjuction. When the conclusion is affirmed, the argument is one

of the modus ponens type, and when the conclusion is negated (or denied), it is one of the modus

tollens type. However, if the wrong part of the subjunction is affirmed or negated, this gives rise to

one of the fallacies, as we shall see.

Modus Ponens Modus Ponens translates, roughly, as ‘the method of affirming’ and therefore, what our implication

will do, ultimately, is affirm something. This means that the conclusion must be affirmative.

There are two types of modus ponens implications:

1. Modus (Ponendo) Ponens.

This is the ‘pure’ modus ponens, which involves only affirmation. We start from a subjunction, affirm

the antecedent, and then affirm the consequent as the conclusion. Here is an example:

If Socrates is a man, then Socrates is mortal. Socrates is a man. Therefore, Socrates is mortal.

Let a stand for “Socrates is a man” Let b stand for “Socrates is mortal”

Our implication then is:

Modus (Ponendo) Ponens: A → B,, A B

It is important to remember that in modus ponens, the antecedent (A) is the second premise while

the consequent (B) is affirmed as the conclusion. If we switch these around, we end up with a

fallacy (see below).

2. Modus Tollendo Ponens:

There is another form of modus ponens, which is not purely affirmative, but involves a negation.

Modus Tollendo Ponens can be translated as ‘the method of affirming by denying.’ Therefore,

ultimately we are going to affirm something as the conclusion, and we will do this by denying

something else.


Again, it is the consequent which must be affirmed as the conclusion, and the antecedent will be

negated as the second premise. In other words, if the antecedent is A, and the consequent B, our

second premise will be A and the conclusion will be B. The implication will look like this:

(premise 1),, A B.

In ‘mixed’ ponens (and mixed ‘tollens’) implications, we have to change the first premise. For modus

tollendo ponens, simply take the implication involving the second premise and the conclusion ( A

B ) and turn it into a subjunction: A → B.

Here’s an example:

If he’s not careful, he will waste all his money. He wasn’t careful and so, he wasted all his money.

Let a stand for “He was not careful” Let b stand for “He wasted all his money”

Modus Tollendo Ponens: A → B,, A B

The Fallacy of Aff irming the Consequent (as the premise) As mentioned above, if we try to affirm the consequent as the premise and the antecedent as the

conclusion, we end up with a fallacy. This is known as the fallacy of affirming the consequent, but it

is worth emphasising that the fallacy occurs when the consequent is affirmed as the premise. If the

consequent is affirmed as the conclusion, then what we get is one of the valid modus ponens

implications, as we saw above.

Here is an example which we have already met:

If it rains, the street will get wet. The street got wet. Therefore, it rained.

Let a stand for “it rained” Let b stand for “the streets got wet”

Fallacy of Affirming the Consequent: A → B,, B A As we saw in lesson 11, the streets might have got wet for another reason, apart from the rain, and

therefore we cannot conclude that the antecedent is true (that it rained) from the truth of the

consequent (that the streets got wet). In other words, if we affirm the consequent as a premise and

then conclude that the antecedent is true, we make a mistake in reasoning, and end up with an

invalid argument.

Modus Tollens


Modus Tollens can be understood as ‘the method of negating’ and therefore, what our implication

will do, ultimately, is negate (or deny) something. This means that the conclusion must be negative.

There are two types of modus tollens implications:

1. Modus (Tollendo) Tollens.

This is the ‘pure’ modus tollens, which involves mainly negation. We again start from a subjunction,

negate the consequent, and then negate the antecedent as the conclusion. Here is an example:

If it rains, the streets will get wet. The streets didn’t get wet. Therefore, it didn’t rain.

Let a stand for “It rains” Let b stand for “The streets get wet”

Our implication then is:

Modus (Tollendo) Tollens: A → B,, B A

It is important to remember that in modus tollens, the consequent (B) is the premise while the

antecedent (A) is negated as the conclusion. If we switch these around, we end up with a fallacy

once again (see below).

2. Modus Ponendo Tollens:

This is the second form of modus tollens, which can be translated as ‘the method of negating by

affirming.’ Therefore, ultimately we are going to negate something as the conclusion, and we will do

this by affirming something else.

Again, it is the antecedent which must be negated as the conclusion, and the consequent will be

affirmed as the second premise. In other words, if the antecedent is A, and the consequent B, our

second premise will be B and the conclusion will be A. The implication will look like this:

(premise 1),, B A.

Again, we have to change the first premise. One way of doing this is by contraposition. Using the two

steps of contraposition:

1. Negate the primary formulae B A

2. Reverse the implication: A B

And re-write this as a subjunction: A → B

We now have the first premise and can write the entire modus ponendo tollens form:

Modus Ponendo Tollens: A → B,, B A.

Here’s an example:


If he were on time, we wouldn’t have had to wait so long. However, we had to wait long, which means he was not on time.

Let a stand for “He was on time” Let b stand for “We had to wait long”

A → B,, B A

The Fallacy of Denying the Antecedent (as the premise) As mentioned above, if we try to negate the antecedent as the premise we end up with a fallacy,

known as the fallacy of denying the antecedent.

Here is an example:

If it rains, the street will get wet. It didn’t rain. Therefore, the streets did not get wet.

Let a stand for “it rained” Let b stand for “the streets got wet”

Fallacy of Denying the Antecedent: A → B,, A B Again, the streets might have got wet for another reason, even though it didn’t rain, and therefore

we cannot conclude that the consequent is false (that the streets did not get wet) from the falsity of

the antecedent (it didn’t rain). If we negate the antecedent as a premise and then conclude that the

consequent is false, we make a mistake in reasoning, and end up with an invalid argument.

Exercise Riolo, pp 94 - 95. Ex. 17 - 22 (second edition only)

Summary: Standard Implications

Modus Ponendo Ponens: A → B,, A B

Fallacy of Affirming the Consequent: A → B,, B A

Modus Tollendo Tollens: A → B,, B A

Fallacy of Denying the Antecedent: A → B,, A B

Modus Tollendo Ponens: A → B,, A B

Modus Ponendo Tollens: A → B ,, B A


17: Equivalence Two formulae are said to be equivalent when each one implies the other. Let’s say we have two

formulae, call them F1 and F2. We know that the implication F1 F2 is valid and also that F2 F1 is

valid. This means our two formulae are equivalent and we can represent this as F1 F2. Notice that

the symbol for equivalence () is made up of two implication signs, one facing each direction. This

should be intuitive since equivalence, as we have seen, means that an implication is valid in both

directions (F1 implies F2 and F2 implies F1).

How to check for equivalence val idity us ing truth -tables

Example: (A ⨆ B) (A ↔ B)

One way to determine whether an equivalence is valid or not is to work out two truth-tables. That is,

we need to check whether the implication is valid in both directions. This means we need to check

whether

1. (A ⨆ B) (A ↔ B) is valid

2. (A ↔ B) (A ⨆ B) is valid

There are no counterinterpretations here, and therefore

the first implication is valid.

Again, there are no counterinterpretations, and so the

second implication is valid.

Since both implications are valid, this means that our

equivalence is valid.

An equivalence is valid, that is, when each formulae implies the other.

There is a quicker way of working out whether an equivalence is valid. As the term ‘equivalence’

implies, the results of each formula must be identical. That means we can simply work out the table

once and check to see whether the results match each other. If they do, if there is no case where

one formula is True and the other is False, then the equivalence is valid.

Exercise: Check whether the following equivalences are valid.

i. (A B) C (A C) B

ii. (A → B) (B → A)

A B (A ⨆ B) (A ↔ B)

T T T T F T T F T F T F F T F F F F F F F T T T

A B (A ↔ B) (A ⨆ B) T T T T T F T F F T F T F T F F F F F F T F T T

Make sure you have understood the following: - An equivalence is valid when the two formulae both imply each

other, i.e. there is a valid implication in both directions.


18: Properties of Implications and Equivalences: (Reflexivity, Symmetry and Transitivity)

Equivalence relations have three properties; they are reflexive, symmetric and transitive.

Implications only have two of these properties; they are reflexive, transitive but not symmetric. We

can prove these using truth tables.

The Ref lexive Property of Implicat ions and Equivalences Reflexivity refers to a relation something has with itself. The implication sign is reflexive because

any formula will always imply itself.

Examples:

i. A A

ii. A B A B

iii. (A ⨆ B) → ( A B) (A ⨆ B) → ( A B) In other words, whatever formula we come up with, if we work out an implication with the same

formula on both sides of the implication sign, it will always be valid.

This implication is valid, as there are no counterinterpretations.

Again, there are no counterinterpretations, so the implication is valid.

This implication is valid too.

It should be evident that since we have the same formula on both sides of the implication sign, we

will have the same results and therefore there cannot be a counterinterpretation (T F). Thus we have

proved that the implication is reflexive.

That the equivalence relation is also reflexive should be easy to see. The equivalence relation is

reflexive because any formula is always equivalent to itself.

A A A T T T F F F

A B A B A B T T T T T F F F F T F F F F F F

A B (A ⨆ B) → (A B) (A ⨆ B) → (A B)

T T F T F T T T T F T F F F F F F T T T T T T T F F F T T T F T


Recall that one way to prove an equivalence valid, is to work out two truth-tables; an implication in

both directions. Since we have the same formula on both sides of the equivalence sign this will turn

out to be the same table.

Examples:

i. A B A B

ii. (A → B) ⨆ (A B) (A → B) ⨆ (A B)

i. Strictly speaking, we need to work out two implications - A B A B and A B A B. Since the

formulae are the same it should be clear that the implication is valid in both directions. We can do it

the short way working the table once, and checking whether the results are identical.

Since the results match, the equivalence is valid

ii. Again, we could work out two tables, but since the formulae are the same, the implication must

be valid in both directions. Thus, the equivalence is valid.

Thus we can see that the equivalence relation is reflexive – any formula is always equivalent to itself. Exercise:

1. Without using a truth-table explain how you know that (A → B) ⨆ (A B) (A → B) ⨆ (A B) is valid

Transit ivity Transitivity can be defined as “passing over to or affecting something else.” It describes a relation in

which one element is related to a second, the second is related to a third, and thus, the first element

is also related to the third.

For example:

A B ,, B C A C Hypotheses conclusion/ result

Remember the sign, which we use to set up rules? In order to prove the relation of transitivity we

must prove that this rule is admissible. That is, we must show that if A B and B C then A C.

A B A B A B T T T T T F T T F T T T F F F F


The key term here is “if.” We must assume that the implications before the sign (the hypotheses)

are valid. Given the validity of the hypotheses, we want to find out whether the implication after the

sign (the conclusion or result) is also valid in this case. If so, then our rule is admissible. Since we

are assuming that the hypotheses are valid, we rule out any interpretations which turn out to be

counterinterpretations of them.

Hyp. 1 Hyp. 2 Concl. A B C A B B C A C T T T T T T T T T Four interpretations are ruled out because: T T F T T T F 1. B < C (Hypothesis 2) is not valid T F T T F 2. A < B (Hypothesis 1) is not valid T F F T F 3. A B (Hypothesis 1) is not valid F T T F T T T F T F T F F T T F 4. B < C (Hypothesis 2) is not valid F F T F F F T F T F F F F F F F F F

The conclusion (A < C) does not have any counterinterpretations. Thus, the rule is admissible, because, given that the hypotheses are valid, the conclusion turns out to be valid too. We have proved, in other words, that the implication sign is transitive. Of course, we could have used more complex formulae, e.g.

(A B) A ,, A < (A B) (A B) (A B) Hyp. 1 Hyp 2 Concl. Exercise: 2. Prove that the above rule is admissible. Why don’t we rule out any interpretations in this

case? As mentioned above, the equivalence relation is also transitive. This means that if one formula is

equivalent to a second, and the second is equivalent to a third, the first must also be equivalent to

the third. In other words:

A >< B,, B >< C A >< C

Again, we have to assume the truth of the hypotheses, and rule out any interpretations which turn

out to be counterinterpretations of them. Since we have an equivalence, this means we either need

to work out implications in two directions, i.e:

1. A < B,, B < C A < C (This is the table worked above.)

2. B < A,, C < B,, B >< C A >< C

When we prove these rules admissible, we will have proved the transitivity of the equivalence sign. I

will use the short method here, i.e. rule 3.

Hyp. 1 Hyp. 2 Concl. A B C A B B C A C T T T T T T T T T Six interpretations are ruled out because: T T F T T T F 1. B><C (Hyp. 2) is not valid (T F) T F T T F 2. A> B (Hyp. 1) is not valid (F T) F F T F F F T 6. B>< C (Hyp. 2) is not valid (F T) F F F F F F F F F Because the results are identical in the interpretations of the conclusion that are left, the rule is

admissible. Thus, we have proved that the equivalence relation is transitive.

Symmetry Symmetry occurs when the relation between two formulae can be reversed. For example, in maths,

the = sign and the + sign are symmetric, but the – sign is not. That is, if a=b, then b=a; 1 + 4 gives the

same result as 4 + 1. On the other hand, 4 – 1 is not the same as 1 – 4.

In logic, the equivalence sign is symmetric, but the implication sign is not. Again, we prove symmetry

by showing that certain rules are admissible. If the rule is inadmissible, as we shall see for the

implication sign, then the implication sign is not symmetric.

Symmetry of Equivalence:

A >< A is admissible

Again, we need to assume the validity of the hypothesis, by ruling out any interpretations where the

equivalence in not valid.

A B A B B A T T T T T T Two interpretations ruled out because: T F T F A>< B is not valid (F T) F F F F F F


The interpretations of the conclusion that are left are both valid (the results are the same, either

both T or both F) and thus the rule is admissible. Therefore, we have proved that the equivalence

relation is symmetric.

Asymmetry of Implication:

A < B B < A is inadmissible

A B A B B A T T T T T T One interpretation ruled out because: T F T F A<B is not valid (T F) F T F T T F counterinterpretation in conclusion! F F F F F F

Because of the counterinterpretation in the conclusion (interpretation 3) we can see that the

implication is not symmetric. That is, even though we assume the validity of the hypothesis, our

conclusion is still not valid.

The Relation between , → , and between , ↔ ,

Recall the rules for the subjunctor (→) and compare these with an implication () :

As we can see the only interpretation where the subjunction is false (namely where the antecedent

is true, and the consequent false) is the very same interpretation which makes the implication

invalid (i.e. the counterinterpretation, where the premise is true and the conclusion false).

This allows us to re-write subjunctions as implications and vice-versa, taking the antecedent as the

premise and the consequent as the conclusion. That is, “If A then B” can be re-written as an

argument of the form “A therefore B” and vice-versa. If the subjunction is true then the argument is

valid, and vice-versa.

Similarly, if we compare the rules for the bi-subjunctor (↔) and compare these with an equivalence

():

A B A B

T T T T T F T F F T F T F F F F

a B a → B T T T T F F F T T F F T

a B a ↔ b T T T T F F F T F F F T

A B A B

T T T T T F T F F T F T F F F F


Again, the two interpretations where the bi-subjunction is false are the same which make the

equivalence invalid. Therefore, bi-subjunctions can be re-written as equivalences and vice-versa; if

the equivalence is valid, the bi-subjunction will be true.

It should be intuitive that the double arrow we use for rules (, ) can also be interpreted in such a

way.

An admissible rule A B can be seen as saying “if the hypothesis A is valid, then the conclusion B is

also valid.” Thus this arrow can also be seen as a subjunction or “if” statement.

Similarly the double arrow facing both directions can be seen as a bi-subjunction. The admissible

rule A B can be seen as “If A is valid, then so is B and if B is valid, then so is A.” In other words, it is

similar to an “if and only if.”

Thus, it can be said that the subjunctor and bi-subjunctor have the same properties as the

implication and equivalence respectively.

It is important, however, not to confuse the symbols, and to use the right one in the right context.

Use and for implications and equivalence, → and ↔ for propositions or formulae, and or

for rules.

Exercises: No. 1 and 2 on p 49, 50 above

Make sure you have understood the following: - The equivalence relation is reflexive, transitive and symmetric

- The implication sign is reflexive, transitive, but NOT symmetric

- How to prove reflexivity, transitivity and symmetry.

- Why we rule out certain interpretations when proving that rules are admissible.

- The relation between , < and

- The relation between , >< and


19: Generalisation, Instantiation and Partial Replacement Rules

1. Generalizat ion

Riolo (p. 67): “Replacing every small letter of a valid implication [or equivalence] by its capital

counterpart transforms it into a valid implication-form [or equivalence-form].”

E.g. a →b,, a < b A → B,, A < B

2. Instant iat ion

Riolo (p. 67): “ An implication [or equivalence] obtained by replacing every capital letter of a valid

implication-form [or equivalence-form] by any formula as long as it is the same formula for every

occurrence of that letter is valid.”

E.g: A ˄ B >< ¬ (¬ A ˅ ¬ B) x ˄ y >< ¬ ( ¬ x ˅ ¬ y)

3. Partial Replacement Rule

Riolo (p. 71): “For any formula-form, if one substitutes a partial formula-form of it, at any one of the

positions where it occurs, by an equivalent one, the resulting formula-form is equivalent to the

original one.”

E.g. A ˄ B >< ¬ (¬ A ˅ ¬ B) A ˄ B ˄ C >< ¬ (¬ A ˅ ¬ B) ˄ C


20: Deriving Equivalent Formulae: De Morgan’s and Double Negation

Equivalence

The next four lessons will return to implications and equivalences, starting with De Morgan’s

equivalences. De Morgan’s equivalences, as the name suggests, are a set of equivalent formulae, or

propositions, which were discovered by Augustus De Morgan in the 19th Century. What his

discovery essentially entails is that any proposition which includes only conjunctors () and negators

() can be re-written using adjunctors () and negators () and vice-versa. In other words, he

discovered that there is an equivalent way of writing an ‘and’ proposition which uses ‘or’, and that

‘or’ in turn, can be expressed in terms of ‘and.’

The First Two De Morgan Equivalences There are four De Morgan Equivalences, but if you can remember any one, it is easy to derive the

others from that. Here’s one of De Morgan’s equivalences:

De Morgan 1: A B ( A B)1

The first thing to notice is that there is a conjunctor () on one side of the equivalence sign and an

adjunctor () on the other side. This is an important point to remember and is true of all four De

Morgan equivalences. Remember, the whole point of De Morgan’s equivalences is to change a

conjunction into an adjunction or an adjunction into a conjunction. Therefore there must be a

conjunctor on one side and an adjunctor on the other side of the equivalence sign. There will not be

any other junctors except for negators ().

Of course, we should first of all see whether this is valid or not. Here, I will do it the quick way, using

one table. Remember, for exams you need to draw two tables, checking the implication in both

directions (See lesson 16).

Equivalence is valid

From this first De Morgan equivalence, we can see that to derive an adjunction from a conjunction

(A B) what we need to do is the following:

Original conjunction: A B

1. Negate each primary formula: A becomes A; B becomes B A B

2. Change the junctor: becomes A B

3. Negate the entire formula: ( A B) The final formula is then equivalent to the original giving us the first De Morgan equivalence:

1 This is not necessarily the first De Morgan equivalence, I am just calling it that because it’s the first one we’re

looking at. There is no need to remember De Morgan’s equivalences in any particular order.

A B A B ( A B)

T T T T F F F T F F F F T T F T F F T T F F F F F T T T * *


De Morgan 1: A B ( A B)

The same rules can be used to derive the other De Morgan equivalences. Suppose we start with an

adjunction (A B):

Original adjunction: A B

1. Negate each primary formula: A becomes A; B becomes B A B

2. Change the junctor: becomes A B

3. Negate the entire formula: ( A B) The second De Morgan equivalence then is:

De Morgan 2: A B ( A B)

The first two De Morgan equivalences differ only in one way: one has an adjunctor on the left and a

conjunctor on the right, while the other has the conjunctor on the left and the adjunctor on the

right.

The Double Negation Equivalence Before moving on to the next De Morgan’s equivalences, we need to formally introduce another

equivalence, which is called the Double Negation equivalence. Of course, many of you will have

already understood that this equivalence is valid, however, as this is constructive logic, we cannot

just take it for granted.

The Double Negation Equivalence is the following:

Double Negation: A A

What this equivalence basically says is that two negators () which are next to each other cancel

each other out, so that when you have a primary formula negated twice, this is equivalent to the

primary formula.

Important: This equivalence is not valid when (only) one of the negators is inside brackets.

Therefore, ( A B) is NOT equivalent to A B.

The Third and Fourth De Morgan Equivalences We obtain the final De Morgan equivalences in the same way except we start from a negated

conjunction and a negated adjunction:

Original conjunction: (A B)

1. Negate each primary formula: A becomes A; B becomes B ( A B)

2. Change the junctor: becomes ( A B)

3. Negate the entire formula: ( A B)

4. By the Double Negation equivalence we get: A B

De Morgan 3: (A B) A B


Original adjunction: (A B)

1. Negate each primary formula: A becomes A; B becomes B ( A B)

2. Change the junctor: becomes ( A B)

3. Negate the entire formula: ( A B)

4. By the Double Negation equivalence we get: A B

De Morgan 4: (A B) A B

Deriving Equivalent / / Formulae

In general, the De Morgan equivalences can be used to derive an equivalent / / formulae from

any / / formula whatsoever , and we follow the same procedure by which we obtained De

Morgan equivalences.

Note that by a “ / / formula (or implication)” we mean a formula (or implication) which contains

nothing but primary formulae and the junctors , and .

The same four steps can be applied to any formula at all, as long as it contains nothing but negators

(), conjunctors () and adjunctors (), and the result will be an equivalent formula. So for instance,

suppose we start with the following:

( a b) ( c a)

Original formula: ( a b) ( c a)

1: Negate primary formulae: ( a b) ( c a)

2: Replace junctors: becomes ; becomes ( a b) ( c a)

3: Negate the entire formula: [( a b) ( c a)]

4: By Double Negation Equivalence we get: [(a b) (c a)]

Thus ( a b) ( c a) [(a b) (c a)]

Exercise: 1. Translate the following propositions and derive equivalent propositions changing conjunctors

into adjunctors and vice-versa. Give your answer as an equivalent English sentence:

i. It is not the case that Mary is at work or John is at work ii. Mary and John are at work iii. Mary or John is at work

Practice deriving 3 equivalent / / formulae starting from any formula you like.

- De Morgan Equivalences

- Double Negation Equivalence


21: Deriving Valid Implications I: The Duality Principle

The Duality Principle deals, again, with implications involving only the junctors / /

By applying a similar procedure to the one seen in lesson 20, we can obtain a valid implication from

another valid implication.

Recall that we could derive a formula which is equivalent to another by negating the primary

formulae, changing the junctors, and negating the entire formula.

Note that this time, we are deriving an implication from another. If the first is valid, then the second

will be valid too.

To emphasise this point, when we derive equivalent formulae, we start off with one formula (call it

A) and obtain another (call it B). Then, A B is valid.

What the Duality Principle allows us to do is, starting from an implication (e.g. A B) which is valid,

we apply a few steps to end up with another valid implication. Thus we are going to end up with a

rule A < B _________

Suppose we start with this valid implication:

A B B C

Original Implication: A B B C

1. Negate primary formulae: A B B C

2: Replace junctors: becomes ; becomes A B B C

3. Reverse implication: premise becomes conclusion and vice-versa B C A B This implication is valid, therefore one way of dualising is to apply these three steps. We could however simplify this. If we

4. Negate primary formulae again: B C A B

5: By Double Negation Equivalence we get: B C A B

Therefore, our result B C A B is also valid.

Of course, you will have realised that by the Double Negation Equivalence, we can leave out steps 1

and 4 (and consequently 5) to get a much simpler method. Rather than negating primary formulae,

and then negating them again, we can simply follow two steps:

A B B C

Original Implication: A B B C

1: Replace junctors: becomes ; becomes A B B C

2. Reverse implication: premise becomes conclusion and vice-versa B C A B The Duality Principle, therefore, can be summarized as follows:


If one dualises a valid / / implication – i.e. replaces the conjunctors occurring in it by adjunctors and vice-versa, and reverses the implication sign – the resulting implication is also valid (Riolo 2001, 80).

Exercise:

Practise deriving valid implications. Use the De Morgan equivalences to obtain 3 valid implications

and then practice dualising them.

Summary:

- Duality Principle (Dualising a valid / / implications)

1. Negate primary formulae (optional) 2: Replace junctors 3. Reverse implication


22: Deriving Valid Implications II: Contraposition

Contraposition Contraposition is a second way of obtaining a valid implication from another. Suppose we have the

implication A B. There are two steps to obtaining another valid implication, similar, but not the

same as the steps we have been following in the past two lessons:

Original implication: A B

1. Negate the primary formulae A B

2. Reverse the implication: B A

Our result is that if the original implication is valid, so will the second be. Notice the emphasis on if;

here we need to assume that A B is valid, in other words it is our hypothesis (see lesson 17. p. 21).

Therefore, we need to write our result as a rule:

Contraposition 1A: A B B A

Notice we have a double arrow pointing in two directions. This means we have to check the rule

twice, first taking A B as the hypothesis, to see whether the conclusion, B A, turns out to be

valid. Then we take B A as our hypothesis, to see whether A B turns out to be valid. If both

turn out to be valid, then our rule is admissible.

First Direction: A B B A Second Direction: B A A B

A B B A A B

T T F F T T T F T F F T F T F T F F T T F F

The rule is admissible in both directions thus A B B A is admissible.

Re-writ ing the rule of contraposition as an equivalence:

As we saw in lesson 17, the bi-directional rule arrow () is similar to the bi-subjunctor (↔) and bi-

subjunctions can be re-written as equivalences (). The implication () can also be written as a

subjunction (→). Contraposition therefore can also be written as an equivalence between

subjunctions, i.e. by changing to →, and changing to

Contraposition 1A: A B B A

becomes

Contraposition 1B: A → B B → A

Let’s see whether this is valid (using the quick method):

A B A B B A T T T T F F T F T F F T F T F T F F F F T T


The results are the same and therefore the equivalence is

valid.

Contraposition with Conjunctions So far we have seen how contraposition works when the original implication is made up of two

primary formulae; one as the premise and one as the conclusion. We can also do contraposition with

complex formulae. Contraposition can be done with a conjunction as the premise of the

implication. Suppose we start with the following implication:

A B C

To obtain another valid implication from this, the method is slightly more complicated. This time we

do not negate all the primary formulae, instead, we negate the conclusion, and one part of the

conjunction (this can be either A or B). Rather than reversing the implication, we exchange the

conclusion and that part of the conjunction we have negated:

Original Implication: A B C

1. Negate conclusion and one part of conjunction: A B C

2. Exchange conclusion with negated part of conjunction: A C B Our rule then, says that

Contraposition 2A: A B C A C B

Again, this can be written as an equivalence, by replacing with → and with

Contraposition 2B: (A B) → C (A C) → B

Contraposition with Adjunctions We can also do contraposition with an adjunction. It is important to note that while a conjunction

needs to be the premise, contraposition also works with an adjunction as the conclusion of the

implication. For example, we might start with the following:

A B C

The method for obtaining another valid implication is similar to the previous. Instead of the

conclusion, we negate the premise, and one part of the adjunction (this can be either B or C). We

then exchange the premise with that part of the adjunction we have negated:

Original Implication: A B C

1. Negate premise and one part of adjunction: A B C

2. Exchange premise with negated part of adjunction: B A C Our rule then, says that

A B A → B B → A T T T F T F T F F T F F F T T F T T F F T T T T * *


Contraposition 3A: A B C B A C

Again, this can be written as an equivalence, by replacing with → and with

Contraposition 3B: A → (B C) B → ( A C)

Exercise: Explain, in not more than 10 lines, why we can’t do contraposition with a conjunction as the

conclusion of the implication, or with an adjunction as the premise.

Summary: - The Relation between

, →,

, ↔, - Simple Contraposition:

Rule: A B B A

Equivalence: A → B B → A - Contraposition with conjunctor:

Rule: A B C A C B

Equivalence: (A B) → C (A C) → B - Contraposition with adjunctor:

Rule: A B C B A C

Equivalence: A → (B C) B → ( A C)


23: Deriving Valid Implications III: Transportation Rule

A third and final way of deriving a valid implication from another involves a combination of the two

procedures above, i.e. contraposition and dualisation.

Contraposition, recall, can involve a conjunction for the premise, or an adjunction for the conclusion:

i) A B < C _________? ii) A < B C ________________?

1. Contraposition: A C B B < A C OR OR

B C < A C < A B

2. Duality: B < A C A C < B OR OR

A < B C A B < C Thus we get the following transportation rules:

i) A B < C B < A C ii) A < B C A C < B OR OR

i) A B < C A < B C ii) A < B C A B < C Notice how we still have the basic pattern of conjunctions in the premise and adjunctions in the

conclusion. With transportation, we start with a conjunction in the premise [A B < C] and get an

adjunction in the conclusion [A < B C]. Or else, we start with an adjunction in the conclusion [A <

B C] and get a conjunction for the premise [A B < C].

Exercise: Practise deriving more valid implications, using De Morgan’s. Then practise contraposition and dualising them to obtain more valid implications through transportation.

Summary:

- Transportation with conjunctor:

A B < C B < A C

A B < C A < B C

- Transportation with adjunctor:

A < B C A C < B

A < B C A B < C



24: A Complete System of Junctors

We have so far introduced six junctors and seen their truth-tables, however, it should be clear that

these are not the only possible truth-tables that there are. If we suppose that there is a two-place

junctor which we do not know, (let’s symbolize this with *), we can be sure that this has one of 16

possible truth-tables.

How do we know there are 16? The formula A * B has four interpretations, and each of these can

have one of two truth-values, True or False. Thus we get 2 4 = 16.

Here are the 16 possible truth-tables:

A B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 T T T T T T T T T T F F F F F F F F T F T T T T F F F F T T T T F F F F F T T T F F T T F F T T F F T T F F F F T F T F T F T F T F T F T F T F

We have already seen tables for 2, 5, 7, 8 and 10; they correspond to the junctors , →, ↔, , and ⨆

respectively.

We could define our junctor * to have rules corresponding to any of these sixteen tables. Therefore,

it is important to understand that there are more possible junctors than those we have introduced.

If you see such an unknown junctor in an exam paper, do not panic; it will be defined for you in

terms of the junctors which you do know, and therefore, its table can be worked out.

/ / are a complete system of junctors With a little thought we can see that in fact, many of the junctors we have defined are redundant, and have little or no role to play in formal logic. For instance, the disjunctor can easily be reduced to the conjunctor, adjunctor and negator (since the exclusive OR means ‘A or B, but not both’).

1. A ⊔ B >< (A B) (A B) In fact, all of the other junctors can be reduced to these three. Thus, the bisubjunctor, recall, is

equivalent to a negated disjunctor, [(A ↔ B) >< ( A ⊔ B)] and so:

2. A ↔ B >< [ (A B) (A B)] Remember that the bisubjunction is also equivalent to the conjunction of two subjunctions

[(A↔ B) >< (A B) (B A)] and therefore if we define the subjunctor as:

3. A B >< A B Another way of reducing the bisubjunction would be:

4. A ↔ B >< ( A B) ( B A)


The junctors / / are known as a complete system of junctors, because we can obtain all 16 possible truth-tables using these three junctors alone. Working out what a formula is , based o n the truth-table

Any of the sixteen truth-tables listed above can be obtained with just the use of / / . Suppose we take numbers 4, 9 and 12, and call them P, Q and R: A B P Q R T T T F F T F T T T F T F T F F F F T F There is a simple method which allows us discover what the formulae P, Q, and R are. A formula with any truth-table at all can be written in terms of the primary formulae it contains (A and B in this

case) and nothing but the junctors / / . Let’s start with R which is the simplest because it has a single intepretation which is a model - i.e. a single T: A B R T T F T F T* F T F F F F We can see that R is true precisely when A is True and B is False and thus we can write the formulae

as A B. When starting from interpretations that are models, for each interpretation that results as True, write a formula containing those primary formulae which are true (in that interpretation), the

negation of those primary formulae that are false and the junctor . In this case we only have one interpretation which is a model, and therefore, our formula is complete:

R >< A B. P is a little more complicated. We see that there are two models here, so we write two conjunctions using the same method as before: A B P T T T* T F T* F T F F F F

(A B),, (A B)


Since our proposition P is true when either (A B) or (A B) is true, we can obtain the formula by

combining them with an adjunctor (A B) (A B) When there is more than one model, insert each formula obtained in brackets and join them together with adjunctors.

P>< (A B) (A B) Starting from Intepretations which are NOT models

Suppose we now have a truth-table with three T’s as in Q above. We can use the above system to

get an adjunction with three adjuncts [(....) (....) (....)]. If there are more T’s than F’s it is

advisable to start from F’s, since we will get a simpler and more elegant formula.

If we think about the rules of our conjunctor and adjunctor, we will see that while the conjunctor has

precisely one interpretation which is a model, the adjunctor has only one interpretation which is not

a model. Thus when we were trying to obtain a formula from a T, we used a conjunctor. Now that

we are trying to obtain our formula based on that interpretation which is NOT a model, we are going

to use an andjunctor.

In other words, a b has only one possibility of being true (only one model), whereas a b has only

one possibility of being false (only one non-model). Therefore, if we have fewer T’s in our truth-

table, it is easiest to use the conjunctor, whereas if we have fewer F’s, it is easier to write an

adjunction. So to go back to our formula Q:

A B Q T T F* T F T F T T F F T

We see that Q is False, precisely when both A and B are both true. Since we’re going to write an

adjunction this time, and since a b is false when both a and b are false, this time we must negate

any primary formulae that are true to obtain A B:

When starting from interpretations that are not models, for each interpretation which is false, write a formula containing those primary formulae which are false, the negation of those primary

formulae which are true, and the junctor .

Since there is only one interpretation which is false in Q, we have our answer:

a b a b a b T T T* T T F F T F T F T F F F F*


Q >< A B

What if there were two F’s, as in proposition P above? We would write two adjunctions for each

instance of F and get:

A B P A B A B T T T T T T F T T T F T F* F T F F F* T F

When there is more than one interpretation which results as F, insert each formula obtained in brackets and join them together with conjunctors.

P >< ( A B) (A B)

Summary

From our rules we see that a b is true, precisely when both a and b are true. Therefore if we want

to move that T into any of the other interpretations, we must negate any of the primary formulae

which are F in that interpretation.

Now, a b is false, precisely when both a and b are false. Therefore if we want to move that F into

any other interpretation, we must negate any of the primary formulae which are T in that

interpretation.

Exercise: We found two formulae for the truth -table P:

P>< (A B) (A B)

P >< ( A B) (A B)

Without using truth-tables, explain why (A B) (A B) >< ( A B) (A B) is valid

a b A B a b a b a b T T T T T F F F T F F T T T F F F T F F T T T F F F F F F T T T

a b A B a b a b a b T T T T T F F F T F T T F F F T F T T F F F T T F F F F F T T T


Practise obtaining formulae for random truth-tables. Do at least three formulae with eight interpretations (i.e. three primary formulae)

Summary:

- There are 16 possible truth-tables for a two-place junction.

- / / are a complete system of junctors.

- How to work out formulae based on a truth-table.


References:

Copi, Irving M, Cohen, Carl, and Kenneth McMahon. 2011. Introduction to Logic. New York: Pearson. Hurley, Patrick J. 2008. A Concise Introduction to Logic. 10th Edition. Belmont CA, Thomson Wadsworth. Riolo, V. 2001. Introduction to Logic. 2nd edition. Msida, MUP.

Documents

Philosophy A/ Int. Level: Logic Notes and Readings · PDF filePhilosophy A/ Int. Level: Logic Notes and Readings Colette Sciberras, PhD (Dunelm) 2016 Please do not distribute without