Philosophy 148

Chapter 6

Truth-Functional Logic• Chapter 6 introduces a formal means to determine

whether arguments are valid, so that there is never any guesswork where validity is involved.

• From our earlier work on argument forms, we have seen that the structure of an argument (the way that propositions are used and joined) determines validity, not the content of the propositions themselves.

• This form of logic is called truth-functional because the truth of a set of propositions joined together is a function of 1) the truth values of the individual propositions and 2) the way that those propositions are joined.

How truth tables work:• The leftmost columns are called reference columns

and contain each individual propositional variable (or sentence), usually in alphabetical order.

• There is one remaining column for each connective (&, v, ~, →) used.

• Each row of a truth table corresponds to one possible state of affairs.

• Every possible state of affairs is represented on a truth table. The number of rows is 2n where n is the number of reference columns.

Conjunction:

• The ‘&’ symbol represents when two propositions (or sets of propositions) are joined together and asserted to both be true (e.g. ‘John runs and Bob jumps’ = ‘p & q’)

• Note that the & is not identical to the English word ‘and’– Sometimes ‘and’ is in a nonpropositional context (e.g. ‘The

priest married John and Mary’ is not ‘p & q’, it’s just ‘p’.)– Sometimes conjunction happens without an and (e.g. ‘That

ring is beautiful, but exprensive.’ asserts two things that are both true, that the ring is beautiful and that the ring is expensive, hence it is symbolized ‘p & q’)

The truth table for ‘&’

p q p & q

T T T

T F F

F T F

F F F

Disjunction

• Disjunction is symbolized by the ‘v’ sign and occurs when at least one of two propositions (or sets of propositions is asserted to be true)

• Disjunction is similar to but not identical with the English word ‘or’

Exclusive vs. Inclusive ‘or’• Sometimes when a person says something of the form “p or q”

they mean “p or q or both” and sometimes they mean “p or q and not both”. The former is an inclusive ‘or’ and the latter is exclusive.

• Most logicians default to the inclusive ‘or’. Some even claim that all uses of ‘or’ are inclusive, and it is conversational implication that makes some of them exclusive.

• In any case, it is important to examine cases where ‘or’ is used to determine which is which, because it will affect the validity of any argument that ‘or’ is used in.

• ‘v’ is by default inclusive. An exclusive ‘or’ is symbolized ‘(p v q) & ~(p & q), that is, ‘p or q and not both’

Truth Table for ‘v’

p q p v qT T TT F TF T TF F F

Negation

• It is tempting to say that “Smurfs are blue” and “Smurfs are not blue” are sentences that express two propositions.

• That is not the case. What is going on is that the same proposition is involved, and in one case the proposition is negated.

• If ‘s’ stands for “Smurfs are blue” and ‘~’ is our symbol for negation, then “Smurfs are not blue” is formalized as “~s”.

Be careful with Negation• Sometimes ‘not’ is syntactically ambiguous. Translating ‘~’ as

“it is not the case that…” can help to disentangle ambiguity.• Be careful with opposites.

– “nobody owns Mars” is the negation of “somebody owns Mars” because “it is not the case that somebody owns Mars” means the same thing as “nobody owns Mars”

– However, some opposites are not binary. Consider “Cheering for the Yankees is moral”. The negation of this should just be “It is not the case that cheering for the Yankees is moral”. Resist the temptation to translate the negation as “Cheering for the Yankees is immoral”. This is because actions that are not moral could be either amoral or immoral (but not both).

– The point is, just be strict in translating ‘~’ as “it is not the case that…”

Truth table for ‘~’

P ~p

T F

F T

Conditionals

• Conditionals are statements of the form ‘If _______ then _______’ where the blanks are filled with sentences that express propositions.

• Again, the first blank is called the antecedent and the second blank is called the consequent.

• However, the kind of conditional that ‘->’ symbolizes is called a material conditional and only refers to indicative mood conditionals in the present tense.

The (seemingly) wacko truth table for conditionals:

• Assume p is “The pitcher throws a fastball” and q is “The batter hits a home run”

• Line 2 is very straightforward. If Biff bets you that if the pitcher throws a fastball then the batter hits a home run, Biff will lose his bet if things turn out as on Line 2. But what about the others?

p q p → q

T T ?

T F F

F T ?

F F ?

Conditional Truth Table• Line 1 seems equally

straightforward. Biff wins his bet by virtue of saying something true, just as on line 2 he would lose his bet by virtue of saying something false.

• But what happens when the antecedent is false?

p q p → q

T T T

T F F

F T ?

F F ?

Conditional Truth Table• Imagine you’re watching the game

and Biff makes his bet. You accept, and you see the pitcher throw a curve ball (i.e. NOT a fastball, making the proposition ‘p’ false) yet the batter still hits a home run (making the proposition q true, as on line 3).

• The best way to interpret this is that the bet is neither won nor lost, and no money changes hands. This would also be the case if the batter has swung at and missed the curveball (line 4). But since we still have to assign one or the other truth values to ‘if p then q’ what do we do?

p q p → q

T T T

T F F

F T ?

F F ?

Do I really need to type the title for this slide again?

• We give the conditional phrase the benefit of the doubt. We have better reasons for saying that the conditional is not false than we have reasons to say it’s not true.

• Also there are more weird problems that result from taking the conditional to be false in lines 3 and 4 than result from taking it to be true.

• This comes through more clearly when dealing with conditionals that are not predictions. “If it is raining then the ground is wet” is a true conditional even if it’s not raining.

p q p → q

T T T

T F F

F T T

F F T

One of those reasons: Material Implication

• If we focus on the second line of the truth table for conditionals, which was a clear case, we can see that having a true conditional must mean that it is not the case that the antecedent is true and the consequent false. Formalized, that looks like this: – a true conditional (p → q) implies that it is not the case

that (p is true and q is false)– or: ~(p & ~q)– By DeMorgan’s Law, ~p v q (read as “It is not the case that

p unless q is the case”) is equivalent to the above

If p → q, ~(p & ~q), and ~p v q are equivalent:

p q p → q ~(p & ~q) ~p v q p & ~q ~q ~p

T T T T T F F F

T F F F F T T F

F T T T T F F T

F F T T T F T T

Necessary and Sufficient conditions:

• Necessary– the consequent of a conditional always lays down

a necessary condition for the antecedent.

• Sufficient– the antecedent of a conditional always lays down

a sufficient condition for the consequent.

interesting notes:

• A is sufficient for B if and only if B is necessary for A• If A is sufficient for B, then ~A is necessary for ~B• If A is necessary for B, then ~A is sufficient for ~B.• The totality of necessary conditions is sufficient.• A necessary AND sufficient condition is represented

with a biconditional.

If and Only If

• notice that “p if q” is q -> p• also see that “p only if q” is p -> q

– “if not q, then not p” is ~q -> ~p– by contraposition, this is equivalent to p -> q

(check it on a truth table) • “p if and only if q” means (p -> q) & (q -> p)

– sometimes a person might imply “if and only if” by only saying “only if”

Unless

• “Unless” can be tricky too.• “p unless q” could plausibly be p v q, but most

people look at ‘unless’ as a conditional, which would be ~q -> p or ~p -> q. But check them out on a truth table.

• Sometimes when people use conditionals they conversationally imply biconditionals (this is a nice way of saying they are sloppy with their language).

Pay attention to parentheses• Notice that ~a & g means something different than

~(a & g).• Substitute “Annie is rich” for ‘a’ and “Gina is happy”

for ‘g’.• The first phrase translates to “It is not the case that

Annie is rich and it is the case that Gina is happy.”• The second phrase translates to “It is not the case

that both Annie is rich and Gina is happy.”• How about ~a & ~g?

Equivalence of ~a & g, ~(a & g), ~a & ~g

a g ~a ~g a & g ~a & g ~(a & g) ~a & ~g

T T F F T F F F

T F F T F F T F

F T T F F T T F

F F T T F F T T

Exercise (assume A,B,C are T, Z is F) A v ((~B & C) v ~(~B v ~(Z v B)))

T v ((~T & T) v ~(~T v ~(F v T))) T v ((~T & T) v ~(~T v ~T)) T v ((~T & T) v ~(~T v ~T)) T v ((F & T) v ~(F v F)) T v ((F & T) v ~(F v F)) T v (F v ~F) T v (F v ~F) T v (F v T) T v (F v T) T v T T v T T

Determining Validity with Truth Tables:

• Remember, a truth table lists in each row, each possible state of affairs.

• Since the definition of validity is that it is not possible for the conclusion to be false when the premises are true, the truth table will show each instance of true premises.

• If in each of those instances the conclusion is also true, it is demonstrated that whenever the premises are true, the conclusion is, hence the argument is valid.

Setting up the table:

1. Set up the reference columns (one column for each propositional variable in the argument, in alphabetical order)

2. Create a column for each connective in the argument (and everything in the scope of that connective)

3. Label the columns which represent premises and conclusion

4. Fill out the appropriate T and F values5. Check for validity

Validity for ‘&’

conclusion premisep q p & qT T TT F FF T FF F F

How about this one?

conclusion premisep q p & qT T TT F FF T FF F F

What about this?

premise conclusionp q p & qT T TT F FF T FF F F

Notice:

premise conclusionp q p & qT T TT F FF T FF F F

How about:

premise premise conclusionp q p & qT T TT F FF T FF F F

So any substitution instances of the following will ALWAYS be valid.

p & qp

p & qq

pq___p & q

Disjunctive Syllogism

• Consider the argument:

p v q ~pq

C P2 P1p q ~p p v q

T T

T F

F T

F F

Disjunctive Syllogism

• Consider the argument:

p v q ~pq

VALID

C P2 P1p q ~p p v qT T F TT F F TF T T TF F T F

Consider the Argument:

p v qp___~q

P2 C P1p q ~q p v q

T T

T F

F T

F F

Consider the Argument:

p v qp___~q

INVALID

P2 C P1p q ~q p v qT T F TT F T TF T F TF F T F

the MODUSES

• Modus Ponens

p → qp____q

• Modus Tollens

p → q~q___~p

Valid Moduses:

P2 C P1

p q p → q

T T T

T F F

F T T

F F T

P1 P2 C

p q p → q ~q ~p

T T T F F

T F F T F

F T T F T

F F T T T

A couple common fallacies: (and trouble with conditionals in general)

• Affirming the consequent

p → qq____p

• Denying the antecedent

p → q~p___~q

Many examples in economic debates…

Hypothetical Syllogism (Chain Argument)

p → qq → rp → r

Valid