Embed Size (px)
Citation preview
Formal Methods: Sentential Logic
2.2 Semantics
Johns Hopkins University, Fall 2017
Beyond Syntax
We are now familiar with the syntax of LSENT . We know its lexicon. Weknow how these basic symbols can be combined into wffs. We know howto translate an English argument into LSENT and thereby make itssentential form explicit.
Let us now give a semantics for LSENT and imbue its wffs with meaning.
When we were doing translations, we said that ‘¬’ roughly corresponds to‘not’, ‘∧’ to ‘and’, ‘∨’ to ‘or’, ‘⊃’ to ‘if...then...’, and ‘≡’ to ‘...iff...’. Sowe have already gestured towards a semantics for these connectives. Butwe can give a more systematic semantics for LSENT using truth tables.
Truth Tables
It is commonplace in philosophy and linguistics to identify the meaning ofa sentence in context with its truth conditions. To understand asentence is to know the conditions under which it is true.
The sentential connectives in LSENT are truth functional: once thetruth values of the sentence letters in a wff ϕ are fixed, the truth value ofthe entire wff is fixed.
In a slogan: the truth value of the whole depends only on, or is afunction of, the truth value of the parts.
A truth table can exhibit this dependence; it can show how the truthvalue of ϕ depends on the truth values of the sentence letters within it.
Negation
ϕ ¬ ϕT F TF T F
A negation has the opposite truth value of the negated sentence.
The reference columns to the left of the vertical lines keep track oftruth value assignments to constituent wffs (usually sentence letters) thatdetermine the truth value of the target wff. Each row of the tablecorresponds to a different truth value assignment and all possibleassignments are considered.
The columns to the right of the vertical line are for the complex wffunder consideration. In each row, we evaluate its truth value based onthe truth value assignment to the constituent wffs in that row.
Conjunction
ϕ ψ ϕ ∧ ψT T T T TT F T F FF T F F TF F F F F
A conjunction is true just in case both of its conjuncts are true.
Note that this table has 4 rows. How many rows do we need if we havereference columns for n constituent wffs? 2n.
Truth tables get big fast. They become impractical when working withlarge formulae.
Disjunction
ϕ ψ ϕ ∨ ψT T T T TT F T T FF T F T TF F F F F
A disjunction is true just in case at least one of its disjuncts are true.
Disjunction
ϕ ψ ϕ ∨ ψT T T T TT F T T FF T F T TF F F F F
A disjunction is true just in case at least one of its disjuncts are true.
Note that the bold T is crucial. If this entry were F, then disjunctionwould have an exclusive rather than an inclusive reading.
Material Conditional
ϕ ψ ϕ ⊃ ψT T T T TT F T F FF T F T TF F F T F
A material conditional sentence is true just in case either its antecedentis false or its consequent is true.
Intuition: When is a material conditional sentence false? When truth isnot preserved from its antecedent to consequent. In all other cases, thiswff is true.
Material Biconditional
ϕ ψ ϕ ≡ ψT T T T TT F T F FF T F F TF F F T F
A material biconditional sentence is true just in case both sides of thisbiconditional have the same truth value.
Putting Truth Tables to Use
You can think of the above truth tables as giving the meaning of thesentential connectives ‘¬’, ‘∧’, ‘∨’, ‘⊃’, and ‘≡’.
Method of evaluation for sentences in LSENT : construct a truth table forϕ with reference columns for each of the sentence letters appearing in it(we will henceforth call this the truth table of ϕ).
To fill in this table, we can work from the bottom up:
We will first fill in the truth values for sentential connectives that applyonly to sentence letters.
After that, we will work on sentential connectives that apply tosubsentences whose truth values have been filled in.
And so forth.
Example
A B (A ⊃ B) ∨ (A ⊃ ¬ B)T T T T T T T F F TT F T F F T T T T FF T F T T T F T F TF F F T F T F T T F
It is a convention to boldface or circle the final column of the table.
The sentential connective above the final column is the mainconnective. Even though the wff under consideration involves othersentential connectives, it is a disjunction.
Example
A B C ¬ (A ∧ (¬ A ∨ (B ∧ C )))T T T F T T F T T T T TT T F T T F F T F T F FT F T T T F F T F F F TT F F T T F F T F F F FF T T T F F T F T T T TF T F T F F T F T T F FF F T T F F T F T F F TF F F T F F T F T F F F
Formal Logical Concepts
Truth tables are a nice tool to have in one’s toolkit. Importantly, theyallow us to introduce formal analogues of our informal logical notions like‘logical validity’, ‘logical truth’, ‘logical equivalence’, and so forth.
Having a satisfactory formal explication of validity (and related logicalconcepts) would certainly be useful. By working with the formal notion,we would have a precise way of determining when an argument has theinformal property of logical validity.
So far, we have been considering truth tables for single wffs. So we willbegin by discussing logical properties of single claims. We will then turnto logical relations between claims and logical properties of arguments.
Boolean-Necessities
Recall the definition of logical truth:
Def 2.1.5. A claim ϕ is a logical truth just in case it is impossible for ϕto be false by virtue of its logical form.
Here is the formal analogue in sentential logic:
Def 2.2.1. A sentence ϕ of LSENT is a Boolean-Necessity ortautology just in case the final column of its truth table contains only Ts
N.B. We can call a natural language sentence ϕ a tautology if asuitable translation of this sentence into LSENT that captures itssentential form is a tautology.
N.B. We are replacing the imprecise ‘impossible for ϕ to be false’ in ouroriginal definition with the more precise ‘the final column of the truthtable for ϕ contains only Ts’.
Boolean-Necessities
(1) Pugsley is a member of the Addams Family or he isn’t.
A A ∨ ¬ AT T T F TF F T T F
Boolean-Impossibilities
Recall the definition of logical falsehood:
Def 2.1.6. A claim ϕ is a logical falsehood just in case it is impossiblefor ϕ to be true by virtue of its logical form.
Here is the formal analogue in sentential logic:
Def 2.2.2. A sentence ϕ of LSENT is a Boolean-Impossibility just incase the final column of its truth table contains only Fs.
Boolean-Impossibilities
(2) Pugsley is a member of the Addams Family and he isn’t.
A A ∧ ¬ AT T F F TF F F T F
Boolean-Possibilities
Recall the definition of logical possibility:
Def 2.1.7. A claim ϕ is logically possible just in case it is possible forϕ to be true by virtue of its logical form.
Here is the formal analogue in sentential logic:
Def 2.2.3. A sentence ϕ of LSENT is Boolean-Possible just in case thefinal column of its truth table contains at least some Ts.
More Examples
A B (A ⊃ B) ∧ (A ∧ ¬ B)T T T T T F T F F TT F T F F F T T T FF T F T T F F F F TF F F T F F F F T F
The target wff is a Boolean-Impossibility.
More Examples
A B (A ≡ B) ⊃ ((A ∧ B) ∨ (¬ A ∧ ¬ B))T T T T T T T T T T F T F F TT F T F F T T F F F F T F T FF T F F T T F F T F T F F F TF F F T F T F F F T T F T T F
The target wff is a Boolean-Necessity. It is also Boolean-Possible.
Joint Truth Tables
We have been looking at single claims and sentences in isolation. But todevelop formal analogues of logical concepts like logical validity andlogical equivalence, we need to construct joint truth tables for multiplesentences at the same time.
A ¬ A ¬ ¬ A ¬ ¬ ¬ AT F T T F T F T F TF T F F T F T F T F
We now have a different section to the right of the reference columns foreach wff under consideration. Each section has its own final column.
Tautological Equivalence
Recall the definition of logical equivalence:
Def 2.1.8. Claims ϕ and ψ are logically equivalent just in case it isimpossible for their truth values to differ by virtue of logical form.
Here is the formal analogue in sentential logic:
Def 2.2.4. Sentences ϕ and ψ of LSENT are tautologically equivalentjust in case there is no row in their joint truth table where the truthvalues differ in their final columns.
Tautological Equivalence
(3) Gomez is married to Morticia.
(4) It is not the case that Gomez is not married to Morticia.
M M ¬ ¬ MT T T F TF F F T F
Tautological Equivalence
(5) If Fester is home, then Thing is home.
(6) Either Fester is not home or Thing is home.
F T F ⊃ T ¬ F ∨ TT T T T T F T T TT F T F F F T F FF T F T T T F T TF F F T F T F T F
Tautological Validity and ConsequenceRecall the definitions of logical validity and consequence:
Def 2.1.2. The argument from ϕ1, ..., ϕn to ψ is logically valid just incase it is impossible for each of ϕ1, ..., ϕn to be true and for ψ to be falseby virtue of logical form.
Def 2.1.4. A claim ψ is a logical consequence of the possibly emptyset of claims {ϕ1, ..., ϕn} just in case the argument with premisesϕ1, ..., ϕn and conclusion ψ is logically valid.
Here are their formal analogues in sentential logic:
Def 2.2.5. The argument from ϕ1, ..., ϕn to ψ in LSENT istautologically valid just in case there is no row in the joint truth tablefor the premises ϕ1, ..., ϕn and conclusion ψ where the final column foreach of ϕ1, ..., ϕn contains T but the final column for ψ contains F.
Def 2.2.6. A sentence ψ of LSENT is a tautological consequence ofthe possibly empty set of sentences {ϕ1, ..., ϕn} of LSENT just in case theargument from ϕ1, ..., ϕn to ψ is tautologically valid.
Tautological Validity and Consequence
(7) If Pugsley is a member of the Addams Family, then Pugsley delightsin the macabre. Puglsey is a member of the Addams Family. So,Pugsley delights in the macabre.
A D A ⊃ D A DT T T T T T TT F T F F T FF T F T T F TF F F T F F F
Tautological Validity and Consequence
(8) Gomez is married to Morticia. After all, Gomez is married toMorticia or Fester, and he is not married to Fester.
M F (M ∨ F ) ∧ ¬ F MT T T T T F F T TT F T T F T T F TF T F T T F F T FF F F F F F T F F
Tautological Invalidity
An argument is tautologically invalid just in case its truth table has atleast one bad row where the final column for each of the premisescontains T but the final column for the conclusion contains F.
(9) Wednesday is a member of the Addams Family if and only ifWednesday delights in the macabre. Wednesday delights in themacabre. So, Wednesday is not a member of the Addams Family.
A D A ≡ D D ¬ AT T T T T T F T ← bad rowT F T F F F F TF T F F T T T FF F F T F F T F
Tautological Invalidity
(10) Lurch is home or both Fester and Thing are home. Hence, Thing ishome and either Lurch or Fester is home.
L F T L ∨ (F ∧ T ) T ∧ (L ∨ F )T T T T T T T T T T T T TT T F T T T F F F F T T T ← bad rowT F T T T F F T T T T T FT F F T T F F F F F T T F ← bad rowF T T F T T T T T T F T TF T F F F T F F F F F T TF F T F F F F T T F F F FF F F F F F F F F F F F F
ModelsBefore moving on, we are going to reformulate the semantics for LSENT .This will connect our work here with material later in the course wherewe develop more sophisticated logical systems.
Roughly, a model M for a language L is the basic information requiredto determine the truth values of all sentences in this language.
What is a model for LSENT ?
AtLSENT= {A,B, ...} is the set of atoms in LSENT .
SLSENTis the set of all sentences (wffs) in LSENT .
Def 2.2.7. A model M = V for LSENT consists of a valuation functionV : AtLSENT
→ {T ,F} mapping each sentence letter p ∈ AtLSENTto a
truth value.
As we have seen, if you know the truth values of all sentence letters inAtLSENT
, you can determine the truth values of all wffs in SLSENT. Why?
Because the connectives ‘¬’, ‘∧’, ‘∨’, ‘⊃’, and ‘≡’ are truth functional.
Models
A B (A ⊃ B) ∨ AT T T T T T TT F T F F T TF T F T T T FF F F T F T F
Each row of a truth table corresponds to the equivalence class of modelsfor LSENT that assign the sentence letters in the table the truth values inthat row.
Row 1: Models where V1(A) = V1(B) = T .
Row 2: Models where V2(A) = T but V2(B) = F .
And so forth.
Models
Def 2.2.8. A recursive specification of truth in model M = V extendsV to the full interpretation function J KM : SLSENT
→ {T ,F} for LSENT
mapping each sentence ϕ ∈ SLSENTto a truth value:
JpKM = T iff V(p) = TJ¬ϕKM = T iff JϕKM = F
Jϕ ∧ ψKM = T iff JϕKM = T and JψKM = TJϕ ∨ ψKM = T iff JϕKM = T or JψKM = TJϕ ⊃ ψKM = T iff JϕKM = F or JψKM = TJϕ ≡ ψKM = T iff JϕKM = JψKM
These clauses capture the above truth table semantics in a concise way.
Example
Consider a model M = V where V(A) = T , V(B) = T , and V(C ) = F .
J(A ∨ B) ∧ (B ∨ C )KM = ?
Example
Consider a modelM = 〈V〉 where V(A) = T , V(B) = T , and V(C ) = F .
J(A ∨ B) ∧ (B ∨ C )KM = T. Why?
By the semantic clause for sentence letters, JAKM = T , JBKM = T , andJCKM = F .
By the semantic clause for ∨, JA ∨ BKM = T and JB ∨ CKM = T .
By the semantic clause for ∧, J(A ∨ B) ∧ (B ∨ C )KM = T .
Logical Concepts Redux
With this notion of truth in/satisfaction by a model in hand, we canreformulate our definitions of Boolean-Necessity, tautological equivalence,and so forth.
Def 2.2.1. A sentence ϕ of LSENT is a Boolean-Necessity ortautology just in case JϕKM = T for each model M of LSENT .
Def 2.2.2. A sentence ϕ of LSENT is a Boolean-Impossibility just incase JϕKM = F for each model M of LSENT .
Def 2.2.3. A sentence ϕ of LSENT is Boolean-Possible just in caseJϕKM = T for some model M of LSENT .
Logical Concepts Redux
Def 2.2.4. Sentences ϕ and ψ of LSENT are tautologically equivalentjust in case JϕKM = JψKM for each model M of LSENT .
Def 2.2.5. The argument from ϕ1, ..., ϕn to ψ in LSENT istautologically valid just in case there is no model M for LSENT whereJϕ1KM = ... = JϕnKM = T but JψKM = F .
Such a model where each of the premises are true but the conclusion isfalse is a counter-model to the argument.
Truth Functional Completeness
We have now seen that a great many English sentences can be translatedinto LSENT with the five connectives ‘¬’, ‘∧’, ‘∨’, ‘⊃’, and ‘≡’. But arethese connectives enough?
There are non-truth functional expressions in English like ‘...because...’and ‘It is necessarily the case that...’ that cannot be captured in LSENT .But is there any truth functional expression that cannot be translatedinto LSENT using the sentential connectives? If so, this is an unfortunatelimitation of LSENT .
How can we even answer this question?
Well, the meaning of any truth functional expression is given by a truthtable. So we must examine whether all possible truth tables can berecovered in LSENT .
Truth Functional Completeness
There are four possible 1-place connectives:
ϕ F ϕT T TF T F
ϕ � ϕT T TF F F
ϕ N ϕT F TF T F
ϕ � ϕT F TF F F
Note that ‘N’ is effectively ‘¬’. But what about the other three 1-placeconnectives? Can we express them with the existing connectives?
Fϕ is equivalent to ϕ ∨ ¬ϕ.
�ϕ is equivalent to ϕ.
�ϕ is equivalent to ϕ ∧ ¬ϕ.
Thus, every 1-ary truth functional connective can be captured in LSENT .
Truth Functional Completeness
Let us turn to the 2-place connectives.
ϕ ψ ϕ • ψT T T T/F TT F T T/F FF T F T/F TF F F T/F F
There are 16 possible truth tables. The sentential connectives ‘∧’, ‘∨’,‘⊃’, and ‘≡’ cover 4 of these options. But what about the remaining 12?
Truth Functional Completeness
ϕ ψ (ϕ ∧ ¬ ϕ) ∧ (ψ ∧ ¬ ψ)T T T F F T F T F F TT F T F F T F F F T FF T F F T F F T F F TF F F F T F F F F T F
The table with all Fs is covered by (ϕ ∧ ¬ϕ) ∧ (ψ ∧ ¬ψ).
Truth Functional Completeness
ϕ ψ ϕ ∧ ψ ϕ ∧ ¬ ψ ¬ ϕ ∧ ψ ¬ ϕ ∧ ¬ ψT T T T T T F F T F T F T F T F F TT F T F F T T T F F T F F F T F T FF T F F T F F F T T F T T T F F F TF F F F F F F T F T F F F T F T T F
The 4 tables with exactly one T are covered by ϕ ∧ ψ, ϕ ∧ ¬ψ, ¬ϕ ∧ ψ,and ¬ϕ ∧ ¬ψ.
Truth Functional Completeness
ϕ ψ (ϕ ∧ ψ) ∨ (ϕ ∧ ¬ ψ)T T T T T T T F F TT F T F F T T T T FF T F F T F F F F TF F F F F F F F T F
The tables with exactly two Ts are covered by disjunctions of theprevious sentences.
And so forth for the truth tables with three and four Ts.
In fact, arbitrary n-place truth functional connectives can be handled in asimilar fashion.
N.B. We are effectively arguing that every sentence in LSENT istautologically equivalent to one in disjunctive normal form (seesupplementary slides for more on normal forms).
Truth Functional Completeness
Since we have appealed only to ‘¬’, ‘∧’, and ‘∨’, we have the followingresult:
Lem 2.2.1. The set of connectives {¬,∧,∨} is truth functionallycomplete.
That is, every truth functional connective can be captured with theseconnectives.
Truth Functional Completeness
Note that ‘∧’ is definable in terms of ‘¬’ and ‘∨’:
ϕ ∧ ψ ⇔T ¬(¬ϕ ∨ ¬ψ).(notation: ⇔T designates tautological equivalence)
Together with Lem 2.2.1, this implies the following:
Lem 2.2.2. The set of connectives {¬,∨} is truth functionally complete.
Moreover, ‘∨’ is definable in terms of ‘¬’ and ‘∧’:
ϕ ∨ ψ ⇔T ¬(¬ϕ ∧ ¬ψ).
Together with Lem 2.2.1, this implies the following:
Lem 2.2.3. The set of connectives {¬,∧} is truth functionally complete.
There are more results in the vicinity. Some are left as an exercise.
Supplementary Material
Substitution of Equivalents
Knowing that two sentences ϕ and ψ are tautologically equivalent(notation: ϕ⇔T ψ) can be very useful. Why?
Suppose that some complex sentence χ of LSENT contains ϕ as asubsentence.
Now suppose that we substitute ψ for ϕ in χ to obtain the sentence ξ.
Since ϕ⇔T ψ, χ⇔T ξ as well.
This can be proven rigorously but we will just take the following principlefor granted:
Substitution of Tautological Equivalents (STE): Substitutingtautologically equivalent sentences preserves tautological equivalence.
Double Negation
Let us focus on a few important tautological equivalences.
ϕ ϕ ¬ ¬ ϕT T T F TF F F T F
Double Negation (DN): ϕ⇔T ¬¬ϕ.
DeMorgan’s First Law
ϕ ψ ¬ (ϕ ∨ ψ) ¬ ϕ ∧ ¬ ψT T F T T T F T F F TT F F T T F F T F T FF T F F T T T F F F TF F T F F F T F T T F
DeMorgan’s First Law (DeM1): ¬(ϕ ∨ ψ)⇔T ¬ϕ ∧ ¬ψ.
DeMorgan’s Second Law
ϕ ψ ¬ (ϕ ∧ ψ) ¬ ϕ ∨ ¬ ψT T F T T T F T F F TT F T T F F F T T T FF T T F F T T F T F TF F T F F F T F T T F
DeMorgan’s Second Law (DeM2): ¬(ϕ ∧ ψ)⇔T ¬ϕ ∨ ¬ψ.
First Distribution Law
ϕ ψ χ ϕ ∧ (ψ ∨ χ) (ϕ ∧ ψ) ∨ (ϕ ∧ χ)T T T T T T T T T T T T T T TT T F T T T T F T T T T T F FT F T T T F T T T F F T T T TT F F T F F F F T F F F T F FF T T F F T T T F F T F F F TF T F F F T T F F F T F F F FF F T F F F T T F F F F F F TF F F F F F F F F F F F F F F
Distribution of ∧ over ∨ (Dist1): ϕ ∧ (ψ ∨ χ)⇔T (ϕ ∧ ψ) ∨ (ϕ ∧ χ).
Second Distribution Law
ϕ ψ χ ϕ ∨ (ψ ∧ χ) (ϕ ∨ ψ) ∧ (ϕ ∨ χ)T T T T T T T T T T T T T T TT T F T T T F F T T T T T T FT F T T T F F T T T F T T T TT F F T T F F F T T F T T T FF T T F T T T T F T T T F T TF T F F F T F F F T T F F F FF F T F F F F T F F F F F T TF F F F F F F F F F F F F F F
Distribution of ∨ over ∧ (Dist2): ϕ ∨ (ψ ∧ χ)⇔T (ϕ ∨ ψ) ∧ (ϕ ∨ χ).
Disjunctive Normal Form
By applying these principles, we can convert every sentence of LSENT
into certain special forms.
Def. A literal or basic sentence of LSENT is a sentence letter or anegated sentence letter.
Examples: A and ¬A are literals but ¬¬A and (A ∧ B) are not.
Def. A sentence ϕ of LSENT is in disjunctive normal form (DNF) justin case ϕ is a disjunction of one or more conjunctions of one or moreliterals.
Note that we are here allowing for ‘disjunctions’ with one disjunct and for‘conjunctions’ with one conjunct.
Disjunctive Normal FormAre the following sentences in DNF?
(A ∧ B) ∨ (¬C ∧ ¬D) Yes.
(A ∨ B) ∧ (¬C ∨ ¬D) No.
A ∧ B Yes.
A ∨ B Yes.
Each sentence of LSENT is tautologically equivalent to one in DNF.
Example: (A ∨ B) ∧ (¬C ∨ ¬D) is not in DNF.
By Dist1, this is equivalent to ((A ∨ B) ∧ ¬C ) ∨ ((A ∨ B) ∧ ¬D).
By STE and Dist1, this is equivalent to((A ∧ ¬C ) ∨ (B ∧ ¬C )) ∨ ((A ∨ B) ∧ ¬D).
By STE and Dist1, this is equivalent to(A ∧ ¬C ) ∨ (B ∧ ¬C ) ∨ (A ∧ ¬D) ∨ (B ∧ ¬D) in DNF.
Conjunctive Normal Form
Def. A sentence ϕ of LSENT is in conjunctive normal form (CNF) justin case ϕ is a conjunction of one or more disjunctions of one or moreliterals.
Note that we are here allowing for ‘conjunctions’ with one conjunct andfor ‘disjunctions’ with one disjunct.
Conjunctive Normal FormAre the following sentences in CNF?
(A ∨ B) ∧ (¬C ∨ ¬D) Yes.
(A ∧ B) ∨ (¬C ∧ ¬D) No.
A ∧ B Yes.
A ∨ B Yes.
Note that the last two sentences are in both DNF and CNF.
Each sentence of LSENT is tautologically equivalent to one in CNF.
Example: (A ∧ B) ∨ (¬C ∧ ¬D) is not in CNF.
By Dist2, this is equivalent to ((A ∧ B) ∨ ¬C ) ∧ ((A ∧ B) ∨ ¬D).
By STE and Dist2, this is equivalent to((A ∨ ¬C ) ∧ (B ∨ ¬C )) ∧ ((A ∧ B) ∨ ¬D).
By STE and Dist2, this is equivalent to(A ∨ ¬C ) ∧ (B ∨ ¬C ) ∧ (A ∨ ¬D) ∧ (B ∨ ¬D) in CNF.
