Proofs for Boolean Logic

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Soundness and completeness

Logik f¨ ur Informatiker Logic for computer scientists

Till Mossakowski

WiSe 2013/14

Till Mossakowski Logic 1/ 35

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Proofs for Boolean Logic Conditionals Soundness and completeness

Proofs for Boolean Logic

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Summary of Rules

Propositional rules (

F^T

)

Conjunction Introduction (∧ Intro)

P₁

⇓ P_n

...

. P₁∧. . .∧P_n

Conjunction Elimination (∧ Elim)

P₁∧. . . ∧P_i∧. . .∧P_n ...

. P_i

Disjunction Introduction (∨ Intro)

P_i ...

. P₁∨. . .∨P_i∨. . .∨P_n

Disjunction Elimination (∨ Elim)

P₁∨. . . ∨P_n ...

P₁ ... S

⇓ Pn

... S ...

. S

557

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Summary of Rules

Propositional rules (

F^T

)

P1

⇓ P_n

...

. P₁∧. . .∧P_n

P1∧. . . ∧Pi∧. . .∧Pn

... . P_i

Pi

...

. P₁∨. . .∨P_i∨. . .∨P_n

P1∨. . . ∨Pn

... P₁

... S

⇓ Pn

... S ...

. S

557

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Summary of Rules

Propositional rules (

F^T

)

Conjunction Introduction (∧Intro)

P1

⇓ P_n

...

. P1∧. . .∧Pn

P1∧. . . ∧Pi∧. . .∧Pn

... . Pi

Disjunction Introduction (∨Intro)

P_i ...

. P1∨. . .∨Pi∨. . .∨Pn

P₁∨. . . ∨P_n ...

P₁ ... S

⇓ P_n

... S ...

. S

557

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Summary of Rules

Propositional rules (

F^T

)

P₁

⇓ Pn

...

. P1∧. . .∧Pn

P₁∧. . . ∧P_i∧. . .∧P_n ...

. Pi

P_i ...

. P1∨. . .∨Pi∨. . .∨Pn

P₁∨. . . ∨P_n ...

P1

... S

⇓ P_n

... S ...

. S

557

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Negation Introduction (¬ Intro)

P ...

⊥

. ¬P

Negation Elimination (¬ Elim)

¬¬P ...

. P

⊥Introduction (⊥ Intro)

P...

¬P ...

. ⊥

⊥Elimination (⊥Elim)

⊥...

. P

Conditional Introduction (→ Intro)

P ... Q

. P→Q

Conditional Elimination (→Elim)

P→Q ... P...

. Q

Summary of Rules

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558 /Summary of Rules

Negation Introduction (¬Intro)

P ...

⊥

. ¬P

¬¬P ...

. P

⊥Introduction (⊥Intro)

P...

¬P ...

. ⊥

⊥Elimination (⊥ Elim)

⊥...

. P

P ... Q

. P→Q

Conditional Elimination (→ Elim)

P→Q ... P...

. Q

Summary of Rules

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P ...

⊥

. ¬P

¬¬P ...

. P

P...

¬P ...

. ⊥

⊥...

. P

P ... Q

. P→Q

P→Q ... P...

. Q

Summary of Rules

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P ...

⊥

. ¬P

¬¬P ...

. P

P...

¬P ...

. ⊥

⊥...

. P

Conditional Introduction (→Intro)

P ... Q

. P→Q

P→Q ... P...

. Q

Summary of Rules

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Strategies and tactics in Fitch

1 Understandwhat the sentences are saying.

2 Decidewhether you think the conclusion follows from the premises.

3 If you think it does not follow, or are not sure, try to find a counterexample.

4 If you think it does follow, try to give aninformal proof.

5 If aformal proofis called for, use theinformal proof to guideyou in finding one.

6 In giving consequence proofs, both formal and informal, don’t forget the tactic ofworking backwards.

7 In working backwards, though, always check that yourintermediate goals are consequencesof the available information.

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Strategies in Fitch, cont’d

Always try to matchthe situation in your proof with the rules in the book (see book appendix for a complete list)

Look at the main connectivein a premise, apply the corresponding elimination rule (forwards)

Or: look at themain connective in the conclusion, apply the corresponding introduction rule(backwards)

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Conditionals

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Conditionals

178 / Conditionals

do, however, make it much easier to say and prove certain things, and so are valuable additions to the language.

Section 7.1

Material conditional symbol: →

The symbol → is used to combine two sentences P and Q to form a new sentence P → Q, called a material conditional. The sentence P is called the antecedentof the conditional, andQis called theconsequentof the conditional.

We will discuss the English counterparts of this symbol after we explain its meaning.

Semantics and the game rule for the conditional

The sentence P →Q is true if and only if either P is false or Q is true (or both). This can be summarized by the following truth table.

P Q P →Q

t t T

t f F

f t T

f f T

truth table for →

A second’s thought shows that P → Q is really just another way of saying

¬P∨Q. Tarski’s World in fact treats the former as an abbreviation of the latter. In particular, in playing the game, Tarski’s World simply replaces a game rule for →

statement of the formP → Q by its equivalent ¬P∨Q.

Remember

1. IfP and Q are sentences of fol, then so is P →Q.

2. The sentence P → Q is false in only one case: if the antecedent P is true and the consequentQ is false. Otherwise, it is true.

English forms of the material conditional

We can come fairly close to an adequate English rendering of the material conditionalP → Q with the sentence If P then Q. At any rate, it is clear that if . . . then

Chapter 7

Game rule: P →Q is replaced by¬P ∨Q.

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Formalisation of conditional sentences

The following English constructions are all translatedP →Q:

IfP thenQ;Q ifP;P only ifQ; andProvidedP,Q.

Unless P,Q andQ unlessP are translated: ¬P →Q. Q is a logical consequence ofP₁, . . . ,P_n if and only if the sentence (P1∧ · · · ∧Pn)→Q is a logical truth.

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P ...

⊥

. ¬P

¬¬P ...

. P

P...

¬P ...

. ⊥

⊥ Elimination (⊥Elim)

⊥...

. P

Conditional Introduction (→Intro)

P ... Q

. P→Q

P→Q ... P...

. Q

Summary of Rules

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P ...

⊥

. ¬P

¬¬P ...

. P

P...

¬P ...

. ⊥

⊥...

. P

P ... Q

. P→Q

P→Q ... P...

. Q

Summary of Rules

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Biconditionals

182 / Conditionals

means of the biconditional: P ↔ Q. A sentence of the form P ↔ Q is true if and only if P and Q have the same truth value, that is, either they are both true or both false. In English this is commonly expressed using the expression if and only if. So, for example, the sentence Max is home if and only if Claire if and only if

is at the library would be translated as:

Home(max)↔ Library(claire)

Mathematicians and logicians often write “iff” as an abbreviation for “if iff

and only if.” Upon encountering this, students and typesetters generally con- clude it’s a spelling mistake, to the consternation of the authors. But in fact it is shorthand for the biconditional. Mathematicians also use “just in case” as a just in case

way of expressing the biconditional. Thus the mathematical claims n is even iff n² is even, and n is even just in case n² is even, would both be translated as:

Even(n)↔ Even(n²)

This use of “just in case” is, we admit, one of the more bizarre quirks of mathematicians, having nothing much to do with the ordinary meaning of this phrase. In this book, we use the phrase in the mathematician’s sense, just in case you were wondering.

An important fact about the biconditional symbol is that two sentences P and Q are logically equivalent if and only if the biconditional formed from them, P ↔Q, is a logical truth. Another way of putting this is to say that P ⇔ Q is true if and only if the fol sentence P ↔Q is logically necessary.

So, for example, we can express one of the DeMorgan laws by saying that the following sentence is a logical truth:

¬(P∨Q)↔ (¬P∧ ¬Q)

This observation makes it tempting to confuse the symbols↔ and⇔. This

↔ vs. ⇔

temptation must be resisted. The former is a truth-functional connective of fol, while the latter is an abbreviation of “is logically equivalent to.” It is not a truth-functional connective and is not an expression of fol.

Semantics and the game rule for ↔

The semantics for the biconditional is given by the following truth table.

P Q P↔ Q

t t T

t f F

f t F

f f T

truth table for ↔

Chapter 7

Game rule: P ↔Q is replaced by (P →Q)∧(Q →P).

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Biconditionals and logical equivalence

P and Q are logically equivalent (P ⇔Q) if and only if

the sentenceP ↔Q is a logical truth.

Note thatP ⇔Q is ameta statement, not a sentence of PL1.

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Conversational implicature

“Max is home unless Claire is at the library” can be formalised as

¬Library(claire)→Home(max) but many people would formalise it as

¬Library(claire)↔Home(max) The part

¬Library(claire)←Home(max)

is calledconservational implicature. It is possibly, but not necessarily meant by the English sentence.

An addition cancancelthe implicature:

“On the other hand, if Claire is at the library, I have no idea where Max is.”

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Truth-functional completeness

Definition

A logical connective istruth-functional, if the truth value of a complex sentence built up using these connectives depends on nothing more than the truth values of the simpler sentences from which it is built.

Truth-functional: ∧,∨,¬,←,↔

Not truth-functional: because, after, necessarily Definition

A set of logical connectives istruth-functionally complete, if it suffices to express every truth-functional connective.

Theorem

The set{∧,∨,¬} is truth-functionally complete.

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Example: a binary truth-functional connective

P Q Neither P nor Q

T T F

T F F

F T F

F F T ¬P∧ ¬Q

Neither P nor Q can be expressed as¬P ∧ ¬Q.

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Example: a ternary truth-functional connective

P Q R ♣(P,Q,R)

T T T T P∧Q∧R

T T F T P∧Q∧ ¬R

T F T F

T F F F

F T T T ¬P∧Q∧R

F T F F

F F T T ¬P∧ ¬Q∧R

F F F F

♣(P,Q,R) can be expressed as

(P∧Q∧R)∨(P ∧Q∧ ¬R)∨(¬P ∧Q∧R)∨(¬P∧ ¬Q∧R).

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Logical equivalences of (bi)conditionals

P →Q ⇔ ¬Q → ¬P P →Q ⇔ ¬P ∨Q

¬(P →Q) ⇔ P ∧ ¬Q

P ↔Q ⇔ (P →Q)∧(Q →P) P ↔Q ⇔ (P∧Q)∨(¬P∧ ¬Q)

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First-order rules (F)/ 559

Biconditional Introduction (↔ Intro)

P ... Q Q ... P

. P↔Q

Biconditional Elimination (↔ Elim)

P↔Q(or Q↔P) ...

P...

. Q

Reiteration (Reit)

P...

. P

First-order rules (

F

)

Identity Introduction (=Intro)

. n=n

Identity Elimination (= Elim)

P(n)... n=m

...

. P(m)

First-order rules (F)

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First-order rules (F)/ 559

P ... Q Q ... P

. P↔Q

P↔Q(orQ↔P) ...

P...

. Q

Reiteration (Reit)

P...

. P

First-order rules (

F

)

. n=n

Identity Elimination (=Elim)

P(n)... n=m

...

. P(m)

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P ... Q Q ... P

. P↔Q

P↔Q(orQ↔P) ...

P...

. Q

Reiteration (Reit)

P...

. P

First-order rules (

F

)

. n=n

Identity Elimination (=Elim)

P(n)... n=m

...

. P(m)

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Soundness and completeness

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Object and meta theory

Object theory = reasoningwithina formal proof system (e.g. Fitch)

Meta theory = reasoningabout a formal proof system

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Tautological consequence

A sentenceS is a tautological consequenceof a set of sentences T, written

T |=T S,

if all valuations of atomic formulas with truth values that make all sentences inT true also makeS true.

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Propositional proofs

S isFT-provable fromT, written T `T S,

if there is a formal proof ofS with premises drawn from T using the elimination and introduction rules for∨,∧,¬,→,↔and ⊥. Again note: T may be infinite.

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Soundness

Theorem 1. The proof calculusFT is sound, i.e. if T `T S,

then

T |=T S.

Proof: Book: by contradiction, using the first invalid step.

Here: by induction on the length of the proof.

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Soundness

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Completeness

Theorem 2(Bernays, Post). The proof calculusFT is complete, i.e. if

T |=T S, then

T `T S.

This theorem will be proved later in the lecture.

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