Proofs for Boolean Logic

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Formal Proofs and Booelan Logic

Logik f¨ ur Informatiker Logic for computer scientists

Till Mossakowski

WiSe 2013/14

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Proofs for Boolean Logic

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Logical consequence

1 Q is a logical consequence ofP₁, . . . ,P_n, if all worlds that

makeP₁, . . . ,P_n true also makeQ true.

2 Q is a tautological consequenceofP₁, . . . ,P_n, if all valuations of atomic formulas with truth values that make P1, . . . ,Pn

true also make Q true.

3 Q is a TW-logical consequence ofP1, . . . ,Pn, if all worlds from Tarski’s world that makeP₁, . . . ,P_n true also makeQ true.

The difference lies in theset of worlds that is considered:

1 all worlds (whatever this exactly means . . . )

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Proofs

With proofs, we try to show (tauto)logical consequence Truth-table method can lead to very large tables, proofs are often shorter

Proofs are also available for consequence in full first-order logic, not only for tautological consequence

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Limits of the truth-table method

1 truth-table method leads to exponentially growingtables 20 atomic sentences⇒more than 1.000.000 rows

2 truth-table method cannot be extended to first-order logic model checkingcan overcome the first limitation (up to 1.000.000 atomic sentences)

proofscan overcome both limitations

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Proofs

A proof consists of a sequence ofproof steps Each proof step is known to be valid and should

be significant but easily understood, ininformalproofs, follow someproof rule, in formalproofs.

Some valid patterns of inference that generally go unmentioned in informal (but not in formal) proofs:

FromP∧Q, inferP.

FromPandQ, inferP∧Q.

FromP, inferP∨Q.

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Proof by cases (disjunction elimination)

To proveS fromP₁∨. . .∨P_n, proveS from each of P₁, . . . ,P_n. Claim: there are irrational numbers b andc such thatb^c is rational.

Proof: √ 2

√2

is either rational or irrational.

Case 1: If √ 2

√

2 is rational: takeb=c =√ 2.

Case 2: If √ 2

√

2 is irrational: takeb =√ 2

√

2 andc =√ 2.

Then b^c= (√ 2

√2

)

√ 2=√

2⁽

√2·√ 2) =√

2² = 2.

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Proof by contradiction

To prove¬S, assume S and prove a contradiction ⊥. (⊥may be infered from P and¬P.)

AssumeCube(c)∨Dodec(c) andTet(b).

Claim: ¬(b=c).

Proof: Let us assumeb =c.

Case 1: If Cube(c), then by b=c, also Cube(b), which contradictsTet(b).

Case 2: Dodec(c) similarly contradictsTet(b).

In both case, we arrive at a contradiction. Hence, our assumption b=c cannot be true, thus¬(b=c).

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Arguments with inconsistent premises

A proof of a contradiction⊥from premisesP1, . . . ,Pn (without additional assumptions) shows that the premises areinconsistent.

An argument with inconsistent premises is alwaysvalid, but more importantly, alwaysunsound.

Home(max)∨Home(claire)

¬Home(max)

¬Home(claire)

Home(max)∧Happy(carl)

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Arguments without premises

A proof without any premises shows that its conclusion is alogical truth.

Example: ¬(P ∧ ¬P).

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Formal Proofs and

Boolean Logic

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Formal proofs in Fitch

Well-defined set offormal proof rules

Formal proofs in Fitch can be mechanically checked For each connective, there is

anintroduction rule, e.g. “fromP, inferP∨Q”.

anelimination rule, e.g. “fromP∧Q, inferP”.

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Proofs for Boolean Logic Formal Proofs and Booelan Logic

Summary of Rules

Propositional rules ( F

^T

)

Conjunction Introduction (∧ Intro)

P₁

⇓ Pn

...

. P1∧. . .∧Pn

Conjunction Elimination (∧ Elim)

P₁∧. . . ∧P_i∧. . .∧P_n ...

. Pi

Disjunction Introduction (∨ Intro)

P_i ...

. P1∨. . .∨Pi∨. . .∨Pn

Disjunction Elimination (∨ Elim)

P₁∨. . . ∨P_n ...

P1

... S

⇓ P_n

... S ...

. S

Till Mossakowski Logic 13/ 24

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Summary of Rules

Propositional rules ( F

^T

)

P₁

⇓ P_n

...

. P1∧. . .∧Pn

P₁∧. . . ∧P_i∧. . .∧P_n ...

. Pi

P_i ...

. P1∨. . .∨Pi∨. . .∨Pn

P₁∨. . . ∨P_n ...

P₁ ... S

⇓ P_n

... S ...

. S

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Summary of Rules

Propositional rules ( F

^T

)

Conjunction Introduction (∧Intro)

P₁

⇓ P_n

...

. P₁∧. . .∧P_n

P₁∧. . . ∧P_i∧. . .∧P_n ...

. P_i

Disjunction Introduction (∨Intro)

P_i ...

. P₁∨. . .∨P_i∨. . .∨P_n

P₁∨. . . ∨P_n ...

P₁ ... S

⇓ P_n

... S ...

. S

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Summary of Rules

Propositional rules ( F

^T

)

P₁

⇓ P_n

...

. P₁∧. . .∧P_n

P₁∧. . . ∧P_i∧. . .∧P_n ...

. P_i

P_i ...

. P₁∨. . .∨P_i∨. . .∨P_n

P₁∨. . . ∨P_n ...

P₁ ... S

⇓ Pn

... S ...

. S

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The proper use of subproofs

In the following two exercises, determine whether the sentences are consistent. If they are, use Tarski’s World to build a world where the sentences are both true. If they are inconsistent, use Fitch to give a proof that they are inconsistent (that is, derive⊥from them). You may useAna Con in your proof, but only applied to literals (that is, atomic sentences or negations of atomic sentences).

6.15

➶ ¬(Larger(a,b)∧Larger(b,a))

¬SameSize(a,b)

6.16

➶

Smaller(a,b)∨Smaller(b,a) SameSize(a,b)

Section 6.4

The proper use of subproofs

Subproofs are the characteristic feature of Fitch-style deductive systems. It is important that you understand how to use them properly, since if you are not careful, you may “prove” things that don’t follow from your premises. For example, the following formal proof looks like it is constructed according to our rules, but it purports to prove thatA∧Bfollows from (B∧A)∨(A∧C), which is clearly not right.

1. (B∧A)∨(A∧C) 2.B∧A

3.B ∧Elim: 2

4.A ∧Elim: 2

5.A∧C

6.A ∧Elim: 5

7.A ∨Elim: 1, 2–4, 5–6

8.A∧B ∧Intro: 7, 3

The problem with this proof is step 8. In this step we have used step 3, a step that occurs within an earlier subproof. But it turns out that this sort of justification—one that reaches back inside a subproof that has already ended—is not legitimate. To understand why it’s not legitimate, we need to think about what function subproofs play in a piece of reasoning.

A subproof typically looks something like this:Till Mossakowski Logic 17/ 24

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The proper use of subproofs (cont’d)

In justifying a step of a subproof, you may cite any earlier step contained in the main proof, or in any subproof whose

assumption is still in force. You may never cite individual steps inside a subproof that has already ended.

Fitch enforces this automatically by not permitting the citation of individual steps inside subproofs that have ended.

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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

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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

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558 /Summary of Rules

P ...

⊥

. ¬P

¬¬P ...

. P

P...

¬P ...

. ⊥

⊥...

. P

P ... Q

. P→Q

P→Q ... P...

. Q

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P ...

⊥

. ¬P

¬¬P ...

. P

P...

¬P ...

. ⊥

⊥...

. P

Conditional Introduction (→Intro)

P ... Q

. P→Q

P→Q ... P...

. Q

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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.

In working backwards, though, always check that yourintermediate

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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)