Truth, Deduction, ComputationLecture 7Boolean Logic and Formal Proofs

Vlad PatryshevSCU2013

Natural Deduction

A proof calculus that:● has inference rules● models actual human reasoning

Natural Deduction

● introduction rules● elimination rules● shortcuts (“rule defaults”)

Conjunction Rules

Elimination (∧ Elim) P

1∧P

2∧...P

i∧...

∧Pn

Pi

Do we need n here?!

Conjunction Rules

Introduction (∧ Intro) P

1

. . .

Pn

P1∧P

2∧...P

i∧...

∧Pn

Do we need n here?!

Disjunction Rules

Introduction (v Intro) P

i

P1vP

2v...P

iv...vP

n

Do we need n here?!

Disjunction Rules

Elimination (v Elim) P

1vP

2v...vP

n

P1

S

. . .

Pn

S

S

Do we need n here?!

} subproof

Example

Negation Rules

Elimination (¬ Elim) ¬¬P … P

Negation Rules

Introduction (¬ Intro)

P

⊥

¬P

⊥ Rules

Introduction (⊥ Intro) P ¬P ⊥

⊥ Rules

Elimination (⊥ Elim)

⊥

P

All These Negation Rules

More ⊥ Rules in Fitch

● Taut Con: If you can deduce a TT-contradiction, you can deduce ⊥

● FO Con: If you can deduce a logical contradiction, you can deduce ⊥

● Ana Con: If you can deduce sentences that are contradictory due to the meaning of predicates, you can deduce ⊥

Rules - be careful

Bad Example

Now that you know the rules...

● Remember the meaning of sentences● Trust the past● Find a counterexample if you don’t● To formalize your proof, track back your

arguments - “where did this come from?”● … use Fitch…

At times you don’t need premises!

E.g.

That’s it for today