Truth, Deduction, ComputationLecture 9The Logic of Conditionals, and more

Vlad PatryshevSCU2013

Why do we need →

● they make proofs more meaningful● e.g.

○

New Rules

● Modus Ponens (conditional elimination)● Biconditional Elimination● Contraposition

Modus Ponens

P

P → Q

Q

Biconditional Elimination

P

P ↔ Q

Q

Contraposition

P → Q

¬Q → ¬P

These are Tautologies

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

Conditional Proof

To prove P → Q, assume P and prove Q.

E.g., prove transitivity of →: ((P → Q)∧(Q → R)) → (P → R)

P

P → Q

Q

Q

Q → R

R

Conditional Proof

Or use contraposition (prove by contradiction): To prove P → Q, assume ¬Q and prove ¬P.E.g., prove that Even(n2) → Even(n):

1. Suppose n=2*m+1 2. then n2=4*...+1 - it is odd.3. We got ¬Even(n2) → ¬Even(n)4. Apply contraposition

Biconditional Proof

To prove P↔Q, prove P → Q and Q → P, then use ∧-Intro, since P↔Q ⇔ (P→Q)∧(Q→P).

More, since → is transitive, and you have Q1→Q

2, Q

2→Q

3,..., Q

n→Q

1, you can prove

Qi→Qj for each i and j, and so have Qi↔Q

j.

Examples (8.1) (not all are good)

● Affirming the Consequent From A → B and B, infer A.

● Modus Tollens From A → B and ¬B, infer ¬A.

● Strengthening the Antecedent From B → C, infer (A ∧ B) → C.

● Weakening the Antecedent From B → C, infer (A ∨ B) → C.

● Strengthening the Consequent From A → B, infer A → (B ∧ C).

● Weakening the Consequent From A → B, infer A → (B ∨ C).

● Constructive Dilemma From A∨B, A→C,and B→D,infer C∨D.

● Transitivity of the Biconditional From A ↔ B and B ↔ C, infer A ↔ C.

And now… Formal rules for → and ↔

● → Elim● → Intro● ↔ Elim● ↔ Intro

Conditional Rules

P→QP

Q

→ Elim → Intro P

Q

P→Q

Biconditional Rules

P↔QP

Q

↔ Elim ↔ Intro P

Q

Q

P

P→Q

Soundness and Completeness

● Logical system is sound if any sentence that can be deduced in this system, using sound arguments, is true (in the world’s semantics)

● Logical system is complete if any sentence that is true (in the world’s semantics) can be deduced

Two More Symbols

● P1..P

n ⊢ Q - a proof exists for Q from premises P

1..P

n● P

1..P

n ⊨ Q - P

1..P

n, taken together, semantically entail Q

Examples

A → B ⊢ ¬B → ¬A⊥, A ⊢ ¬B

Round(x) ⊨ ¬Cube(x)Home(clara) && InTheLibrary(clara) ⊨ LivesInTheLibrary(clara)

Soundness of FT

(do we really need to prove it?)

● FT - a subsystem of F that consists of intro/elim for ¬, ∨, ∧, →, ↔, and ⊥.

● P1..P

n ⊢

T Q - a proof exists for Q from premises P

1..P

n

Soundness Theorem for FT. If P

1,...,P

n ⊢

T S then S is a tautological consequence of P

1,...,

Pn.

Completeness of FT

Is FT complete, really?!

An Interesting Example

That’s it for today