Download - Truth, deduction, computation lecture 9
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Truth, Deduction, ComputationLecture 9The Logic of Conditionals, and more
Vlad PatryshevSCU2013
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Why do we need →
● they make proofs more meaningful● e.g.
○
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New Rules
● Modus Ponens (conditional elimination)● Biconditional Elimination● Contraposition
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Modus Ponens
P
P → Q
Q
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Biconditional Elimination
P
P ↔ Q
Q
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Contraposition
P → Q
¬Q → ¬P
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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)
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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
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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
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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.
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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.
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And now… Formal rules for → and ↔
● → Elim● → Intro● ↔ Elim● ↔ Intro
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Conditional Rules
P→QP
Q
→ Elim → Intro P
Q
P→Q
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Biconditional Rules
P↔QP
Q
↔ Elim ↔ Intro P
Q
Q
P
P→Q
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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
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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)
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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.
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Completeness of FT
Is FT complete, really?!
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An Interesting Example
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That’s it for today