Truth, deduction, computation lecture 9
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This document summarizes a lecture on the logic of conditionals. It introduces conditionals and why they are useful in proofs. It then outlines new rules for conditional elimination (modus ponens), biconditional elimination, and contraposition. Examples of these rules and tautologies involving conditionals and biconditionals are provided. Methods for conditional and biconditional proofs are described. The concepts of soundness and completeness of logical systems are briefly covered, along with symbols used in proofs.
Truth, deduction, computation lecture 9
- 1. Truth, Deduction, Computation Lecture 9 The Logic of Conditionals, and more Vlad Patryshev SCU 2013
- 2. Why do we need → ● they make proofs more meaningful ● e.g. ○
- 3. New Rules ● Modus Ponens (conditional elimination) ● Biconditional Elimination ● Contraposition
- 4. Modus Ponens P P → Q Q
- 6. Contraposition P → Q ¬Q → ¬P
- 7. These are Tautologies ● ● ● ● ● P→Q ⇔ P→Q ⇔ ¬(P→Q) P↔Q ⇔ P↔Q ⇔ ¬Q→¬P ¬P∨Q ⇔ P∧¬Q (P→Q)∧(Q→P) (P∧Q)∨(¬P∧¬Q)
- 8. Conditional Proof To prove P → Q, assume P and prove Q. E.g., prove transitivity of →: ((P → Q)∧(Q → R)) → (P → R) P Q P → Q Q → R Q R
- 9. 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. 2. 3. 4. Suppose n=2*m+1 then n2=4*...+1 - it is odd. We got ¬Even(n2) → ¬Even(n) Apply contraposition
- 10. 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→Q2, Q2→Q3,..., Qn→Q1, you can prove Qi→Qj for each i and j, and so have Qi↔Qj.
- 11. 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.
- 12. And now… Formal rules for → and ↔ ● ● ● ● → → ↔ ↔ Elim Intro Elim Intro
- 15. 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
- 16. Two More Symbols ● ● P1..Pn ⊢ Q - a proof exists for Q from premises P1..Pn P1..Pn ⊨ Q - P1..Pn, taken together, semantically entail Q Examples A → B ⊢ ¬B → ¬A ⊥, A ⊢ ¬B Round(x) ⊨ ¬Cube(x) Home(clara) && InTheLibrary(clara) ⊨ LivesInTheLibrary(clara)
- 17. Soundness of FT ● ● FT - a subsystem of F that consists of intro/elim for ¬, ∨, ∧, →, ↔, and ⊥. P1..Pn ⊢T Q - a proof exists for Q from premises P1..Pn Soundness Theorem for FT. If P1,...,Pn ⊢T S then S is a tautological consequence of P1,..., P n. (do we really need to prove it?)
- 18. Completeness of FT Is FT complete, really?!
- 20. That’s it for today