Truth, deduction, computation lecture 7
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This document discusses natural deduction and the rules of Boolean logic and formal proofs. It introduces natural deduction as a proof calculus that uses inference rules to model human reasoning. The document then explains introduction and elimination rules for conjunction, disjunction, negation and the contradiction symbol ⊥. It notes that the rules allow deducing ⊥ from a tautological, logical or meaning-based contradiction. The document emphasizes being careful with the rules, finding counterexamples if deductions don't make sense, and formalizing proofs by tracking arguments back to their sources. It also notes that at times deductions can be made without premises.
Truth, deduction, computation lecture 7
- 1. Truth, Deduction, Computation Lecture 7 Boolean Logic and Formal Proofs Vlad Patryshev SCU 2013
- 2. Natural Deduction A proof calculus that: ● has inference rules ● models actual human reasoning
- 3. Natural Deduction ● introduction rules ● elimination rules ● shortcuts (“rule defaults”)
- 4. Conjunction Rules Elimination (∧ Elim) P1∧P2∧...Pi∧... ∧Pn Pi Do we need n here?!
- 5. Conjunction Rules Introduction (∧ Intro) P1 . . . Pn P1∧P2∧...Pi∧... ∧Pn Do we need n here?!
- 6. Disjunction Rules Introduction (v Intro) Pi P1vP2v...Piv...vPn Do we need n here?!
- 7. Disjunction Rules Elimination (v Elim) Example P1vP2v...vPn P1 } S subproof . . . Pn S S Do we need n here?!
- 8. Negation Rules Elimination (¬ Elim) ¬¬P … P
- 9. Negation Rules Introduction (¬ Intro) P ⊥ ¬P
- 10. ⊥ Rules Introduction (⊥ Intro) P ¬P ⊥
- 11. ⊥ Rules Elimination (⊥ Elim) ⊥ P
- 12. All These Negation Rules
- 13. 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 ⊥
- 14. Rules - be careful Bad Example
- 15. 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…
- 16. At times you don’t need premises! E.g.
- 17. That’s it for today