Truth, Deduction, Computation Lecture D Quantifiers, part 3 (almost there
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My logic lectures at SCU Quantifiers, part 3
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Truth, Deduction, Computation Lecture D Quantifiers, part 3 (almost there
- 1. Truth, Deduction, Computation Lecture D Quantifiers, part 3 (almost there) Vlad Patryshev SCU 2013
- 2. Can we express “there’s only one”? ● ∃ same as ● ∃ ∧ ∀ → “quantifier” - a tool for measuring quantity; “some”, “all”, “just one” - look like ancient ways of counting people or stuff… or Gods.
- 3. Translating English → FOL English Each cube is to the left of a tetrahedron No cube to the right of a tetrahedron is to the left of a tetrahedron Every farmer who owns a donkey beats it. ∀ ∃ → ∃ ∧ ∃ ∃ ∧ ∧ ∧ ∀ ∧ ∃ then what? bad… try again! ∀ → ∀ ∧ ∧ → Only large objects have nothing in front of them Every minute a man is mugged in NYC. We will interview him tonight. ∀ or ∃ → ∃ ∧ ∀ ∧ ∧ →
- 4. Try 11.27
- 5. Function ---> Predicate can be represented as or ∀ ∀ →
- 6. Prenex Normal Form of WFF A sentence is in prenex form if all its quantifiers are at the beginning of it. But is it possible?! source: https://en.wikipedia.org/wiki/Prenex_normal_form
- 7. Steps to Convert a WFF to PNF 1. Conjunction ● ∀ ∧ ⇔ ∀ ∧ ● ∃ ∧ ⇔ ∃ ∧
- 8. Steps to Convert a WFF to PNF 2. Disjunction ● ∀ ∨ ⇔ ∀ ∨ ● ∃ ∨ ⇔ ∃ ∨
- 9. Steps to Convert a WFF to PNF 3. Implication ● ● → ∀ → ∃ ⇔ ∀ ⇔ ∃ Really? → →
- 10. Steps to Convert a WFF to PNF 4. Implication ● ∀ → ⇔ ∃ → ● ∃ → ⇔ ∀ → Really?
- 11. Steps to Convert a WFF to PNF 5. Negation (follows from 4, actually.) ● ● ∀ ∃ ⇔ ∃ ⇔ ∀
- 12. PNF Example “if a cube is to the left of a tet, it’s behind a dodec” “if a cube is to the left of tet, it’s behind a dodec”
- 13. Proofs in FOL Universal Elimination ∀ ⊢
- 14. Proofs in FOL Existential Introduction ⊢ ∃ (aka generalization)
- 15. Proofs in FOL Existential Elimination (aka Instantiation) 1. Suppose ∃ 2. Invent a name (e.g. ) for such an 3.
- 16. Proofs in FOL Existential Elimination (aka Instantiation)
- 17. Proofs in FOL General Conditional Proof ● To prove ⊢ ∀ → ● Introduce a new name, e.g. , to denote anything satisfying ● Prove ⊢ ● Profit (this is not a “rule”, this is a trick with substitutions)
- 18. Proofs in FOL Universal Introduction 1. ∀ 2. ∀ 3. ∀ (aka Generalization) → do you see modus ponens and prenex form transformations?
- 19. Example (12.2) Twas brillig, and the slithy toves Did gyre and gimble in the wabe: All mimsy were the borogoves, And the mome raths outgrabe.
- 20. How about mixing quantifiers?...
- 21. That’s it for today