Truth, deduction, computation lecture d
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Transcript of Truth, deduction, computation lecture d
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Truth, Deduction, ComputationLecture DQuantifiers, part 3 (almost there)
Vlad PatryshevSCU2013
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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.
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Translating English → FOL
EnglishEach 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∃ ∧ ∀ →
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Try 11.27
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Function ---> Predicate
can be represented as
or∀ ∀ →
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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
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Steps to Convert a WFF to PNF
1. Conjunction
● ∀ ∧ ⇔ ∀ ∧
● ∃ ∧ ⇔ ∃ ∧
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Steps to Convert a WFF to PNF
2. Disjunction
● ∀ ∨ ⇔ ∀ ∨
● ∃ ∨ ⇔ ∃ ∨
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Steps to Convert a WFF to PNF
3. Implication
● → ∀ ⇔ ∀ →● → ∃ ⇔ ∃ →
Really?
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Steps to Convert a WFF to PNF
4. Implication
● ∀ → ⇔ ∃ →
● ∃ → ⇔ ∀ →Really?
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Steps to Convert a WFF to PNF
5. Negation (follows from 4, actually.)
● ∀ ⇔ ∃● ∃ ⇔ ∀
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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”
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Proofs in FOLUniversal Elimination
∀ ⊢
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Proofs in FOLExistential Introduction (aka generalization)
⊢ ∃
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Proofs in FOLExistential Elimination (aka Instantiation)
1. Suppose ∃2. Invent a name (e.g. ) for such an 3.
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Proofs in FOLExistential Elimination (aka Instantiation)
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Proofs in FOLGeneral 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)
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Proofs in FOLUniversal Introduction (aka Generalization)
1. ∀ →2. ∀3. ∀
do you see modus ponens and prenex form transformations?
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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.
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How about mixing quantifiers?...
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That’s it for today