Truth, Deduction, ComputationLecture DQuantifiers, part 3 (almost there)

Vlad PatryshevSCU2013

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.

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∃ ∧ ∀ →

Try 11.27

Function ---> Predicate

can be represented as

or∀ ∀ →

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

Steps to Convert a WFF to PNF

1. Conjunction

● ∀ ∧ ⇔ ∀ ∧

● ∃ ∧ ⇔ ∃ ∧

Steps to Convert a WFF to PNF

2. Disjunction

● ∀ ∨ ⇔ ∀ ∨

● ∃ ∨ ⇔ ∃ ∨

Steps to Convert a WFF to PNF

3. Implication

● → ∀ ⇔ ∀ →● → ∃ ⇔ ∃ →

Really?

Steps to Convert a WFF to PNF

4. Implication

● ∀ → ⇔ ∃ →

● ∃ → ⇔ ∀ →Really?

Steps to Convert a WFF to PNF

5. Negation (follows from 4, actually.)

● ∀ ⇔ ∃● ∃ ⇔ ∀

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”

Proofs in FOLUniversal Elimination

∀ ⊢

Proofs in FOLExistential Introduction (aka generalization)

⊢ ∃

Proofs in FOLExistential Elimination (aka Instantiation)

1. Suppose ∃2. Invent a name (e.g. ) for such an 3.

Proofs in FOLExistential Elimination (aka Instantiation)

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)

Proofs in FOLUniversal Introduction (aka Generalization)

1. ∀ →2. ∀3. ∀

do you see modus ponens and prenex form transformations?

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.

How about mixing quantifiers?...

That’s it for today