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**(Recursive) Definition of Well-Formed Proposition of FOL:**Remember:

- "terms" (or NPs) name or describe
*objects*in the domain; - "predicate" (or VPs) name
*properties*of objects or*relations*among ≥2 objects - "variables" (or pronouns) are like variables in programming languages

- Base Cases:
- Atomic propositions (
*p*,*q*,*r*, …) of propositional logic are**well-formed (atomic) propositions**of FOL. -
If
*t*_{1},…,*t*are terms (NPs)_{n}

& if R is an*n*-place predicate,

then R(*t*_{1},…,*t*) is a_{n}**WF (atomic) proposition**of FOL (a "subatomic" proposition)

- Atomic propositions (
- Recursive Cases:
- If
*A*,*B*are WF (atomic or molecular) propositions of FOL,

& if*v*is a variable,

then:- ¬
*A* - (
*A*∧*B*) - (
*A*∨*B*) - (
*A*⊕*B*) - (
*A*→*B*) - (
*A*↔*B*) - ∀
*v*[*A*] - ∃
*v*[*A*]

are

**WF (molecular) propositions**of FOL - ¬

- If

- "terms" (or NPs) name or describe
**Definition of Term of FOL:**- If
*v*is a variable, then*v*is a**term**of FOL.- E.g.,

*x*,*y*,*z*, … are terms - If
*c*is a constant, then*c*is a**term**of FOL.- E.g., "fred", "evelyn", 1, √2, …
are terms

- (another clause to be presented later, maybe…)
- Nothing else is a term.
- In particular, no predicate is a term!

I.e., if R is a predicate, then R is not a term.So: predicates can never appear in term-position;

i.e., in FOL, predicates cannot be terms of other predicates;

i.e., predicates cannot appear*inside*other predicates

- In particular, no predicate is a term!

- If
**Syntax & Semantics of the Quantifiers:**NAME SYNTAX ENGLISH E.G. SEMANTICS universal

quantifier∀ *v*[*A*],

where*v*is any variable

& where*A*is*any*proposition,

including:- all of the "old" ones

& any of these "new" ones;if

*A*is atomic,

the brackets can be omitted;- e.g., ∀

*x*R(*x*)Everything in the domain satisfies *A*.For all objects

*v*in the domain,*A*is the case.All humans are mortal ≡ ∀

*x*[Human(*x*) → Mortal(*x*)]Note the use of →!

tval(∀ *vA*)=T

iff

all objects in the domain are such that tval(*A*)=Te.g., tval(∀

*x*R(*x*))=T

iff

all objects in the domain have the property named by Rexistential

quantifier∃ *v*[*A*],

as abovee.g., ∃

*x*R(*x*)Something satisfies *A*There is (or: there exists) an object

*v*in the domain for which*A*is the case.Some human is mortal ≡ ∃

*x*[Human(*x*) ∧ Mortal(*x*)]Note the use of ∧!!

tval(∃ *vA*)=T

iff

at least one object in the domain is such that tval(*A*)=Te.g., tval(∃

*x*R(*x*))=T

iff

at least one obj in the domain has the property named by R- The truth value of a quantified proposition depends on the domain:
proposition Domain: **W**={1,2,3,…}**N**={0,1,2,3,…}**Z**={…–3,–2,–1,0,1,2,3,…}∀ *x*[*x*> 0]T F

(because ¬(0>0))F ∀ *x*[*x*≥ 0]T T F Notes:

- "∀

*x*[*x*> 0]" can also be written (slightly more grammatically) as: ∀*x*[>(*x*, 0)]"∀

*x*[*x*≥ 0]" can also be written (slightly more grammatically) as: ∀*x*[>(*x*, 0) ∨ =(*x*, 0)] **Finite domains:**Consider the domain be {0,1,2,3}.

Then tval(∀*x*R(*x*)) = T iff, for all values of*x*in the domain, tval(R(*x*)) = T

iff tval(R(0))=T & tval(R(1))=T & tval(R(2))=T & tval(R(3))=T∴ ∀

*x*R(*x*) ≡ R(0) ∧ R(1) ∧ R(2) ∧ R(3)Similarly, we can show that: ∃

*x*R(*x*) ≡ R(0) ∨ R(1) ∨ R(2) ∨ R(3)- ∀
*x*R(*x*) is like a for-loop! - ∃
*x*R(*x*) is like a search procedure!

- ∀
**De Morgan's Laws for Quantifiers (or: How to Negate a Quantifier):**In a finite domain, say {0,1}:

∀

*x*P(*x*) ≡ P(0) ∧ P(1).∴ ¬∀

*x*P(*x*) ≡ ¬(P(0) ∧ P(1))

≡ (¬P(0) ∨ ¬P(1)), by DeMorgan for the negation of ∧

≡ ∃*x*[¬P(*x*)]Not only is this also true in an

*infinite*domain, but so is this:¬∃

*x*P(*x*) ≡ ∀*x*[¬P(*x*]and these:

∃

*x*P(*x*) ≡ ¬∀*x*¬P(*x*)

∀*x*P(*x*) ≡ ¬∃*x*¬P(*x*)

Text copyright © 2010 by William J. Rapaport (rapaport@buffalo.edu)

http://www.cse.buffalo.edu/~rapaport/191/F10/lecturenotes-20100920.html-20100920