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- In English, every (atomic) sentence consists of a
noun phrase (NP) & a verb phrase (VP)
- The sentence "My computer is a Mac" consists of the NP "my computer" and the VP "is a Mac".
- "Fido barks" consists of the NP "Fido" & the VP "barks".

- The NP names or describes an object;

the VP says something*about*the object. - The NP is called the "subject" of the sentence;

the VP is called the "predicate" of the sentence. - The subject is what the sentence is about;

it refers to an*object*in the "world"

(more generally, it refers to an object in the*domain of discourse*,

i.e., the world that we are talking about) - The predicate says something about the subject;

it refers to a*property*or*relation*

**Propositional functions:**- "
*x*is a Mac" is not a sentence;

it is neither true nor false.- Nor is it a "predicate", contrary to what some textbooks say!
- Only "is a Mac" is a predicate.

- Rather, it is a
**propositional function**- As a (mathematical) function, its input is
a value for
*x*

& its output is a proposition with a truth value

- As a (mathematical) function, its input is
a value for
- Here's another propositional function:
2 +

*y*< 4- Its subject is
*y*; - its predicate is "2 +
- this predicate names the property of "being such that when added to 2, the sum is < 4"

- We can denote the predicate by a capital
letter,

& write it in mathematical-function notation:- P(
*y*) stands for: 2 +*y*< 4

Here,

*y*is called P's "variable" or "parameter" or "argument" or "term".A

**term**is a NP that names or describes an object in the domain - P(
- We can make a table showing
possible input values to this propositional
function, its output values, and—because
the output is a proposition—we can show
the truth value of the output:
*y*P( *y*)tval(P( *y*))… … … 0 2+0<4 T 1 2+1<4 T 2 2+2<4 F 3 2+3<4 F … … …

- Its subject is
- Preds can apply to 1
*or more*terms:- If a predicate applies to two or more terms,

then it names a**relation**among the terms.- E.g., in "John bought a book",

there is a 2-place predicate "bought" that has 2 terms: "John" and "a book"

- E.g., in "John bought a book",
- E.g., in
*x*+*y*< 4,

there are 2 terms:*x*,*y*

& a 2-place predicate: "- This predicate names "the relation between two numbers such that when added together their sum is < 4"
- We can denote a 2-place predicate by a capital letter

& use functional or relational notation:- Q(
- Note that P(
*y*) above stands for the same propositional function as Q(2,*y*)

*x*,*y*) can stand for:*x*+*y*< 4 - Note that P(

- We can make a table for Q, similar to the one above for P:
*x**y*Q( *x*,*y*)tval(Q( *x*,*y*))… … … … 0 0 0+0<4 T … … … … 6 3 6+3<4 F … … … …

- If a predicate applies to two or more terms,

- "
**(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

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

http://www.cse.buffalo.edu/~rapaport/191/F10/lecturenotes-20100917.html-20100919