Lecture Notes, 17 Sep 2010
Last Update: 19 September 2010
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§1.3: Predicates & Quantifiers (cont'd)
- 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
- 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
- Here's another propositional function:
2 + y < 4
- Its subject is y;
- its predicate is "2 + < 4"
- 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
& write it in mathematical-function
- 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
- 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:
- Preds can apply to 1 or more terms:
- If a predicate applies to two or more terms,
then it names a relation among the
- 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 x + y <
there are 2 terms: x, y
& a 2-place predicate:
" + … < 4"
- 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(x, y) can stand for:
x + y < 4
- Note that P(y) above stands for the same
function as Q(2, y)
- We can make a table for Q, similar to the one above for P:
- (Recursive) Definition of Well-Formed Proposition of FOL:
- "terms" (or NPs) name or describe objects in the domain;
- "predicate" (or VPs) name properties of objects or relations among
- "variables" (or pronouns) are like variables in programming
- Base Cases:
- Atomic propositions (p, q, r,
…) of propositional logic are well-formed (atomic) propositions of FOL.
are terms (NPs)
& if R is an n-place predicate,
R(t1,…,tn) is a WF (atomic)
proposition of FOL (a "subatomic" proposition)
- Recursive Cases:
Text copyright © 2010 by William J. Rapaport
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