Function Symbols & Arithmetic

PHIL012 January 22, 2001

Outline

• Announcements

• function symbols, terms, complex terms

• The first order language of arithmetic

• Sample problems

• Assignment

Terms

• A “Term” is another name for the arguments that predicates take.

• Examples Terms PredicateCube(a)

Between(a,b,c)

Likes(Mary, Tom)

Tall(Father(Max)

a Cube

a, b, c Between

Mary, Tom Likes

Father(Max) Tall

Function Symbols

• A Function Symbol looks like a predicate (since it takes arguments):

• Predicate: Cube(a)• Function Symbol: Father(Max)• However, a Predicate with its arguments form

a sentence.• Whereas, a Function Symbol with its

argument is simply a term.

Function Symbols vs. Predicates

• Cube(a) is a sentence because it makes a claim. It says “a is a cube”

• Father(Max) does not make a claim. It simply picks out a person, “Max’s Father.”

• Father(Max) is a term because it doesn’t say anything about Max’s father.

• Cube(a) has a truth value. Father(Max) does not.

Terms and Function Symbols

• In the two expressions, Cube(a) and Tall(Father(Max)),

• “a” is like “Father(Max)”• “Tall” is like “Cube”• “a” and “Father(Max)” behave like names in

picking out objects• “Tall” and “Cube” are predicates, specifying

properties of objects.

Complex Terms

• Father(Max) is an example of a Complex Term.

• A Complex Term is formed by putting a function symbol in front of either a name or another complex term.

• Complex Terms are used just like names (simple terms) in forming atomic sentences.

Sample Problem

• Suppose we have 2 languages for talking about employment.

• Language 1 is functional, since it contains the function symbol, “Employer”. Employer(Claire) means “Claire’s Employer”

• Language 2 is relational. It uses the predicate Employs(Tony,Claire) to say that Tony is Claire’s employer.

Language 1

• Names: Tony, Claire, Max

• Function Symbol: Employer

• Predicates: EarnsMoreThan, =

Language 2

• Names: Tony, Claire, Max

• Predictates: EarnsMoreThan, =, Employs

Translate these Language 2 sentences into Language 1

Lanuage 2 Language 2

Employs(Claire,Max) Employer(Max) = Claire

Employs(Max,Tony) Employer(Tony) = Max

EarnsMoreThan(Claire,Tony)

EarnsMoreThan(Claire,Tony)

Language 1 into Language 2

Language 1 Language 2

Employer(Max) = Claire Employs(Claire,Max)

Employer(Max) = Employer(Tony) untranslatable

EarnsMoreThan(Employer(Max),

Employer(Tony))untranslatable

Summary

• Terms are simple or complex.

• A Simple Term is a Name

• A Complex Term is a Function Symbol followed by some number of Simple Terms or Complex Terms.

• Terms pick out objects.

• Predicates specify properties of objects.

Function Symbols & Arithmetic

• We can specify all of the numbers and operations of arithmetic using a simplified set of symbols:

• Names: The numbers 0, 1

• Predicates: =, <

• Function Symbols: +, *

Inductive Definition of Terms

• The language includes an infinite number of complex terms:

• 0, 1, (1+1), (1+1)+1, ((1+1)+1)+1, …

• We need to have a routine way of determining whether an expression is a term in the language or not.

• We do this by inductive definition.

Inductive Definition

• To form an inductive definition,

• We set up initial set of terms,

• Definition 1: 0 and 1 are terms

Inductive Definition Continued

• We specify a set of rules that must be followed to get additional terms. In this case, these are:

• Definition 2: If t1 and t2 are terms, then

(t1 + t2) and (t1 * t2) are also terms.

• Definition 3: An expression is a term IFF it was derived by repeating definitions 1 and 2.

Sample Problems

• Show that the following are expressions in our language:

1. ((1 + 1) * ((1+1) +1))

Sample Problems

2. ((0 * (1 + 0)) + 1)

Assignment• For Wednesday:

– Read 2.7– Try to work through problems 14-17.

• By Midnight Thursday:– Turn in problems 14-17 (Homework 3)

• For Friday:– Read 2.8

• Our first test (Chapter 2) will be on Feb 5

• We will have a review on Feb 2.