Function Symbols & Arithmetic

1. Function Symbols & Arithmetic PHIL012 January 22, 2001

2. Outline • Announcements • function symbols, terms, complex terms • The first order language of arithmetic • Sample problems • Assignment

3. Terms • A “Term” is another name for the arguments that predicates take. • Examples Terms Predicate Cube(a) a Cube Between(a,b,c) a, b, c Between Likes(Mary, Tom) Mary, Tom Likes Tall(Father(Max) Father(Max) Tall

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. Language 1 • Names: Tony, Claire, Max • Function Symbol: Employer • Predicates: EarnsMoreThan, =

10. Language 2 • Names: Tony, Claire, Max • Predictates: EarnsMoreThan, =, Employs

11. 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)

12. 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

13. 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.

14. 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: +, *

15. 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.

16. Inductive Definition • To form an inductive definition, • We set up initial set of terms, • Definition 1: 0 and 1 are terms

17. 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.

18. Sample Problems • Show that the following are expressions in our language: 1. ((1 + 1) * ((1+1) +1))

19. Sample Problems 2. ((0 * (1 + 0)) + 1)

20. 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.