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. ExamplesTermsPredicate Cube(a) Between(a,b,c) Likes(Mary, Tom) Tall(Father(Max) aCube a, b, cBetween Mary, TomLikes 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 2Language 2 Employs(Claire,Max)Employer(Max) = Claire Employs(Max,Tony)Employer(Tony) = Max EarnsMoreThan(Claire, Tony)

12. Language 1 into Language 2 Language 1Language 2 Employer(Max) = ClaireEmploys(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 t 1 and t 2 are terms, then (t 1 + t 2 ) and (t 1 * t 2 ) 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.