Atomic Sentences Chapter 1 Language, Proof and Logic

Overview 1.0 FOL is not really a particular language but rather a family of languages. The members of this family share the same grammar and certain important (“logical”) vocabulary terms, known as the connectives and quantifiers. Languages of this family can differ, however, in the specific (“nonlogical”) vocabulary used in their most basic sentences, called atomic sentences. Atomic sentences are formed by combining names (also called individual constants) and predicates. The differences between different FOLs will thus be in what particular names and predicates they contain.

Individual constants 1.1 Individual constants are simply symbols that are used to refer to some fixed individual object. The individual constants of our blocks language are the letters a through f plus n 1,n 2,... Every individual constant must name an actually existing object (unlike English, where Zeus or Santa Claus name non-existing things). Conditions: No individual constant can name more than one object (unlike English, where there are many Toms and Bobs). An object can have more than one name, or no name at all (just as in English, where Morning Star and Venus name the same object, while most asteroids have no names at all).

Predicate symbols 1.2 Predicate symbols are used to express some property of objects or some relation between objects, called its arguments. Predicate symbols are also often called relation symbols. Each predicate symbol has a fixed number of arguments, called the arity of the predicate (symbol). A 1-ary predicate is also said to be unary, a 2-ary predicate is said to be binary, and a 3-ary predicate is said to be ternary. Some of the predicates of the blocks language are: 1-ary: Cube, Small, Large 2-ary: Smaller, Larger, = 3-ary: Between FOL assumes that every predicate is interpreted as a determinate property, i.e., a property for which, given any object, there is a definite fact of the matter whether the object has the property. E.g., while in English “small” may have various and unclear “degrees of truth”, in the blocks language every object is just either Small or not.

Atomic sentences 1.3 Atomic sentences look like P(c 1,...,c n ) where P is an n-ary predicate symbol and the c i are names. An exception is the case with the identity symbol, where we put the two required names on either side of the predicate, as in a=b. This is called infix notation, since = appears between its two arguments. With the other predicates we use prefix notation: the predicate precedes the arguments. Each sentence makes a claim --- something that is true or false; which of these it is we call its truth value. E.g., White(snow) expresses a true claim while Black(snow) expresses a false claim. Do “You try it” on page 24

General first-order languages 1.4 FOLs differ in the names and predicates they contain. Sometimes you deal with a predefined language, as in Tarski’s World. Sometimes you need to design a language of your own. Considerations when designing languages: Universality Should be able to say everything you may potentially need to say Flexibility WifeOfBob(Jane) vs. Wife(Jane,Bob) Economy If you have Wife, no need for having Husband

Function Symbols 1.5 Function symbols allow us to form name-like terms from other name-like terms. In English, to function symbols correspond noun phrases: Bob’s mother, the square of 5, etc. Tarski’s world has no function symbols. Function symbols, just like predicate symbols, have arities. Using function symbols, we can form arbitrarily complex terms: father(mother(mother(bob))) You cannot do the same with predicates: Father(tom,Mother(mary,bob)) is nonsense Do not confuse function symbols with predicates!

The first-order language of set theory 1.6 No function symbols Two predicates: = and . Both binary, both infix Let a be the name of 2 b be the name of {2,4,6} c be the name of {x | x is an integral root of x 2 =5} d be the name of {x | x is an elephant that can fly} True or false? a  a a  b b  b c  b c = d

The first-order language of arithmetic 1.7 Individual constants: 0,1 Predicate symbols: =,< (binary, infix) Function symbols: +,  (binary, infix) Terms: 1. 0,1 are terms 2. if t an r are terms, then so are (t+r) and (t  r) 3. nothing is a term unless it can be obtained by repeated application of (1) and (2) Atomic sentences: t=r or t<r, where t,r are terms. What are the following? Express number 3 Express number 5 1. ((1+1)  (1+(1  (1+0)))) Express number (1+1) = (1  (1+0)) Express number (1+1) < (0  (1+1)) 4. (1+1) = (1+(0<1))