Philosophy 2340
Symbolic Logic

Notes on Barwise and Etchemendy,
Language, Proof, and Logic, Chapter 9

Curtis Brown

With Chapter 9, we move from (mainly) propositional logic to predicate logic. We will need to extend and complicate our language. Here is a quick list of the main changes to the syntax of the first-order language we're developing.

We add two new pieces to our language: variables and quantifiers.

Up until now, the primitive symbols of our language have all been constants, predicates, and connectives.

Now, however, we will treat constants as only one kind of term. Variables are also terms. (So are function symbols with terms in their argument places, but we have mostly not been using function symbols.)

We use letters early in the alphabet for constants: a, b, c, d, e, f, g.

We use letters late in the alphabet for variables: t, u, v, w, x, y, z

Variables can appear anywhere constants do. So our syntax is now more complex.

We will define the notion of a well-formed formula, or wff for short.

1. If F is a predicate of arity n, and t₁, . . ., t_n are terms, then F(t₁, . . ., t_n) is an atomic wff. 
2. If P is a wff, then so is ~P.
3. If P and Q are wffs, then so is (P & Q)
4. If P and Q are wffs, then so is (P v Q)
5. If P and Q are wffs, then so is (P -> Q)
6. If P and Q are wffs, then so is (P <-> Q)
7. If P is a wff, and v is a variable, then Ev P is a wff
8. If P is a wff, and v is a variable, the Av P is a wff

The scope of a quantifier is the first complete wff that follows it

A quantifier binds any variable which (a) is within its scope, (b) matches the quantifier's variable, and (c) isn't already bound by another quantifier.

A variable that is not bound by any quantifier is free.

A sentence is a wff with no free variables.