LOGICAL CONNECTIVES

We shall look in more detail at each connective in the coming chapters, but here are our five sentential connectives:

if … then …

and

not

or

if and only if

Since we shall replace all English expressions with symbols in our logical language, here are the symbols we shall use to denote the connectives:

if … then … = →

and = &

not = ~

or = ∨

if and only if = ↔

The technical terms for these connectives are: material conditional or arrow (→), conjunction (&), negation (~), disjunction (∨), and biconditional (↔).

Simple sentences

A simple sentence is defined as one that contains none of our five connectives. So the following English sentences are all simple:

1. Bill is bald.

2. Fred is fat.

3. All men are fat.

4. Some people vote Liberal.

Since it would be unwieldy to keep repeating English sentences, we abbreviate simple sentences using capital letters. In each simple sentence we shall write one word in capital letters. The first letter of that word can then stand for the entire sentence in our symbolic notation. Thus, for example:

Bill is BALD = B

Fred is FAT = F

ALL men are fat = A

Some people vote LIBERAL = L

All sentences in our formal or symbolic language are called formulae. B, F, A and L are all simple or atomic formulae.

Complex sentences

Complex sentences are defined as those sentences that contain one or more of our five connectives.