CONJUNCTION

The first system of formal logic that we will examine concerns proposi- tional (or sentential) connectives. Propositional connectives are terms that allow us to build new propositions from old ones, usually combining two or more propositions into a single proposition. For example, given the propositions “John is tall” and “Harry is short,” we can use the term “and” to conjoin them, forming a single compound proposition: “John is tall and Harry is short.”

Let us look carefully at the simple word “and” and ask how it functions. “And” is a curious word, for it does not seem to stand for anything, at least in the way in which a proper name (“Churchill”) and a common noun (“dog”) seem to stand for things. Instead of asking what this word stands for, we can ask a different question: What truth conditions govern this connective? That is, under what conditions are propositions containing this connective true? To answer this question, we imagine every possible way in which the component propositions can be true or false. Then, for each combination, we decide what truth value to assign to the entire proposition. This may sound complicated, but an example will make it clear:

John is tall. Harry is short. John is tall and Harry is short.

T T T

T F F

F T F

F F F

Here the first two columns cover every possibility for the component propo- sitions to be either true or false. The third column states the truth value of the whole proposition for each combination. Clearly, the conjunction of two propositions is true if both of the component propositions are true; other- wise, it is false.