# A calculus of the absurd

##### 18.3.3 The material conditional

This one can be really confusing. The basic idea that we seek to express using the “material conditional” is “if $$A$$, then $$B$$”. Unfortunately natural language is often quite confusing, because when we say “if $$A$$, then $$B$$” many people, not at all unreasonably, interpret this to mean that if $$A$$ is false and $$B$$ is true then it is not at all true to say “if $$A$$, then $$B$$”. However, in mathematics “if” is a one-way street; all we are saying is that should $$A$$ be true, then $$B$$ is also true (for “if $$A$$, then $$B$$ to hold”.

• Definition 18.3.1 The “material conditional”, i.e. the logical operator denoted by $$\implies$$ has the following truth table

 $$A$$ $$B$$ $$A \implies B$$ F F T F T T T F F T T T
• Theorem 18.3.1 The (mathematical) natural language statement “if $$A$$, then $$B$$” corresponds to $$A \implies B$$ in logic.

• Technique 18.3.1 To prove a statement in the form “if $$A$$, then $$B$$” or “$$A$$ if $$B$$” one must

• 1. Assume that $$A$$ is true.

• 2. Show that therefore $$B$$ is also true.

Note that if $$A$$ is false it doesn’t matter what happens, as $$A \implies B$$ is true in this case.