Conditional Statements

The compound proposition implication

p q

is a conditional statement, and can be read as ''if p then q'' or ''p implies q''. Its precise definition is given by the following truth table

Let us briefly see why the above definition via the truth table is ''reasonable'' and is consistent with our day to day understanding of the notion of implications. We observe that the only explicit contradiction to ''if p then q'' comes from the case when p is true but q is false, and this explains the only ''F'' entry in the p q column. We also note that some people would never use ''p implies q'' to refer to p q; they would instead use ''p implies q'' to exclusively refer to p q, i.e. p q is a tautology. More details on '''' can be found in one of the later lectures.


  1. Let p denote ''I buy shares'' and q denote ''I'll be rich''. Then p q means ''If I buy shares then I'll be rich''. Let us check row by row the ''reasonableness'' of the truth table for p q given shortly before.
    Row 1: ''I buy shares'' (p true) and ''I'll be rich'' (q true) is certainly consistent with (p q) being true.
    Row 2: ''I buy shares'' and ''I won't be rich'' means ''If I buy shares then I'll be rich'' (i.e. p q) is false.
    Row 3 and 4: ''I don't buy shares'' won't contradict our statement p q, regardless of whether I'll be rich, as obviously there are other ways to get rich.

p q ( p) q.

This can easily be proved by the use of the truth table.

Note Obviously a string like p)) qr is not a legitimate logical expression. In this unit, we always assume that all the concerned strings of logical expressions are well-formed formulas, or wffs, i.e. the strings are legitimate.