Conditional Connectives

The conditional – “p implies q” or “if p, then q”

Here p is the condition and q is the conclusion.

p | Q | p→q |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

The logic behind the truth table can be illustrated with examples.

Example : “If you get an A, then I’ll give you a dollar.”

Let it consider as a promise, and this particular example will be true, if this promise is kept

First row

Condition says: p is true and q is true

The conditional statement is saying that if p is true, then q will immediately follow and thus be true. So, the first row naturally follows this definition.

It’s *true* that you get an A and it’s *true* that
I give you a dollar. Since I kept my promise, the implication is *true*.

### Second row

Condition says: p is true and q is false

Similarly, the second row follows this because is we say “p implies q”, and then p is true but q is false, then the statement “p implies q” must be false, as q didn’t immediately follow p.

It’s *true* that you get an A but it’s *false* that
I give you a dollar. Since I *didn’t* keep my promise,
the implication is *false*.

### Third row

Condition says: p is false, q is true.

If p is false, then p→q is true, no matter whether q is true or not. For instance:

if it’s false that you get an A? Whether or not I give you a dollar, I haven’t broken my promise. Thus, the implication can’t be false, so (since this is a two-valued logic) it must be true

Fourth row

Condition says: p is false, q is false.

If p is false i.e. you do not get A and q is false i.e. I do not give you a dollar. It directs that i haven’t broken the promise. So it must be true.

The biconditional uses a double arrow because it is really saying “p implies q” and also “q implies p”. Symbolically, it is equivalent to:

(p⇒q)∧(q⇒p)

This form can be useful when writing proof or when showing logical equivalencies.

p | Q | p→q | p→q | p ↔ q |

T | T | T | T | T |

T | F | F | T | F |

F | T | T | F | F |

F | F | T | T | T |

Summary

To help you remember the truth tables for these statements, you can think of the following:

The conditional, p implies q, is false only when the condition is true but the conclusion is false. Otherwise it is true.

The biconditional, p iff q, is true whenever the two statements i.e. condition as well as the conclusion have the same truth value. Otherwise it is false.