# Every Logical Expression is either a Tautology or Contradiction

The question ask if the above claim is True or False. if true I Must prove that and give a counter example if it is false.

I prefer the claim to be false.

since looking at every logical expression either a conditional P then Q statement or bi-conditional statement the last column will always be T=TRUE. therefor I conclude it is false since the last column of every logical expression will always be true.

I'm I correct or there is a law or example to prove it is true.

• Consider the expression $p$ for any propositional variable $p$. It's neither a tautology, nor a contradiction. – Stefan Mesken Mar 31 '16 at 23:09
• I am confused about your explanation above, would you be able to clarify? – Inazuma Mar 31 '16 at 23:21
• I agree (it's true) that the claim is false (as @Stefan notes, consider the formula $p$); but I don't understand the reasons you give. In fact, I don't understand your second to last paragraph at all. – BrianO Mar 31 '16 at 23:22
• I am referring to the last column of the truth table . in which is true for every expression. – Surdz Mar 31 '16 at 23:27
• The truth table values p -> q (conditional) or a p <-> q (biconditional) is not all true, and if it was, that would make it a tautology. – Inazuma Mar 31 '16 at 23:31

## 1 Answer

First, the easy answer is that any proposition which gives you a mixture of true and false as your result will be neither a tautology (which requires that the end result is always true) or a contradiction (which requires that the end result is always false.) So it is simple to give a counter-example: say,p, as Stefan suggested, or p implies q or ......lots.

To get deeper into this, you want to be clear why the values of True and False are used when talking about contradictions and tautologies, and what role truth tables have. From there you can show the statement false in a number of ways.

The values of True and False usually depend on the interpretation given to the symbols. If the interpretation "fits", then we say the expression is true under that interpretation, and false otherwise. Let's assume for neatness that the interpretation does not assign both truth values to an expression; in this case we call the interpretation a model of the logical expressions concerned. The only instances in which we do not have to consult our model to see if an expression is true or false occurs when we have an expression which is true under all models, or false under all models. The former are tautologies, the latter are contradictions. Truth tables then tell you what the truth value is after you have determined the truth of component sentences. So, anything that is true or false according to which interpretation you pick will be a counter-example. Say, "it's nice weather here" will depend on your location, the time of year, and your personal tastes, as to whether it is true or false; it is definitely not a tautology.

(You can also get into some fun topics about statements that have fuzzy truth values, but that would get us too far afield.)