# Will a contradiction always be false?

Will a contradiction always have only false values?

A tautology will always have true vales, is the opposite true for contradictions?

For example: $(p \ \& \ q) \ \& \ (p \ \& \ \neg q)$ has the truth table:

• YES : a contradiction is a formula that is always false, like $p \land \lnot p$ – Mauro ALLEGRANZA Oct 2 '16 at 18:30
• Yes: That it is always false is in fact the definition of a contradiction. – lemontree Oct 2 '16 at 19:20
• @lemontree yup, and if you look at the table of 16 operators, you'll see F aka contradiction with false in every row. – The Great Duck Oct 2 '16 at 22:56
• See the principal of explosion en.m.wikipedia.org/wiki/Principle_of_explosion – Awn Jun 4 '18 at 18:46

Contradictions will always end in all entries of the rightmost column of a truth table being only "F".

$a \land b\land c \land .... \land p\land \lnot p$ will never be true, because $p \land \lnot p$ is False, and no matter what you "and" to that to form a new statement, it will still be false. Any statement for which no matter the assignment of truth-values to the propositions, ends in the evaluation of False, is a contradiction.

• Note also that a tautology is always true. For example, say we have $p\lor q\lor r \lor \lnot p$. Here, the truth-value of q and that of r are irrelevant to the truth value of the proposition. $p\lor \lnot p$ is always true, because $p$ is true, or $\lnot p$ is true. If any proposition is transformed to a truth-table has ALL T's in the last column, you can be assured the proposition is a tautology.: – Namaste Oct 2 '16 at 19:33
• What if the "..." in a $\land$ b $\land$ c $\land$ ... p $\land$ $\lnot$p consisted of a sequence with more variables and $\land$ symbols than could ever get written in the physical universe? In such a case, the assignment of truth values to the propositions would never end. Though, evaluating from right to left we would quickly reach a falsity, making the entire conjunction false. – Doug Spoonwood Oct 2 '16 at 21:57
• You seem to have in mind some physical, mechanical process that computes truth values. But that's not intrinsic to mathematics. In the (perhaps Platonic) universe of mathematical propositions, the number of particles in our universe is irrelevant, as is any notion of time that would make an evaluation begin or "end". The propostion simply exists, as do its truth values, as an immutable consequence of the logic defining the propositional universe. – Greg Martin Oct 2 '16 at 22:19

A contradiction is something that is always false, regardless of it's truth values.