A Tautology is a statement that is always true because of its structure-it requires no assumptions or evidence to determine its truth. A tautology gives us no genuine information because it only repeats what we already know. Thus, tautologies are usually worthless as evidence or argument for anything; the exception being when a tautology occurs in testing the validity of an argument.
In mathematics, 'A = A' is a tautology. In formal two-valued logic (i.e. logic based on the two principles: (1) that nothing can be both true and false at the same time and in the same way, and (2) that every statement is either true or false), the statements 'P → P' (interpreted in English as 'If P then P' or sometimes and less accurately as 'P implies P'), 'P v ~P' (in English, 'P or not P' or 'Either P is true or not P is true'), and 'P ↔ P' (interpreted in English as 'P if and only if P' or sometimes and less accurately as 'P is logically equivalent to P') are all tautologies. Each of them is always true.
Some people consider definitions to be tautologies. For example, 'bachelor' is defined as 'unmarried male."Bachelor' and 'unmarred male' mean the same thing, so, according at least to this understanding of definitions, defining 'bachelor' as 'unmarried male' does not give us any new information; it merely links together two terms that are identical.
Tautologies versus valid arguments
In formal logic, an argument is a set of statements, one or more of which (the premise or premises) is/are offered as evidence for another of those statements (the conclusion). An argument is deductively valid if and only if it is truth-conferring, meaning that it has a structure that guarantees that if the premise(s) are true, then the conclusion will necessarily be true.
Some but not all arguments, then, are tautologies. The argument form Modus Ponens, for example, is valid but is not a tautology. Modus Ponens has the form:
- (First or major premise): If P then Q.
- (Second or minor premise): P is true.
- (Conclusion): Thus Q is true.
It is impossible for both premises of that argument to be true and for the conclusion to be false. Any argument of this form is valid, meaning that it is impossible for the premises to be true and the conclusion to be false. But this argument is not a simple tautology because the conclusion is not a simple restatement of the premise(s).
But the following argument is both valid and a tautology:
- Premise: (Any statement) P.
- Conclusion (That same statement) P.
The argument has the form, 'If P, then P.' It is indeed a valid argument because there is no way that the premise can be true and the conclusion false. But it is a vacuous validity because the conclusion is simply a restatement of the premise.
In fact, all circular arguments have that character: They state the conclusion as one of the premises. Of course, the conclusion will then necessarily follow, because if a premise is true and the conclusion is simply a restatement of that premise, the conclusion will follow from the premise. But, although it is technically valid, the argument is worthless for conveying any information or knowledge or proof. That is why circular arguments should be rejected, and why showing that an argument is circular is sufficient to show that it is no good: Circular arguments are trivially valid, but are worthless for establishing their conclusion(s).
Statements as tautologies, and discovering tautologies
Some statements, especially logical statements or expressions, can be understood as being tautologies. This means that, under any interpretation of truth or falsity of its constituent parts, the entire statement is always true.
For example, the logical statement: “It is not the case that the conjunction of P and not-P is true,” symbolized by '~(P • ~P)' (where ~ is the symbol for negation and • is the symbol for conjunction) is a tautology. This can be shown by a truth table:
- ~ (P • ~ P)
- T (T F F T)
- T (F F T F)
Meaning that whether P is true or false, the conjunction of P and not-P is always false, so the negation of that conjunction is always true. (Shown in the above table by having 'T' under the leftmost negation sign, which is the major operator in this logical formula.)
An inconsistent statement is one that, whatever the truth or falsity of the constituent parts, the entire statement is always false: the simplest example of an inconsistent statement is any of the form 'P and not-P.' So the negation of an inconsistent statement is always true, meaning that the negation of an inconsistent statement is a tautology.
Similarly, the negation of a tautology is inconsistent, meaning that it is always false.
It is also the case that a valid argument, if expressed in a conditional with the conjunction of its premises as the antecedent of the conditional and the conclusion as the consequent of the conditional, is a tautology. In fact, this is one method for testing the validity of arguments in sentence-logic form: Construct a conditional with the conjunction of the premises as the antecedent and the conclusion as the consequent, and then use a truth table to see whether the entire thing becomes always true under every possible interpretation of truth and falsity for its constituent parts.
Such a construction would have the form, "(Premise 1 • Premise 2 •… Premise N i.e., however many premises the argument has) → (Conclusion)”
We can use the example of Modus Tollens, which has the form:
- (Major Premise) If P then Q
- (Minor Premise) Not Q
- (Conclusion) Not P
Making a conjunction of the argument, as stated above, we would get: (P → Q) • (~Q) → ~P
Constructing a truth table would give us:
- (P → Q)• (~Q) → ~P
- (T T T)F (FT) T FT
- (T F F)F (TF) T FT
- (F T T)F (FT) T TF
- (F T F)T (TF) T TF
In every case, the truth value under the major operator - which is the truth-value for the entire expression (in this example it is the right arrow joining together the left hand and right hand parts of the formula) - is true, meaning that any interpretation of truth or falsity for P or Q will yield truth for the entire logical formula, so the entire formula is a tautology, which shows that the original logical form of modus tollens is valid.
The problem with constructing truth tables for arguments having more than a few variables is that truth tables are constrained by the fact that the number of logical interpretations (or truth-value assignments) that have to be checked increases as 2k, where k is the number of variables in the formula. So a truth table for three variables will have eight lines and one for four variables will have 16 lines, meaning that it will get cumbersome.
Thus natural deduction or other methods of checking formulas quickly become a practical necessity to overcome the "brute-force," exhaustive search strategies of tabular decision procedures.
Tautologies also exist for quantification logic. The expression, "For all x, the conjunction of Fx and not Fx is false" is a tautology. In a similar way, the expression, "There is no x such that Fx and not Fx is true" is also a tautology. Further exploration of this would require study and development of quantification logic.
Almost all logic textbooks-and there are now hundreds of them-contain a section or sections on tautologies.
Three such representative textbooks are:
- Copi, Irving M., and Carl Cohen. Introduction to Logic. Prentice Hall. (Many editions; the latest, from 2004, is the 12th.)
- Hurley, Patrick J. A Concise Introduction to Logic. Belmont, CA: Wadsworth/Thompson Learning. (Many editions; the latest is the 9th.)
- Johnson, Robert M. Fundamentals of Reasoning: A Logic Book. Belmont, CA: Wadsworth. (Latest is the 4th edition.)
- Reese, William L. "Tautology," in Dictionary of Philosophy and Religion, New and Enlarged Edition. Atlantic Highlands, NJ: Humanities Press, 1996.
All links retrieved November 17, 2015.