Difference between revisions of "Tautology" - New World Encyclopedia
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In mathematics, ‘A = A’ is a tautology. In formal logic, the statements ‘P → P’ (interpreted in English as ‘If P then P’ or sometimes and less accurately as 'P implies P') 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 both tautologies. | In mathematics, ‘A = A’ is a tautology. In formal logic, the statements ‘P → P’ (interpreted in English as ‘If P then P’ or sometimes and less accurately as 'P implies P') 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 both tautologies. | ||
| + | Some people consider definitions to be tautologies. For example, 'bachelor' =df. 'inmarried 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 validities == | ||
| + | In formal logic, an argument is a set of statements, one or more of which (the premises) 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. | ||
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| + | 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: | ||
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| + | *(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 nad 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). | ||
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| + | 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 knolwedge or proof. That is why circular arguments are worthless. | |
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==Discovering tautologies== | ==Discovering tautologies== | ||
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==See also== | ==See also== | ||
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===Normal forms=== | ===Normal forms=== | ||
* [[Algebraic normal form]] | * [[Algebraic normal form]] | ||
Revision as of 02:59, 3 July 2006
Tautologies are statements that are always true because of their structure — they require no assumptions or evidence to determine their truth. A tautology gives us no genuine information because it only repeats what we already know.
In mathematics, ‘A = A’ is a tautology. In formal logic, the statements ‘P → P’ (interpreted in English as ‘If P then P’ or sometimes and less accurately as 'P implies P') 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 both tautologies.
Some people consider definitions to be tautologies. For example, 'bachelor' =df. 'inmarried 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 validities
In formal logic, an argument is a set of statements, one or more of which (the premises) 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 nad 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 knolwedge or proof. That is why circular arguments are worthless.
Discovering tautologies
An effective procedure for checking whether a propositional formula is a tautology or not is by means of truth tables. As an efficient procedure, however, 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. Algebraic, symbolic, or transformational methods of simplifying formulas quickly become a practical necessity to overcome the "brute-force", exhaustive search strategies of tabular decision procedures.
See also
Normal forms
- Algebraic normal form
- Conjunctive normal form
- Disjunctive normal form
Related topics
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