Argument

Tautology (logic)

Overview

In mathematical logic, a tautology (from Ancient Greek: ταυτολογία) is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. It is a logical truth. For example, a formula that states "the ball is green or the ball is not green" is always true, regardless of what a ball is and regardless of its colour. Tautology is usually, though not always, used to refer to valid formulas of propositional logic.

The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false.

Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be made either true or false based on the values assigned to its propositional variables.

The double turnstile notation

⊨

{\displaystyle \vDash S}

is used to indicate that S is a tautology. Tautology is sometimes symbolized by "Vpq", and contradiction by "Opq". The tee symbol

⊤

{\displaystyle \top }

is sometimes used to denote an arbitrary tautology, with the dual symbol

⊥

{\displaystyle \bot }

(falsum) representing an arbitrary contradiction; in any symbolism, a tautology may be substituted for the truth value "true", as symbolized, for instance, by "1".

Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (equiv., whether its negation is unsatisfiable).

The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers—a feature absent from sentences of propositional logic. Indeed, in propositional logic, there is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic, and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (i.e., sentences that are true in every model).

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

In the original texts

Passages from public-domain books where Tautology (logic) is discussed.

And if the reader rebels at the iteration of such obvious tautology, I can only reiterate that the realization of the _meaning_ of this tautology is the key to the whole question of the objectivity of experience, as that stands to the empiricist. The lines of _that_ experience _are_ divergent; not merely _seem_ so.
John Dewey, The Influence of Darwin on Philosophy, and other essays in contemporary thought
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The determinations of a substance, which are only particular modes of its existence, are called accidents. They are always real, because they concern the existence of substance (negations are only determinations, which express the non-existence of something in the substance). Now, if to this real in the substance we ascribe a particular existence (for example, to motion as an accident of matter), this existence is…
Immanuel Kant, The Critique of Pure Reason
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If the intuition is _a priori_ this is called construction; but if it is empirical, then the Object is displayed by means of which objective reality is assured to the concept. Thus we say of an anatomist that he demonstrates the human eye, if by a dissection of this organ he makes intuitively evident the concept which he has previously treated discursively. It hence follows that the rational concept of the…
Immanuel Kant, Kant's Critique of Judgement
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The faculty of being acquainted with things other than itself is the main characteristic of a mind. Acquaintance with objects essentially consists in a relation between the mind and something other than the mind; it is this that constitutes the mind's power of knowing things. If we say that the things known must be in the mind, we are either unduly limiting the mind's power of knowing, or we are uttering a mere…
Bertrand Russell, The Problems of Philosophy
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Similarly, it is a tautology to say that matter in its natural state has no exchange value, because it does not contain any labor, and that exchange value as such does not contain matter. But when William Petty calls “labor the father and earth the mother of wealth,” or when Bishop Berkeley asks “whether the four elements and man’s labour therein, be not the true source of wealth,”[7] or when the…
Karl Marx, A Contribution to the Critique of Political Economy
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Tautology (logic)

Overview

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Philosophers who discussed Tautology (logic)

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