Contradiction
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact.
Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect.
"[1] In modern formal logic and type theory, the term is mainly used instead for a single proposition, often denoted by the falsum symbol
By creation of a paradox, Plato's Euthydemus dialogue demonstrates the need for the notion of contradiction.
I have often heard, and have been amazed to hear, this thesis of yours, which is maintained and employed by the disciples of Protagoras and others before them, and which to me appears to be quite wonderful, and suicidal as well as destructive, and I think that I am most likely to hear the truth about it from you.
This is called the "principle of explosion", or "ex falso quodlibet" ("from falsity, anything follows").
The use of this fact forms the basis of a proof technique called proof by contradiction, which mathematicians use extensively to establish the validity of a wide range of theorems.
[6] Each of these extensions leads to an intermediate logic: In mathematics, the symbol used to represent a contradiction within a proof varies.
In fact, this often occurs in a proof by contradiction to indicate that the original assumption was proved false—and hence that its negation must be true.
Moreover, it seems as if this notion would simultaneously have to be "outside" the formal system in the definition of tautology.
When Emil Post, in his 1921 "Introduction to a General Theory of Elementary Propositions", extended his proof of the consistency of the propositional calculus (i.e. the logic) beyond that of Principia Mathematica (PM), he observed that with respect to a generalized set of postulates (i.e. axioms), he would no longer be able to automatically invoke the notion of "contradiction"—such a notion might not be contained in the postulates: The prime requisite of a set of postulates is that it be consistent.
Since the ordinary notion of consistency involves that of contradiction, which again involves negation, and since this function does not appear in general as a primitive in [the generalized set of postulates] a new definition must be given.
[8]Post's solution to the problem is described in the demonstration "An Example of a Successful Absolute Proof of Consistency", offered by Ernest Nagel and James R. Newman in their 1958 Gödel's Proof.
They observed that: The property of being a tautology has been defined in notions of truth and falsity.
Therefore, the procedure mentioned in the text in effect offers an interpretation of the calculus, by supplying a model for the system.
In a thorough manner, Post demonstrates in PM, and defines (as do Nagel and Newman, see below) that the property of tautologous – as yet to be defined – is "inherited": if one begins with a set of tautologous axioms (postulates) and a deduction system that contains substitution and modus ponens, then a consistent system will yield only tautologous formulas.
On the topic of the definition of tautologous, Nagel and Newman create two mutually exclusive and exhaustive classes K1 and K2, into which fall (the outcome of) the axioms when their variables (e.g. S1 and S2 are assigned from these classes).
A system will be said to be inconsistent if it yields the assertion of the unmodified variable p [S in the Newman and Nagel examples].In other words, the notion of "contradiction" can be dispensed when constructing a proof of consistency; what replaces it is the notion of "mutually exclusive and exhaustive" classes.
Some dialetheists, including Graham Priest, have argued that coherence may not require consistency.
An inconsistency arises, in this case, because the act of utterance, rather than the content of what is said, undermines its conclusion.
[14] In dialectical materialism: Contradiction—as derived from Hegelianism—usually refers to an opposition inherently existing within one realm, one unified force or object.
This contradiction, as opposed to metaphysical thinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not cancelling each other out, but actually defining each other's existence.
According to Marxist theory, such a contradiction can be found, for example, in the fact that: Hegelian and Marxist theories stipulate that the dialectic nature of history will lead to the sublation, or synthesis, of its contradictions.
Marx therefore postulated that history would logically make capitalism evolve into a socialist society where the means of production would equally serve the working and producing class of society, thus resolving the prior contradiction between (a) and (b).
