Contradiction

In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that 'One cannot say of something that it is and that it is not in the same respect and at the same time.'... I in my astonishment said: What do you mean Dionysodorus? 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. The dictum is that there is no such thing as a falsehood; a man must either say what is true or say nothing. Is not that your position?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 a new definition must be given.The property of being a tautology has been defined in notions of truth and falsity. Yet these notions obviously involve a reference to something outside the formula calculus. Therefore, the procedure mentioned in the text in effect offers an interpretation of the calculus, by supplying a model for the system. This being so, the authors have not done what they promised, namely, 'to define a property of formulas in terms of purely structural features of the formulas themselves'. ... proofs of consistency which are based on models, and which argue from the truth of axioms to their consistency, merely shift the problem.For example, given a formula such as ~S1 V S2 and an assignment of K1 to S1 and K2 to S2 one can evaluate the formula and place its outcome in one or the other of the classes. The assignment of K1 to S1 places ~S1 in K2, and now we can see that our assignment causes the formula to fall into class K2. Thus by definition our formula is not a tautology.Definition. A system will be said to be inconsistent if it yields the assertion of the unmodified variable p . In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that 'One cannot say of something that it is and that it is not in the same respect and at the same time.' By creation of a paradox, Plato's Euthydemus dialogue demonstrates the need for the notion of contradiction. In the ensuing dialogue Dionysodorus denies the existence of 'contradiction', all the while that Socrates is contradicting him: Indeed, Dionysodorus agrees that 'there is no such thing as false opinion ... there is no such thing as ignorance' and demands of Socrates to 'Refute me.' Socrates responds 'But how can I refute you, if, as you say, to tell a falsehood is impossible?'. In classical logic, particularly in propositional and first-order logic, a proposition φ {displaystyle varphi } is a contradiction if and only if φ ⊢ ⊥ {displaystyle varphi vdash ot } . Since for contradictory φ {displaystyle varphi } it is true that ⊢ φ → ψ {displaystyle vdash varphi ightarrow psi } for all ψ {displaystyle psi } (because ⊥ → ψ {displaystyle ot ightarrow psi } ), one may prove any proposition from a set of axioms which contains contradictions. This is called the 'principle of explosion' or 'ex falso quodlibet' ('from falsity, whatever you like'). In a complete logic, a formula is contradictory if and only if it is unsatisfiable. For a proposition φ {displaystyle varphi } it is true that ⊢ φ {displaystyle vdash varphi } , i. e. that φ {displaystyle varphi } is a tautology, i. e. that it is always true, if and only if ¬ φ ⊢ ⊥ {displaystyle eg varphi vdash ot } , i. e. if the negation of φ {displaystyle varphi } is a contradiction. Therefore, a proof that ¬ φ ⊢ ⊥ {displaystyle eg varphi vdash ot } also proves that φ {displaystyle varphi } is true. The use of this fact constitutes the technique of the proof by contradiction, which mathematicians use extensively. This applies only in a logic using the excluded middle A ∨ ¬ A {displaystyle Avee eg A} as an axiom. In mathematics, the symbol used to represent a contradiction within a proof varies. Some symbols that may be used to represent a contradiction include ↯, Opq, ⇒⇐ {displaystyle Rightarrow Leftarrow } , ⊥, ↔   {displaystyle leftrightarrow !!!!!!!} / , and ※; in any symbolism, a contradiction may be substituted for the truth value 'false', as symbolized, for instance, by '0'. It is not uncommon to see Q.E.D. or some variant immediately after a contradiction symbol; this occurs in a proof by contradiction, to indicate that the original assumption was false and that its negation must therefore be true. A consistency proof requires (i) an axiomatic system (ii) a demonstration that it is not the case that both the formula p and its negation ~p can be derived in the system. But by whatever method one goes about it, all consistency proofs would seem to necessitate the primitive notion of contradiction; 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: