User:Logicist/LEM
- In logic, The Law of excluded middle, also known as the Principle of excluded middle or Excluded middle is the principle that any proposition is either true, or its negation is. The principle can be expressed in either a logical or a semantical form. The semantical form uses the non-logical word 'true', as above. The logical form uses only logical expressions 'either', 'or'[1]. Thus the law can be expressed by the formula "P ∨ ¬P": "either P or not P", where 'P' is schematic [1] for any proposition such as 'snow is white', 'Socrates is running' and so on.
- The earliest extant form of the law is in the book On Interpretation by Aristotle[2], where he says that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false[3]. He also states it as a principle in the Metaphysics book 3, saying that it is necessary in every case to affirm or deny[4], and that it is impossible that there should be anything between the two parts of a contradiction[5]
- The law should not be confused with the principle of bivalence, which states that every proposition is either true or false, and only has a semantical formulation.
- The principles would be equivalent, however, if 'it is false that P' were equivalent to 'not P', and 'it is true that P' is equivalent to 'P'. In this case 'P or not-P' is obviously equivalent to 'it is true that P or it is false that P'[6].
- However they are not equivalent because bivalence does not hold when there is a third truth-value, meaning that 'not P' is not the same as 'it is false that P'. See three-valued logic.
- In traditional logic the principle of bivalence was not distinguished from the Law of excluded middle[7].
- Historical formulations of the law. Christian Wolff said that between contradictory propositions there is no medium: of contradictory propositions it is necessary that one or the other is true[8]. Baumgarten said that every possible object is either A or not A [9]. Hegel said that "Everything is essentially different; or as it is also expressed, - Of two contradictory attributes only the one belongs to anything and there is no third" [10]
- The law is closely connected with the law of double negation. In Principia Mathematica, Whitehead and Russell deduce double negation from excluded middle. Assume 'p or not-p' (excluded middle, formula *2.11). Substitute 'not-p' for p, giving 'not-p or not-not-p'. The definition of implication[11] then gives 'p implies not-not-p'. A slightly more involved argument gives double negation[12].
- However, Peter Geach has argued that double negation is more fundamental than excluded middle[13], and that excluded middle is derived from it.
- He argues that a negative term is not any more complex than a non-negative term: the meaning of 'not male' is no more complex than 'male' [14]. A predicate is analogous to a closed line on the surface of a sphere, and predicating it is like placing an object on one or the other side of this line. Thus 'there can be no question of logical priority as between the inside and the outside of the line, which inseparably coexist'. A predicate is no more definite than its negation is. Thus double negation (which in effect places an object first on one side of the line, then on the other) is necessarily equivalent to affirmation, and is evident from first principles.
- Excluded middle, by contrast, can be derived from double negation and principle of contradiction as follows. According to the principle of contradiction, a proposition and its negation cannot be true at the same time. Thus 'not (p and not-p)'. But according to one of De Morgan's laws, 'not (p and q)' is definitionally equivalent to 'not-p or not-q'. Thus, substituting 'not-p' for 'q', gives 'not-p or not-not-p'. Assuming double negation gives 'not-p or p' i.e. excluded middle.
- There are two forms of the law, corresponding to the two forms of negation: sentential negation [2] and predicate negation - [3]. In predicate negation, the negative operator is applied to the predicate. Thus the formulation "For any x, either x is F or x is not F" uses predicate negation. With sentential negation, the operator precedes the whole sentence. Thus the formulation 'either p or not-p' uses sentential negation. Propositional negation is a familiar concept in modern logic, and is a more sophisticated concept than predicate negation, since in ordinary languate negation is almost always applied to some part of a sentence [note that in the actual article the proof of the irrationality of root 2 uses the predicate form]. The negation Aristotle discusses is predicate negation.
Note the red link. A few articles would already link to this, and 'search' suggests there would be a few more if the term was wikified. Quite hard to find a good definition of the term, however.
I suggest #REDIRECTAxiom schema. This contains a reasonable definition.
In Logic, a schematic variable is a metalinguistic expression which stands in for any term or subformula of an axiomatic system.
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Semantics
[edit]- ^ Geach,P. "The Law of Excluded Middle", Logic Matters p. 74
- ^ Geach p. 74
- ^ On Interpretation, c. 9
- ^ Metaphyics 2, 996b 26-30
- ^ Metaphyics 7, 1011b 26-27
- ^ Kneale p. 47
- ^ See Łukasiewicz, appendix to 'Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls', Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, 1930
- ^ "Inter contradictoria non dari medium; propositionum contradictoriarum altera necessario vera" (Log. ~532)
- ^ "omne possibile aut est A aut non A" (Metaphysics 10)
- ^ (Log. ~41)
- ^ *1.01 p implies q = not-p or q
- ^ As follows. Substituting ~p for p again, gives '~p -> ~~~p' (*2.13). Therefore 'p v ~p' implies 'p v ~~~p'. From excluded middle, 'p v ~p' is true, therefore so is 'p v ~~~p'. By transposition, and from the definition, we get (*2.14) '~~p -> p'. [5]
- ^ Geach pp 80-81
- ^ Geach p. 79