The first known formulation is found in Aristotle`s discussion of the principle of non-contradiction, first proposed in On Interpretation,[3] where he says that of two contradictory propositions (i.e. when one proposition is the negation of the other), one must be true and the other false. [4] He also explains it as a principle in Book 3 of Metaphysics, saying that in any case it is necessary to affirm or deny,[5] and that it is impossible for there to be anything between the two parts of a contradiction. [6] In classical mathematics, there are non-constructive or indirect proofs of existence that intuitionists do not accept. For example, to prove that there exists an n such that P(n), the classical mathematician can derive a contradiction from the hypothesis for all n, not for P(n). In classical and intuitionistic logic, this leads by reductio ad absurdum not for all n, not for P(n). Classical logic converts this result to n so that P(n), but not generally into an intuitionist. the classical sense that somewhere in the complete infinite totality of the natural numbers, an n occurs, such that P(n), is not available to it, since it does not include the natural numbers as a completed totality. [2] (Kleene 1952: 49 – 50). Kronecker insisted that there could be no existence without construction.
For him, as for Paul Gordan [another older mathematician], Hilbert`s proof of the finiteness of the basis of the invariant system was simply not mathematical. Hilbert, on the other hand, insisted throughout his life that if it can be proved that the attributes attributed to a concept never lead to a contradiction, the mathematical existence of the concept is thus established (Reid p. 34) In logic, the law of the excluded middle (or the principle of the excluded middle) states that for each sentence, either that sentence is true or its negation is true. It is one of the three laws of thought, along with the law of non-contradiction and the law of identity. The law of the excluded middle is logically equivalent to the law of non-contradiction by De Morgan`s laws; However, no logical system is based solely on these laws, and none of these laws contain rules of inference, such as the laws of Modus Ponens or De Morgan. Laws of logic: (1) the law of contradiction, (2) the law of the excluded (or third) middle, and (3) the principle of identity. The three laws can be symbolically formulated as follows. (1) For all p-sentences, it is impossible for p and not p to be true, or: ∼(p. The transfer of the classical principle of non-contradiction (NC) and its double law: the principle of the excluded medium (EM) to the framework of fuzzy logic leads to a well-known functional equation with appropriate aggregation and negation functions. If the implied negation is the natural negation of a fuzzy implication function, the NC principle becomes particularly important because it is then a necessary condition for satisfying the Ponens modus inequality as well as for the residual property for the residual implications.
In this thesis, the functional equation corresponding to the NC principle is examined in detail if the aggregation function and the fuzzy negation are as general as possible. In addition, the results are applied to the case where fuzzy negation is the natural negation of a fuzzy implication function, thus extending this study to the most common classes of fuzzy involvement functions. Thus, Hilbert said, “If p and ~p are both proven true, then p does not exist,” referring to the law of the excluded middle, which was formulated in the form of the law of contradiction. Modern mathematical logic has shown that the excluded medium leads to a possible self-contradiction. In logic, it is possible to make well-constructed sentences that can be neither true nor false; A common example of this is the “liar`s paradox”,[11] the statement “this statement is false,” which itself can be neither true nor false. The law of the excluded middle still applies here, since the denial of this statement “This statement is not false” can be classified as true. In set theory, such a self-referential paradox can be constructed by examining the set “the set of all sets that do not contain”. This set is clearly defined, but leads to a Russellian paradox.[12][13] In modern Zermelo–Fraenkel set theory, however, this kind of contradiction is no longer allowed.
The law is also known as the law (or principle) of the excluded middle, in Latin principium tertii exclusi. Another Latin name for this law is tertium non datur: “no third [possibility] is given”. It is a tautology. The proof of ✸2.1 is something like this: “primitive idea” 1.08 defines p → q = ~p ∨ q. Replacing q by p in this rule gives p → p = ~p ∨ p. Since p is true → p (this is theorem 2.08, which is proved separately), then ~p ∨ p must be true. Let`s call things that are immediately known in sensation “sensory data”: things like colors, sounds, smells, hardness, roughness, etc. We will call the experience of being immediately aware of these things “sensation.” The color itself is a sensory date, not a sensation. (page 12) The following sheds light on the deep mathematical and philosophical problem behind what “knowing” means and also helps explain what “law” implies (i.e.
what the law actually means). Their difficulties with the law are highlighted: they do not want to accept as true implications drawn from what is unverifiable (unverifiable, unknowable) or the impossible or false. (All quotes are from van Heijenoort, emphasis added). ✸2.1 ~p ∨ p “This is the law of the excluded middle” (PM, p. 101). That is, when we judge (say) “this is red”, what happens is a relationship of three terms, the spirit and “this” and “red”. On the other hand, when we perceive “the redness of it”, there is a relationship of two concepts, namely the mind and the complex object “the redness of it” (pp. 43-44). Some systems of logic have different but analogous laws. For some logics with n finite values, there is an analogous law called the n+1 excluded law. If the negation is cyclic and “∨” is a “max operator”, then the distribution can be expressed in object language as (P ∨ ~P ∨ ~~P ∨.
∨ ~…~P), where “~…~” n−1 sign of negation and “∨. ∨” n−1 sign of disjunction. It is easy to check whether the sentence should have at least one of the n logical values (and not a value that is not one of the n). Gödel`s approach to the law of the excluded middle was to assert that objections to “the use of `unpredative definitions`” “carry more weight” than “the law of the excluded middle and related propositional calculus” (Dawson, p. 156). He proposed his “System Σ. And he concluded by mentioning several applications of his interpretation. Among them was a proof of conformity with the intuitionistic logic of the ~ principle (∀A: (A ∨ ~A)) (despite the inconsistency of hypothesis ∃ A: ~ (A ∨ ~A)… (Dawson, S.
157) ✸2.11 p ∨ ~p (permutation of claims is permitted by axiom 1.4) ✸2.12 p → ~(~p) (principle of double negation, part 1: If “this rose is red” is true, then it is not true that “`this rose is not red` is true”.) ✸2.13 p ∨ ~{~(~p)} (Lemma used with 2.12 to derive 2.14) ✸2.14 ~(~p) → p (principle of double negation, Part 2) ✸2.15 (~p → q) → (~q → p) (One of the four “implementation principles”. Similar to 1.03, 1.16 and 1.17. A very long demonstration was needed here.) ✸2.16 (p → q) → (~q → ~p) (If it is true that “If this rose is red, then this pig flies”, then it is true that “If this pig does not fly, then this rose is not red.”) ✸2.17 ( ~p → ~q ) → (q → p) (Another of the “implementation principles”.) ✸2.18 (~p → p) → p (Called “The complement of reductio ad absurdum.
