The Dictionary of Logics provides many wordings of this principle:
Onthologic wordings: “at the same time and under the same circumstances, it is impossible that the same thing to exist or not to exist” or “at the same time and under the same circumstances, it is impossible that a thing to have or to not have a property”;
Semantic wordings: “at the same time and under the same circumstances, it is impossible that a sentence to have or to have not a logical value W, “a sentence is impossible to be true and not true at the same time”, it is impossible that a sentence to be true together with its negation”;
Aristotle’s wordings: “it is impossible that contradictory assertions to be both true” and “it is impossible that something to belong and to not belong to a thing in the same sense”.
A similar definition but with an additional specification is included in the law of the excluded tertium from the same dictionary: “at the same time and under the same circumstances, a thing either exists or does not exist, the third possibility being excluded”.
One may observe that these definitions have a common component made-up from two elements: the dichotomy and simultaneity concepts. As we have previously find-out, the dichotomy is able to create a dichotomic classification into a set of objects based on the criterion of a property existence, that is a splitting of the set (base) into objects which have and do not have that particular property. The evaluation of the existence or non-existence of the property is made by an IPS by following a clear rule: if the existential attribute of the property has the value under the perception threshold of IPS, that property does not exist, and if the value is over the threshold, the property exists and the extent of this existence is the value of that particular existential attribute. Attention! This binary-type evaluation method of a property existence deals only with the fact that the property is able to exist or not. In case that it exists, its evaluation is no longer a binary one, but all the possible values of the existential attribute have a common component - they are different from zero.
Therefore, as a result of the dichotomic classification of the base’s objects, we have obtained two complementary classes of objects which have or do not have a specific property, that is the affiliation of the existential attribute value of those objects only to one of the two adjacent-disjoint intervals which were created as a result of the bipartition of the basis domain.
These objects which make-up (by means of union) the base, must also have an essential property, which is the simultaneous existence. The simultaneous existence of the objects is also validated by the perceiving IPS, and that is because of the existence of more perception channels in that IPS, which are able to run both in a parallel and synchronic way (as we saw in the chapter 8).
After all the aspects presented so far, it results that this principle is exclusively applied to those abstract objects which have a simultaneous existence and which are the subject of a dichotomic classification. If generally speaking, an abstract object means either abstract concrete objects or more general abstract objects (sentences, reasonings etc. at which the complementary properties may be their true values) this principle may be defined as follows:
Thus, all the definitions mentioned at the beginning of this section can be covered and the two principles can be even unified, unification which depends on an essential aspect - the existence of a dichotomic classification. The non-existence of this classification directly determines an incorrect application of the law of the excluded tertium, which is no longer valid in case of a politomic classification (for example, see the polyvalent logics), as well as of the non-contradiction principle. But, as regards the dichotomic classifications, these two principles (unified) shall be applied with no exceptions, and the non-contradiction principle alongside other principles stated in this paper is considered as a basic element of the objectual philosophy structure.
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