When teaching logic or other courses that bear on the subject, some of my students have expressed their doubts regarding the law of contradiction (also known as the law of non-contradiction). But it is very difficult to outright deny the truthfulness of the LNC once this law (or principle) is rightly understood. Even opponents of the famed "law" usually have not been able to reject it in toto: they too still hold to remnants of it in their formal logistic schemas. Nevertheless, those who normally oppose strict adherence to the LNC think that there are exceptions to the law(such as the liar's paradox or the famed "cat" of quantum mechanics). One writer refers to the law as a "principle" instead.
My interest in the law has been shaped by my studies on the Trinity doctrine: the Trinity appears to be contradictory. If the word "God" is an identity marker (A = A) and not a predicative signifier, then the Trinity seems to be incoherent and logically impossible. I submit that ELOHIM/QEOS (used within a given context of utterance) is a marker of identity and that the proposition, "The Father is God," is an identity statement and not a predicative one. Thus, if there is one person to whom the identity marker "God" applies and this word "God" speaks to His identity simpliciter et simpliciter--then there cannot be anyone else called God/god who has all properties in common with this singular person such that we could rightly apply the law of indiscernibility of identicals to this person and say that A = B or that X = Y (known as the law of Leibniz). Simply put, Ps. 90:2 states that from OLAM to OLAM, Jehovah is God. That one God of Israel is identified as Father in the Hebrew Bible. ELOHIM is therefore not something merely predicated of Jehovah the Father: it tells us Who He is. Cf. the Shema in Deut. 6:4, which certain scholars interpret as a monotheistic confession about one divine person.
Eph. 4:4-6 also declares that there is one God and Father, who is over all and through all and in all. If God is an identity statement, and since there is only one God, no other entity or person can be identified as God without contradicting a major supposition of Trinitarianism, namely, the proposition that there is only one true God. But one question about this whole line of reasoning involves the very LNC itself.
From the Western standpoint, I can safely say that Aristotle was not the first person to employ the LNC, since both Zeno and Parmenides utilized the law during the Presocratic era. And even when Aristotle invokes, formulates and amplifies the law, he is simply abstracting from observations of the empirical world. For minstance, the Philosopher knew that p and ~p could be observed in everyday life. Aristotle evidently realized that a woman cannot be both pregnant and not pregnant simultaneously (at the same time and in the same sense or in the same circumstances); a house cannot be white and non-white simultaneously; no object can be red and green all over simultaneously, and as far as we can tell--no entity is able to subsist in two natures at one time (the stories about minotaurs and centaurs must be false). The LNC also makes me suspect the logical possibility of the Incarnation (Christ existing in two natures simultaneously).
Some Potential Objections to the LNC and My Replies:
"(1) "Transition states: when I leave the room, for an instant I am both in it and not in it."
This example is debatable. If you leave a room, the transition has been completed, so that you're not still in the room. There is no genuine refutation here. This objection is playing upon vague predication.
"(2) "Some of Zeno's paradoxes: the moving arrow is both where it is, and where it is not."
Read Aristotle for a sound refutation of Zeno's paradoxes. The latter's famed puzzles only work if one accepts the major presupposition of his argument, namely, that space is composed of discrete units. Furthermore, the results of Zeno's reasoning are not that desirable--according to that philosopher, motion is illusory and impossible. Do you agree with Zeno in this regard? At any rate, Aristotle refuted the sophisms of Zeno.
"(3) "Borderline cases of vague predicates: an adolescent is both an adult and not an adult."
The term "adolescent" simply denotes an artifical distinction that is contrived by certain societies (not all societies). You cannot legitimately use an arbitrary or constructed distinction to overthrow what seems to be a necessary and immutable truth of the cosmos.
"(4) "Certain quantum mechanical states: a particle may go through two slits simultaneously, even though this is not possible."
The problems that attend QM may be more epistemological than ontological (Mortimer Adler). What is more, QM emphasizes ontological contingency (not ontological necessity): QM may allow for circumstances in which the LNC is circumvented, but using QM for this purpose seems like a stretch to me. As one physicist remarked: "I think it is safe to say that no one understands quantum mechanics."
"If you think you can talk about quantum theory without feeling dizzy, you haven't understood the first thing about it" (Niels Bohr).