Axiom: The Law of Non-Contradiction

Aristotle formulated the law of non-contradiction (LNC) inMetaphysics, 1005b18-20:

Evidently then such a principle is the most certain of all; which principle this is, let us proceed to say. It is, that the same attribute cannot at the same time belong and not belong to the same subject and in the same respect;

When fitted in EUO, LNC can be formulated as: one property cannot be instantiated and not instantiated by the same particular at the same time in the same respect. As temporal stages of the Universe (TSUs) are particulars, it follows from LNC that the TSUs are non-contradictory. E.g. the attributeroundcannot at the same time and in the same respect belong and not belong to the subjectball, where the ball is a physical object. The attribute round can be translated as a determinable range of properties

(§4.10): a particular can instantiate exactly one of these properties at one time in one respect.

LNC applies also for mental properties of particulars: one mental property cannot belong and not belong to the same particular at the same time in the same respect. E.g. the propertyround mental object cannot at the same time and in the same respect belong and not belong to the same particular. Saying that the propertyround cannot at the same time and in the same respect belong and not belong to themental propertyball, is an indirect way of saying that the propertyround mental object cannot at the same time and in the same respect belong and not belong to the same particular.

According to Gottlieb [156] the above version of LNC is the ontological version which

“concerns things that exist in the world” and this is also its intended scope here.⁵⁴ Got-tlieb maintains that altogether three versions of LNC can be found from Aristotle: on-tological, doxastic and semantic versions, where the doxastic version is not immediately important with respect to the topics of this thesis. Aristotle formulates the semantic version as follows: “opposite assertions cannot be true at the same time” (Metaphysics 1011b13-20). The semantic version seems to be very close to thecoherence theorem of correspondence truths(§6.1) —a theorem of ontological realism and the ontological ver-sion of LNC— where correspondence truths are the assertions, i.e., propositions realized in minds of human beings. This is in line with Tahko’s [389, p. 27] remark that “Aristo-tle’s line of thought suggests that the link that is often taken to exist between language or grammar, and logic, is in fact between reality and our thoughts.” Gottlieb [156] notes that the semantic version results from the ontological version: “the idea that opposite assertions cannot be true at the same time suggests that this third version is better inter-preted as a variant of the first formulation.” That is, the semantic version follows from the ontological version when it is coupled with the notion that “any assertion involves predicating one thing of another” (ibid); and this is very close to saying that the coher-ence theorem follows from the fusion of ontological realism and the ontological version of LNC. Form here on, LNC denotes the ontological version. Consider three arguments for LNC.

(1) A violation of LNC is inconceivable. If a particular would violate LNC, we could not conceive how that violation is realized, for everything that is in principle conceiv-able conforms to LNC: “The law of non-contradiction provides the ‘formal’ criterion of conceivability: anything that violates it is inconceivable” (Stang [373, p. 191]). In other words, perception can never confirm the existence of a contradiction, because it is impos-sible to simultaneously perceive (or measure) and not to perceive something in the same respect. For instance, it is impossible to measure that the temperature is 10 Celsius and not 10 Celsius in the same place at the same time in the same respect.

(2) If something is interpreted to violate LNC, the same thing can be interpreted so that LNC is not violated, i.e., we are dealing with mutually exclusive metaphysical commitments. The commitment that there are some particulars that violate LNC and some that do not (those which are perceived), is more complex than the commitment that all particulars conform to LNC, and therefore economy favours committing to LNC.

54For comparison, Kutach [212, ch. 1.6] postulates something very close to LNC: “Fundamental reality is as determinate as reality ever gets. . . . Fundamental reality is consistent.” Sami Pihlstr¨om commented:

“Contradictions can be found from expressions of language or in logical systems, but hardly in the world itself; it is not wholly clear that it is even sensible to attribute contradictoriness or non-contradictoriness to nature.” Certainly, if one commits to LNC, then one commits to non-contradictoriness of nature.

LNC may seem to be self-evident, but it is important to explicitly postulate this self-evident axiom, so that there would be no ambiguities about the case.

(3) The rejection of LNC is either meaningless or the rejection implicitly commits to LNC.

Suppose that an extreme relativist argues that LNC is a vague or a subjective statement that does not have a definite meaning; if the relativist’s argument has a definite meaning, then he commits to LNC, for his argument does not simultaneously have and not have a meaning in the same sense; if the relativist’s argument does not have a definite meaning, then there is nothing to worry about. Gottlieb [156] says the same, where PNC denotes LNC: “Anyone asking for a deductive argument for PNC, as Aristotle points out, is missing the point, or, rather, is asking for something that is impossible without using PNC. You cannot engage in argument unless you rely on PNC.” As the rejection of LNC requires committing to LNC, it follows that the willingness to reject LNC is a signal of either a failure in understanding the meaning of LNC, or of self-deception. Self-deception occurs when a person implicitly relies on what he explicitly denies. When a relativist travels in an aeroplane at 30 000 feet, he implicitly relies on the technology that is used in building the aeroplane, but he explicitly denies the validity of LNC which is a prerequisite for the technology (cf. Norris [294, pp. 249, 314]). Norris (ibid, p. 249) refers to an example from Richard Dawkins [100, p. 32]: “show me a cultural relativist at 30 000 feet and I’ll show you a hypocrite.” Likewise, when a relativist is seriously ill and accepts the medicine which was developed by holding LNC as a premise, he implicitly relies on what he explicitly denies.

many-valued logic, paraconsistent logics, dialetheism. LNC is compatible with the existence of propositions which violate thelaw of the excluded middleand the principle of bivalence, i.e., LNC does not exclude the application of many-valued logics (§7.3). LNC is also compatible with paraconsistent logics, whose basic idea is in non-trivial inconsistency: although the surface structure of a sentence is incoherent, you can deduce something intelligible from it, but you cannot deduce everything from it, i.e., paraconsistent logic accepts incoherent statements but denies that everything can be deduced from them. To illustrate, even though the surface structure of the proposition S=“Mary is happy and sad” is incoherent, S makes perfect sense together with the propositions “Mary won the lottery” and “Mary’s house burned down.” Mary is happy in the sense that she won the lottery and sad in the sense that her house burned down, i.e., LNC is not violated here as Mary is not happy and sad at the same time and in the same respect.

Whether LNC anddialetheismare compatible or incompatible depends on how LNC is interpreted and how the range of dialetheism is interpreted. According to Priest and Berto [316] a “dialetheia is a sentence, A, such that both it and its negation, ¬A, are true.” E.g. “Mary is happy” seems to be a dialetheia in the sense that it is true and its negation is true, in the context of the above paragraph: again, not in the same sense and therefore LNC is saved. Priest and Berto maintain that dialetheism violates a certain version of LNC, which is denoted here as thelinguistic LNC (LLNC): “for anyA, it is impossible for bothA and¬A to be true.” Again, Priest and Berto talk solely about sentences (propositions) and they do not indicate anything about the correspondence of sentences to mind-independent reality. Therefore, they do not claim that the ontological version of LNC is violated, but they remain strictly in the linguistic realm.

Priest and Berto [316] note that dialetheism draws its orientation from the paradoxes of self-reference. Also the self-reference industry is purely linguistic, or if it is metaphysical then it is uneconomical: genuine self-reference is explicitly rejected in§4.18 on the basis that it violates finite divisibility, which is an axiom of EUO. Therefore, there are no paradoxes of self-reference to be resolved in EUO: if a seemingly self-referring proposition has a meaning —other than the intention of expressing a paradox— then the meaning can

be explained also without self-reference. The order is the same as always in economical unification: ontology first, semantics second. It would be against economical unification to take a paradoxically formulated surface structure of a sentence so seriously as to propagate it into a violation of LNC. Although Priest and Berto maintain that other

“cases involve contradictions affecting concrete objects and the empirical world” they still remain in the linguistic realm and discuss violations of LLNC, not LNC. To illustrate, they consider transition states of leaving a room, maintaining that there must be “a precise instant in time, call it t, at which I leave the room. Am I inside the room or outside at timet? . . . if I am neither inside not outside the room, then I am not inside and not-not inside” and that this is “a dialetheic situation.” This does not violate LNC, for nothing in the temporal stage of the Universe which is realized att violates LNC, although examples that violate LLNC can be invented.