Contradictions arise quite naturally in several areas of thought and it is not easy to resolve them.  [1] It is generally felt that contradictions are unacceptable for two reasons: (a) contradictions entail anything and (b) they violate the logical law of non-contradiction. But some have questioned these reasons for rejecting contradictions, developing what is called “para-consistent logic”. There are even those who stoutly maintain that contradictions can be true (as opposed to necessarily false). For example, in the case of the Liar paradox a sentence can be both true and false (not true); and an author might describe a fictional character in contradictory ways (knowingly or unknowingly) both of which must be accepted as true. The point I want to make is that even accepting that kind of view we can still insist on the universal applicability of the logical law of non-contradiction, properly understood.

            Suppose we allow that a proposition of the form “x is F and x is not F” is true. We are then saying that an object x has both the property of being F and the property of being not-F, so that these two properties don’t logically exclude each other (of course, traditional logicians deny this, holding that these properties do exclude each other). What we are clearly not saying is that x lacks one of these properties: we are saying that x has both of them. Suppose now that we say explicitly that x lacks one of the properties, say the property of being F: “x is F and xlacks the property of being F”. That surely is impossible: no object can both have a property and lack it—if the property is present, it is necessarily not absent. We could call this law “the Law of Non-Exclusion”: if an object has a property, that fact excludes lacking that property. I don’t think there are any areas in which we are forced, or even tempted, to accept any exception to that law. In the case of the Liar paradox, accepting the contradiction as true is a matter of accepting that a proposition can be both true and false—it is not a matter of accepting that a proposition can be true and yet lacks the property of being true (similarly for the property of being false or not true). In other words, saying that an object is not F is not the same as saying that it lacks the property of being F, since the latter logically excludes being F while the former does not (if we accept true contradictions).

            Suppose an author describes a character, Ned, as tall early in her novel but as not tall later in the novel, having forgotten her earlier description. We then seem entitled to say that Ned is both tall and not tall—at least one can sympathize with the temptation to say that. But this is not tantamount to saying that Ned both has and lacks the property of being tall: he has both, though they contradict each other. It would be quite wrong to say that Ned lacks the property of being tall because he is later described as not tall, since the author’s earlier description is sufficient to confer tallness on Ned (like her later description of Ned as not tall). That is, accepting such contradictions as true, as some recommend, is not the same as accepting that objects can have properties but also lack them: it is accepting that objects can actually have contradictory properties. If this is right, then accepting contradictions as true is not in violation of the correct formulation of the logical law of non-contradiction, since that law states that objects cannot lack properties that they have. To accept that objects can have contradictory properties is not the same as accepting that they can have properties and also lack them; on the contrary, it is accepting precisely that they have both properties. So we can accept contradictions without violating the law of non-contradiction, correctly understood, i.e. the law of non-exclusion. Or perhaps we should say that there are really two laws of non-contradiction, one of which has exceptions (if we choose to go that way) and one that does not. To accept that there can be true contradictory statements is not to accept that there can be states of affairs in which objects both have and lack the same properties.

We thus have some wiggle room when it comes to areas that easily generate contradictions: it becomes easier to accept the existence of true contradictions—for example, accepting that the Liar sentence is both true and false. This does not require us to give up on the law of non-exclusion, which is arguably what the law of non-contradiction was saying all along. The disturbance is nowhere near as great as would be occasioned by accepting that something can both have and lack a given property. However, there is no reason to contemplate that possibility stemming from any of the areas that generate contradictions, and it is surely logically ruled out. We are naturally led to entertain propositions of the form “x is F and x is not F”, but nothing requires us to contemplate the (alleged) possibility expressed by “x is F and x lacks the property of being F”. In the problematic cases these two propositions come apart.

            Contradictions don’t arise everywhere but only in special cases; maybe they could not arise everywhere. Contradictions are restricted in scope. This means that we need not tinker with the law of non-contradiction for the majority of cases, since there cannot be naturally occurring contradictions in these cases. It is only in certain special cases that we might contemplate accepting true contradictions: but even here we need not contemplate abandoning a universal law of logic, namely the Law of Non-Exclusion. We can tolerate contradictions without tolerating objects simultaneously having and lacking properties.  So the para-consistent logician need not be quiteas radical as he appears to be.  


  [1] I won’t discuss any of these areas in detail, but I have in mind not only the semantic and set-theoretic paradoxes but also the following: contradictions arising in fiction, contradictions arising through vague concepts, and Kripke’s argument that our practices of belief ascription license contradictory ascriptions of belief. In all these cases there are apparently cogent reasons to endorse contradictions—reasons to accept contradictions as true. My question is how drastically revisionary it would be to succumb to those reasons.

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