An antilogism is an inconsistent triad: a set of three propositions that cannot all be true. The most interesting antilogisms are those in which the constitutent propositions are each of them plausible. If they are not merely plausible but self-evident or undeniable, then we are in the presence of an aporia in the strict sense. (From the Greek a-poros, no way.) Aporiai are intellectual impasses, or, to change the metaphor, intellectual knots that we cannot untie. Here is a candidate:
1. Being is independent of knowledge: what is or is the case is not made so by anyone's knowledge of it.
2. Knowledge is knowledge of being: we cannot know what is not or what is not the case.
3. Knowledge requires an internally available criterion or justification.
Each of the limbs of this aporetic triad is exceedingly plausible if not self-evident.
Ad (1). If a thing exists, its existence is not dependent on someone's knowledge of it. It is rather the other around: knowledge of thing presupposes the logically antecedent existence of the thing. And if a proposition is true, it not true because someone knows it. It is the other way around: the proposition's being true is a logically antecedent condition of anyone's knowing it.
Ad (2). 'Knows' is a verb of success: what one knows cannot be nonexistent or false. There is no false knowledge. What one 'knows' that ain't so, as the saying goes, one does not know. Necessarily, if S knows x, then x exists; necessarily, if S knows that p, then p is true. The necessity is broadly logical.
Ad (3). If I believe that p, p a proposition, and p happens to be true, it does not follow that I know that p. There is more to knowledge than true belief. If I believe that Jack is at home, and he is, it does not follow that I know that he is. Justification is needed, and this must be internalist rather than externalist. If I see a cat, it does not follow that I know a cat exists or that the cat I see exists. For I might be dreaming or I might be a brain in a vat. There are dreams so vivid that one literally sees (not imagines, or anything else) what does not exist. If I know a cat just in virtue of seeing one, then I need justification, and this justification must be available to me internally, in a way that does not beg the question by presupposing that there exist things external to my consciousness. Note that 'I see a cat' and 'No cat exists' express logically consistent propositions. They both can (logically) be true. For in the epistemologically primary sense of 'see,' seeing is not existence-entailing. In its epistemologically primary sense, 'see' is not a verb of success in the way 'know' is. 'False knowledge' is a contradictio in adiecto; 'nonexistent visual object' is not.
The limbs of our antilogism, then, are highly plausible and for some of us undeniable. Speaking autobiographically, I find each of the propositions irresistable. But I think most philosophers today would reject (3) by rejecting internalist as opposed to externalist justification.
The propositions cannot all be true. Any two, taken together, entails the negation of the remaining one. Thus, corresponding to this one antilogism, there are three valid syllogisms. That is true in general: every antilogism* sires three valid syllogisms.
The first takes us from (1) & (2) to ~(3). If what exists is independent of knowledge, and knowledge is of what exists, then it is not the case that knowledge requires an internally available criterion.
The second syllogism takes us from (1) & (3) to ~(2). If being is independent of knowledge, and knowledge requires a purely internal criterion, then being is inaccessible to knowledge: what we know are not things themselves, but things as they appear to us. To solve the antilogism by rejecting (2) would put us in the vicinity of Kant's epistemology according to which there are things in themselves but we know only phenomena.
The third syllogism takes us from (2) & (3) to ~(1). If knowledge is of what exists, and knowledge is knowledge only if justified internally, then being is not independent of knowledge, and we arrive at a form of idealism.
Is our antilogism insoluble? In one sense, no aporetic polyad is insoluble: just deny one of the limbs. In the above case, one could deny (3). To justify that denial one would have to work out an externalist theory of epistemic justification. An aporetically inclined philosopher, however, will expect that the resulting theory will give rise to aporetic polyads of its own.
And so we descend into a labyrinth from which there is no exit except perhaps by a confession of the infirmity of reason, a humble admission of the incapacity of the discursive intellect to solve problems that it inevitably and naturally poses to itself.
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*The term and the theory was introduced by Christine Ladd-Franklin.
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