Logical and Spiritual REFLECTIONS
Book 3. In Defense of Aristotle’s Laws of Thought
Chapter 6. Not on the geometrical model
Since (or insofar as) the “geometrical model” of theory justification involves arbitrary axioms, it is ultimately conventional. If the first principles (“axioms”) of a body of alleged knowledge cannot apparently be justified by experience, but have to be based on mere speculation (“arbitrary”), such principles must be admitted to be without proof (“conventional”). If the axioms are unproven, then logically so are all claims based on them.
This is freely admitted in the case of geometry (where for instance Euclid’s fifth postulate may be replaced by alternative assumptions), and similarly in other mathematical disciplines. Here, the apparent conventionality of certain axioms gives rise to the possibility of alternative systems, all of which might eventually be found useful in specific empirical contexts. But such a liberal attitude is impossible with regard to the science of Logic.
If we accept the geometrical model for Logic, then Wittgenstein’s claim that “The propositions of logic are tautologies… [and] therefore say nothing” is made to seem true. But if we follow him, and admit that logic is meaningless babbling, then we must regard his own statement as meaningless – for, surely, it is itself intended as a “proposition of logic”, indeed as the highest principle of meta-logic! Granting that, it is as if he has said nothing, and we can well ignore him and move on.
Similarly, some critics have accused Aristotle of ‘begging the question’ in his defense of the laws of non-contradiction and of the excluded middle, i.e. of arguing in a circular manner using the intended conclusion(s) as premise(s). Here again, we can more reflexively ask: does that mean that the fallaciousness of such petitio principii is an incontrovertible axiom of logic? If the speaker is convinced by this rational principle as an irreducible primary, why not also – or even more so – by the second and third laws of thought? Can he justify his antipathy to circularity without committing circularity?
If Logic is not solidly anchored in reality through some more rigorous process of validation, then all knowledge is put in doubt and thus effectively invalidated. If all knowledge is without validity, then even this very claim to invalidity is without validity. The latter insight implies that this skeptical claim is itself invalid, like all others, note well. Therefore, since this skeptical claim is paradoxical, i.e. self-denying, the opposite claim (which is not inherently paradoxical) must be admitted as necessarily true. That is to say, we must admit that Logic has undeniable validity. Only given this minimal admission, does it become possible to admit anything else as true or false.
I have said all this before again and again, but must keep repeating it in view of the ubiquity of statements I encounter these days in debates to the effect that Aristotle’s three Laws of Thought are mere conventions. To make such a statement is to imply one has some privileged knowledge of reality – and yet at the same time to explicitly suggest no such knowledge is even conceivable. Thus, any such statement is self-contradictory, and those who utter it are either fools or knaves, kidding themselves and/or others.
The said laws of thought must not be viewed as axioms of knowledge within a geometrical model. The very idea of such a model is itself an offshoot of Aristotle’s logic – notably his first-figure syllogism, where a broad principle or general proposition (the major premise) is used to derive a narrower principle or particular proposition (the conclusion). It follows that such a model cannot be used to justify Logic, for in such case we would be reasoning in circles and obviously failing to anchor our truths in reality.
The only way out of this quandary is to notice and understand the inductive nature of all knowledge, including deductive knowledge. The ground of all knowledge is experience, i.e. knowledge of appearances (material, mental and spiritual appearances of all sorts). Without cognition of such data, without some sort of given data whatever its ultimate status (as reality or illusion), no knowledge true or false even arises.
There is no such thing as “purely theoretical” knowledge: at best, that would consist of words without content; but upon reflection, to speak even of words would be to admit them as experienced phenomena. To attempt to refer, instead, to wordless intentions does not resolve the paradox, either – for intentions that do not intend anything are not. There has to be some experiential basis to any knowledge claim. Whether the knowledge so based is indeed true, or the opposite of it is true, is another issue, to be sorted out next.
Logic comes into play at this stage, when we need to discriminate between true and false theoretical knowledge. We are always trying to go beyond appearances – and that is where we can go wrong (which does not mean we cannot sometimes be right). If we stayed at the level of pure appearance – the phenomenological level – we would never be in error. But because we existentially need to surpass that stage, and enter the rational level of consciousness, we are occasionally evidently subject to error.
Moreover, it is very difficult for us to remain at the purely phenomenological level: we seem to be biologically programmed to ratiocinate, conceptualize and argue; so we have little choice but to confront logical issues head on. The principles of Logic, meaning the laws of thought and the specific logical techniques derived from them, are our tools for sorting out what is true and what is false. We do not infer truth from these principles, as if they were axioms containing all truth in advance. Rather, these principles help us to discern truth from falsehood in the mass of appearances. Without some appearance to work with, logic would yield no conclusion – it would not even arise.
The validity of Logic is, thus, itself an inductive truth, not some arbitrary axiom. Logic is credible, because it describes how we actually proceed to distinguish truth from falsehood in knowledge derived from experience. No other logic than the standard logic of the three laws of thought is possible, because any attempt to fancifully propose any other logic inevitably gets judged through standard logic. The three laws of thought are always our ultimate norms of discursive conduct and judgment. They point us to an ideal of knowledge we constantly try to emulate.
This logical compulsion is not some deterministic force that controls our brain or mind. It is based on the very nature of the ratiocination that drives our derivation of abstract knowledge from concrete appearances. The primary act of ratiocination is negation: thinking “not this” next to the “this” of empirical data. That act is the beginning of all knowledge over and above experience, and in this very act is the secret of the laws of thought, i.e. the explanation as to why they are what they are and not other than they are.
For, whereas the law of identity (A is A) is an acknowledgment of experience as it presents itself, the law of non-contradiction (nothing can be both A and not A) and the law of the excluded middle (nothing can be neither A nor not A) both relate to things as they do not present themselves. These two laws define for us what denial of A means – they set the standard for our imagination of something not presented in experience at the time concerned. Note this well, for no one before has noticed it that clearly.
Negation is the beginning of the “big bang” of conceptual and argumentative knowledge, the way we pass from mere experience to concepts and principles; and the only way to test and ensure that our rational framework remains in reasonable accord with the givens of experience is to apply the laws of thought. Negations are never directly positively experienced: they are only expressions that we have not experienced something we previously imagined possible. There is no bipolarity in concrete existence; bipolarity is a rational construct.
The concept or term ‘not X’ can be interpreted to mean ‘anything except X’ (whether X here intends an individual thing or a group of things). To deny the law of non-contradiction is to say that this “except” is not really meant to be exclusive – i.e. that ‘not X’ can sometimes include ‘X’ (and similarly, vice versa). Again, to deny the law of the excluded middle is to say that this “anything” is not really meant to be general – i.e. that besides ‘X’ there might yet be other things excluded from ‘not X’. Thus, to deny these laws of thought is to say: “I do not mean what I say; do not take my words seriously; I am willing to lie”.
 In Tractatus, 6 (quoted in A Dictionary of Philosophy).
 For instance, in Chapter IV of his Metaphysics, Gamma.
 This is made clear if we consider what we mean when we say, for example, neither the dog nor the cat is in the room we are in. The absence of the dog and the absence of the cat look no different to us; what we actually see are the positive phenomena only, i.e. the carpet, the desk, the chairs, etc. We do not see a non-dog and a non-cat, or anything else that “is absent” from this room, as if this is some other kind of “presence”. (However, it does not follow that non-dog and non-cat are equivalent concepts – for the cat may be present when the dog is absent and vice versa.)