Q&A: Initial Reflections on the First Being
Initial Reflections on the First Being
Question
Hello,
I started your book last night, and I’m enjoying it very much.0
In the first chapter, you argue that a person who disputes certain conclusions drawn from certain premises necessarily indicates thereby that he does not agree with the premises from which we began (you gave as an example “all tables have four legs”).
It seems as though you are ignoring the fact that systems of inference themselves are not absolute, and one can challenge their validity, which of course is not equivalent to challenging the premises you are talking about. Even the claim in the sentence above, before the example, uses the contrapositive, which is equivalent to the law of the excluded middle in reasonable logical systems (and here I am deliberately sidestepping the lack of an absolute definition of a reasonable system, since I would argue that there is no such thing, for the simple reason that the only candidate is the human system of inference and its mathematical interpretation, and even that is disputed), and there are those who do not accept it. How do these things fit together? Is it not impossible to conduct any discussion at all with a person whose logical system of inference differs in validity from ours?
The main reason I’m asking is that I myself had trouble in a logic course putting my finger on the real difference between rules of inference and axioms.
Thank you,
Answer
Hi A.,
You’re quick.
I don’t think systems of inference are not absolute. Even if mathematicians can formally define systems without the law of the excluded middle, it is clear that in our ordinary thinking it exists, alive and kicking. Moreover, the discussion of those systems themselves (such as the system of iewicz) is conducted within the framework of binary logic. So I relate to systems that do not include the law of the excluded middle as formal systems that do not represent an alternative logical inference. Our philosophical thinking operates within the law of the excluded middle.
Furthermore, it seems to me that without the law of the excluded middle, proofs by contradiction cannot be established. Is anyone really prepared to give up all the statements in mathematics that were proven in that way? I’m not talking about some philosophical proposal or other (paper can absorb anything), but someone who actually thinks that way. I find it hard to believe that such a person exists. So inference without that rule, in my view, has nothing to do with the question of how we think. It is a formalization of a certain relation, and that is all.
An example of something I once wrote about is the lack of distributivity in quantum logic. All the games called “quantum logic” are attempts to provide an explanation for quantum theory and its baffling features within a formal framework. In my view, these are not explanations but descriptions of the problematic system in a different language (a pseudo-logical one). The fact that something is formalized does not mean that it has thereby been explained. The logic within which quantum theory is discussed is ordinary logic. Even quantum logic itself is discussed (in a meta-language) within the framework of ordinary logic (what is true there excludes its opposite. After all, if quantum theory says something, it necessarily excludes its opposite. No one would claim that if we proved the uncertainty principle by contradiction, that does not mean it is true). Therefore, quantum logic cannot really be interpreted as a proposal for a different logic (as though scientific measurements were supposed to affect our logic, which cannot be, because logic is an a priori condition for interpreting measurements). At most, it is a formalization of the results in quantum theory, and they call it “quantum logic” because it is a logic-like system. For example, I am not willing to accept statements that the electron is both a particle and a wave. That contradicts the law of non-contradiction (because a wave is not a particle). You can say that it is in a mixed state of particle and wave (what is called superposition), and there is no logical contradiction in that. By the way, I showed this in a logical analysis of a Talmudic and legal topic, and there it is completely clear.