And and Not

And and Not

Sharp thinkers (Frege, Russell, Wittgenstein, among others) have felt that there is something special about the classical logical connectives, and, or, not, and if. I will list the features commonly attributed to these concepts. They are truth-functional and referentially transparent. They are disquotational in the manner formulated by Tarski’s so-called recursion clauses (“p and q” is true if and only if “p” is true and “q” is true). They are inter-definable and permit reduction to just two primitive concepts (and and not are the most intuitive—I set aside the Scheffer stroke). They can be iterated indefinitely. They are logically central in that they form the structure of logical arguments. They are topic-neutral. They can be used to define the quantifiers all and some (more or less). They are constructive in that they build new propositions from old. They are universal in belonging to any language or system of thought worthy of the name. They have been thought to be unique in that no other linguistic constructions have the properties just listed, despite some superficial syntactic resemblance (“because”, “necessarily”, “believes”); and it is true that other locutions don’t have the full range of properties that mark the classical connectives. They do appear special. So, how are we to understand their special role in our thought and language—what do they do for us? I think the answer is as follows. The connective and (i.e., conjunction) operates so as to create totalities: these can be propositions or facts or objects (Bob and Ted and Carol and Alice). It is accumulative (additive, aggregative). This is why it is so closely connected to all, which also collects things together in thought. That is, and is the basis of set-theoretic thinking, which is the basis of mathematics and much of science. It is the foundation of classes and categories. With and we go from the particular to the general. We can even formulate ideas such as that of the whole world (the totality of facts—this fact and that fact and the other fact, etc.). Conjunction is a fecund operator. In the case of not we get the possibility of unreality—of not being. If you negate a proposition, you create a representation of something (purportedly) unreal—something that is not the case. Snow is white, but it might not be white. Thus, possibility enters our thoughts, our understanding of reality: reality contrasts with unreality, truth with error, presence with absence, life with death (not-aliveness). The existentialists were right to see in negation an expression of the human condition—our awareness that reality is suspended over an abyss of unreality. The false is as real as the true in the sense that error really exists. It is doubtful that other animals grasp this contrast; they live in the not-not, i.e., the is. So, and gives us the idea that reality forms collections and not gives us the idea that reality has alternatives: it isn’t that reality consists of nothing but unrelated particulars and what you can see with your eyes. There are totalities and non-actualities, wholes and absences. Logic is built around both ideas. So, andand not have a claim to being conceptually basic and indispensable to reason—hence as the essence of logic. Those sharp thinkers were onto something, even if they couldn’t quite say what it was. They gazed in wonder at conjunction and negation, and we can join them in that. As Frege might say, without and and not thought would be crippled, but with them it soars. They form the backbone of logic, which is the backbone of reason, which is the mark of humanity: and and not make us special.[1]

[1] I don’t mean they are projections of human nature—that would be psychologism. I mean our grasp of these objective operations is what sets us apart from the rest of nature, though I wouldn’t rule out some primitive grasp of them in other species, especially primates.  

Share
0 replies

Leave a Reply

Want to join the discussion?
Feel free to contribute!

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.