Mathematical Logic

Mathematical Logic

It has always been felt that logic and mathematics have a lot in common. This led to the idea that mathematics reduces to logic. I am going to argue that the converse is true: logic is basically mathematical; logic “reduces” (expands) to mathematics. It’s more complex than it looks in the textbooks. The first and vital point to get clear about is that logic is not concerned with language—words, symbols. It is concerned with thought—logical thought. If we imagine logical thought before language existed, logic is about that. There is no guarantee that the grammatical forms of language will correctly represent the structure of thought; indeed, that is unlikely to be true, given the prime purpose of spoken language, i.e., efficient communication. Logical thought (reasoning) takes place within the individual and has whatever complexity it needs. Logical concepts are not elements of speech. So, you need to open your mind to the possibility that logic de re is quite unlike the formulas you are familiar with from “symbolic logic” class (an oxymoron in my book). I am talking about conceptual logic—logic in the head. Be prepared, then, for some startling suggestions. Logic needs to be re-written from the ground up.[1]

My thesis then is that what we know as propositional and predicate calculus are species of arithmetic. This is easier to see for predicate logic because it is quantificational—it is about quantity, number. It is about “all” and “some”, or the concepts that underlie these words. These are part of a whole system of quantitative concepts corresponding to “a few”, “several”, “many”, “most”, “more”, “nearly all”, “innumerably many”, “infinitely many”, “one”, “two”, “twenty- seven”, and so on. These are all number words, words for counting, adding, etc. They answer the question “How many?”. It is to be noted that this quantificational logic is second-order: it concerns the number of things falling under a concept, as in “The concept dog has many things falling under it”. The thought is captured by that verbal formulation: we are thinking about concepts and assigning numbers to them. So, we are not doing the same thing in thought as we are when we ascribe a property to an object, as expressed by “That dog is brown”. This has long been recognized by philosophical logicians (or logical psychologists). The logical form of a quantificational thought is given by a property and a number: F (n)—e.g.,Dog (many). In predicate logic (quantification theory) we go up a level and think about concepts and their multiplicity; we don’t think about objects and their properties. Logic is meta.

But how does so-called propositional (or sentential) logic fit this description? In this logic we encounter the ideas of a sentence connective, a truth function, an operator on propositions. Natural language is taken very seriously as a guide to logical structure. There is scarcely a linguistic turn; the subject matter is already assumed to be linguistic (not that people are very clear what this means). How do these logical words and concepts assume a mathematical shape? The answer is not far to seek: “and” means “both”, to put it simply. The conjunction “p and q” means “Both p and q are true”; not one of them, or none, but both. This is a quantity word: the mind is working quantitatively when it thinks conjunctively—numerically, arithmetically. If both are true, then not just one is true—truth here requires a number of separate truths. If the conjunction is long, it will require that the corresponding number of propositions be true. The word “and” works like the word “plus”: it adds to the number of conjuncts. So, conjunctions in thought are mathematically conceived; you can’t form them unless you have some basic mathematical competence. Conjunction is often compared to universal quantification and the comparison is apt; but “all” is a number word, and so conjunction is a mathematical concept too, perhaps the simplest one. Disjunction works similarly: “or” means “at least one of the disjoined propositions is true”. It doesn’t have to be all disjuncts, so long as it is some; one will do. All this we grasp when we use the word “or”. We are thinking quantificationally. Hence “or” is often compared to “some”, aptly enough. Propositional logic is thus a species of quantificational logic, as complex and mathematical as it is. Moreover, it refers to propositions, so it is second-order, meta. But what about “not”? That little word has always been a thorn in the side of philosophers of logic (negative facts anyone?); it seems like a thing apart, curious in its connotation. What is this “not-ness”? It doesn’t yield easily to quantificational analysis—where is the “some” and “all”? What does “Not-p” mean? What do I mean when say that snow is not black? I suggest that I mean something like this: “There is no fact corresponding to the proposition that snow is black”, or “None of the facts in the world is the (putative) fact that snow is black”. It is a negative quantifier like “no one”. We could paraphrase our target sentence as “Nothing fits the description ‘black snow’”; or again, “There is no object such that it is both snow and black”. Thus, “not” is a simple word with a complex meaning; it takes a bit of brains to comprehend (animals are probably not up to it). You see why I said logical thought may not be properly reflected in ordinary language? The negative thought is a convoluted thought raising conceptual puzzles. In it the number zero presumably has its beginnings. It goes along with the word “false”: we can equally say “It is false that snow is black”, which is also second-order and meta. To think negatively is to entertain a counterfactual state of affairs and repudiate it as actual. In a way it lies at the very heart of logic, because logic is about necessity and possibility. The concept of possibility is itself tacitly quantificational and hence mathematical: “In some possible world, p”. We can naturally say “In hundreds of possible worlds, p” or “In no possible world, p”. Modal logic is likewise mathematical in character. Logic with negation is also logic with quantification. Similarly for tense logic in which we have quantification over times. All of these are mathematical, though primitively so (actually not so primitive). The logical mind is at bottom a mathematical mind. We would do well to render it more perspicuously in mathematical terms, thereby revealing its affinity with other types of mathematics. It is a kind of pre-school arithmetic.

I end with identity. Logic is sometimes said to include identity, and not without reason because identity is implicit in it. How does identity fit the mathematical analysis? As follows: to say that Hesperus is identical to Phosphorus is to say that Hesperus and Phosphorus are one. There are not two objects here but one object. We thought there were two (or more) and now we learn there is just the one. We counted wrong, as it turns out: if you put Hesperus and Phosphorus together in the same room, you find only one object there; there is one less object in the world than you supposed. In this thought we find the roots of the idea that a perceived plurality might reduce to an objective unity. Identity forces us to think numerically. It is an essentially mathematical concept. Accordingly, even elementary logic (with or without identity) is shot through with mathematical notions, though not particularly advanced ones; we might as well declare it a branch of mathematics. But remember it is not the symbolism that makes it mathematical but the underlying thought content. Mathematical logic is not a branch of logic but its very nature.[2]

[1] Compare my paper “Mathematical Ethics”.

[2] There is an irony in this in that the formulas of symbolic logic were supposed to lay bare the structure of logical thought, but actually it is the structure of logical thought that reveals the real content of those formulas. The thought is mathematical upon closer analysis, and this constitutes the meaning of the symbolic system. The hidden logical form of the formulas is given by the mathematical analysis of the underlying thought. Logic as we have it is a kind of baby math.