In simple terms, what is the difference between logic in mathematics and philosophy?

Question

I want to understand the difference between mathematical and philosophical logic. I actually thought they were the same till I read this post. Concisely speaking, what is the difference between how a philosopher conceives as logic vs how a mathematician does?

Answer 1

The definitions of 'logic' and 'mathematics' are themselves subject to dispute. In particular, the word 'logic' is used in different senses. At its narrowest, it is concerned with the relationship of consequence between propositions or sentences. In a wider sense it is sometimes used as a synonym for good reasoning, i.e. it has a cognitive component. In a wider sense still, it sometimes means what is rational and prudent: according to Mr Spock, it is illogical to cause the extinction of a species. Logicians themselves tend to stick to the narrow sense.

Also, the term has evolved over time. Originally logic was concerned with what distinguishes a good argument from a bad one. It had a strong normative aspect. You ought to reason like this; you ought not to reason like that. Today, whether logic is normative or descriptive is a debated question in the philosophy of logic. Much of what used to be part of logic is now considered to be epistemology.

But I would say that the short answer to your question is that mathematical logic is simply logic done with mathematical rigour. Which is to say, with a high degree of use of symbols and strict formalisms, and typically expressed using formal languages, axioms, and rules. Logic as used by philosophers is often expressed in natural language, because many arguments are difficult to formalise precisely. As a result, such logic is often informal in nature, although many philosophers use formal logic wherever feasible, just because it is less error prone and helps to avoid ambiguities and other problems.

If you pick up an introductory textbook of mathematical logic it will start by teaching you classical propositional logic and first order logic. It will most likely include some proof theory, model theory, recursion theory, and set theory. Also, some meta level material about compactness, completeness, undecidability, definability, and computability.

If you pick up an introductory textbook aimed at philosophers, it will also teach you classical propositional and first order logic. You will get some proof theory, including formal methods of natural deduction. You will probably get less material on completeness and computability, etc., but you will likely get some coverage of common fallacies in reasoning, and maybe some material about causal reasoning and some elementary probability theory.

Textbooks aimed at computer scientists, particularly those specialising in AI, will also teach you classical propositional and first order logic. Again, you will get some proof theory and model theory. And usually some Bayesian probability theory and maybe an introduction to default logic.

As such, there is a great deal of overlap, and mostly the difference is one of emphasis.

Answer 2

Caveat

The definitions of 'logic' and 'mathematics' are themselves subject to dispute. In particular, the word 'logic' is used in different senses. - Bumble

Bumble's answer is great, and mine is meant to supplement his with links of convenience.

Short Answer

Philosophical logic encompasses both informal and formal types. Mathematical logic is a rigorous use of formal logic to do proof and models. There are no rigorous divisions between philosophical logic and mathematical logic, except in how universities are organized to teach these topics. Likewise, it would be difficult to draw a sharp line between logic and math.

If one were to caricature it, philosophical logic is done to understand the nature of logic and further logical arguments, whereas mathematical logic is used to make claims about mathematics and theorems. Of course, many philosophers, particularly in the analytical tradition, look to mathematics to make philosophical arguments.

Long Answer

While one may not be able to provide a definition of necessity and sufficiency to definitively lay out criteria for the two terms, 'mathematics' and 'logic', one can establish what is known as a prototypical definition in the spirit of what is known in philosophy as family resemblances of meanings. This second method of definition requires a broad familiarity with what is generally held to be criteria or characterizations.

Mathematics can be seen as the study of the interrelation of study of certain topics such as numbers and number sense, shape, direction, relations, and operations. Logic on the other hand is concerned with truth, rhetoric, logical proof, fallacies, argumentation, logical formalisms like propositional calculus, and questions about language like the difference between syntax and semantics or the difference between an utterance and proposition.

Today, the intersection of both might be understood as mathematical logic which uses formal systems to study proof systems and mathematical models, inter alia.

From "mathematics":

Mathematics (from Ancient Greek μάθημα (máthēma) 'knowledge, study, learning') is an area of knowledge, which includes the study of such topics as numbers (arithmetic and number theory),1 formulas and related structures (algebra),5 shapes and spaces in which they are contained (geometry),1 and quantities and their changes (calculus and analysis).35 There is no general consensus about its exact scope or epistemological status.6

If you haven't studied mathematics at an undergraduate level, it's difficult to get a sense of what modern mathematics encompasses, but a peek at all of its branches can be seen here at The Map of Mathematics (YT).

From logic:

Logic is the study of correct reasoning or good arguments. It is often defined in a more narrow sense as the science of deductively valid inferences or of logical truths. In this sense, it is equivalent to formal logic and constitutes a formal science investigating how conclusions follow from premises in a topic-neutral way or which propositions are true only in virtue of the logical vocabulary they contain. When used as a countable noun, the term "a logic" refers to a logical formal system. Formal logic contrasts with informal logic, which is also part of logic when understood in the widest sense. There is no general agreement on how the two are to be distinguished. One prominent approach associates their difference with the study of arguments expressed in formal or informal languages. Another characterizes informal logic as the study of ampliative inferences, in contrast to the deductive inferences studied by formal logic. But it is also common to link their difference to the distinction between formal and informal fallacies.

If you haven't studied mathematics at an undergraduate level, it's difficult to get a sense of what modern philosophical methods encompass, but here is The Map of Philosophy (YT) at about 1:40 to 8:00.

Answer 3

There are many logics, and thus philosophers and mathematicians may conceive of "logic" similarly or differently in a large variety of cases.

So the question as it stands is somewhat vague. But perhaps this will clarify:

Philosophical logic has "philosophical"- whatever that means - concerns. For a sampling, these include modal logics, which are indispensable to modern day philosophy, as well the nature of truth, logical pluralism, etc. Mathematical logic has "mathematical" concerns- given some logical system, we wish to see if certain properties hold, eg completeness, soundness, decidability.

This is not to say that mathematical logicians may not study, say modal logic. Indeed, working in either subfield usually requires that one have some understanding of both the mathematical and philosophical aspects of logic. The difference is that when one publishes in a math journal, reviewers wish to see mathematical tools and methods applied to the logical system in question. And likewise for philosophy.

Answer 4

Logic in general is a study of arguments: how they work, how they are categorized, which ones are strong or valid. Logicians often construct rules about what makes a good or valid argument.

Mathematical logic is not a study of arguments; it is a study of logic itself. That is, it is a study of the rules that logicians make, how the rules interact, what they mean, to what extent they can be reduced to mere meaningless structures that can be employed automatically.

For example, a logician might define the rule of modes ponens: if A is true and A implies B, then B is true. Notice that this rule is about propositions, sentences with meanings that are either true or false.

Mathematical logic might then reduce this to a formal rule:

A, A->B
-------
   B

This rule is not about literal propositions (although they are called propositions); it is about symbols and formulas. There is a set of formulas that are derivable. This rule says that if A is a derivable formula and A->B is a derivable formula, then B is also a derivable formula. It doesn't matter whether A or B is true; they don't even have a meaning. They are just formulas.

Answer 5

Basically, the logic used in mathematical proofs is more informal than that of pure symbolic logic. In the latter, you have to explicitly specify and document/annotate every minute step of a derivation/proof; in a typical mathematical proof, on the other hand, you can be more loose or free-style about it.

For general logic, I recommend "Schaum's outline of Logic" by Nolt and Rohatyn (the best introductory logic book). For logic as applied to math, I recommend "How to prove it", by Velleman. There's also "Introduction to Logic", by Copi, which has been a standard intro textbook for a long time. But I would always start with Schaum's.

One of the most important concepts in logic is "proof by contradiction", aka "reductio ad absurdum" or RAA. Very powerful.

Answer 6

I believe that your question can be at least partially answered by considering the original research program that led to the creation of modern mathematical logic. Much of proof theory stems from research on the foundations of mathematics that was carried out by Hilbert, Bernays, Gödel, and others during the first few decades of the twentieth century. After the discovery in the late nineteenth century that Frege’s foundational approach was hindered by Russell’s paradox, which in turn had problematic implications for Cantorian set theory as a foundation for mathematics, mathematicians became alarmed at the possibility that some of the assumptions and methods underlying their research might also entail a hidden logical contradiction. Hence the objective of what came to be known as Hilbert’s program was to formally axiomatize mathematics, and to provide a rigorous proof of its logical consistency. The highly formalized, symbolically-oriented proof theory that one encounters in modern texts on mathematical logic was invented in this context about a century ago by the people who were working on Hilbert’s program.

The other major branch of mathematical logic - model theory - was arguably invented with the same considerations in mind. In his famous Grundlagen der Geometrie, published in 1899, David Hilbert was interested in establishing the consistency of Euclidean geometry, and he went about this by providing what might now be considered a model of Euclid’s axioms in two-dimensional real space. Hence much of the original inspiration for model theory was similar to that of proof theory, namely a perceived need to establish the consistency of a mathematical theory that was thought to have already been demonstrated as true. Of course, Hilbert's approach here implicitly assumed Gödel’s completeness theorem which establishes the equivalence of satisfiability and consistency in first-order logic, which itself was a product of the research program undertaken several decades later that was described above in the first paragraph.

Obviously, the scope of mathematical logic has expanded over the past century or so. Current research in logic that could be considered distinctly mathematical, as opposed to philosophical (whatever that means) might include model-theoretic algebra and the formalization of proof in homotopy type theory. But even in these instances, there are people who are employed by philosophy departments who are also making contributions. In my view, it is generally difficult to draw a meaningful distinction between “mathematics” and “philosophy” in the context of foundational subjects like formal logic.

Answer 7

I know mathematical logic, I only assume w.r.t. philosophical logic.

Math builds its logic model on a few core principles, e.g.:

1. Something can be either true, or false. There is no "maybe". (For some things, we cannot say. That's another thing, whether something can be deduced etc. But in the end, the result, IF there is any, it can ONLY be true or false.)
2. If something is true, it's opposite is not. Or: Two things can't be true at the same time and contradict each other.

It's a black and white world, e.g. a (natural) number is either even, or odd: " for any x of the natural numbers, odd(x) is true if (x+1) is a multiple of 2." And it's a rather timeless world, if something can be deduced from basic axioms, it remains true (as long as the axioms hold - and they usually do :)).

Philosophy, from my understanding, is not that strict. Strong points in arguments can be "right" at a specific point in the argument and "wrong" in another. Another argument following may turn this upside-down, a new system of premises may lead to a complete reevaluation. Things that were right yesterday because of various assumptions, may be wrong tomorrow.