Logic

What it asks

Some arguments hold and some do not, and whether they hold has nothing to do with whether anyone finds them convincing. Logic asks what that difference consists of.

The central insight, and it is Aristotle's, is that validity depends on FORM rather than content. If all A are B and all B are C, then all A are C - and this holds whatever A, B and C are. That one move made it possible to study reasoning as a structure rather than as a psychological habit, and everything since has been an extension of it.

The recurring problems

What the formal system is answerable to. A logic can be built to do anything; whether it captures actual valid reasoning is a separate question, and not a formal one.

The limits of formalisation. Gödel showed in 1931 that any system rich enough to express arithmetic contains truths it cannot prove. That is a mathematical result, and it ended a programme that had run for fifty years.

Which logic. Classical logic assumes every statement is true or false. Intuitionist logic denies it, relevance logic objects that a contradiction should not entail everything, and modal, temporal and many-valued logics each add machinery for something classical logic cannot say. There is no neutral ground from which to referee.

The paradoxes. The sentence that says it is false has resisted every treatment for two and a half thousand years, and each proposed fix costs something.

How it developed

Aristotle's syllogistic was the whole of the subject for two thousand years, and it was thought to be complete. The Stoics developed a propositional logic alongside it that was in some ways more powerful and was largely lost.

The medieval logicians did substantial original work on consequence and reference, and then the subject went quiet for three hundred years.

Frege restarted it in 1879. His Begriffsschrift introduced quantifiers and variables and made it possible to represent statements of arbitrary complexity, which the syllogism could not do. It is the single largest advance the field has ever had and it was almost entirely ignored for twenty years.

Russell and Whitehead then attempted to derive mathematics from logic, an effort that ran to three volumes and failed instructively. Gödel's incompleteness theorems in 1931 showed why it could not succeed, and Tarski's work on truth in the same decade gave the field its semantic half. From there the subject supplied the theoretical basis of computation, which is why a branch of philosophy is a required course in computer science.

The positions within it

Classical logic is the default. Intuitionism rejects the law of excluded middle and with it the standard proof by contradiction. Logicism, formalism and intuitionism are also positions about mathematics, and are covered under Schools and under philosophy of mathematics.

Dialetheism holds that some contradictions are true, which is a minority position taken seriously for the reason that the alternatives to it are also uncomfortable.

Where it stands now

Formal logic is now largely done in mathematics and computer science departments, and philosophical logic has concentrated on the systems that model something philosophy cares about: necessity, time, obligation, knowledge, vagueness.

The other current work is in the opposite direction. Informal logic and argumentation theory study reasoning as it is actually conducted, where the premises are unstated and the conclusion is a matter of degree - which is most reasoning, and which formal systems handle badly.