Logic

The big questions

1. What makes an argument valid, and why can no valid argument lead from truths to a falsehood?
2. Is there one correct logic, or can rival logics each be right?
3. Why do the truths of logic seem necessary, and how do we come to know them?
4. Can reasoning from what has been observed to what has not ever be justified?
5. Are the laws of logic laws of thought, of language, or of the world?
6. Can a contradiction ever be true?

If it rained last night, the road is wet. The road is wet. So it rained last night. Something is wrong here even if it did rain: a water tanker may have passed. Keep the shape and change the subject. If Delhi is in Kerala, Delhi is in India. Delhi is in India. So Delhi is in Kerala. The premises are true and the conclusion false, which settles the matter for every argument of this shape, whatever it is about.

Logic is built on that observation: some arguments succeed or fail by their form, the pattern left when their subject matter is taken away. The pattern above is called affirming the consequent. Its neighbour (if it rained, the road is wet; it rained; so the road is wet) is modus ponens, and no argument of that form leads from true premises to a false conclusion. Formal logic, symbolic logic and, in Sanskrit and Hindi, tarkaśāstra (the science of reasoning) all name the study of such patterns; “logic” comes from the Greek logos, reason or discourse.

What logic studies

An argument is a set of premises offered in support of a conclusion. A deductive argument is valid when its premises cannot be true while its conclusion is false, and sound when it is valid and its premises are in fact true. Validity is not truth: a valid argument can run from false premises to a false conclusion, and an invalid one can end in a truth, as the wet road shows.

Most reasoning outside mathematics is not deductive. Inductive arguments generalise from observed cases; abductive ones infer the best explanation of the evidence, as a doctor does from symptoms (deduction and induction, abduction). Their premises support a conclusion without guaranteeing it, and Hume argued, in his sections on induction, that no reasoning can show the future will resemble the past.

Formal logic studies valid inference in languages built for the purpose: propositional logic, which tracks “not”, “and”, “or” and “if” and can be checked mechanically by a truth table, and first-order logic, which adds quantifiers such as “every” and “some”. Informal logic studies arguments as they are actually made, in courts, newspapers and quarrels, and with them the fallacies: errors of reasoning that persuade anyway.

Is there one right logic?

The standard answer is classical logic, the system most commonly used today, in which every statement is true or false. Its first-order proof system is sound and, as Kurt Gödel proved in 1929 (published in 1930), complete: it derives all and only the valid arguments of its language.

Rivals came with serious motives. The intuitionists L. E. J. Brouwer and Arend Heyting held that mathematics is mental construction, so that nothing guarantees that every statement or its negation can be established; excluded middle is not a general law. Michael Dummett argued from the theory of meaning that if truth cannot outrun the evidence for it in some area, bivalence and with it classical logic should be given up for that area. Relevant and paraconsistent logicians reject the classical principle that a contradiction entails everything, so that inconsistent information, which databases often contain, can be reasoned with instead of exploding into triviality. Dialetheists go further and hold that some contradictions are true, appealing to paradoxes of self-reference such as the liar; paraconsistency does not by itself commit anyone to that view. Logical pluralists, prominently JC Beall and Greg Restall, answer that more than one logic can be correct, because what counts as a “case” in “true in every case in which the premises are true” can be made precise in more than one legitimate way. Monists reply that two logics that disagree about the same argument cannot both be right.

Beneath this lies a question about what logic is. The traditional laws of thought presented it as the science of how we must think. Wittgenstein’s Tractatus treated the propositions of logic as tautologies that say nothing about the world, and the logical positivists held the truths of logic and mathematics to be analytic, true by the rules of language. Quine cast doubt on the line between truths of meaning and truths of fact, and held that no statement is immune to revision, logic included: it keeps its place by what it contributes to our theory as a whole.

A short history

Aristotle’s logical works, later gathered as the Organon (“instrument”), contain the earliest formal study of logic that survives. In the Prior Analytics he defines a syllogism as “discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so” (in A. J. Jenkinson’s translation), and works out which pairs of premises such as “every A is B” and “some A is not B” yield a conclusion: the theory of the syllogism. The Stoics, above all Chrysippus, built a rival logic of whole propositions joined by “if”, “or”, “and” and “not”, but his works were lost, and Aristotle’s logic passed to the Arabic and Latin traditions. Ibn Sīnā reshaped it so thoroughly that later Arabic logic divides into the periods before and after him. In Latin Europe Boethius translated it and William of Ockham’s Summa Logicae is a large, systematic treatment of logic and semantics. In 1787 Kant could say of logic that “since Aristotle, it has been unable to advance a step” (Preface to the Critique of Pure Reason, in Meiklejohn’s translation).

Within a century the claim was overtaken. George Boole applied algebra to logic in The Mathematical Analysis of Logic (1847) and The Laws of Thought (1854). Frege’s Begriffsschrift (1879) is often credited with inventing modern quantificational logic, though just what was new in it, compared with the work of Boole, Peirce and others, is debated. Frege set out to show that arithmetic is logic; in 1902 Russell wrote to him with a contradiction in his system, and Russell and Whitehead’s Principia Mathematica tried again on other foundations. In 1931 Gödel showed that any consistent formal system in which enough arithmetic can be carried out leaves some statements neither provable nor refutable, and cannot prove its own consistency.

Logic in India and China

Logic also grew up in India, inside the theory of knowledge and the practice of debate. The Nyāya-sūtra of Akṣapāda Gautama, founding text of the Nyāya school (often placed around the second century CE, though estimates range from the sixth century BCE to the third century CE), counts inference, anumāna, among the means of knowledge and sets out a five-membered argument, which the later tradition illustrates with a stock example:

1. There is fire on the hill (thesis).
2. Because there is smoke on the hill (reason).
3. Wherever there is smoke there is fire, like a kitchen hearth and unlike a lake (example).
4. This hill is likewise smoky (application).
5. So there is fire on the hill (conclusion).

The weight falls on the third member, which states the pervasion, vyāpti, of the sign by what it signals, and anchors it in a case both parties accept. A reason that fails is a hetvābhāsa, a “specious reason”. Since a false premise and a bad form both end in false cognition, Nyāya did not stress the line between validity and soundness.

The Buddhist Dignāga (c. 5th–6th century) distinguished inference for oneself from inference for another, and devised a “wheel of reasons”, a three-by-three table sorting good reasons from bad by where the reason occurs among similar and dissimilar cases. Scholars disagree whether his canonical argument form was deductive, inductive or defeasible. Dharmakīrti (c. 7th century) worked within his framework, and a deductively valid form of the argument came to be adopted by virtually every classical Indian thinker. In the fourteenth century Gaṅgeśa founded Navya-Nyāya, the “new” Nyāya. Jaina thinkers developed syādvāda, a theory of conditioned predication in seven modes, each marked syāt, “in some respect”, so that nothing is asserted absolutely.

In China the later Mohists, heirs of Mozi, studied disputation (biàn, “distinguishing”) in the Mohist Canons. They treated argument as analogical at root and did not study logical form for its own sake, though they applied versions of non-contradiction and excluded middle. The “disputers” later grouped as the School of Names became notorious for paradoxes such as Gongsun Long’s “a white horse is not a horse”.

What logic is for now

Logic now lives mainly in three departments. Mathematicians study formal systems through model theory, proof theory and computability; computer scientists build it into circuits (Claude Shannon showed in 1937 that Boolean algebra describes and simplifies switching circuits), programming languages and automated provers; philosophers use it to make arguments precise across epistemology, metaphysics and the philosophy of language.

In India, logic is a full unit of the UGC NET syllabus and runs through the UPSC optional’s topics on Russell, early Wittgenstein and logical positivism; the UGC NET and UPSC guides map it.

How to begin

Logic is learned like a language, by working examples. Start with Lewis Carroll’s Game of Logic, then read the first books of the Prior Analytics and the Nyāya-sūtra side by side, and Leibniz’s two great principles. In Carroll’s What the Tortoise Said to Achilles, Achilles promises that “Logic would take you by the throat, and force you to do it!”, and the Tortoise shows why it cannot: a rule of inference written down as one more premise needs a further rule, without end. The reading path below sets these in order; for exercises, forall x: Calgary is a free textbook with solutions.