What Is Logic? Reasoning, Fallacies and the Types of Logic

Logic is the branch of philosophy that studies correct reasoning: what makes an argument valid, what follows from what, and why. It ranges from formal logic, which tests inferences by their structure alone, through informal logic, which studies arguments and fallacies in everyday language, to the philosophy of logic, which asks what logic itself is.

On The Coordinates, logic is Root IV: 16 question cards, from “What makes an argument work, regardless of its subject?” to “Is logic discovered or invented?” It is the root the others lean on, since every philosophical position is only as strong as the reasoning that supports it. One card, on non-classical logics, carries the contested mark (△): logicians disagree about whether classical logic should be revised at all, and for which problems.

What is logic in philosophy?

The word comes from the Greek logos, which can mean word, reason or account. A working definition: logic is the study of the principles that separate good inference from bad. Logic does not tell you whether your premises are true. It tells you what you are committed to if they are.

Two terms carry most of the weight. An argument is valid when its conclusion follows from its premises, so that the premises cannot all be true while the conclusion is false. It is sound when it is valid and its premises are in fact true. “All philosophers are mortal; Socrates is a philosopher; so Socrates is mortal” is valid and sound. “All philosophers are fish; Socrates is a philosopher; so Socrates is a fish” is just as valid, though unsound. The form does the work, whatever the subject, which is the point of the first card, Formal Logic.

Deductive and inductive reasoning

In a deductive argument the conclusion is meant to be guaranteed by the premises. In an inductive argument it is only supported: the sun has risen every morning on record, so it will rise tomorrow. However strong the evidence, an inductive conclusion could be false while every premise is true, which is why Hume’s problem of induction belongs to epistemology as much as to logic.

Many textbooks add a third pattern, abductive reasoning, or inference to the best explanation: the street is wet and the sky is grey, so it has probably rained. Most everyday reasoning, and much scientific reasoning, is inductive or abductive. Informal logic calls such arguments defeasible: new information can overturn them, which it cannot do to a valid deduction.

Types of logic: formal vs informal

The broadest division is between formal and informal logic. Formal logic studies valid inference in artificial languages with precise rules, so that validity can be checked step by step. Informal logic studies arguments as people actually make them, in ordinary language, with premises left unstated and conclusions supported rather than proved. Its card asks a pointed question: why does formal logic tell you so little about whether an argument is any good?

Beyond these sit modal logic (necessity, possibility and their relatives), the non-classical logics that revise classical assumptions, and the philosophy of logic. Here is every card in the root. The 131 cards across all six roots come from The Coordinates of Thought by Syed Raheel Shahzad.

Formal logic: syllogistic, propositional and predicate

Syllogistic Logic, Aristotle’s system, reasons from two premises that share a term, using four kinds of statement: all S are P, no S are P, some S are P, and some S are not P. For about two thousand years it was, in Europe, the dominant form of formal logic.

Propositional Logic treats whole statements as units that are true or false and studies how they combine. The Stoic Chrysippus developed an early version; George Boole gave it an algebra in the nineteenth century. Its connectives are few:

Symbol	Read as	True when
¬P	not P	P is false
P ∧ Q	P and Q	both are true
P ∨ Q	P or Q	at least one is true
P → Q	if P then Q	P is false, or Q is true
P ↔ Q	P if and only if Q	both have the same value

The “if” row hides a well-known puzzle. On this reading, “if the moon is made of cheese, then Paris is in France” comes out true, which is one reason some logicians developed relevance logics. Predicate Logic, built by Gottlob Frege and refined by Bertrand Russell and others, looks inside statements at objects, properties and quantifiers (“for all”, “for some”). Its card asks why two thousand years of logic could not handle “every boy loves some girl”. The answer is scope: the ability to fix which quantifier governs which variable.

Logical fallacies: when is a bad-looking argument actually bad?

A fallacy is a pattern of reasoning that appears to support its conclusion but does not. Lists of logical fallacies are everywhere; what they rarely say is that most of these patterns are sometimes perfectly good reasoning. The card sets out both sides:

Pattern	When it is a fallacy	When it can be fair reasoning
Ad hominem	Rejecting a claim because of who said it	Questioning testimony when the speaker’s reliability or interest is genuinely at issue
Appeal to authority	Citing someone outside their competence, or where experts disagree	Deferring to real expertise in its own field, which is how most knowledge works
Slippery slope	Asserting a chain of consequences with no mechanism	Arguing a chain where each step has demonstrable precedent or pressure
Straw man	Attacking a version nobody holds	Attacking a version someone does hold, however weak
False dilemma	Offering two options when more exist	Offering two options when the alternatives really are exhausted
Post hoc	Inferring cause from sequence alone	Combining sequence with a mechanism and controls

This is where informal logic earns its place. Argumentation Theory asks what an argument has to do besides be valid, with models from Stephen Toulmin and the pragma-dialectics of Frans van Eemeren. The Nyāya tradition in India long ago distinguished debate aimed at truth from debate aimed at winning. Rhetoric asks whether persuading someone is different from giving them a reason, a question Plato and Aristotle answered differently.

Modal logic: necessity, possibility, obligation and time

Modal Logic extends formal logic to reason about what must be and what merely might be. It adds two operators: □P (necessarily P) and ◇P (possibly P). Aristotle discussed modal syllogisms; Ibn Sina reworked them in detail, paying close attention to how necessity and time interact within a statement. In the twentieth century C. I. Lewis built modern modal systems and Saul Kripke gave them a semantics in terms of possible worlds.

The same machinery reaches well beyond necessity. Deontic Logic handles obligation and permission (“What follows logically from ‘you must’?”). Epistemic Logic handles knowledge and belief (“If you know something, do you know that you know it?”). Temporal Logic, developed by Arthur Prior, handles past and future (“Is it already true that it will rain tomorrow?”), a question Aristotle raised about a sea battle.

Non-classical logics and the philosophy of logic

Classical logic rests on assumptions that look obvious: every statement is either true or false, no contradiction is true, and from a contradiction anything follows. The contested card on Non-Classical Logics △ asks which of these, if any, should be revised. Intuitionists such as L. E. J. Brouwer doubt that the law of excluded middle holds for unproved mathematical statements. Many-valued and fuzzy logics allow statements to be partly true, a tool for vague words like “tall”. Paraconsistent logics let a theory contain a contradiction without collapsing, and dialetheists such as Graham Priest go further, holding that some contradictions are true; the liar sentence (“this sentence is false”) is their leading example.

Behind all of this lies the Philosophy of Logic: is logic discovered or invented? Realists think it describes the structure of the world. Psychologism, which Frege attacked, treats it as a description of how minds think. Conventionalists treat it as rules we adopted, pluralists hold that more than one logic can be correct, and instrumentalists judge logics by their usefulness. W. V. O. Quine and Rudolf Carnap disagreed sharply here, and the question remains open.

Logic across traditions: Greek, Indian, Buddhist, Chinese and Islamic

Logic was developed in more than one civilisation, and the cards show it:

• Greek: Aristotle’s syllogistic and the Stoics’ logic of propositions, rival systems in antiquity that modern logic eventually brought together.
• Indian (Nyāya): the Nyāya school, from the sūtras attributed to Gautama, analysed inference in a five-step form: thesis, reason, example, application, conclusion. The standard illustration is “the hill has fire, because it has smoke, as in a kitchen”. Gaṅgeśa, in the fourteenth century, gave the tradition its later, highly technical form. See Indian philosophy.
• Jain: syādvāda, a sevenfold scheme of conditional predication, which some modern writers compare with many-valued logic.
• Buddhist: Dignāga set out the conditions a good reason must meet and classified reasons accordingly; Dharmakīrti, who followed him, developed this into a full account of inference and knowledge. Nāgārjuna’s four-cornered reasoning (it is, it is not, both, neither) is discussed today alongside paraconsistent logic. See Buddhist philosophy.
• Chinese: the later Mohists studied disputation (bian), including how names and descriptions can mislead, in the same period as the School of Names, whose Gongsun Long argued that “a white horse is not a horse”. See Chinese philosophy.
• Islamic: Ibn Sina’s modal and temporal syllogistic was studied and argued over for centuries, and the jurists’ grading of acts from obligatory to forbidden is often cited as a forerunner of deontic logic. See Islamic philosophy.

These are the philosophers most often named on the logic cards:

How to start learning logic

Logic rewards practice more than most of philosophy. A sensible order: learn validity and soundness, then propositional logic with truth tables, then predicate logic, while reading real arguments in newspapers and debates to spot the patterns in the fallacies table. Then try the philosophy of logic, where the subject turns back on itself:

The Book corner lists books on logic and reasoning, beginner-friendly first:

The site’s Logic, Language & Mathematics category is a good place to test an argument on other people. Here are the latest conversations:

Where do you stand?

Every question here has serious answers on more than one side. Pick a card, read the rival answers, then say which you find strongest and why. Members ask, reply and take sides; reading is free for everyone.

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