What Is Logic?

A Working Definition

Logic is the study of which inferences are truth-preserving: if the premises are true, the conclusion must be true. The word "must" is doing all of the work. Logic is the discipline that turns that must into something the reader can check.

This is narrower than the colloquial use. In ordinary speech, "logic" can mean reasonableness, rigor, calculation, or just the absence of obvious mistakes. The philosophical use is sharper: an argument is logically valid when there is no way for the premises to be true and the conclusion false. Whether the premises actually are true is a separate question, one for the world, not for logic.

The Central Question

Given a proposed inference from premises to conclusion, how can we tell whether the conclusion follows? The history of logic is the history of progressively sharper answers to that question.

• Aristotle (4th c. BCE) gave the first systematic answer: identify the form of the inference, distinct from its content. All As are Bs; all Bs are Cs; therefore all As are Cs is valid for any A, B, C. The form is what guarantees truth-preservation.
• Frege (1879) extended formal analysis to inferences involving quantifiers (all, some) and relations (x is greater than y) that Aristotle's syllogistic could not capture. Begriffsschrift is the founding document of modern logic.
• Tarski (1933, 1936) gave a precise mathematical definition of truth in a structure, separating the syntactic form of a sentence from its semantic interpretation.¹
• Gödel (1931) showed that any formal system rich enough to encode arithmetic contains true statements it cannot prove. Formal systems are necessary; they are not enough.²

Each step kept the same target, what makes inference valid, but raised the level of precision.

Why It Matters

Three concrete cases.

Case 1: a code review. A reviewer writes: if the function is pure and the inputs are bounded, the test will be deterministic. The function is pure, but the inputs are not bounded; therefore the test is non-deterministic. The argument looks fine. It is invalid. From "if A and B then C" plus "A and not-B" you cannot conclude "not C"; you can only conclude that the consequent has not been guaranteed. The reviewer has confused not entailing C with entailing not-C. Logic catches that.

Case 2: a legal brief. A brief argues: since the contract terminates upon material breach, and the buyer's late payment was a material breach, the contract terminated. Valid in form. Whether the premises hold is the question for the trial. Logic separates the argumentative structure from the empirical question, which is exactly what the appellate court needs.

Case 3: a model evaluation. A team claims that because their model passed all benchmarks, it will perform reliably in production. The form is if benchmark-passing, then reliability, but this conditional is exactly the Humean inductive license the empirical evidence does not establish. Logic does not let the team smuggle the conditional past the reader.

In each case, logic is not a substitute for domain expertise. It is the discipline that prevents domain expertise from being used to launder a bad inference.

Formal Logic vs Informal Logic

Two complementary practices.

Formal logic studies inference at the level of syntactic structure. It abstracts away from natural-language ambiguity by translating arguments into a formal language with precise rules. The primary systems are propositional logic (inference among whole statements connected by and, or, not, if-then) and predicate logic or first-order logic (inference involving quantifiers and relations). Both are taught in any introductory logic course; both are the substrate of modern mathematical and computer-scientific reasoning.

Informal logic studies inference in natural language: how arguments are actually built and how they typically fail. It catalogs fallacies, patterns of bad reasoning common enough to deserve names (ad hominem, straw man, false dichotomy, circular reasoning). Informal logic is what most people practice without ever opening a logic textbook.

The two are not rivals. Formal logic gives the standards; informal logic gives the field guide. A serious treatment of an argument moves between them.

The Main Systems

Modern logic is plural. The phrase the logic is misleading; there are several formal systems, each capturing different intuitions about inference.

Choosing a system is itself a philosophical decision: it commits the reasoner to particular answers about what truth and inference are. The choice is not arbitrary, but it is also not predetermined.

A Worked Validity Check

Take the argument:

Premise 1. If the model is overfit, the validation loss exceeds the training loss. Premise 2. The validation loss exceeds the training loss. Conclusion. Therefore, the model is overfit.

The form is:

This is the fallacy of affirming the consequent. From $p \to q$ and $q$, you cannot conclude $p$. The conditional says that overfitting is sufficient for the loss gap; it does not say it is necessary. The loss gap could be produced by distribution shift, noisy labels, a bad train/validation split, an evaluation bug, or a number of other causes.

The argument is invalid. To repair it, the reviewer would need to either (a) replace $P_1$ with the biconditional $p \leftrightarrow q$ (overfit if and only if loss gap), which is empirically false, or (b) gather additional evidence that rules out the alternatives. Logic flagged the gap; the empirical work fills it.

Common Confusions

Confusion 1: validity vs truth. A valid argument can have false premises and a false conclusion. All cats are reptiles; all reptiles are mammals; therefore all cats are mammals is logically valid. The form is correct. The premises are false. See Validity vs Soundness for the full distinction.

Confusion 2: logic vs persuasion. A logically valid argument is not necessarily a persuasive one. A persuasive argument is not necessarily logically valid. Rhetoric optimizes for persuasion; logic optimizes for truth-preservation. The two diverge often enough that conflating them is the most common single source of bad public discourse.

Confusion 3: deduction vs induction. Deductive arguments are evaluated by validity. Inductive arguments, the sun has risen every day so far, so it will rise tomorrow, are evaluated by strength: how much support the premises give the conclusion, given background assumptions. Logic primarily studies deduction; the inductive case requires probability, statistics, and the philosophical apparatus around the problem of induction.

Confusion 4: classical logic is the only logic. Classical logic is the default for working scientists and mathematicians, but it is not the only formal system. Intuitionistic logic dominates constructive mathematics and parts of theoretical computer science. Modal logic is the substrate of contemporary metaphysics and epistemic reasoning. Paraconsistent logic matters wherever inconsistent data must be reasoned about without explosion.

Three Exercises

For each argument, decide whether it is valid (the conclusion follows by form) or invalid. Then check the answers in the next section.

Exercise 1. If the system is fair (in the metric sense), it equalizes false-positive rates. The system equalizes false-positive rates. Therefore the system is fair.

Exercise 2. All transformers are sequence models. Some sequence models are autoregressive. Therefore some transformers are autoregressive.

Exercise 3. If the agent has a world model, it can plan. The agent cannot plan. Therefore the agent does not have a world model.

Answers

Exercise 1: invalid. Affirming the consequent. The form is $p \to q$, $q$, therefore $p$. Equalizing false-positive rates is a necessary condition on metric-fairness in the sense given but not a sufficient one (a model can equalize FPR while violating other fairness criteria the metric requires).

Exercise 2: invalid. This is a quantifier mistake, not a propositional one. From "all $A$ are $B$" and "some $B$ are $C$," it does not follow that "some $A$ are $C$." (Counterexample structure: all dogs are mammals; some mammals are whales; therefore some dogs are whales, clearly false.) The autoregressive sequence models could be a different subset of sequence models than the transformers occupy.

Exercise 3: valid. Modus tollens. The form is $p \to q$, $\neg q$, therefore $\neg p$. If having a world model entails being able to plan, and the agent cannot plan, then it does not have a world model. (The argument is valid; whether the conditional premise is true is a separate question, see the philosophical literature on world models.)

Prerequisites and Next Pages

• Prerequisite: What Is Philosophy?, for the methodological frame argument operates within.
• Next: Validity vs Soundness, the technical distinction that makes argument-evaluation precise.
• Next: What Is a Symbolic System?, for the formal-systems hub the modern logical tradition built.
• Forward link: Syntax vs Semantics, the structural distinction Tarski's truth definition rests on.

References

Primary texts:

• Aristotle. Prior Analytics. Robin Smith translation, Hackett, 1989. The foundational text of formal logic.
• Frege, Gottlob. Begriffsschrift. 1879. English translation by Stefan Bauer-Mengelberg in van Heijenoort, ed., From Frege to Gödel, Harvard, 1967.
• Tarski, Alfred. "The Concept of Truth in Formalized Languages." 1933. In Logic, Semantics, Metamathematics, Oxford, 1956.
• Gödel, Kurt. "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I." 1931.

Modern reference:

• Hurley, Patrick, and Lori Watson. A Concise Introduction to Logic. Cengage, 13th ed. 2017. Standard introductory textbook.
• Smith, Peter. An Introduction to Formal Logic. Cambridge, 2nd ed. 2020. Open-access at logicmatters.net.

Stanford Encyclopedia entries (link, do not paraphrase):

• "Classical Logic."
• "Informal Logic."
• "Intuitionistic Logic."
• "Modal Logic."
• "Paraconsistent Logic."