Logic

Plato does not use the word "logic" and would perhaps have been suspicious of it. What he offers instead is dialectic, described in the Republic as the highest intellectual discipline, the one that distinguishes the philosopher from the mathematician. The mathematician takes his axioms as given and reasons downward to conclusions; the dialectician ascends from hypotheses to the unconditioned first principles, then descends again to their consequences.

The distinction is important because it connects method to the nature of the objects known. The objects of mathematical reasoning are, in a sense, hypothetical: given these axioms, these conclusions follow. Dialectic presses further, asking whether the axioms themselves are true, and this requires ascending toward the Forms, which are not merely posited but grasped as the actual nature of things. Reasoning that cannot account for its own premises is not, on Plato's view, genuinely scientific. The questions this raises are treated from a different angle in the chapter on DIALECTIC.

Plato's contribution to the tradition of logic is twofold. He provides Aristotle with the material for the theory of classification and definition, for the root notion of the syllogism, and for the general outlines of a method that Aristotle will call dialectic in a more restricted sense. But Plato's deeper contention is that formal correctness is not sufficient. A valid argument from false premises produces false conclusions. Logic as an instrument must serve truth, and the question of how premises are secured is, in his view, the most fundamental question in the theory of method.

The question Plato raises, whether formal rules can guarantee truth or whether logic must reach beyond form to the nature of things, runs through every subsequent account of logic. Aristotle answers by separating the form of argument from its material content, and the question of whether this separation is fully adequate remains a matter of debate in the tradition.

Aristotle is the first writer to produce a systematic account of valid inference, and his Organon covers the basic elements of discourse (terms, propositions, syllogisms), the forms of valid and invalid reasoning, the conditions for scientific demonstration, and the rules of dialectical argument. The scope of the Organon is treated more fully in the chapters on REASONING and SCIENCE.

The theory of the syllogism is central to Aristotle's logic. "A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so." The Prior Analytics analyzes all valid syllogistic forms. The Posterior Analytics shows what additional conditions are required if a syllogism is to constitute scientific knowledge: its premises must be true, primary, immediate, better known than the conclusion, and explanatory of it. The Categories classifies the basic types of predication; the Topics develops rules for dialectical reasoning from probable premises.

Aristotle conceives of logic as an organon, an instrument, rather than a science in its own right, because it does not study some portion of reality but the forms of discourse applicable to any subject. Logic is formal in a precise sense: the validity of an argument depends on its structure, not its content. This formal character, Aristotle holds, allows logic to serve all the sciences impartially. It trains the mind before it engages a particular subject matter, so that the student is not distracted by logical confusions when he turns to a special study.

Aristotle's Organon remains the dominant framework for logic for nearly two millennia. The difficulty that later writers, particularly Bacon, identify is that the syllogism is an instrument of demonstration from premises already known, not of discovery from data not yet organized. When the ambition of early modern science shifts from systematizing established knowledge to investigating what is not yet known, the adequacy of Aristotelian logic as the sole instrument of inquiry comes into question.

Aquinas situates logic within the medieval university's organization of learning. As one of the three arts of the trivium, alongside grammar and rhetoric, logic is propaedeutic to all the higher sciences. Its function is to form the mind in the discipline of reasoning before it encounters theology, physics, or metaphysics. Without that formation, Aquinas holds, the student will reason carelessly in every field. The relation of logic to the other liberal arts is treated in the chapter on EDUCATION.

Following Aristotle, Aquinas identifies three operations of the intellect to which logic corresponds: simple apprehension (which the Categories addresses by classifying predicables), composition and division of concepts (which On Interpretation addresses by analyzing the proposition), and discursive reasoning (which the Analytics addresses by analyzing inference). This tripartite structure organizes the entire Organon into a coherent pedagogical sequence.

What is distinctive in Aquinas's account is his attention to the peculiar object of logic. The other sciences study real beings; logic studies what Aquinas calls "beings of reason," the intentional structures that the mind produces in reflecting on its own operations. A universal concept, a proposition, a syllogism: these exist only as objects of reflection, not in the world. This makes logic formally distinct from metaphysics, even though both concern very general features of things. Metaphysics studies being as being; logic studies the patterns of discourse applicable to any subject whatsoever, as discussed in the chapter on METAPHYSICS.

Aquinas transmits Aristotle's logic to the medieval and early modern world with a precision and comprehensiveness that make it available for further development and criticism. When Bacon and Descartes direct their objections against "scholastic logic," it is the tradition that Aquinas consolidates that they have principally in view.

Bacon opens the Novum Organum with the contention that the logic inherited from Aristotle is defective at its root as an instrument of discovery. The syllogism, for all its formal correctness, commands assent from the mind without, as Bacon puts it, gripping the subtlety of nature. It works with notions already formed by the understanding, notions that are themselves confused and ill-defined. To build a system of knowledge on such foundations is, in his view, to erect a structure on unsound ground.

His critique has two principal parts. First, the syllogism proceeds too quickly: it moves from a few particulars to the most general axioms and then deduces intermediate conclusions, without pausing to examine whether the intermediate axioms are themselves well-grounded. Second, the categories employed in Aristotelian science are inadequate to nature. Terms like "heat," "cold," "heavy," and "light" are, Bacon argues, as opaque and unstable as the perceptions from which they are derived. The bearing of this criticism on the questions treated in the chapter on SCIENCE is evident.

The new organon that Bacon proposes proceeds by a different path: systematic induction through elimination. By tabulating the presence, absence, and degrees of a phenomenon across a wide range of instances, one can progressively exclude false candidates for its cause until the true cause is isolated. This method does not require genius; it requires patience, discipline, and the systematic interrogation of nature by experiment. The relation of this method to the questions treated in the chapter on INDUCTION is direct.

Bacon's critique of Aristotelian logic as an instrument of discovery sets the terms for the seventeenth-century debate about method. Descartes, who shares Bacon's dissatisfaction with the syllogism, proposes a rationalist alternative; Locke and Hume pursue the empiricist implications of Bacon's position; and Mill later undertakes to provide the systematic inductive logic that Bacon described in outline.

Descartes shares Bacon's dissatisfaction with the traditional logic of the syllogism as an instrument of discovery, but proposes a different remedy: not induction from experience, but a purified deduction modeled on mathematics. The syllogism, Descartes argues, merely makes explicit what is already implicit in the premises. It cannot produce new knowledge; it can only display the logical consequences of what is already known. Genuine discovery requires the intuition of simple natures and the deduction of their necessary connections.

The four rules stated in the Discourse on Method express Descartes's positive account: accept only what presents itself so clearly and distinctly to the mind that there is no occasion to doubt it; divide each difficulty into as many parts as possible; conduct one's thoughts in order, beginning with the simplest objects and ascending to the most complex; and review everything so completely that nothing is omitted. This is not logic as a formal discipline of the schools but a method for the actual conduct of inquiry, applicable, Descartes believes, to any subject. The bearing of this method on the questions treated in the chapter on MATHEMATICS is considerable.

Descartes holds that the mind possesses a natural light, the capacity for clear and distinct perception, that traditional logic has obscured rather than cultivated. His early Rules for the Direction of the Mind develops the conception of a "mathesis universalis," a universal science of order and measure applicable to any subject, not merely to quantities. This anticipates the later project, developed by Leibniz and others, of a formal language adequate to all reasoning.

Descartes establishes for the rationalist tradition the thesis that the formal structure of reasoning is independent of experience. Kant takes up this thesis in a transformed version, arguing that the a priori forms of the understanding constitute the structure of experience rather than being abstracted from it.

Locke is skeptical of formal logic as a practical instrument for the extension of knowledge. In the Essay Concerning Human Understanding he argues that the syllogism contributes little to actual reasoning. Men reasoned well long before Aristotle formulated the syllogism, and the syllogism itself provides no assurance of the truth of its premises. God, Locke writes, "has not been so sparing to men to make them barely two-legged creatures, and left it to Aristotle to make them rational."

His deeper objection is epistemological. The syllogism does nothing to secure the truth of its premises, and the premises are everything. An argument may be formally valid while its conclusion is entirely false. The real work of the understanding, forming ideas from experience, combining them into propositions, and testing those propositions against reality, is completed before formal logic enters the picture. What Locke offers in place of the syllogistic logic is a genealogy of ideas, showing how all the materials of the intellect derive from sensation and reflection alone. The bearing of this account on the questions treated in the chapter on EXPERIENCE is direct.

The limits of knowledge, on Locke's view, follow from the limits of ideas. We can know the agreement or disagreement of ideas with certainty only where they can be directly compared; in all other cases, what we possess is at best probability. This restriction is not, in Locke's account, a defect but a safeguard: it protects the mind from the pretension to know what lies beyond its cognitive reach. Logic, so understood, functions as a therapy as much as a method, correcting the mind's susceptibility to speculative systems built on words rather than on ideas grounded in experience.

Locke's critique of formal logic reinforces Bacon's but adds an epistemological dimension: the question of ideas. Before one can evaluate the validity of an inference, one must know the content of its terms. The theory of ideas is thus, on this account, prior to the theory of inference. Hume takes this further, subjecting the ideas themselves to an analysis that calls into question the claims of rational metaphysics.

Hume draws a distinction between two kinds of reasoning that has a bearing on the whole subsequent discussion of logic: relations of ideas (comprising logic and mathematics) and matters of fact (comprising empirical knowledge). The former is knowable by reason alone, and its denial involves a contradiction. The latter is knowable only through experience, and its denial is always logically possible.

From this distinction Hume derives his challenge to the pretensions of rational inquiry. Relations of ideas, being known a priori, say nothing about the world. Matters of fact, being known through experience, can yield no necessary or universal conclusions. When we reason about causal connections, the basis of all empirical science, we are not performing a logical operation at all but following a habit of the imagination that constant conjunction has produced in us. There is no logical connection between cause and effect; there is only the trained expectation of the mind. The questions this raises about causation are treated more fully in the chapter on CAUSE.

Hume's contribution to the discussion of logic consists in part in the separation of two things that previous writers had tended to conflate: the laws of thought, which describe how propositions formally relate to one another, and the laws of association, which describe how ideas actually move in the mind. The latter belongs to empirical psychology, not to logic. Logic as a formal science is, on Hume's account, secure but limited; what it secures is the validity of inference, not the truth of conclusions about the world.

Hume's critique poses a problem that Kant undertakes to resolve. If formal logic can guarantee only analytic truths, and if empirical knowledge can yield only contingent conclusions, then the synthetic a priori knowledge that Newtonian physics appears to presuppose becomes a question requiring explanation. Logic itself must be reconceived to answer it.

Kant draws a distinction between general (formal) logic and transcendental logic that has consequences for the entire field. General logic, he holds, abstracts from all content of knowledge and concerns itself solely with the form of thought. It is, in Kant's view, essentially complete, having gained little in extent since Aristotle. It sets down the a priori rules for the correct use of the understanding, without reference to any particular subject matter.

Transcendental logic, by contrast, investigates the pure concepts of the understanding, the categories, that make experience possible. These concepts are not derived from experience but are a priori; yet they are applicable only to experience. They include substance, causality, and community, among others, and they organize the raw manifold of intuition into an experience of determinate objects. Without them, Kant argues, no coherent experience of events in time would be possible. The relation of this doctrine to the questions treated in the chapter on EXPERIENCE is direct.

This distinction opens a dimension of logical inquiry that had not previously been recognized. Formal logic determines which forms of inference are valid. Transcendental logic asks what a priori conditions must be satisfied for these logical forms to apply to anything real. The answer, on Kant's account, is that the forms of sensibility (space and time) and the categories of the understanding must cooperate to constitute the objects of experience. Logic, so understood, is not merely a set of rules for reasoning but a constitutive element in the possibility of knowledge itself. The bearing of this on the questions treated in the chapter on KNOWLEDGE is considerable.

Kant's reconception of logic generates divergent responses. Hegel develops a dialectical logic that seeks to reunify the form and content that Kant separated. Mill, proceeding from empiricist premises, rejects the a priori altogether and develops an inductive logic grounded in experience. The question of whether logic is purely formal or has a constitutive role in determining what can be known remains a central issue in the philosophy of logic.

Mill's A System of Logic, published in 1843, offers the most comprehensive treatment of inductive method in the English-speaking tradition. Mill rejects the Kantian claim that any synthetic knowledge is a priori: what Kant took for necessary truths are, on Mill's view, well-confirmed inductive generalizations. Even the axioms of logic and mathematics, he maintains, are grounded in experience, though confirmed by so many instances as to be practically indubitable.

Mill also reinterprets the syllogism. The classical account treats the major premise ("All men are mortal") as a universal truth from which the particular conclusion is derived. Mill argues, however, that the real inference proceeds from particular to particular: the major premise is a convenient summary of observed cases, and the syllogism does not teach us anything new but helps us check and communicate inferences already made by induction. The questions this raises about the nature of REASONING are treated in the chapter on that idea.

His five methods of experimental inquiry, agreement, difference, the joint method, residues, and concomitant variation, systematize the logic of causal inference that, in Mill's account, underpins all the natural sciences. They are not mere rules of thumb but the logical form of all discovery in natural history, physics, and the moral sciences. Logic, so conceived, is not formal in Kant's sense but empirical: it is the organized practice of reasoning from experience. The relation of these methods to the questions treated in the chapter on INDUCTION is direct.

Mill thus provides the detailed inductive logic that Bacon had called for but left in outline. A difficulty remains, however, in his reliance on the uniformity of nature as a principle that itself cannot be established by induction without circularity. This gap, and the broader question of whether logic is fundamentally empirical or a priori, continues to be debated in the philosophy of science.