Reasoning

Plato distinguishes two modes of reasoning in the figure of the divided line. Mathematical reasoning, which he calls dianoia, begins from hypotheses and draws out their consequences, but it does not examine the starting points from which it sets out. Dialectical reasoning, or noesis, treats those same hypotheses not as beginnings but as stepping stones, and ascends by way of them to a first principle which is itself unhypothetical, the Form of the Good, from which the mind may then descend again with understanding. The distinction thus drawn is between reasoning that proceeds within assumptions and reasoning that undertakes to ground the assumptions themselves. Whether the second sort of thinking is properly called reasoning at all, or is rather an intuitive vision toward which reasoning only prepares the way, is a question the later tradition takes up under the contrast between the discursive and the intuitive.

The Meno presents reasoning in its working form. A slave boy, questioned by Socrates and taught nothing, arrives at a geometrical truth he had never been shown. Socrates infers from this that the process is not instruction from without but recollection, the soul calling back what it has always in some manner possessed. On this view the movement of thought from confusion to clarity is a movement of recovery rather than acquisition. The Socratic elenchus is reasoning employed in the same office negatively: propositions are tested by drawing out what follows from them until a contradiction appears, and the false opinion is thereby exposed. The connection between reasoning and the doctrine of the soul's pre-existence is treated more fully under the idea of Knowledge, as the theory of the Forms belongs to the idea of Idea.

Reasoning for Plato is not confined to the speculative order. In the Republic the training in dialectic is prescribed for those who are to govern, on the ground that the man who cannot give an account of the good will not know how to choose it; and the dialogues repeatedly treat the failure of reasoning, in the form of sophistical argument, as a political and moral danger rather than a merely logical one. The distinction between reasoning that yields knowledge and argument that yields only persuasion is drawn again in the chapters on Dialectic and Rhetoric.

Plato thus leaves the tradition a conception of reasoning that is at once a method and an account of what the mind attains. Aristotle separates the two questions, assigning the analysis of valid inference to the Analytics and the question of what the mind ultimately apprehends to the treatises on soul and on first philosophy. Plotinus, taking the other course, ranks reasoning below intuitive intellection and intellection below the One, holding that the need to move step by step is itself a mark of deficiency.

Aristotle defines the syllogism as a discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. The Prior Analytics considers this structure apart from any question of the truth of its parts. Using the middle term, which appears in the premises but not in the conclusion, Aristotle classifies syllogisms into three figures according to the position that term occupies, and within each figure he distinguishes the moods, or formally correct patterns, that the quantity and quality of the premises permit. No conclusion follows from two particular premises, or from two negative ones; if one premise is negative the conclusion must be negative, and if one is particular the conclusion must be particular. Underlying these rules is the principle later called the dictum de omni et nullo: whatever is predicable of a predicate is predicable also of its subject.

The Posterior Analytics turns from the form of inference to its use in science. A demonstration must proceed from premises that are true, primary, immediate, better known than the conclusion, and causally related to it, so that the reasoning shows not merely that something is the case but why it is. Since demonstration proceeds from principles, and principles cannot themselves be demonstrated without regress, Aristotle holds that the starting points are supplied by induction, the exhibiting of the universal as implicit in the clearly known particular. The relation of induction to demonstration is treated more fully under the idea of Induction, and the status of indemonstrable first principles under the idea of Principle.

Not all reasoning is demonstrative. The Topics treats dialectical argument, which proceeds from premises that are merely probable and yields opinion rather than science, and Aristotle further distinguishes both from the sophistical argument that only appears to proceed from probable premises. It is equally foolish, he remarks, to accept probable reasoning from a mathematician and to demand scientific proof from a rhetorician. He also recognizes a practical syllogism, in which the major premise is a general rule of conduct and the minor a particular perception, and whose conclusion is not an assertion that something is true but a judgment that something is to be done.

The order of the three treatises, formal analysis in the Prior Analytics, demonstration in the Posterior Analytics, probable argument in the Topics, sets the divisions within which the later tradition works. Aquinas adapts the theory of demonstration to theology, adding the distinction between proving that a thing is and proving what it is. Bacon and Mill charge the syllogism with sterility in discovery, and Kant objects to the classification by figures and moods as a mistaken subtlety while retaining the syllogism itself. Against such criticism it may be observed that Aristotle presents the syllogism as a means of expounding and testing arguments rather than of finding them.

Aquinas places reasoning within a scale of intellectual beings. The human intellect, he says, comes to know truth by a kind of movement and discursive operation, advancing from one thing known to another. The angels, in the truths they know naturally, at once behold whatever can be known in them, so that they are called intellectual and men are called rational; and in the divine knowledge there is no discursiveness at all, God seeing all things together and not successively. Recourse to reasoning on the part of men thus betrays, in his phrase, the feebleness of their intellectual light, for if they possessed its fullness they would in the first grasping of principles comprehend all that those principles imply. Reason (ratio) is on this account not a faculty distinct from intellect (intellectus) but a mode of its operation: the intellect apprehends first principles, and reason draws out what lies implicit in them.

Within human reasoning Aquinas distinguishes two directions, according to the part played by causes. Demonstration may be made through the cause, which is called a priori and argues from what is prior absolutely, or through the effect, which is called a posteriori and argues from what is prior only relatively to us. The former, the demonstration propter quid, shows why a thing is as it is; the latter, the demonstration quia, establishes only that it is. The proof that God exists belongs to the second kind, since it proceeds from effects, and Aquinas denies that anything of God's essence can be demonstrated in the first way. The bearing of this distinction on the knowledge of causes is treated under the ideas of Cause and of God.

Reasoning in the practical order, according to Aquinas, does not simply repeat the pattern of the speculative. Something may be derived from the natural law in two ways, he writes in the Treatise on Law: as a conclusion drawn from premises, after the manner of the demonstrated sciences, or by way of determination of generalities, after the manner in which general forms are particularized to details by the arts. Only the second mode is peculiar to practical thinking. The question how far moral reasoning admits of demonstration is pursued further in the chapters on Law and on Prudence.

The contrast between discursive and intuitive knowing, drawn by Aquinas along the hierarchy of angels, men, and God, becomes a commonplace of the schools and survives the theology that framed it. Descartes and Locke retain the opposition of intuition to deduction while dropping the scale of intellects, and both attempt to reduce the distance between the two by requiring that every step of a demonstration be itself intuitively certain. Whether reasoning is on this account a defect or a distinctive power is a question that recurs wherever the modes of knowledge are compared, as in the chapter on Knowledge.

Descartes reduces the operations of the knowing mind to two, intuition and deduction. Intuition is the immediate and undoubting apprehension of a truth, and it differs from deduction, he says, in that into the conception of the latter there enters a certain movement or succession, into that of the former there does not. First principles are given by intuition alone; remote conclusions are furnished only by deduction, which he understands as all necessary inference from other facts known with certainty. Deduction is nevertheless at no stage independent of intuition, for each step in the chain must itself be seen. Asked to consider that 2 and 2 amount to the same as 3 and 1, we must see intuitively not only that 2 and 2 make 4 and that 3 and 1 make 4, but further that the third statement follows necessarily from the first two.

If both the premises and the act of inference are matters of intuition, it may be asked in what way deduction adds anything. Descartes answers that although the mind has a clear vision of each step, it cannot comprehend in one intuition all the connections involved in a long chain. Only by taking the steps successively can we know that the last link is joined to the first, remembering that we have reviewed the intermediate links even though we do not survey them in a single act of vision. The remedy for this weakness of memory is practice, by which a chain rehearsed often enough may at length be run through in one continuous movement.

Descartes accordingly finds little use in the syllogistic forms, which in his opinion are of no aid in perceiving the truth about objects, and he prescribes instead four rules of method: to accept nothing that is not clearly and distinctly perceived, to divide each difficulty into parts, to proceed from the simplest objects to the more complex, and to enumerate so completely that nothing is omitted. He also distinguishes two ways of proof, the analytic, which shows the way by which a thing was discovered, as it were effect from cause, and the synthetic, which proceeds in the opposite order. For both mathematical and metaphysical reasoning he prefers analysis. The relation of these two orders to induction and deduction is treated further under the ideas of Induction and Method.

In requiring that the logical connection between premises and conclusion be itself intuitively perceived, Descartes shifts the test of good reasoning from formal validity to certainty at every step. Locke follows him in this, contrasting intuitive with demonstrative knowledge and holding that nothing is demonstrated unless each link has intuitive certainty. Aquinas and Aristotle, in contrast, place the emphasis rather on the containment of the conclusion in the premises, so that the advance reasoning appears to make consists in coming to know actually what was already known potentially. Both parties agree that demonstration must rest at last on truths that are not demonstrated, whether these are called axioms, immediate propositions, or self-evident maxims.

Hobbes regards reasoning as a kind of calculation performed with names. When a man reasons, he says, he does nothing else but conceive a sum total from the addition of parcels, or a remainder from the subtraction of one sum from another; and what arithmetic does with numbers, reasoning does with words. The addition of "body" and "animated" yields "living creature," and the subtraction of "rational" from "man" leaves what remains. The operation being one upon names, its soundness depends wholly upon the determination of their meanings, so that Hobbes places definition at the beginning of every train of reasoning. Where the names are settled by agreement the reckoning may be trusted; where they are equivocal, no correctness of form will save the conclusion. He also prefers to describe the relation of premises to conclusion in terms of antecedent and consequent rather than of cause and effect, as Aristotle does.

On this account there is no appeal to intellectual intuition, to innate principles, or to any apprehension of essences. Definitions state the meanings that men have agreed to give their words and not, as Aristotle held, the natures of things; a definition is therefore a convention entered into at the outset rather than, as it may be for Aristotle, the conclusion of a demonstration giving essential nature. The difference in the two theories of definition carries with it a difference in what reasoning can accomplish, a matter treated further under the idea of Definition. Since reasoning for Hobbes is an operation upon signs, its power is that of drawing consequences from consequences, and its liability to error lies in the failure of memory and the misuse of words rather than in any defect of the understanding.

The same conception governs Hobbes's political writing. He holds that the science of civil government admits of demonstration because its terms, being of men's own making, can be defined with precision, whereas disputes about justice and right persist so long as men use words to which they have attached no settled significations. The claim that the moral and political sciences are demonstrable in the manner of geometry is considered again in the chapters on Science and on Law.

The reduction of reasoning to reckoning stands at the opposite pole from the Platonic ascent through dialectic to a first principle, and it dispenses as well with the Cartesian requirement of intuitive vision at each step. Locke refuses to identify reasoning with computation upon words, but shares the insistence that clear and settled definitions are the condition of sound argument, and devotes much of the third book of the Essay to the abuse of words. Leibniz, taking the suggestion in the other direction, proposes a universal characteristic in which disputes might be settled by calculation.

Locke, like Descartes, contrasts intuitive with demonstrative knowledge. Sometimes the mind perceives the agreement or disagreement of two ideas immediately by themselves, without the intervention of any other, and this he calls intuitive knowledge. When it cannot bring its ideas together in that way, it is fain by the intervention of other ideas to discover the agreement it seeks, and this is what we call reasoning. Demonstration is therefore a chain of intermediate ideas, and it produces certain knowledge only if every step in it has intuitive certainty. Beside these two Locke sets a third and lesser sort, the sensitive knowledge we have of the existence of particular things without us, which falls short of demonstration though it is more than opinion.

Asked what need there is of reason, Locke answers that it is necessary both for the enlargement of our knowledge and for the regulating of our assent, since sense and intuition reach but very little of the way and the greatest part of what we know depends on deductions and intermediate ideas. In those cases where we must substitute assent for knowledge, and take propositions for true without being certain of them, we have need to find out, examine, and compare the grounds of their probability. Much of the fourth book of the Essay is accordingly given to the degrees of assent that different sorts of evidence will bear, and to the weighing of testimony. The treatment of probability as the province in which most of our reasoning is actually conducted is carried further under the ideas of Opinion and of Judgment.

Locke's estimate of the syllogism follows from this account. He grants that all right reasoning may be reduced to Aristotle's forms, but denies that they are the best way of reasoning for those who wish to find the truth, since the rules of syllogism serve not to furnish the mind with the intermediate ideas that show the connection of remote ones. Syllogism, he writes, is but the art of marshalling and ranging the proofs we already have; a man knows first, and is afterwards able to prove syllogistically. He holds, further, that demonstration of the same type is possible in morals as in mathematics, a view that appears to rest on his doctrine that all knowledge consists in the comparison of ideas, and that stands opposed to theories which make the mode of reasoning differ with the subject matter.

Locke and Descartes agree with Aquinas and Aristotle that demonstration depends on truths that are themselves indemonstrable, though they differ from them in stressing that the connection between premises and conclusion must be intuitively perceived rather than shown to be contained in the premises. His account of assent and probability passes to Hume, who takes the opposite view of what reasoning about matters of fact can establish, and to the later inquiry into the grounds of inductive inference treated in the chapter on Induction.

Hume divides the objects of human inquiry into relations of ideas and matters of fact, and he assigns a different sort of reasoning to each. Concerning relations of ideas, as in geometry and arithmetic, demonstration proceeds with certainty, since the contrary of any such proposition implies a contradiction. Concerning matters of fact it is otherwise, for the contrary of every matter of fact remains conceivable. All reasoning of this second kind, he holds, is founded on the relation of cause and effect, by which we pass from something present to the senses or memory to something not present. What he calls experimental reasoning is founded, in his words, on a species of analogy which leads us to expect from any cause the same events we have observed to result from similar causes.

Asked what warrants that expectation, Hume answers that it is warranted by neither demonstration nor experience. Demonstration cannot supply it, since no contradiction attends the supposition that the course of nature may change. Experience cannot supply it without circularity, since experience acquaints us only with constant conjunction in the past, and the inference to the future presupposes the very uniformity in question. What remains is custom or habit: after the repeated conjunction of two objects the mind is determined by usage to expect the one upon the appearance of the other. Belief so produced is a species of natural instinct which no reasoning is able either to produce or to prevent. Hume does not present this as a defect to be remedied, since without such propensities we could not conduct our lives at all.

The account has a consequence for the comparison of human with animal reasoning. Beasts, Hume observes, learn from experience as men do, yet it is impossible that their inference should be founded on any process of argument. The experimental reasoning which we possess in common with them, and on which the whole conduct of life depends, is nothing but a species of instinct or mechanical power that acts in us unknown to ourselves, and is not directed in its chief operations by any comparison of ideas. That animals reason without syllogisms bears on the question, treated under the ideas of Animal and of Habit, whether reasoning is one thing in kind or many.

Hume's distinction between a priori and a posteriori reasoning, that is, between reasoning from principles and reasoning from experience, depends on his understanding of what matters must be submitted to experience and of the manner in which experience generates belief. Kant, who reports that this analysis interrupted his dogmatic slumber, undertakes to show that the principle of causality is a condition of experience rather than a product of it. Mill accepts the empirical character of all inference while attempting to justify induction by the uniformity of nature, a principle which Hume's argument had already placed in question.

Kant separates the understanding, which applies concepts to what is given in intuition and so produces judgments, from reason, which seeks to unify judgments under principles of ever greater generality. Reason in this office pursues the unconditioned: if every judgment rests upon a more general one, thought is carried toward a ground that rests on nothing further. The syllogism is the form this movement takes, for it subsumes a judgment under a condition and then seeks the condition of that condition. In the Transcendental Analytic Kant argues that the categories of the understanding, among them substance and cause, structure experience a priori, and thus answers Hume by making causality a condition of experience rather than a habit derived from it.

In the Transcendental Dialectic the same drive is shown to generate ideas which outrun experience altogether, the soul, the world as a totality, and God, and with them the antinomies, in which contradictory conclusions can be argued with equal force because reason has left the field where its concepts have application. Reasoning is on this account indispensable in its regulative use, where it directs the understanding toward systematic unity, and misleading in its constitutive use, where it pretends to determine objects lying beyond possible experience. Kant's undertaking is to mark the boundary between the two rather than to curtail reason as such.

On the formal side Kant retains the syllogism while objecting to what he calls the mistaken subtlety of its classification into figures and moods. He does not deny all distinction among syllogisms, holding that they are threefold, like all judgments, according as they express the relation of knowledge categorically, hypothetically, or disjunctively. In the Introduction to Logic he calls the dictum de omni, that whatever is universally affirmed of a concept is affirmed of everything contained under it, the supreme principle of affirmative syllogisms, and the dictum de nullo its negative counterpart; and he derives both from the more general rules that an attribute of an attribute is an attribute of the thing, and that whatever is inconsistent with the attribute of a thing is inconsistent with the thing. Whether the hypothetical and disjunctive forms are distinct kinds of reasoning or special cases is considered in the chapter on Hypothesis.

The distinction between the legitimate and illegitimate employments of reason governs the discussion that follows Kant. Hegel holds that the contradictions which Kant treated as signs that reason had overstepped its limits are rather the movement of thought itself, and so denies the boundary. Mill takes the contrary course, denying that any reasoning is a priori and treating the axioms of mathematics as generalizations from experience. The status of the principles on which reasoning proceeds is considered again in the chapters on Principle and on Metaphysics.

Mill denies that the syllogism is a genuine form of inference. From the conclusion of a syllogism, he holds, one learns nothing that was not already asserted in the premises: when we say that all men are mortal and that Socrates is a man, the mortality of Socrates has been affirmed in advance by the major premise, and the appearance of advance from the known to the unknown is an illusion. The inferential work was done earlier, when observation of particular deaths was generalized into the proposition that all men are mortal. That generalization is an induction, and induction is for Mill the only reasoning that adds to knowledge. All inference, he concludes, is from particulars to particulars; the universal proposition is a memorandum in which the results of past inductions are recorded, together with a formula for reading the record correctly.

The criticism is not directed against the validity of syllogistic forms but against the claim that they are instruments of discovery. In this Mill continues a line of objection that runs through Bacon, Descartes, and Locke, each of whom held in one way or another that syllogism comes after knowledge rather than before it. Against them it may be urged that Aristotle himself presents the syllogism as a method of expounding and testing arguments rather than of finding them, and that his own account of demonstration requires the principles to be furnished by induction. Whether the difference between the parties concerns the nature of reasoning or only its uses is a question treated further under the idea of Logic.

Mill's positive doctrine turns on the canons of experimental inquiry, the methods of agreement, difference, residues, and concomitant variations, by which causal connections are to be established. All of these presuppose the uniformity of nature, which is itself a generalization from experience, and Mill acknowledges the circle. He answers that the principle is warranted by the accumulated success of the inductions that assume it, and he extends the same empirical account to matters that had been thought exempt: the axioms of geometry, the principles of arithmetic, and the law of causation are for him generalizations from experience differing in degree of confirmation, not in kind, from the observation that swans are white. On this last point his opposition to Kant is direct.

Mill's account states one extreme of the issue that begins with Aristotle's separation of induction from demonstration. Those who follow him treat all reasoning as at bottom inductive and all principles as provisional; those who follow Aristotle and Aquinas maintain that demonstration must rest on truths that experience cannot establish. The difficulty Hume raised remains at the center of the dispute, since the uniformity of nature which Mill invokes to justify induction is itself an inductive conclusion. Whether that circle is vicious or merely marks the point at which justification comes to an end is examined further in the chapter on Induction.