Deduction is defined as reasoning by which logically necessary conclusions are derived from certain premises; truth or judgment arrived at by this process | in com. usage, the act of inferring, of arguing; what is inferred, inferred: arbitrary deductions.

Since the deductive method always starts from a postulate or an axiom, that is, from an absolute truth that does not need to be verified, from which it deduces, through reasoning, particular facts, the validity of what has been demonstrated would collapse if it were shown that the starting statement was false or arbitrary. In this way the very premises on which the reasoning itself was based would collapse. And this is often taken as a criticism of the deductive method by proponents of the inductive method. However, the debate between deductivists and inductivists is still open among philosophers of science.

A typical discipline that makes use of deductive thinking is mathematics: deduction is synonymous, from time to time, with derivation, demonstration, and inference. However, the most frequent use is as a synonym for derivation. In this sense by deduction of a formula F in a logical calculus L is meant a finite succession of formulas of L, which satisfies the following conditions: first, each formula of the deduction is either a logical axiom, an assumption, or the result of applying one of the rules of L to one or more previous formulas; second, the final formula of the succession must be precisely F. It should be noted that the peculiar character of deduction is given by the fact that it refers not only to the logical axioms of L, but also to assumptions (which can be infinite) that are made hypothetically. The set of axioms and assumptions of L constitutes the premises of the deduction and the derived formula is its conclusion.

History of deductive method

The introduction of the concept of deduction is due to Aristotle (384 BC-322 BC), who identified it substantially with the syllogism. From this identification derives the traditional interpretation, accepted until modern times, according to which the procedure of deduction allows to start from a universal law to reach particular conclusions. The opposite procedure is called induction, which instead moves from the particular to the universal.

An example of Aristotelian syllogism is the following: “All men are mortal; Socrates is a man; therefore Socrates is mortal”. It can be seen that the conclusion (particular) is derived from two more general statements: this is an exact reasoning from the point of view of logical consistency, which, however, can not in any way guarantee the truth of the first principles, since these must start the deduction. This is why Aristotle reserved the task of establishing the validity and universality of the premises, from which the syllogism will draw only conclusions necessarily consistent, intellectual intuition (or noùs), as distinct from simple reason (diànoia). Intuition is for Aristotle a supra-rational faculty that has the ability to penetrate the essence of reality under investigation, making it pass to the act, that is, grasping the true and unchanging aspect, regardless of its external and contingent particularities.

The intuitive intellect also makes use of empirical induction (epagoghé), which, however, unlike the meaning that will assume contemporary epistemology, does not have for Aristotle the ability to arrive at the universal essences of reality, but is only a preparatory degree of initiation towards intuition. Basing itself on single particular cases, in fact, the inductive method can only obtain purely arbitrary knowledge, devoid of that binding universality which is instead proper to the deductive method: the main feature of the latter is given precisely by its necessity, by its logical consequentiality.

From the Middle Ages to the Modern Age

The Aristotelian gnoseology – passed through the medieval scholasticism and made its own by metaphysical and neoplatonic logic, which saw in deduction the method par excellence with which to reproduce reality from the supreme intuition of the Idea – will remain valid at least until the seventeenth century. Since then, with the gradual abandonment of Aristotelian essentialism, which strictly linked logic to ontology, deduction will tend more and more to take the form of a relationship between purely syntactic objects, regardless of the content of the propositions we are talking about.

Galileo Galilei (1564-1642) first renounced to the knowledge of qualities and essences of reality in favor of an analysis limited to its quantitative aspects. Galilei however, next to the new inductive-experimental method, continued to use the Aristotelian deductive method. He distinguished two moments: knowledge for him starts from experience, during which, by induction, the intellect accumulates data (Galileo will speak of sensible experiences); then, reworking with reason these data, it comes to the formulation of universally valid laws and that, as such, exceed the moment of particular and sensitive experience; from these universal laws will then be possible, in turn, to deduce other particular determinations (process that Galileo calls necessary demonstrations).

Philosophers who instead will keep well separated the two processes, in modern science, were Bacon, who preferred only induction and Descartes, who relied instead on deduction, but he too renounced to the essences and focused only on the search for a method; he will come to consider animals as pure machines and on the other hand to him we owe the invention of the “Cartesian plane”, a fundamental element for mathematics and its applications especially in physics and economics. To the methodology of Descartes will head the rationalism of Spinoza, who, however, recovered the value of intuition as the supreme foundation of the scientific method-deductive.

Bacon’s inductivism was succeeded instead by Locke’s empiricism and then by David Hume, who pushed it to its extreme consequences to the point of resolving it in skepticism. Hume in fact questioned the validity of scientific laws that are assigned to nature because he attributed to them an inductive origin and therefore arbitrary. To him reacted Kant (1724-1804) who proposed to demonstrate the deductive origin (and not inductive) or a priori of scientific laws, to safeguard them from Hume’s skepticism.

Kant used the term Deduction precisely in the sense of demonstrating the universal and necessary character of the so-called synthetic a priori judgments that science uses: synthetic because they unify and synthesize the multiplicity of perceptions arising from the senses, but a priori because they do not depend on them. With his Transcendental Deduction Kant argued that our reason plays a critical and highly active role in producing science, which is deduced from a supreme principle of the “I think” placed at the foundation of all knowledge. The “I think” uses special categories of the intellect that are transcendental, i.e. they are activated only when they receive information to be processed and justify the character of universality, necessity and objectivity that we give to science; vice versa, without these characteristics there is no true knowledge.

German idealism took up the concept of deduction elaborated by Kant, assigning it a function not only cognitive but also ontological: the “I”, or the Absolute, will be the first principle from which phenomenal reality is produced by dialectical deduction. With Fichte and Schelling there was a reproposition of classical metaphysics, especially neoplatonic. With Hegel instead the deduction was no longer subordinated to a higher principle but became itself Absolute: Hegel rejected those philosophies that placed at the base of the deduction an intuitive act of supra-rational nature and transformed the deductive method in a spiral procedure that finally comes to justify itself. Aristotelian logic was thus abandoned; while the latter proceeded in a linear way from A to B, Hegelian dialectics proceeds in a circular way: from B comes C (synthesis) which is in turn the validation of A.

This new way of understanding deduction – which made the method coincide with the very end of philosophy, also taken up by Marx to justify the theory of class revolt on the basis of the alleged dialectical progress of history – was, however, the subject of numerous criticisms that led, with the advent of positivism, to the abandonment of the deductive method in favor of the inductive one.

Karl Popper

The deductive method has been revised and re-evaluated by Karl Popper (1902-1994), who argued the fallacy of any inductive approach to experience. Referring to Kant and his Copernican revolution of thought, Popper believed that from single particular cases it will never be possible to derive a law valid always and everywhere, precisely because we cannot experience the universal. Universality is instead something a-priori that we project on reality; in fact, according to Popper, every scientific knowledge that we believe to be obtained empirically is actually deduced from our mental schemes and unconsciously conveyed on real data. For intellectual honesty it is necessary to admit that science proceeds only by deduction; it is the so-called “lighthouse theory” or the method of trial and error, also common to animals, which starts from initial assumptions, entirely conjectural, able to predict the tangible consequences that are tested from time to time. From the single facts can never be obtained confirmations of the hypothesized theory, but only denials.