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As used in philosophy and theology, demonstration is a logical and methodological term first employed by Aristotle (Gr. πόδειξις, apodictic) to designate reasoning or proof that is necessarily true and absolutely certain. It was adopted by medieval scholastics (Lat. demonstratio ), notably by St. albert the great and St. thomas aquinas, whose commentaries on Aristotle's Posterior Analytics and Boethius's De Trinitate give significant insights into the concept. Later scholastics also used the notion, modifying it according to their particular views concerning knowledge and science. In modern and contemporary thought, demonstration is commonly equated with any kind of proof that yields certitude.

This article explains the definition of demonstration in the context of Aristotle's theory of scientific reasoning, lists the various types of demonstration, and notes the knowledge prerequisite to it. It then details the usage of demonstration in different scholastic disciplines, and concludes with appraisals of demonstration in the thought of modern philosophers.

Science and Demonstration. The theory of demonstration presupposes Aristotle's notion of scientific knowledge. He says: "We suppose ourselves to possess unqualified scientific knowledge of a thing when we think that we know the cause on which the fact depends, as the cause of that fact and of no other, and, further, that the fact could not be other than it is" (Anal. post. 71b 811). He speaks of unqualified scientific knowledge as edge πιστήμη, the same word that Plato had used for the contemplation of subsistent ideas. The characteristics of such knowledge are the certitude and necessity of some fact through assignment of its proper cause (see causality). Such a requirement is more stringent than that associated with science in modern thought. Aristotle points out that this concept of scientific knowledge is evident, since men do actually desire and claim it (ibid. 71b 12). The logical vehicle for attaining such knowledge is called demonstration. "By demonstration I mean a syllogism productive of scientific knowledge, a syllogism, that is, the grasp of which is eo ipso such knowledge" (ibid. 71b 1719).

As an instrument of scientific knowledge, demonstration can be defined in terms of its conditions: "The premises of demonstrated knowledge must be true, primary, immediate, better known than and prior to the conclusion, which is further related to them as effect to cause" (ibid. 71b 2022). The premises must be known as "true," since doubtful or false premises cannot generate a true and certain conclusion. These premises must be "primary," that is, they are not themselves the result of demonstration, but they are indemonstrable propositions into which a demonstration can be resolved. In a series of demonstrations, the conclusion of one may serve as a premise of the following one, but there must be a first that does not have to be demonstrated. Such a premise is "immediate," that is, no middle term is needed to join subject and predicate, because the mind is able to grasp their mutual implication directly from the meanings of the terms, once these latter have been understood by induction from sensory experience. The premises must also "cause" the conclusion, and this in two ways: (1) as causes of one's knowledge of the conclusion, or the instruments used by the mind in inferring the new truth of the conclusion and (2) as causes, in the ontological order, of the attribute asserted of the subject in the conclusion. As causes, the premises are known "prior to" the conclusion; they are also "better known," as being immediately evident.

Types of Demonstration. A demonstration is had whenever a statement is given together with the reason for its truth, that is, whenever the question "why" is answered. Because there are different senses of "why," there are different kinds of demonstration. Since the reason "why" is expressed by the middle term of the demonstrative syllogism, it can be equivalently said that the types of demonstration correspond to the condition of the middle term that links the subject and predicate of a scientific conclusion. Thus, the general division of demonstration according to Aristotle (ibid. 78a 2279a 16) is into knowledge of the fact (τι, quia ) and knowledge of the reasoned fact (διότι, propter quid ). Only demonstration of the reasoned fact fulfills all the requirements stated in Aristotle's definition of demonstration. Knowledge of fact is called demonstration analogously by reason of its similarity with, and ordination to, demonstration propter quid.

Propter Quid. Demonstration propter quid assigns the proper ontological cause of an attribute's inherence in a subject. Thus, the human soul is immortal because it is incorruptible. In the most perfect type of propter quid demonstration, all the terms of the syllogism are convertible, or commensurately universal. However, as long as the middle term and the attribute are convertible, there is propter quid demonstration, even though the subject of the given demonstration is only a subjective part of the proper subject of the attribute; e.g., every isoceles has three angles equal to two right angles because it is a triangle. What is essential to this sort of demonstration is that the cause of the attribute be proper. This type is sometimes called demonstratio propter quid particularis, or particular demonstration.

Quia. Demonstration quia is had whenever the middle term is not the proper cause of the attribute. This demonstration can be a priori or causal when a remote cause is assigned; e.g., A wall does not breathe because it is not living (the proper cause is that it has no lungs). Demonstration is a posteriori when the middle term is not a cause at all, but an effect of the attribute. If the effect is not adequate or convertible with the cause, the demonstration yields knowledge of the existence of the cause and some of its conditions, e.g., God as known from His creation (St. Thomas Aquinas, In Boeth. de Trin. 6.4 ad 2). If, however, the cause and effect are of commensurate universality, e.g., the intellect as the cause of abstract reasoning, then the demonstration makes known the proper cause, and hence the terms may without circularity be recast as a propter quid demonstration (Anal. post. 78a 39b 10).

Subalternation. When the middle term is a commensurate cause and is defined with principles of the same nature as the major and minor terms (e.g., with sensible matter), then there is proper knowledge of the reasoned fact. If the middle term is defined with other principles (e.g., mathematical or metaphysical), then the conclusion is said to be factual or quia knowledge in its own order, and causal only as subalternated to principles of a higher order. Thus, the practical science of medicine is subalternated to natural science. Physicomathematical sciences demonstrate through remote causes, i.e., mathematical principles, that are formal causes, although Aristotle does refer to them as being in a broad sense demonstrations διότι (Anal. post 79a 3, 12; cf. St. Thomas, In 1 anal. post. 25.6)

Prerequisite Knowledge. All the formal logic of the syllogism is presupposed for demonstration. But only the first figure of the syllogism in the mood of Barbara is perfectly adequate, since it alone concludes to a universal affirmative proposition (see syllogism).

The truth of the premises must be solidly established. The minor premise contains the term that serves as subject of the conclusion; to this term is predicated the middle term, thereby constituting a definition of it. Such a premise, in which a definition is predicated of a subject, is self-evident. The major premise in every demonstration connects a cause and an effect, the nature of the demonstration depending on whether the cause or the effect is the middle term. This premise becomes known through a process of induction (Anal. post. 100b 4), which terminates not merely in empirical correlation, but in un derstanding of, or insight into, the nature of cause and effect and their necessary bond.

Some foreknowledge is required of the terms of the demonstrative syllogism. Usually the existence of the subject, or minor term, must be known in order that attributes may be joined to it. However, in an a posteriori demonstration that establishes the existence of the subject, e.g., proofs for the existence of God, only the nominal meaning of the subject-term can be had. A definition of the subject is necessary as the predicate of the minor premise; this definition represents either a cause in a priori demonstration, or an effect in a posteriori demonstration; in this latter case it is only a nominal definition.

In the process preparatory to propter quid demonstration, some knowledge is already had of the conclusion of the demonstration. For the existence and definition of the subject are known. At least the nominal definition of the predicate is known in the very asking of a question to be demonstrated, e.g., Why is the sky blue on a cloudless day? The fact of the conclusion may be known by observation, as in the example given. Even a definition of the predicate as an accident may be known, but not a definition of the predicate as a property. The definition of a property as such presupposes knowledge of its necessary connection with its proper subject; this is precisely what has to be demonstrated. Therefore, the fore-knowledge of the predicate as a property can be only a nominal definition. There is no foreknowledge of the middle term as such; the finding of the middle term is the demonstrative process.

The self-evident premises of demonstration are called the principles. They deal with definitions and insight into causal connections, and thus their self-evidence becomes apparent only after a careful and sometimes extended investigation. The principles must be proper for propter quid demonstration. There are other more general self-evident principles involved in demonstration, such as the principle of contradiction, the principle of agreement and disagreement, and the principle of the syllogism (dictum de omni and dictum de nullo ). These higher principles, or axioms, are implicit in every demonstration, but they do not function as the content of premises from which conclusions are deduced. (see first principles.)

Demonstration in Various Disciplines. logic, besides being the art of reasoning, is also a science. It establishes true laws of thought and demonstrates them through universal and necessary reasons. Since the subject matter of logic is relations of reason, the definitions used in logical demonstrations will be in the order of formal cause. These are obtained through abstraction by the intellect reflecting on its own operations.

Mathematics. Long considered the perfect model of propter quid demonstrative science, mathematics is similar to logic in that it usually demonstrates from formal cause. In Aristotelian theory mathematical definitions were considered to be obtained through abstraction from sensory matter. Modern theory emphasizes the postulational nature of the principles of mathematics. When principles are only postulated, the conclusions of demonstration share the unproved status of the principles and the proofs reduce to mere formal inference. (see mathe matics, philosophy of.)

Philosophy of Nature. Aristotelian natural philosophy demonstrates according to all four causes (Phys. 194b 16195b 30). The middle terms are defined with sensory matter and in abstraction from individual things, which are contingent. The necessary bond linking the terms in the conclusions of such demonstrations is called conditional. The reason for such necessity rests in natures and essential relations, but the existence of these natures is always contingent and the perfect realization and operation of the natures may be hindered. But if the truth of the conclusion is to be in reality, the cause assigned in the premises will be necessarily involved.

For Aristotle and his medieval followers, philosophy of nature did not stop at general considerations of mobile being, but pursued its investigation to the specific level. At this level demonstrations must be given through efficient and final causes, or by definitions of specific natures in terms of properties, which are signs of the specific nature. A posteriori demonstrations also have a fundamental place in the science of nature, since sensible effects are better known to man than their causes. Modern sciences of nature share this Aristotelian demonstrative method to the extent that they assign proper causes.

Mathematical Physics. Modern sciences of nature that employ mathematical reasoning can in principle give factual demonstration of physical realities, but cannot assign proper causes, which have to be of the physical order. The validity of the demonstrations of these hybrid sciences depends in each case upon the status of the premises, and particularly on whether they have an admixture of constructional or hypothetical matter in them.

Metaphysics. In scholasticism metaphysics has been traditionally considered a science. St. Thomas calls it maxime scientia (In 1 anal. post. 17.5). It treats of "being, its parts, and its passions" (ibid. 20.5) per modum demonstrationis (ibid. 20.6). The subject is ens commune, conceived in abstraction from all matter, but able to exist either in matter or immaterially, therefore excluding God and separated substances (In Boeth. de Trin. 5.4). Whenever one can ask a question about such being, in general or of any of its parts, and can assign a proper reason for it, one has a propter quid demonstration. For example: Why is being contingent? Because it is composed of essence and existence, as two principles really distinct. Note that the reason need not be a physical cause, but can be a prior reason only virtually distinct from the attribute that it explains; e.g., the immutability of God is the reason for His eternity. A posteriori demonstrations are relevant to metaphysics, especially in proving the existence and attributes of God.

Moral Science. Demonstration in ethics and moral theology provides special difficulties, for as practical sciences they have subjects and ends that differ from those of speculative sciences. The subject of moral science generally is human acts, and the end is not knowledge but the morally good operation. There is, however, a phase of such science in which the truth of moral principles is sought in a universal way, and here demonstration is used. If it is asked, for instance, whether a certain medical practice is moral, the answer must apply a proper reason to the case. This will be the proper final cause, which itself is either immediately or mediately connected with the final end of man, the first principle of moral science. The universal truth, however, is not altogether sufficient for moral action; it is in the prudential judgment that the final truth is attained regarding an individual action to be performed in all its contingency.

Sacred Theology. There are demonstrations in sacred theology, insofar as statements are made and proper reasons given for their truth. Thus, the answers to many of the questions asked by St. Thomas in his Summa Theologiae are demonstrations. If the two premises are revealed, then demonstration explicates what was implicit in revelation. Likewise, one premise can be a truth of faith, while the other is a self-evident or demonstrated truth of human science. The middle term is therefore once illuminated by a divine light of faith and once by human intelligence. These middle terms may be causes or effects, proper or remote. Such demonstrated conclusions are said to be virtually contained in the deposit of revelation (see theological conclusion).

Demonstration and Modern Philosophy. Modern philosophy takes an attitude to demonstration corresponding to its acceptance, limitation, or rejection of uni versals, substance, and causality. Galileo galilei, T. hobbes, F. bacon, and many of their successors restricted causal explanation to the material and efficient causes, or to mechanical causes. Hobbes interpreted all rational discourse in a nominalistic spirit, so that science is knowledge, not of the consequences of one thing to another, but of one name of a thing to another name of the same thing (Leviathan, 1.7).

Cartesian Proof. The value of demonstration depends on the status accorded the premises. René des cartes could not admit the validity of intuitions based on sensory experience, but sought clear and distinct conceptions that sprang from the light of reason alone. Such a starting point for deductive reasoning assured him of results of the same order as arithmetic and geometry, the only two sciences that he admitted as free from any taint of falsity or uncertainty (Rules for the Direction of the Mind, 23). For him, the method of proof is two-fold: analytic, or from causes; and synthetic, or from effect to cause. Descartes preferred analysis as the best and truest method; synthesis he considered as not conveniently applicable in metaphysical matters (Objections Urged by Certain Men of Learning Against the Preceding Meditations, Reply to Second Objections, n. 7). Even when a posteriori demonstrations are used, they proceed from effects present within experience, not from effects existing in an objective world. Cartesian demonstration remains more in the realm of ideas and pure essences than in the world of existent things. The triumph of the method can be found in B. spinoza, whose Ethics is evolved more geometrico.

Locke's Theory. John locke resolutely denied the existence of innate ideas. Ideas enter the mind through sensation, yet it is the ideas, not things responsible for producing the ideas, that constitute the object of knowledge. Intuition is immediate perception of the agreement or disagreement among ideas. Demonstration is knowledge of agreement or disagreement among ideas through the mediation of other ideas (Essay Concerning Human Understanding, 4.2.2). The necessary connections are only relations among ideas, not relations of things with their attributes. Locke rigidly restricted the extent of demonstration. While admitting the validity of mathematical demonstration, he denied the possibility of a demonstrative science of the physical world, because the middle terms he sought as causes are unknown motions of minute particles. The existence of God can be demonstrated, but only by starting from intuitive awareness of one's own existence. He admitted in principle the possibility of demonstration in moral matters, but he did not himself elaborate such a demonstrative science (ibid. 3.11.16). Since Locke's doctrine of causality was weak and ambiguous, his concept of demonstration is little more than that of an explicative syllogism, and of the syllogism itself he made little account. Hume's further development of Locke's philosophy and his relegation of causality and universality to mental habits effectively destroyed demonstration.

Kant's Compromise. Immanuel Kant tried to save the prerogatives of reason from the attack of Hume. But the conditions of knowledge necessary to validate demonstration were supplied by Kant through a priori categories of the understanding. Thus, the concepts of substance and accident, existence, universality, causality, and necessity were not conditions of extramental things, but were a priori and constitutive of experience. Reasoning serves merely to unify the manifold knowledge of the understanding. It thus takes on a subjective character, yielding no universal and necessary knowledge of things, or noumena. In fact, Kant limits the use both of definitions and of demonstration to mathematics. He requires that demonstration be intuitive; this is not possible of empirical objects, because no necessity in them can be grasped through intuition, but only subsequently through application of a category of the understanding (Critique of Pure Reason, Transcendental Doctrine of Method, chapter 1, section 1, B756, 76263). Rather than yield the eternal truth claimed by Aristotelian demonstration, Kant's system falls back into the skepticism it attempted to avoid.

Positivism. A. comte and subsequent positivists abandoned the ideal of causal explanation and settled for a notion of science as the laws of relations of phenome na. This noncausal and nonsubstantial phenomenalism constitutes the philosophy that has most influenced modern science. Aristotelian demonstration would henceforth be an alien concept to modern thought, for only in a context of moderate realism does the doctrine of demonstration have relevance.

See Also: argumentation; deduction; dialectics; methodology (philosophy).

Bibliography: aristotle, Anal. post., tr. g. r. g. mure, The Basic Works of Aristotle, ed. r. mckeon (New York 1941). thomas aquinas, Exposition of the Posterior Analytics of Aristotle, tr. p. conway (Quebec 1956). m. a. glutz, The Manner of Demonstrating in Natural Philosophy (River Forest, Ill. 1956). w. a. wallace, The Role of Demonstration in Moral Theology (Washington 1962); "Some Demonstrations in the Science of Nature," Thomist Reader 1 (1957) 90118. p. coffey, The Science of Logic, 2 v. (New York 1912; reprint 1938). e. d. simmons, "Demonstration and Self-Evidence," Thomist 24 (1961) 13962. w. baumgaertner, "Metaphysics and the Second Analytics," The New Scholasticism 29 (1955) 40326. j. f. anderson, "On Demonstration in Thomistic Metaphysics," ibid. 32 (1958) 47694. m. j. adler, ed., The Great Ideas: A Syntopicon of Great Books of the Western World, 2 v. (Chicago 1952) 2:54668. c. filiasi carcano, Enciclopedia filosofica, 4 v. (Venice-Rome 1957) 1:157880.

[m. a. glutz]

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The first amendment guarantees the right of persons to congregate peaceably in large numbers in appropriate public spaces in order to communicate ideas or grievances. In Edwards v. South Carolina (1963) the Court described an assemblage of 187 protesters on the grounds of a state capitol as "an exercise of … basic constitutional rights in their most pristine and classic form." Mass demonstrations cannot be prohibited simply on account of their size or their need to occupy public land.

Constitutional litigation over demonstrations tends to focus on three issues. First is the question of what public spaces must be made available to demonstrators. By virtue of the number of persons involved, mass demonstrations can be disruptive of other activities even when the demonstrators remain peaceable and orderly. When must those other activities give way to the First Amendment claims of persons who wish to engage in a mass demonstration?

The Supreme Court has never given a definitive and comprehensive answer to that question, and probably never could. The Court has indicated, however, that demonstrations in public forums such as streets, sidewalks, and parks cannot be subjected to a blanket prohibition. On the other hand, the Court has upheld regulations that entirely prohibited demonstrations in a jailyard and in areas of a military base otherwise open to the public.

Second, the issue has arisen whether a demonstration can be prohibited or postponed on the ground that audience hostility to the demonstrators threatens to produce a breach of the peace. The Court has inveighed against any such "heckler's veto" in obiter dictum, and has reversed disorderly conduct convictions of speakers who continued their orderly protests in the face of potentially threatening crowds. In language quoted many times in the United States Reports, the Court stated in terminiello v. chicago (1949):

[A] function of free speech under our system of government is to invite dispute. It may indeed best serve its high purpose when it induces a condition of unrest, creates dissatisfaction with conditions as they are, or even stirs people to anger. Speech is often provocative and challenging. It may strike at prejudices and preconceptions and have profound unsettling effects as it presses for acceptance of an idea. That is why freedom of speech … is … protected against censorship or punishment, unless shown likely to produce a clear and present danger of a serious substantive evil that rises far above public inconvenience, annoyance, or unrest.

Despite strong dicta and case outcomes favorable to speakers, it cannot be said with assurance that a hostile audience can in no circumstances provide a basis for disallowing a demonstration. The Court has yet to decide a case in which the regulatory authority was confined by a narrowly drawn statute and the police could not contain the hostile audience by the exercise of due diligence. There is also the unresolved question of whether demonstrators who wish to proceed in the face of a hostile audience have a First Amendment right to do so on repeated occasions, or whether at some point the mounting costs of police protection for the demonstrators might justify a prohibition on the continuation of their expressive activity.

A third set of issues that arise frequently in disputes over demonstrations concerns the doctrine of prior restraint. Demonstrators who wish to assemble in large numbers can be required to obtain permits in advance, despite the general presumption in First Amendment law against licensing. Officials who administer permit systems for marches and rallies are required to rule upon permit requests expeditiously, and to validate denials in court on a strict timetable. Thus, administrative delay is not permitted to serve as an indirect means of prohibiting mass demonstrations. If a permit request is under administrative or judicial consideration by the time a demonstration is scheduled to take place, the demonstrators may be permitted to proceed without a permit and defend against a prosecution on the ground that they exhausted all channels of prior approval and were entitled under the First Amendment to have their permit request granted. However, demonstrators who do not both apply for a permit and pursue all channels of appeal may be prosecuted for holding a march or rally without a permit, despite the fact that had they applied for a permit they would have been entitled under the Constitution to have it issued.

A fourth issue concerning demonstrations that has not generated a great deal of litigation to date but could do so in the future is whether persons who engage in mass demonstrations can be made to pay the costs of municipal services that attend the event. The Court has indicated in dictum that reasonable costs for such services as clean-up, police protection, and the provision of toilets can be assessed against the demonstrators. However, such assessments can be quite large for major events and can be used as a means of discouraging demonstrations. This issue of cost assessment was important in the litigations during the 1970s over the proposed march of American Nazis in the predominantly Jewish community of Skokie, Illinois, and could emerge as a focus of controversy in other cases.

Vincent Blasi


Baker, C. Edwin 1983 Unreasoned Reasonableness: Mandatory Parade Permits and Time, Place, and Manner Regulations. Northwestern University Law Review 78:937–1024.

Bollinger, Lee C. 1982 The Skokie Legacy: Reflections on an "Easy Case" and Free Speech Theory. Michigan Law Review 80:617–633.

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dem·on·stra·tion / ˌdemənˈstrāshən/ • n. 1. the action or process of showing the existence or truth of something by giving proof or evidence. ∎  something that proves or makes evident: the letter was a demonstration of good faith. ∎  the outward showing of feeling: physical demonstrations of affection. ∎  a practical exhibition and explanation of how something works or is performed. ∎  a show of military force. 2. a public meeting or march protesting against something or expressing views on a political issue.

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dem·on·strate / ˈdemənˌstrāt/ • v. 1. [tr.] clearly show the existence or truth of (something) by giving proof or evidence: their shameful silence demonstrates their ineptitude. ∎  give a practical exhibition and explanation of (how a machine, skill, or craft works or is performed). ∎  show or express (a feeling or quality) by one's actions: she began to demonstrate a new-found confidence. 2. [intr.] take part in a public demonstration.

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demonstrate †indicate, exhibit; make evident by proof. XVI. f. pp. stem of L. dēmonstrāre, f. DE- 3 + monstrāre show; see -ATE 3.
So demonstration XIV. — (O)F. or L. demonstrative XIV. — (O)F. — L.