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Hypothesis
	A comparison of their Greek and Latin roots shows that the English words "hypothesis" and "supposition" are synonymous. To hypothesize or to suppose is to place under - to make one thing the basis of another in the process of thought.
	The word "hypothesis" is today often popularly misapplied to mean a guess or hunch. The sleuth in a detective story speaks of having a hypothesis about who committed the crime. The popular notion of what it means to suppose something, or to entertain a supposition, more accurately reflects the meaning of hypothesis in logic, mathematics, and scientific or philosophical method.
	A supposition is generally understood to be something taken for granted, something assumed for the purpose of drawing implications or making inferences. What is supposed is not known to be true; it may be true or false. When we make a supposition, our first concern is to see what follows from it, and only then to consider its truth in the light of its consequences. We cannot reverse this order, when we employ suppositions, and ask first about their truth.
	The word "if" expresses the essence of supposing. The word "then" or the phrase "it follows that" introduces the consequences for the consideration of which we make the supposition. We are not interested in the "if" for its own sake, but for the sake of what it may lead to. In any statement of the "if... then..." sort, it is the if-clause which formulates the supposition or the hypothesis; the other part of the statement, the then-clause, formulates the consequences or implications. The whole complex statement, which makes an if the logical basis for a then, is not a hypothesis. Rather it is what is traditionally called in logic a hypothetical proposition.

There is one use of the word "hypothesis" in mathematics which seems at odds with the foregoing summary. In Euclid's Elements, for example, a hypothesis is that which is given, not as the basis from which the conclusion is drawn or proved, but as a condition of solving the geometric problem under consideration. Let us take Proposition 6 of Book I. It reads: "If in a triangle two angles be equal to one another, then the sides which subtend the equal angles will also be equal to one another." In the demonstration of this theorem, a triangle having two equal angles is regarded as given or granted. That figure or geometric condition is a fact obtained by hypothesis. It is the fact stated in the hypothesis, or the if-clause, of the theorem.
	If the geometric reality of that fact itself is questioned, the answer would have to be obtained by a prior proof that such a figure, conforming to the definition of an isosceles triangle, can be constructed by the use of no other instruments than a straight edge and a compass. The construction is not made, however, as part of the proof of Theorem 6, any more than is the demonstration of an antecedent theorem, which may have to be used in the proof of Theorem 6. In the proof of Theorem 6, the first line, beginning with the word "let," declares that the constructibility of the figure is to be taken for granted as a matter of hypothesis. 
	The whole problem of Theorem 6 is to prove that the then-clause follows from the if-clause. Euclid appears to accomplish this by introducing other propositions-drawn from his axioms, definitions, postulates, or theorems previously demonstrated-which establish this connection and so certify the conclusion as following from the hypothesis. Two points about this procedure should be noted.
	First, the conclusion does not follow from the hypothesis directly, for if that were so, the "if-then" proposition would be self-evident and would need no proof. The mind which sees immediately that the sides opposite to the equal angles in an isosceles triangle are necessarily equal does not need any demonstration of the connection between equal angles and equal sides. The Euclidean demonstration consists in making this connection, which is not immediately evident, mediately evident; that is, evident through the mediation of other propositions. It is not the hypothesis alone which proves the conclusion, but the hypothesis in the company of other propositions which serve to take the mind step by step from the hypothesis granted to the conclusion implied.
	Second, the proposition with the truth of which the reasoning seems to end is not the proposition to be proved. The Q.E.D. at the end of a Euclidean demonstration does not apply to the last proposition in the line of proof, but to the theorem itself, for that is the proposition to be proved. The last proposition in the reasoning is merely the consequent which, according to the theorem, is proposed as following from the hypothesis. When he is able to verify the proposed connection between the hypothesis and its conclusion or consequent, Euclid says Q.E.D. to the theorem as a whole-the whole if-then statement.
	The process of proof seems to be the same when the theorem is stated categorically rather than hypothetically. For example, Theorem 6 might have been stated, as other Euclidean theorems are, in the following manner: "The sides subtended by equal angles in a triangle are also equal to one another." This variation in mode of statement raises a question, not about the meaning of "by hypothesis" in Euclidean proof, but about the difference between hypothetical and categorical propositions, which we will consider later.

The Euclidean use of a given (that is, a constructible) figure as a hypothesis does not seem to be a method of making a supposition in order to discover its implications. Nor does it seem to be a way of testing the truth of a hypothesis by reference to its consequences. Both of these aspects of hypothetical reasoning do appear, however, in Plato's dialogues.
	In the Meno, for example, Socrates proposes, at a certain turn in the conversation about virtue and knowledge, that he and Meno entertain the hypothesis that virtue is knowledge. Socrates immediately inquires about the consequences. "If virtue is knowledge," he asks, "will it be taught?" Since Meno already understands that knowledge is teachable, he answers the question affirmatively. The utility of advancing the hypothesis that virtue is knowledge gradually appears in the next phase of the dialogue, wherein it is discovered that virtue is not teachable at all, or at least not in the way in which the arts and sciences are teachable. The discovery throws some doubt on the truth of the hypothesis that virtue is knowledge; at least it does not seem to be knowledge in the same sense as science or art.
	This mode of reasoning exemplifies the use of a hypothesis to test its truth in terms of its consequences. The underlying logical principle is that the denial of the consequences requires a denial of the antecedent hypothesis, just as an affirmation of the antecedent would require an affirmation of the consequent. Nothing follows logically from a denial of the hypothesis, or from an affirmation of its consequences.
	This example from the Meno also illustrates the difference between Euclid's and Plato's use of hypotheses. Socrates is not here trying to prove that if virtue is knowledge, then virtue is teachable. The validity of the foregoing if-then statement is already understood in terms of the fact that knowledge is teachable. With the if-then statement accepted as valid, Socrates uses it for the purpose of ascertaining whether or in what sense virtue is knowledge. It is not the hypothetical or if-then statement which is proved, but the hypothesis-the antecedent in that statement-which is tested.
	The same general method of employing hypotheses and testing them is found in the empirical sciences. In medical practice the physician, according to Hippocrates, "must be able to form a judgment from having made himself acquainted with all the symptoms, and estimating their powers in comparison with one another"; he should then "cultivate prognosis," since "he will manage the cure best who has foreseen what is to happen from the present state of matters."
	The preliminary diagnosis states a hypothesis (what the disease may be) and the prognosis foresees a set of consequences (what is likely to happen if the diagnosis is correct). Observation of the course of the symptoms and the patient's changing condition will either confirm or invalidate the prognosis. Confirmation leaves the diagnosis a lucky guess, but fails to prove it. If the disease does not run the predicted course, however, the diagnosis on which the prognosis was based can be dismissed as a false hypothesis.

When a hypothesis takes the form of a prediction of what should happen if the hypothesis is true, the failure of the consequences to occur refutes the hypothesis. Though discussions of scientific method frequently speak of "prediction and verification," it would seem as though prediction can only lead to the refutation of a hypothesis rather than to its verification. A hypothesis is overthrown when its prediction fails, but it is not verified when its prediction comes true. To think that it can be verified in this way is to commit the logical fallacy of arguing from the truth of a conclusion to the truth of its premises. How, then, do empirical scientists prove a hypothesis to be true? What do they mean by prediction and verification in relation to the use of hypothesis?
	There seem to be two possible ways in which a hypothesis can be proved by empirical or experimental research. One way can be used when we know that the consequences implied follow only from the truth of the hypothesis. Should the consequences implied be impossible unless the supposed condition exists, then the confirmation of the prediction verifies the hypothesis.
	The other possible method of verification has come to be called "the method of multiple working hypotheses." The validity of this method depends on our knowing that the several hypotheses being entertained exhaust all the relevant possibilities. Each hypothesis generates a prediction; and if upon investigation the observed facts negate every prediction except one, then that one remaining hypothesis is verified. If negative instances have eliminated the false hypotheses, the hypothesis remaining must be true, on the condition, of course, that it is the only possibility which is left. That is why Poincare cautions scientists "not to multiply hypotheses indefinitely."
	Both of these methods seem to be valid only if a prerequisite condition is fulfilled. To verify one of a series of multiple hypotheses through the elimination of the others, the scientist must know that the hypotheses enumerated are truly exhaustive. In the verification of a single hypothesis by the confirmation of its prediction, the scientist must know that the observed consequences can follow from no other supposition. Since such knowledge is often unavailable, probability rather than complete proof results from the testing of hypotheses by observation or experiment.
	In his Treatise on the Vacuum, Pascal offers a summary of the logical situation by distinguishing the true, the false, and the doubtful or probable hypothesis. "Sometimes its negation brings a conclusion of obvious absurdity, and then the hypothesis is true and invariable. Or else one deduces an obvious error from its affirmation, and then the hypothesis is held to be false. And when one has not been able to find any mistake either in its negation or its affirmation, then the hypothesis remains doubtful, so that, in order that the hypothesis may be demonstrable, it is not enough that all the phenomena result from it, but rather it is necessary, if there ensues something contrary to a single one of the expected phenomena, that this suffice to establish its falsity."

In Poincare's view, "every generalization is a hypothesis." Therefore, in science, hypothesis "plays a necessary role, which no one has ever contested." But science requires that hypotheses "should always be as soon as possible submitted to verification." According to Poincare, "some hypotheses are dangerous, - first and foremost those which are tacit and unconscious. And since we make them without knowing them, we cannot get rid of them."
	Both the use of hypotheses and the method of verifying them vary from science to science, according as the character of the science happens to be purely empirical (e.g., the work of Hippocrates, Darwin, Freud), or experimental (e.g., the work of Harvey and Faraday), or a combination of experimentation with mathematical reasoning (e.g., the work of Galileo, Newton, Poincare, Planck, Einstein, Bohr, Dobzhansky). Not all scientific work is directed or controlled by hypotheses, but in the absence of well-formulated hypotheses, the research can hardly be better than exploration.
	A well-constructed experiment, especially what Francis Bacon calls an experimentum crucis, derives its demonstrative character from the hypothetical reasoning which formulates the problem to be solved. The value of such a crucial experiment appears in Bacon's reasoning about the rise and fall of the tides. "If it be found," he writes, "that during the ebb the surface of the waters at sea is more curved and round, from the waters rising in the middle, and sinking at the sides or coast, and if, during a flood, it be more even and level, from the waters returning to their former position, then assuredly, by this decisive instance, the raising of them by a magnetic force can be admitted; if otherwise, it must be entirely rejected."
	In the field of mathematical physics, and particularly in astronomy, the meaning of hypothesis is both enlarged and altered. So far we have considered hypotheses which are single propositions implying certain consequences. But in mathematical physics, a whole theory - a complex system of propositions - comes to be regarded as a single hypothesis.
	In his preface to the work of Copernicus, Andreas Osiander says that the task of the astronomer is "to use painstaking and skilled observation in gathering together the history of the celestial movements; and then - since he cannot by any line of reasoning reach the true causes of these movements - to think up or construct whatever causes or hypotheses he pleases, such that, by the assumption of these causes, those same movements can be calculated from the principles of geometry, for the past and for the future too." The elaborate system constructed by Copernicus and the system constructed by Ptolemy which Copernicus hopes to replace are sometimes called "the Copernican hypothesis" and "the Ptolemaic hypothesis"; and sometimes these two theories are referred to as "the heliocentric hypothesis" and "the geocentric hypothesis."
	A whole theory, regarded as a hypothesis, must be tested in a different way from a single proposition whose implication generates a prediction. As rival hypotheses, one theory may be superior to another in internal consistency or in mathematical simplicity and elegance. Kepler is thus able to argue against Ptolemy by appealing to criteria which Ptolemy accepts, pointing out that Ptolemy himself wishes "to construct hypotheses which are as simple as possible, if that can be done. And so if anyone constructs simpler hypotheses than he - understanding simplicity geometrically - he, on the contrary, will not defend his composite hypotheses."
	But even if the Copernican hypothesis is superior on the grounds of being geometrically simpler, it must meet another test. As indicated in the chapter on ASTRONOMY AND COSMOLOGY, mathematical theories about physical phenomena must be more than ideal constructions of possible universes. They must try to account for this one real world and are therefore subject to the test of their applicability to reality. However elegant it may be mathematically, a hypothesis-when considered from the point of view of physics - is satisfactory only if it accounts for the phenomena it was invented to explain. In the words of Simplicius of Cilicia, it must "save the appearances."
	A hypothesis can therefore be tested for its application to reality by the way in which it fits the observed facts. "In those sciences where mathematical demonstrations are applied to natural phenomena," Galileo writes, "the principles" which are "the foundations of the entire superstructure" must be "established by well-chosen experiments." By such means Galileo chooses between the hypothesis that the uniform acceleration of a freely falling body is proportional to the units of space traversed and the hypothesis that it is proportional to the units of time elapsed.
	According to Dewey, "There is no such thing as experiment in the scientific sense unless action is directed by some leading idea. The fact that the ideas employed are hypotheses, not final truths, is the reason why ideas are more jealously guarded and tested in science than anywhere else . . . as hypotheses, they must be continuously tested and revised, a requirement that demands they be accurately formulated."

To borrow Plato's expression in the Timaeus, the mathematical consistency of a theory makes it "a likely story." The theoretical integrity of the hypothesis makes it credible. But when competing credible hypotheses exist, each saving the relevant appearances equally well, which is to be believed? The fact that one of them, as in the case of the Copernican - Ptolemaic controversy, is mathematically superior cannot decide the question, since the question is, Which is true of reality?
	Sometimes a single fact, such as the phenomenon of the Foucault pendulum, may exercise a decisive influence, if one of the two competing theories finds that fact congenial and the other leaves it inexplicable. Sometimes, as appears in the discussion of the Copernican hypothesis in the chapter on ASTRONOMY AND COSMOLOGY, of two hypotheses which are equally satisfactory so far as purely astronomical phenomena are concerned, one may have the additional virtue of covering other fields of phenomena which that hypothesis was not originally designed to explain.
	As interpreted by Kepler and as developed in Newton's theory of universal gravitation, the Copernican hypothesis brings the terrestrial phenomena of the tides and of falling bodies under the same set of laws which applies to the celestial motions. The hypothesis then has the amazing quality of consilience - a bringing together under one formulation of phenomena not previously thought to be related. This seems to be what Huygens has in mind when he considers the degree of probability that is attainable through experimental research. We have "scarcely less than complete proof," he writes, when "things which have been demonstrated by the principles assumed, correspond perfectly to the phenomena which experiment has brought under observation; and further, principally, when one can imagine and foresee new phenomena which ought to follow from the hypotheses which one employs, and when one finds that therein the fact corresponds to our prevision."
	Then, in common parlance, we say that it is no longer a theory, but has become a fact. Yet the question remains whether the empirical tests which eliminate the less satisfactory hypothesis can ever make the more satisfactory hypothesis more than a likely story.

In the Mathematical Principles of Natural Philosophy, Newton says, "I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy." The context of this passage, and of a similar statement at the end of the Optics, as well as the association in Newton's mind of hypotheses with occult qualities, substantial forms, and hidden causes, seems to indicate a special meaning of "hypothesis."
	Newton criticizes the vortices in the physics of Descartes on the ground that it is unnecessary to appeal to occult or unobservable entities in order to explain natural phenomena. The Cartesian vortices, like the substantial forms of Aristotle, are, for Newton, hypotheses in a very special sense. They are hypothetical entities. They are not inferred from the phenomena. Although treated as if they were realities underlying the phenomena, they are, as Gilbert says of the primum mobile, a "fiction, something not comprehensible by any reasoning and evidenced by no visible star, but purely a product of imagination and mathematical hypothesis."
	There is almost a play on words in this identification of hypotheses with imaginary entities to which reality is attributed; for in their Greek and Latin roots, the words "hypothesis" and "hypostasis," "supposition" and "substance," are closely related. The first word in each of these pairs refers to a proposition which underlies reasoning, the second to a reality which underlies observable qualities or phenomena. To make hypotheses, in the sense in which Newton excludes them from experimental philosophy, is to hypostatize or to reify, that is, to make a thing out of, or to give reality to, a fiction or construction of the mind.
	It has seemed to some critics that, no less than the Cartesian vortices, the ether in Newton's theory of light is a hypothesis in precisely this sense-an imaginary entity. For many cen­turies, the atoms and molecules postulated to explain chemical combinations and changes were attacked as fictions and defended as use­ful hypotheses. On the one hand, there is an issue concerning the theoretical usefulness of such constructions; on the other, a question concerning their counterparts in reality.
	It is sometimes thought that fictions are useful for purposes of explanation even when his their unreality is admitted. Rousseau, for example, explicitly denies any historical reality to the idea of man living in a state of nature prior to the formation of society by the social contract. In this matter, he says, we can lay "facts aside, as they do not affect the question." These related notions-the state of nature and the social contract - are "rather calculated to explain the nature of things, than to ascertain their actual origin; just like the hypotheses which our physicists daily form respecting the formation of the world."
	Similarly Lavoisier posits the existence of "caloric" for its explanatory value. "It is difficult," he writes, "to comprehend these phenomena, without admitting them as the effects of a real and material substance, or very subtle fluid, which, insinuating itself between the particles of bodies, separates them from each other; and, even allowing the existence of this fluid to be hypothetical, we shall see in the sequel, that it explains the phenomena of nature in a very satisfactory manner."
	One other meaning of hypothesis remains to be considered. It is the sense in which postulates or assumptions are distinguished from axioms in the foundations of a science. In Euclid's geometry, as in Descartes's, both sorts of principles appear. The axioms or common notions are those propositions which are immediately seen to be true without proof. The postulates or assumptions are hypotheses in the sense that their truth is taken for granted without proof.
	Both sorts of propositions serve as principles or starting points for the demonstration of theorems, or the conclusions of the science. Both are principles of demonstration in that they are used to demonstrate other propositions without themselves being demonstrated. But axioms are traditionally regarded as intrinsically indemonstrable, whereas hypotheses - postulates or assumptions-may not be indemonstrable. They are simply asserted without demonstration.
	The possibility of demonstrating a hypothesis gives it the character of a provisional assumption. In the Discourse on the Method, Descartes refers to certain matters assumed in his Dioptrics and Meteors, and expresses his concern lest the reader should take "offense because I call them hypotheses and do not appear to care about their proof." He goes on to say: "I have not named them hypotheses with any other object than that it may be known that while I consider myself able to deduce them from the primary truths which I explained above, yet I particularly desired not to do so, in order that certain persons may not for this reason take occasion to build up some extravagant philosophical system on what they take to be my principles."
	The distinction between axioms and postulates or hypotheses raises two issues. The first concerns the genuineness of the distinction itself. Axioms, self-evident propositions, or what William James calls "necessary truths," have been denied entirely or dismissed as tautologies. The only principles of science must then be hypotheses - assumptions voluntarily made or conventionally agreed upon. This issue is more fully discussed in the chapter on PRINCIPLE. The other issue presupposes the reality of the distinction but is concerned with different applications of it in the analysis of science.
	Aristotle, for example, defines scientific knowledge in terms of three elements, one of which consists of the primary premises upon which demonstrations rest. The principles of a particular science may be axioms in the strict sense of being self-evident truths and hence absolutely indemonstrable; or they may be provisional assumptions which, though not proved in this science, can nevertheless be proved by a higher science, as in "the application of geometrical demonstrations to theorems in mechanics or optics, or of arithmetical demonstrations to those of harmonics." The latter are not axioms because they are demonstrable; yet in a particular science they may play the role of axioms insofar as they are used, without being demonstrated, to demonstrate other propositions.
	Reasoning which rests either on axioms or on demonstrable principles Aristotle calls scientific, but reasoning which rests only on hypotheses he regards as dialectical. Reasoning results in scientific demonstration, according to Aristotle, "when the premises from which the reasoning starts are true and primary, or are such that our knowledge of them has originally come through premises which are primary and true." In contrast, reasoning is dialectical "if it reasons from opinions that are generally accepted," and, Aristotle explains, "those opinions are 'generally accepted' which are accepted by everyone or by the majority or by the philosophers - i.e., by all, or by a majority, or by the most notable and illustrious of them." In another place, he adds one important qualification. In defining a dialectical proposition as one that is "held by all men or by most men or by the philosophers," he adds:"provided it be not contrary to the general opinion; for a man would assent to the view of the philosophers, only if it were not contrary to the opinions of most men."
	For Aristotle, dialectical reasoning or argument moves entirely within the sphere of opinion. Even an opinion generally accepted, not only by the philosophers but also by most men, remains an opinion. The best opinions are probabilities - propositions which are not self-evident and which cannot be proved. They are not merely provisional assumptions. Resting on assumptions which cannot ever be more than probable, the conclusions of dialectical reasoning are also never more than probable. Since they lack the certain foundation which axioms give, they cannot have the certitude of science. 
	Plato, on the other hand, seems to think that the mathematical sciences are hypothetical in their foundation, and that only in the science of dialectic, which he considers the highest science, does the mind rise from mere hypotheses to the ultimate principles of knowledge. "The students of geometry, arithmetic, and the kindred sciences," Socrates says in The Republic, "assume the odd and the even, and the figures and the three kinds of angle and the like in their several branches of science; these are their hypotheses, which they and everybody are supposed to know, and therefore they do not deign to give an account of them either to themselves or others." There is a higher sort of knowledge, he goes on, "which reason herself attains by the power of dialectic, using the hypotheses not as first principles, but only as hypotheses - that is to say, as steps and points of departure into a world which is above hypotheses, in order that she may soar beyond them to first principles."
	The issue between Plato and Aristotle may be only verbal - a difference in the use of such words as "science" and "dialectic." Whether it is verbal or real is considered in the chapters on DIALECTIC and METAPHYSICS. In any case, the issue throws light on the difference between a hypothesis as a merely provisional assumption, susceptible to proof by higher principles, and a hypothesis as a probability taken for granted for the purposes of argument, which is itself incapable of being proved.

FINALLY WE COME to the meaning of "hypothetical" in the analysis of propositions and syllogisms. The distinction between the categorical and the hypothetical proposition or syllogism, briefly touched on in Aristotle's Organon, is developed in the tradition of logic which begins with that book.
	In his work On Interpretation he distinguishes between simple and compound propositions. The compound proposition consists of several simple propositions in some logical relation to one another. In the tradition of logical analysis, three basic types of relation have been defined as constituting three different kinds of compound proposition. One type of relation is the conjunctive; it is signified by the word "and." Another is the disjunctive; it is signified by the words "either... or..." The third type is the hypothetical and is signified by the words "if... then..."
	To take an example we have already used, "virtue is knowledge" and "virtue is teachable" are simple propositions. In contrast, the statement, "if virtue is knowledge, then virtue is teachable," is a compound proposition, hypothetical in form. If the proposition were stated in the sentence, "either virtue is knowledge or it is not teachable," it would be disjunctive in form; if stated in the sentence "virtue is knowledge and virtue is teachable," it would be conjunctive in form. In each of these three cases, the compound proposition consists of the two simple propositions with which we began, though in each case they appear to be differently related.
	Whereas Aristotle divides propositions into simple and compound, Kant divides all judgments into the categorical, the hypothetical, and the disjunctive. In the categorical judgment, he says, "we consider two concepts"; in the hypothetical, "two judgments"; in the disjunctive, "several judgments in their relation to one another." As an example of the hypothetical proposition, he offers the statement, "If perfect justice exists, the obstinately wicked are punished." As an example of the disjunctive judgment, "we may say... [that] the world exists either by blind chance, or by internal necessity, or by an external cause." Each of these three alternatives, Kant points out, "occupies a part of the sphere of all possible knowledge with regard to the existence of the world, while all together occupy the whole sphere." The hypothetical judgment does no more than state "the relation of two propositions... Whether both these propositions are true remains unsettled. It is only the consequence," Kant says, "which is laid down by this judgment."
	In the Prior Analytics, Aristotle distinguishes between the categorical and the hypothetical syllogism. The following reasoning is categorical in form: "Knowledge is teachable, virtue is knowledge; therefore, virtue is teachable." The following reasoning is hypothetical in form: "If virtue is knowledge, it is teachable; but virtue is knowledge; therefore it is teachable"; or "If virtue is knowledge, it is teachable; but virtue is not teachable; therefore is not knowledge."
	The basic issue with respect to the distinction between categorical and hypothetical syllogisms is whether the latter are always reducible to the former. One thing seems to be clear. The rules for the hypothetical syllogism formally parallel the rules for the categorical syllogism. In hypothetical reasoning, the consequent must be affirmed if the antecedent is affirmed; the antecedent must be denied if the consequent is denied. In categorical reasoning, the affirmation of the premises requires an affirmation of the conclusion, and a denial of the conclusion requires a denial of the premises.
	With respect to the distinction between the categorical and hypothetical proposition, there is also an issue whether propositions stated in one form can always be converted into propositions having the other form of statement. In modem mathematical logic, for example, general propositions, such as "All men are mortal," are sometimes expressed in hypothetical form: "If anything is a man, it is mortal." Logicians like Russell think that the hypothetical form is more exact because it explicitly refrains from suggesting that men exist; it merely states that if the class 'man' should have any existent members, they will also belong to the class 'mortal.'
	Apart from the question whether a universal proposition should or should not be interpreted as asserting the existence of anything, there seems to be a formal difference between the categorical and hypothetical proposition. This is manifest only when the hypothetical is truly a compound proposition, not when it is the statement of a simple proposition in hypothetical form, as, for example, the simple proposition "All men are mortal," is stated in hypothetical form by "If anything is a man, it is mortal." Because it is truly a compound proposition, and not merely the hypothetical statement of a general proposition, the proposition, "If virtue is knowledge, then virtue is teachable," cannot be restated in the form of a simple categorical proposition.
	A simple proposition, whether stated categorically or hypothetically, may be the conclusion of either a categorical or a hypothetical syllogism. But the hypothetical statement which is really a compound proposition can never be the conclusion of any sort of syllogism, though it may be one of the premises in hypothetical reasoning.