## 14 Hypotheticals (1952)

The view that hypotheticals (or, more broadly, conditionals) are a peculiar *species* of propositions illustrates the inferior way of treating logic — that which, instead of bringing everything under logic, subordinates logic to something else; specifically, to forms of speech, i.e., ultimately, to types of human procedure and relationship. Thus, in classifying propositions, one can go to forms of speech, to the ways in which things are said, or one can go to the sorts of things that can be *meant*; in other words, to the sorts of things that can be.

The former method is, at best, eclectic; no classification could possibly cover all the forms of speech, all the varieties of communication. But all possible forms of speech must fall under forms of being; anything that can be said, or, perhaps better, that can be conveyed, must have some “logical form” — and the first task of logic is to find the types of logical form, even if “putting into logical form” is rendered difficult in particular cases by the confusion of the speaker's thought or by the multifarious purposes which people try to serve by what they say. In other words, the logician's task is to cut through forms of speech to real content; and the main aid to his doing so is absorption in the philosophical tradition. But such absorption depends, again, on his rejection of eclecticism, rejection, in particular, of an external view of philosophers as exponents of this or that (“ideas”, “transcendental unity”, etc.); it depends on his own sense of what is vital in their work, of what is the connecting philosophic *theme*.

In stating this theme as that of objectivism versus subjectivism, of the issue versus the purpose, of truth versus satisfaction, I should have to say that this alone “makes sense” of the course of philosophical inquiry, but also that that “sense” would emerge from a series of studies (*digging out* the contributions to an objective view of things, to a positive conception of truth, made by various philosophers — or, in the first instance, merely becoming capable of *seeing* such issues) and would not be a simple finding or a simple inference from “the philosophical data”.

Nevertheless, I should say that not a great deal of reflection is required to see the force of the contention that “finding the logical form” of any utterance is finding what it purports to convey as truth, as objective, as “the case”, and of the further contention that it is by considering what can be true, or what can be (what is the form of) an issue, a question of truth or falsity, that we determine what *are* the logical forms. Now the issue in its broadest form (the issue as such) is expressed as “Is it so or

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not?” — and it is from this that we find what are the forms of propositions and that they are all “categorical”; that they are, in fact, the A, E, I and O forms of the text-books. It is conceivable, of course, that there could be differences of *notation* not implying any real departure from this classification. But it should be understood that even if the argument to which I am proceeding, an attempted demonstration of “the four forms” as the only logical forms, were open to decisive objections, some argument of this *kind* is essential to logic — the alternative, as I said, being eclecticism, the employment of an unformulated and uncriticised method of selection, the adoption of an unstated “logic” amounting in fact to the subjection of logic to non-logical considerations.

The form of the issue, “Is it *so* or not?” indicates that there is no question of kinds of truth (truth in practice, necessary truth, conditional truth, etc.) or, as I have alternatively put it, that there is only one copula, “the unambiguous ‘is’ of occurrence” or of *being the case*. But to say that this alone is the form of the proposition, that everything else belongs to its material or is part of *what* is the case, would be misleading if it were taken to signify that there is only one propositional form. The fact that a proposition is, or “raises”, an issue already implies the distinction of quality (affirmative and negative); what has further to be brought out is that it implies also the distinction of quantity (universal and particular). The demonstration of this depends on the order of terms in the proposition, on the distinction between subject and predicate, on the expansion of “Is it so?” into “Is *what how*?” or “Is what thing of what character?”. The same point can be expressed by saying that “X is Y” is a different proposition (raises a different issue) from “Y is X”, even if, whenever there is one such issue, there is also the other; that subject and predicate have different “functions”, roughly expressible as “locating” and “describing”. It should still be noted, however, that any term can have either function (whatever can locate can describe or be located, whatever can describe can locate or be described) so that the expressions “things” and “characters” are merely indicative of the functions, respectively, of subject and predicate, and are not indicative of different classes of entities.^{note}

Over and above outright denial of this distinction of functions, we do of course find writers on logic slipping away from it, slipping into *symmetrical* expressions of the import of the proposition (e.g., “connection of contents”). But the necessity of the distinction can be seen from considerations of *implication* — from the difference, for instance, between the two arguments “X is Y, Z is X, therefore Z is Y” and “Y is X, Z is X, therefore Z is Y”. The recognition of the validity of the former and the invalidity of the latter argument depends on the recognition of the distinction of quantity, the distinction, here, between “All X are Y” and “Some X are Y”. (Alternatively, the distinction can be approached from considerations of *contradiction*, from the fact that “X is Y” and “X is not Y” present not a single issue but two issues — or permit a confusion of issues which can be cleared up only by the distinction of quantity.) Broadly,

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the argument is that the distinction of quality requires the distinction of subject and predicate, and this requires the distinction of quantity; and thus we have the four forms, A, E, I and O (XaY, XeY, XiY, XoY), each of which raises a single issue and, of course, presents a settlement of it. These “categorical” forms, then, are *the only* logical forms, the only forms in which we can assert that *something is so*; and thus the logical form of “hypotheticals” (and similarly with “modals”, etc.) must be found among the four forms.

It may be urged in criticism of the above argument that the distinction in validity of the two inferential forms cited depended on the assumption of a certain interpretation of the proposition or on the preference of a type of proposition to another equally admissible; that, if the propositions in question were identities, both arguments would be valid. My contention, against any doctrine of a distinction among copulas, of different “is”-es, including, e.g., “the ‘is’ of identity”, is that it is impossible to *make* such a distinction except by means of an unambiguous copula or absolute “is”, that any real distinction that was in question would be a distinction in the materials, not the forms, of propositions. Thus if we profess to be able to distinguish the identity of X and Y from any other relationship that might hold between these terms, we should have to express it by saying not “X is Y” but “X is identical with Y”; and here we have the ordinary “predicative” form (distinct terms with distinct functions, the predicate “applying to” the subject), we have the predicate “identical with Y” attached to the subject X by the ordinary copula (“the ‘is’ of predication”). But we do not in fact require such terms as “identical with Y”; what is ordinarily meant by identity, what alone at any rate would have any relevance to the treatment of the two arguments as valid, is coextension, and this is formally a pair of propositions, XaY and YaX, each of which is in the predicative form — and, even if we found it convenient to group them together, we should still, in considering their consequences, have to distinguish consequences of XaY from consequences of YaX and recognise where validity would be lost by substituting one for the other.

Similar objections apply to another prevalent doctrine of symmetricality — to the recognition of an “‘is’ of existence” and the doctrine that any proposition (or any “categorical proposition”) asserts either existence or non-existence. If “X is”, meaning “X exists”, is a real issue, then the question is whether X has or has not the character “existing” — which would be a “categorical” question. But to say that there is such an issue is to profess to distinguish a class of existing things from a class of non-existing things, and on the face of it there is not the latter class; it is to profess to find the predicate “non-existing” in a subject which, if *it* were to be found, would not be non-existing. Thus, like the identity theory, the “existence, non-existence” theory of import, whether of all or only of some propositions, is not a consistent departure from the predicative theory, from the logic of the four forms; it cannot consistently treat “existing” as the only predicate or as a predicate at all, and yet it cannot escape from this

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interpretation. And when, for example, it treats XiY as “XY exist”, it cannot, except by returning to a predicative logic, explain what “XY” means; viz., things which *are* X and also *are* Y, or simply *being* X and also *being* Y.

Returning, then, to the necessity of treating “hypotheticals” under the four forms and not as requiring any extension of our classification, we find the question complicated by different *uses* of the “if--then” formula; to show how this supposedly special form fits into a categorical logic, we have to distinguish (A) the hypothetical as a relation between terms, (B) the hypothetical as a relation between propositions, (C) the hypothetical as a relation between “functions”.

A. In the simplest case of the use of the hypothetical form, it represents, or would have as its “logical form” the A or universal affirmative proposition. The matter can be approached from consideration of the “location” formula suggested above — the interpretation of “X is Y” as “X is a *place* where Y is”. Clearly, this could not be taken as an alternative to ordinary predication; it is simply a way of bringing out certain of its features. Now, according to my previous argument, any term can function as a predicate, so that X can describe or be placed as well as placing; in other words, X is not a “pure place” but is, like Y, a descriptive term, and thus we could express “X is Y” by “What is of the description X is (also) of the description Y” (a formula which, as before, raises the question of quantity). But equally we could, from the point of view of location, express “X is Y” by “Where X is, Y is” or, more briefly, “Where X, there Y”; and this in turn could be given the hypothetical form “If X, then Y”. Thus we might have “If man, then mortal” as an alternative to “Where man, there mortal”.

Clearly, however, these are short-hand expressions; in the hypothetical form, in particular, “if man” needs completion, and do does “then mortal”. Hence we encounter such expanded formulae as “If anything is a man, it is mortal” and “If x is a man, x is mortal”. But these are still not formulae in which we can rest; the vagueness of “anything” and the suggestion in “x” of a gap that *could* be filled in one way or another drive us back to the categorical form “All men are mortal” as what is positively said. It might be argued that the hypothetical formula conveys not merely the A proposition but also the possibility of its being the major premise in a number of Barbara syllogisms; but this would equally be conveyed by formulating the A proposition as “Anything that is man is mortal”, i.e., by reminding ourselves that the subject of the proposition is also a predicate, and this “capacity” of the A proposition (for being a major) need not be taken as anything additional to the proposition itself.

The logical form, then, of “If X, then Y” or “If anything is X, it is Y” is simply XaY. And the suggestion, as against this, that the hypothetical form covers the possibility that in fact nothing is X is met by the contention, supported above, that there are no “non-existent” terms. A formula containing a “non-existent” term is meaningless; that is to say, there is a set of words or, more exactly, of marks or noises, but there is no proposition, no issue. This, however, may be somewhat hard to see in the case of complex terms; to say “If anything is XZ, it is Y” (or even,

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categorically, XZaY) would quite commonly be taken to have a meaning when X, Y and Z all exist even if XZ does not. And while, strictly, there is no sense in saying that XZ locates Y when XZ itself has no location — any more than in saying that “X” is Y when there is no “X” — the *appearance* of sense may arise from confusion with something that does have sense, viz., “If any X is Z, it is Y”, which, besides conveying ZaY in the same way as if “anything” were substituted for “any X”, also suggests *arguments* containing the three “existing” terms, X, Y and Z (e.g., XaZ, ZaY, therefore XaY), and thus made up of intelligible propositions, whether they are true or not.

B. This brings us to the second use of the “if--then” formula, viz., for relations of implication among propositions. When the propositions p and q are presented in the relationship “if p, then q” (hereafter, for brevity, “if p, q”), there are the two cases (a) where what is meant is that p itself implies q, (b) where what is meant is that p and some other proposition r, which is “understood” or taken for granted, together imply q. The latter usage is quite common; people often say such things as “If Socrates is a man, he is mortal”, where the passage from “Socrates is a man” to “Socrates is mortal” depends on the assumption that all men are mortal. The relationship in this case, then, is that of the premises of a syllogism to its conclusion; the suggested syllogism is the full “logical form” of “If Socrates is a man, he is mortal”. But the other usage is also encountered; it is exemplified in Swinburne's “If thunder can be without lightning, lightning can be without thunder”, where the form is that of immediate inference even though in fact the inference is fallacious, being the conversion of an O proposition, the inference of YoX from XoY.

In either case, however, we are concerned with inference or implication, and it is here that we find the relevance of the terminology of “antecedent and consequent”; if q in “if p, q” is taken to be consequent upon or to follow from p, p and q must be regarded as propositions, whether or not “if p, q” is also so regarded. This use of the hypothetical form may be connected again with the “method of hypothesis” and its consideration of “consequences”, though there is much confusion in accounts of this method, as may be seen from Burnet's discussion (Greek Philosophy, Part I, ch. IX, pp. 162,3). For if *ta s?µßa????ta* of p are the things which are *contingent on* p, then they are not its consequences but have it as a consequence; the humanity of Socrates is contingent on his mortality, not the other way round. If, on the other hand, it *is* a question of what follows from p, it must be observed that to arrive at a true consequence of p is not to prove p true, any more than to arrive at a construction which we can carry out, and which must be carried out if construction C is to be carried out, establishes our ability to carry out C; hence we are not entitled to say, at the conclusion of our drawing of consequences, that “what was to be demonstrated” has been demonstrated or that “what was to be done” has been done. Confusion in such expositions has, of course, been fostered by the fact that the principal field of their application was geometry, in which the relations studied are frequently

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symmetrical, so that moving in the wrong direction could be overlooked.

But, admitting the existence of such confusions, the fact remains that we can consider consequences, can consider what follows from p or “what would be true if p were true”. Commonly, however, we bring under this head not just immediate inferences from p but propositions which follow from p together with other premises, premises which are taken for granted, are regarded as true by the person or persons engaged in finding p's consequences. This is important in regard to the “falsification of hypotheses” by finding false consequences. We reject a hypothesis when we find to be false something that “would be true if the hypothesis were true”. But if we happen to be mistaken in believing the premise we have combined with p, then we are mistaken in believing the conclusion to be (or, at least, we have not found it to be) a “consequence” of p, something that would be true if p were true, and so we have not falsified p by its consequences.

A more important point here, however, is that of “the consequences of false propositions”; whether we can strictly say that false propositions have any consequences, any more than “non-existing” terms have predicates. We do not, in fact, have to say so. Taking the simplest case of falsification, that of the hypothesis XaY which we have combined with the proposition ZaX, accepted as true, in the syllogism “XaY, ZaX, therefore ZaY”, ZaY then being found to be false, the position is that we have found or believe to be true the two propositions ZoY and ZaX, and that these as the premises of a syllogism imply XoY, which is the contradictory of XaY. The falsification of a proposition, then, is the implication of its contradictory by true propositions; and the syllogism with true premises can be taken as the strict logical form of the argument, a “real implication” to which we have been led by consideration of actual types of connection among propositions, even if this consideration included a “hypothetical” procedure, the setting out of things we were uncertain of as well as things we were certain of.

Taking it, then, that the question of hypotheticals in this second usage is that of relations of implication among propositions, we have particularly to note that it is not a question of our proceedings or attitudes. These are, of course, subject to logic (there can be logical consideration of them), but logic is not subject to them; they do not affect the characters and relations of issues. Thus we can be more or less sure about something, we can “suppose” as well as assert; but this does not entitle us to distinguish *propositions* as certain and uncertain or to recognise “supposals” as forms additional to the forms of assertion. It is from this point of view that we can attack the conventional doctrine of the “mixed hypothetical syllogism”, whose form is “if p, q; p, therefore q”; viz. as an argument with two premises and a conclusion. Since p is a premise and q the conclusion, p and q are propositions; hence “if p, q” is a relation between propositions, and, as it can only be the relation of implication, we have, in what is described as a “syllogism”, merely stated the same implication twice. It would be possible to distinguish between “if p, q” and “p, therefore q”

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only in terms of our attitudes, i.e. according as we are sure of p and infer q or merely consider, from the formal relationship, that being sure of p would lead us to infer q; but logic is not affected by our taking up one attitude or the other — the logical question is that of implication.

On this showing, the so-called major premise has nothing to do with the inferring of q; “p, therefore q” is the whole argument — or, if an additional premise r is “understood”, we still have a “categorical” and not a “hypothetical” argument. If, as against this, it is argued that “if p, q” is the assertion of the formal relationship and that the further *actual* assertion of p is required for the inferring of q, the answer is (a) that, in that case, the “major premise” will be concerned not with p and q but with a type of relationship, so that the connection between the “premises” will not be as represented — an illustration being “Any A proposition implies the I proposition with terms in the opposite order; XaY, therefore YiX”, (b) that this “major premise” is the principle or, more exactly, the form of the inference of YiX from XaY alone. In short, if p and q come into the “major”, we have the same implication twice, and, if they do not, we have it once — “p, therefore q”.

C. These considerations lead on to the third usage mentioned above. Although reason has been found for rejecting the usual presentation of the “mixed” form, this is not all that is to be said about that form or about the hypothetical form in general. It is commonly understood that, while the “major premise” in the mixed form lays down a general connection or connection of kinds, the “minor” refers to a particular case; and that means that the given form is misleading, that the two “p's” (and similarly the two “q's”) are not the same. The determination of the difference may be approached by asking *of what* the special case is a case. It cannot be a case of a proposition (a proposition does not have different cases or instances), and, particularly, we cannot talk about cases of the *truth* of a proposition and cases of its falsity (since a proposition is just true or just false). Nevertheless, this is a common way of rendering the matter; the formulation would be “All cases of the truth of p are cases of the truth of q; this (special case) is a case of the truth of p, therefore, this is a case of the truth of q”, and the argument, if we accepted such terms, would be an ordinary categorical syllogism.

We have to ask, then, what is there which might be confusedly taken as a proposition and yet which could have cases. And the answer is, a “propositional function” — a formula, containing a variable or variables, such that, when a “value” is given to each variable, we have a proposition. In other words, a propositional function “represents” (is the form of, is what is common to) a *class* of propositions. Thus “x is a man”, where x is variable, is a function whose values, corresponding to the values of x, are propositions (“Socrates is a man”, etc.), some of which are true and some of which are false. We have then the distinction between a type of case and cases of that type, and we can see how p could be understood in one premise as “p in general” and in the other as “a case of p”, how

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readily, even if confusedly, “x is a man” and “Socrates is a man” could be represented by the same symbol.

It is still the case that the strict form of “If x is a man, x is mortal; Socrates is a man, therefore Socrates is mortal” is just the categorical syllogism “All men are mortal, Socrates is a man”, etc. And recognition of the major in its hypothetical form, recognition of such relationships as that between “being a man” and “being mortal”, would not give us the third usage; we might speak of the relationship as one between “possibilities”, realised in some cases and not in others, but we should still not have got beyond the first usage, that of the “relationship between terms”, or proposition. In fact the third usage, even in the more complicated form I am about to consider, can always be brought back to the first (a connection of “possibilities” *is* a connection of terms); but the complications are of some interest and are, I should say, one main occasion of adherence to the belief in a peculiar “hypothetical form”.

The question, then, is not just of possibilities, and their realisations and non-realisations, in general — of the occurrence and the non-occurrence of a certain character in this place and that — but of possibilities within a *field*, of a region or genus of things within which various possibilities are variously realised, of a *limited* range of “values”, of the filling of the gap (turning the function into a proposition) by things of a particular kind. Such fields are often left unstated (“understood”), and, while this permits a special hypothetical form to be used, (a) full statement, making the implicit explicit, would still be categorical, (b) the absence of full statement opens the way to ambiguity and thus to the drawing of unwarranted conclusions. If, for example, in a “pure hypothetical syllogism”, the field understood in one premise differed from that understood in the other, then we should have an apparently valid argument, “if p, q; if q, r; therefore, if p, r”, whereas full statement of the premises would show that they had no common term and that the conclusion (whatever *its* field) did not follow; taking F and G as the two fields, we should have the unconnected premises “All F such that p are F such that q” and “All G such that q are G such that r”; where “F such that p” are members (of the field or genus F) in which the possibility p is fulfilled, etc.

The reference to a field renders intelligible “cases” of truth and falsity; the confused notion of cases in which a *proposition* is true and cases in which it is false is replaced by that of instances in a field, or members of a genus, which fulfil a possibility and instances or members which do not, so that it is *different* propositions which are true, and which are false, in the various cases. An example may make the position clearer. Such an assertion as “If the barometer is falling, bad weather is coming” is typical of the “major premises” of hypothetical arguments — and the argument might be completed by “the barometer is falling, therefore bad weather is coming”. The “minor” here is understood to refer to some particular case; but it is not indicated *what* the case is (there is merely the suggestion of “now” or “in this case”) or what it is a case of. Two conditions of exact

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statement of the argument, then, would be (a) the specification of the “field” or class of cases, (b) the specification of the particular case.

These specifications are frequently difficult. But we may suggest as the field in the above example “atmospheric conditions” or perhaps more exactly “atmospheric pressures” or perhaps more exactly again “pressure-regions”. Thus the question would be of *pressures such that*, or *“pressure-regions” in which*, the barometer is falling, etc., and the minor premise would refer to “this” pressure-region (where the speakers are) or to the pressure “here” — the argument would then be “All P such that (or in which) condition A is fulfilled are P such that condition B is fulfilled; this P is a P such that condition A is fulfilled; therefore, this P is a P such that condition B is fulfilled”. But, while this is an ordinary syllogism, the less exact hypothetical form is one which readily arises when we are confronted with diverse possibilities (p and non-p, q and non-q) and with connections among such possibilities (say, between p and q). The field may be left unstated for the sake of brevity, though, as previously noted, the omission makes for ambiguity and bad argument; but it frequently happens that we notice a sequence or concomitance of conditions without being able to specify the field (or without having more than a rough notion of it), so that in seeking and finding such specification we are actually *advancing* in knowledge. Thus where reference to pressures is lacking (in the mind, say, of the uninstructed child) the connections between the barometer and the weather would be magical rather than scientific connections; the specification of the field brings out the point that there is a real continuity, a *series* of connections, with which we can become more and more fully acquainted. (It may be remarked in passing that one reason why the doctrine of “material implication” is a philosophical blind-alley is just its ignoring of such connection; implication then becomes quite arbitrary or “magical” — divorced from inquiry.)

An important point about the “field”, and one of special significance for hypothetical argument, is that it complicates the question of negative cases. If we were concerned with possibilities in general, we could represent “if p, q” by PaQ (where P is “cases in which p is fulfilled”, etc.) and could then recognise its equivalence to “if not q, not p”, represented by (non-Q)a(non-P) (all non-Q are non-P), since PaQ and (non-Q)a(non-P) are equivalents (in fact, contrapositives) of one another. But when we are concerned with possibilities in a field, the categorical forms are PFaQF and (non-Q)Fa(non-P)F, which are not equivalent and are not contrapositives but “virtual contrapositives” of one another (a terminology used in my article, “The Problem of Causality”; A.J.P.P., August, 1938; pp. 133-5* ^{note}*); the point being that PFaQF does not assure us that any F are not Q (that (non-Q)F “exists”). Hence the recognised equivalence, the regular substitution of “if not q, not p” for “if p, q”, the recognition of “denying the consequent” as having the same validity as “affirming the antecedent”, depend on the assumption that the opposite possibilities (non-fulfilment as well as fulfilment) all exist within the field.

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Of course if there were no field, we should be back either at the first, simple usage, at “if--then” as a rough way of expressing the A proposition, or, if p and q were propositions, at the second usage, where, if an argument of the form “p, therefore q” is valid, an argument of the form “not q, therefore not p” is also valid, and if “p, r, therefore q” is valid, “not q, r, therefore not p” is valid. Even in the third usage, where there *is* a field, the “hypothetical form” is only a convenient short-hand; but in using it we have to consider the *conditions* of its use, and one of the most important is that of the existence of the opposite possibilities in the field. Where that assumption is justified, “virtual contraposition” is justified, and so the regular transformation of “if p, q” into “if not q, not p”, and vice versa, is justified.

Another limitation on the use of “conditionals” is to be found in the question of contradiction and particularly in the fact that the contradictory of a hypothetical is not a hypothetical. We can, if we so desire, find a form for such a contradictory, by means of the equivalence of hypotheticals and disjunctives, and the relation between disjunctives and conjunctives. The matter can best be approached from the side of disjunctive and conjunctive *terms*; and here I would contend that there is nothing to be said for the view that the alternatives of a disjunction are exclusive. “Either” means “not neither”; the opposite of being either P or Q is being neither P nor Q, and thus the opposite of the disjunctive term “P or Q” is the conjunctive term (non-P)(non-Q) (not P *and* not Q). “Not both” is quite a different thing, even if it should hold at the same time; it could be true of any X that it is not neither P nor Q *and* that it is not both P and Q, but these would still be different truths, and there would in general be other subjects of which only one of these things could truly be said (i.e., unless P and (non-Q) were coextensive). So, in the case of disjunctive *propositions* (the disjunction of functions or possibilities), “p or q” would be a different proposition from “not p or not q” and would not imply it, though both might hold; it would be equivalent to the hypothetical “if not p, q”, and it would not be equivalent to or even imply the hypothetical “if p, not q”. It could, on the analogy of the conjunctive opposite of a disjunctive *term*, be contradicted by (non-P)(non-Q) (not p *and* not q), a conjunction of possibilities, the non-fulfilment of p along with the non-fulfilment of q; and the equivalent hypothetical would be contradicted by the same conjunction. Or, starting from the hypothetical “if p, q”, we should contradict it by p(non-Q) (p and not q).

The adoption of such a notion, however, would be cumbrous and also confusing. Full statement of the contradictory of a hypothetical, i.e. statement with specification of the field, would be in the form of a particular proposition (I or O); but, in the short-hand formulation, we could not distinguish it from the “particular case”, also unspecified, of the minor premise. Thus, contradicting “If the barometer is falling, bad weather is coming” by “The barometer is falling and bad weather is not coming”, we should have a statement which would be understood as one to be

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supplemented by “now” or “under *these* circumstances” — and thus as *not* the simple contradictory of the given statement, as not a particular but a universal proposition. To get an unambiguous notation we might, if we wanted to avoid going back to categoricals, introduce *modal* expressions; we might give “if (or when) p, it is possible that not q” as the contradictory of “if (or when) p, it is necessary that q”. I do not think, indeed, that consistency could be reached even in this manner; the opening part of my argument shows for what general reasons I should regard “modals” as reducible to “categoricals”, as having A, E, I and O as their strict forms; a detailed reduction would involve complications similar to those we have encountered with “hypotheticals”, but no real difficulties.* ^{note}* But what I am concerned with here are hypotheticals or, more broadly, conditionals, and I think I have said enough to show that a logic of conditionals can have no consistent notation, that it cannot cope with many of the problems, carry out many of the operations, which present no difficulties to a categorical logic, and that, as regards the operations to which it is in some measure adequate, their strict form is still the categorical.

As I contended earlier, there must, if there is to be logic, be a limited number of “logical forms”, and there will be the problem of finding *some* argument which shows what they are. There will, if this problem is solved, be the further problem of carrying out the “reduction” of forms which superficially appear to escape the basic classification. I do not think anyone will deny certain *analogies* between hypothetical and categorical forms; the question, then, is whether there is any real distinction. My contention has been that there is not, that hypotheticals are rough devices for dealing with matters that can be covered in an exact way by a categorical logic, a logic which recognises *only* A, E, I and O propositions, their features and relations.