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Logic as the foundation of method.




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Logic as the foundation of method.

Scientific method consists of procedures for investigating any subject-matter or field of study, for finding out what are the facts in the field (e.g., the field of mental processes, which is investigated by psychology). In order to do this it is necessary to know what are the forms that facts or truths can take, and the study of such “truth-forms” or propositional forms is logic.

To say that logic studies propositions is not to say that it studies words. Words are necessary for communication and thus for any extensive investigation, but what is communicated is that something is the case or is a fact and that it would be a fact whether anyone asserted it or not. At the same time, a person may be mistaken when he takes something to be the case. Thus propositions, which are conveyed by means of words, may be true or false. The use of words may be described as the most general type of method or practical means of investigation; but words can be misleading, a form of words may not clearly distinguish the true from the false. Hence “finding the logical form”, seeing exactly what proposition has been stated, is an essential further step in method, and it depends on a knowledge of what are the forms that propositions can take.

The four forms.

Propositions may be affirmative or negative (distinction of quality), and they may be universal or particular (distinction of quantity). There are thus the universal affirmative or A proposition (e.g., all mathematicians are men), the universal negative or E proposition (e.g., no dogs are horses), the particular affirmative or I proposition (e.g., Some politicians are authors), the particular negative or O proposition (e.g., Some politicians are not authors).




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In each proposition there are four constituents: the two terms (subject and predicate), the copula (are or are not), the sign of quantity (all or some). The E proposition seems to be an exception since the one expression, “no”, conveys both negative quality and universal quantity. But this form is used only to avoid ambiguity, since in ordinary English usage “all X are not Y” means that some X are not Y (if “all that glitters is not gold”). If we were to analyse “no dogs are horses”, we should still take it as saying of all dogs that they are not horses.

The copula is always part of the verb “to be”, because the verb in the ordinary sentence conveys both that something is being said about the subject and part of what is being said about it. These are two different points, and to make the distinction clear we should render, e.g., “all students read books” by “all students are readers of books.” The terms (and, with them, the copula) are given plural form, because any term in principle has various instances; if we take the term “readers of Finnish books in Sydney”, there is nothing in that description to tell us that it does not apply to several persons, though it might in fact apply only to one.

Opposition. Since there are four forms of propositions, there are four propositions with a given subject, X, and a given predicate, Y; the A, E, I and O propositions, referred to for brevity as XaY, XeY, XiY, XoY. These four propositions are such that there must be two of them true and two of them false. To raise the question whether XeY is true or false is to raise the question whether XiY is false or true (as it may be put, “some” means “not none”). Similarly, the truth or falsity of XaY goes with the falsity or truth of XoY.


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The question whether XaY is true or false constitutes an issue, which may equally be expressed as whether XaY or XoY. And whether XeY or XiY is another issue. The distinction between these issues brings out the importance of quantity; if we simply asked whether X is Y or not, without showing whether our propositions were universal or particular, we might not have an issue at all or we might have more than one issue—to the question “Which of the two propositions, XiY and XoY, is true?” the answer might be “Both are true”; and to a similar question about XaY and XeY the answer might be “neither.” When we have an issue, when two propositions are such that one and only one of them is true, the propositions are said to contradict one another. It is an important point in method, then, to clarify issues, to see exactly what is required for the denial or contradiction of a given proposition, and not to give either too much or too little for that purpose.

The distinct issues above referred to, however, are not entirely independent; if we settle the issue between XaY and XoY in favour of XoY, then that leaves it an open question whether XeY or XiY, but, if we settle it is favour of XaY, then by that very fact we determine that XiY is true and XeY is false. We are now able to distinguish the various types of opposition or relation, expressed in terms of truth and falsity, between two propositions with the same subject and the same predicate.

(a) Contradictory opposition: XaY and XoY cannot both be true and cannot both be false—they are “contradictories”; and so are XeY and XiY.

(b) Contrary opposition: XaY and XeY cannot both be true but may both be false—they are “contraries.”


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(c) Subcontrary opposition: XiY and XoY cannot both be false but may both be true—they are “subcontraries.”

(d) Subaltern opposition: XiY cannot be false when XaY is true but may be true when XaY is false—XiY is in subaltern opposition to, or is “the subaltern of”, XaY; similarly, XoY is “the subaltern of” XeY. (This form of opposition, unlike the other three, has a definite order; we cannot express the relationship between XaY and XiY by saying that they are “subalterns”, any more than we can express the fact that A is the mother of B by saying that A and B are mothers.) Another way of expressing this relationship is in terms of implication; we can say that XaY implies XiY, in the sense that XiY can be inferred, or seen to follow, from XaY, but XiY does not imply XaY—in other words, if XaY is true, XiY is also true, but, if XiY is true, XaY may be false.

The relations, in terms of truth and falsity, between the four propositions with a given subject (X) and a given predicate (Y) are commonly illustrated by the “square of opposition” in which the diagonals represent contradiction, the top side contrariety, the bottom side subcontrariety, and the vertical sides subalternation, the subaltern in each case being at the foot of the line. These types of relation are sometimes referred to as “relations in the square”, there being other relations, expressible in terms of truth and falsity, which do not hold between propositions with the same terms in the same order.

Whatever X and Y may be, two of the four propositions are true and two (the contradictories of the former two) false. The three


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possibilities are (a) the two affirmatives true, (b) the two particulars true, (c) the two negatives true; but, in view of subalternation, the different possibilities are adequately referred to as (a) cases where XaY is true, (b) cases where XiY and XoY are true, (c) cases where XeY is true.

Singulars.

The assertion that “whether X is Y or not” does not raise a single, definite issue appears to have exceptions in cases where X is an individual; it would commonly be said, for example, that the propositions “Socrates is a man” and “Socrates is not a man” define a simple issue. At the same time, many logicians regard such singular propositions as universals (referring to the whole of a class which, as it happens, has only one member) and thus as falling within the scheme of “the four forms.” But, if we combine these two contentions, we have to say that a single issue is defined by A and E propositions which not only cannot both be true but cannot both be false; i.e., in such cases A and E are contradictions.

To avoid this confusion in logical theory we have to realise that any logical term not merely “in principle” but actually has various instances (or is “a general term”), that what we call “an individual” is recognised as being of a certain character and this character recurs. Thus, in order to speak of Socrates, we have to be acquainted with what may be called “the Socratic character”; and that character can be found in many different situations (situations in which it can truly be said “that is Socrates”). It is true that these situations together form a single continuous series (“the life of Socrates”), whereas the situations in which we can find, e.g., the human characters do not. But still there is


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in either case a plurality of instances, and hence, with either as subject, the possibility of distinguishing between universal and particular propositions and thus between contrariety and contradiction.

It may still be difficult in particular cases to determine whether universality or particularity is intended. It seems clear that, when we say “Socrates is a man”, we mean that all beings of the Socratic character are of the human character or, briefly, “All Socratic beings are men”—and the contradictory of this would be “Some Socratic beings are not men”. The contrast between universal and particular in such cases is indicated in ordinary speech by adverbs of time (always, never, sometimes, sometimes not), but for logical form we have to attach the indication of quantity to the subject and not to the copula. Thus the strict form of “Socrates is sometimes angry” would be “Some Socratic beings are angry” and, correspondingly, “always” requires the universal form. For brevity, however, the grammatically singular form may be retained in cases where there is no doubt that a universal is meant.

Illustrations of logical form.

On the above showing we are, in understanding any term, understanding it to have both character and instances; i.e., any term can be attached as a predicate to subjects and can as a subject have predicates attached to it. It is the very same term that can characterise and that can be characterised; thus, while grammar distinguishes between substantive (noun) and adjective, logic does not. There is no logical difference between “All organic things are mortal things” and “All organic are mortal”; and, while expressions like “things” or “beings” can always be inserted to remind us of plurality, the use of the adjectival form is convenient for brevity and to remind


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us that any term can be a predicate. The terms in “all organic are mortal” are still understood to be plural or to have instances. Now we have seen that while Socrates, as an instance of humanity, has a certain plurality, he also has a certain unity; and this is connected with the fact that, when we speak of “all X”, we are speaking of each and every X, in other words, that the terms in any proposition have to be understood distributively and not collectively. Thus in propositional form each plural term must have a clear and definite singular; it must be clear what form an example of it would take. A case of non-distributive use of terms is “all men are equal”; this is not, though it appears to be, in strict logical form, for, if it were, it would be intelligible to say “ Socrates is equal.”

A connected point is that logical terms are concrete. In speech we often use “abstract terms”, and such an assertion as “Virtue is incompatible with selfishness” is taken to be quite intelligible. But if we ask what exactly is being talked about, what thing or things we could examine to see whether the statement is true or not, we can see that there is not an individual thing “virtue” which can be subjected to scrutiny and that the real subject is “virtuous beings”; thus the logical form of the statement is “no virtuous (beings) are selfish.” A minor point here is that we should not import into the logical form anything not given in the original statement, and say, e.g., “no virtuous persons are selfish.” It may be true that being virtuous is confined to persons (all virtuous are persons), but that would be additional information; all that is given in the original statement is that those beings (whatever they are) which are virtuous cannot at the same time be selfish. We are apt to add to what is given such things


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as we think may be “taken for granted”, but someone else might possibly contest them. Here again the importance of not running different issues together appears as a point in method.

The misleading appearance of singularity given in the use of “abstract terms” is exemplified also in the use of the word “the”. This is commonly used merely to mean, all (corresponding closely to the use of “the only”); thus, the logical form of “The honest man is humble” is “all honest men are humble”. It is important in this connection to note that the attachment of “the” to a term indicates that that term is the logical subject, even if it is not the grammatical subject; the logical form of “Wrestlers are the healthy members of the community” is “all healthy members of the community are wrestlers”, and the person who asserts it is not even stating that all wrestlers are healthy members; etc., though he might wish to assert this as a further fact. Similarly, assertions of the type “X are the only Y” are to be rendered as YaX and not as XaY; if we say “Students of the classics are the only educated persons”, we are saying that all educated persons are students of the classics, and we could quite consistently believe that some students of the classics are not educated persons. A special case of the use of “the” is that in which it is attached to both terms of a statement. If we say “The virtuous are the happy”, our statement should be rendered logically by two propositions, “all virtuous are happy” and “all happy are virtuous.” These are two quite distinct issues, though they have been conveyed in the first place by a single sentence, and once more the application of logic to assertions takes the form of stating separate issues separately.




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Class-relations.

We saw that, of the four propositions with X as subject and Y as predicate, there must be two true and two false, the true propositions being (a) the two affirmatives or (b) the two particulars or (c) the two negatives. The same may be said of the four propositions with Y as subject and X as predicate. Thus of the eight propositions with X and Y as terms four are true and four are false; and the four time propositions may be (i) the four affirmatives, (ii) the two affirmatives with X as subject and the two particulars with Y as subject, (iii) the two particulars with X as subject and the two affirmatives with Y as subject, (iv) the four particulars, (v) the four negatives.

Using circles to represent the classes X and Y (“the class X” or “the extension of X” being the various things which are X or subjects of which X may be truly predicated) we have the following five distinct possibilities:-

(i)

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Coextension. The four affirmative propositions are true, but, in view of subalternation, the position can be adequately stated by means of two propositions XaY and YaX.

(ii)

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Inclusion of X in Y. XaY, XiY, YiX, YoX are true, but XaY implies XiY, and XiY and YiX are always true together or false together. Hence the position can be adequately stated by means of the two propositions XaY and YoX

(iii)

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Inclusion of Y in X. The position (corresponding to ii) can be adequately stated by XoY and YaX.




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(iv)

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Intersection. The four particular propositions are true, but, as before, XiY and YiX are always true together. Hence the position can be adequately stated by the two O propositions and one I: i.e., either by XiY, XoY, YoX or by XoY, YiX, YoX.

(v)

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Exclusion. The four negative propositions are true, but, in view of subalternation and of the fact that XeY and YeX are always true together, the position can be adequately stated by one E proposition: either by XeY or by YeX.

Any given pair of terms will have one and only one of these possible relations, and the relation in any given case may be described as a class-relation (the relation between the class X and the class Y) or as an extensive relation (the relation between the extension of X and the extension of Y). Thus the class-relation of the terms “mortals” and “men” is that “mortals” includes “men”, the various things that are mortals embracing the various things that are men and other things besides. Such classes do not in general occupy a continuous spatial area, and thus the diagrams may be misleading unless they are taken as merely a convenient reminder of the different possibilities.

As in the case of opposition, there is one class-relation which has order, namely, inclusion. We can say that two terms are co-extensive (have the class-relation of co-extension) or that they intersect or that they are exclusive, but we cannot say that they “are inclusive”, since that would not distinguish between the two incompatible possibilities that Y includes X and that X includes Y.




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Only one class-relation, exclusion, can be indicated by a single proposition; in three other cases two propositions are required, and in the case of intersection three, including two O propositions. The proposition XiY merely indicates that the relation for X and Y is not exclusion; it might be any of the other four.

Extension and distribution.

The extension of a given term consists of the various subjects of which it may be truly predicated—more strictly, of the subjects of the various true A propositions of which it is predicate. (Correspondingly, the intension of a term—to be discussed later—consists of the predicates of the true A propositions of which the term is subject.) The conception of extension is connected with that of distribution. It is commonly said that a term is distributed in any proposition in which it is “taken in its full extension”, and that a universal proposition takes the subject in its full extension (or in all its instances), while a particular proposition does not. A more accurate formulation (though still not quite exact, as will be seen later) would be that, in stating a proposition in which X is distributed, we are saying something of any X we like to take. Thus, in the proposition “all men are mortals”, we are saying of any men we like to take that they are mortals, and so “men” is distributed in this proposition, but we are not saying that all men are any mortals we like to take, e.g., horses, and so “mortals” is undistributed in this proposition. On the other hand, in saying that no men are fish, we are saying that no men are any fish we like to take, e.g., herring, and so “fish”, (as well as “men”) is distributed in “no men are fish.” Arguing in this fashion, we can conclude that the subjects


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of universal propositions and the predicates of negative propositions are distributed, and that the subjects of particular propositions and the predicates of affirmative propositions are undistributed. It should be noted that “distribution” has no meaning with reference to a term by itself; the reference is to a term as it occurs in some specific proposition. Summing up, then:-

XaY has subject distributed, predicate undistributed;

XeY has both terms distributed;

XiY has both terms undistributed;

XoY has subject undistributed, predicate distributed.

In subalternation, in inferring XiY from XaY or XoY from XeY, it appears that we drop a distribution. This is the first illustration of the logical rule that we can argue validly (i.e., in such a way that the conclusion drawn really follows from the information given) from a term distributed to the same term undistributed but not from a term undistributed to the same term distributed.

Logical relations.

In the class-relation of exclusion of X and Y, we noted, XeY and YeX are both true; in all other cases, they are both false. Correspondingly, XiY and YiX are both false in the case of exclusion and both true in all other cases. Propositions which, from their form, must be both true or both false are said to be equivalent, and equivalence is a type of logical relation (relation, expressible in terms of truth and falsity, between propositions) which does not occur “in the square.” It can also be referred to as mutual


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implication; to say that XeY and YeX are equivalent is to say that XeY implies YeX and YeX implies XeY. Similarly, XiY implies YiX and YiX implies XiY.

As a result of these equivalences we can find “outside the square” (i.e., between propositions which do not have the same subject and the same predicate) all the relations, in terms of truth and falsity, that we find “in the square.” For example, since XiY and YiX are true together or false together and since XiY and XeY are contradictories, XeY and YiX cannot both be true and cannot both be false; they are then said to be in contradictory relation. Similarly, XaY and YeX cannot both be true but may both be false; they are in contrary relation. XoY and YiX are in subcontrary relation; they cannot both be false but may both be true. XaY implies YiX but YiX does not imply XaY; this is expressed by saying that YiX is in subaltern relation to XaY. Thus we have examples of general types of logical relation, of which the relations of “opposition” (relations, in terms of truth and falsity, between propositions with the same subject and the same predicate) are particular cases. ‘An exercise for the student would be the working out of all the examples of such relations among propositions with the same terms.’

An additional type of logical relation is found among the A and O propositions with terms in the opposite order; XaY and YaX are indifferent, in the sense that they may both be true, they may both be false, and either may be true and the other false—and this by itself shows that indifference also holds between XaY and YoX, between XoY and YaX, and between XoY and YoX. Adding equivalence


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and indifference to the four types of relation “in the square”, we have six types of logical relation between propositions, and, since subaltern relation has order, there are seven possible “truth-falsity” relations, one and only one of which holds between any two propositions (p and q) we may select.

(i) p and q are equivalent, when they are either both true or both false (when each implies the other).

(ii) p is in subaltern relation to q, when p cannot be false with q true but can be true with q false (when q implies p but p does not imply q—or if q is true, p is true, but if p is true, q may be false).

(iii) q is in subaltern relation to p, when p cannot be true with q false but can be false with q true.

(iv) p and q are in contradictory relation, when they cannot both be true and cannot both be false (when, of the two propositions, one must be true and one must be false).

>(v) p and q are in contrary relation, when they cannot both be true but may both be false.

(vi) p and q are in subcontrary relation, when they cannot both be false but may both be true.

(vii) p and q are indifferent, when they may both be true or may both be false or p may be true with q false or p may be false with q true. (This is, of course, the commonest relation between propositions; there is nothing in the truth or falsity of a proposition with X and Y as terms to tell us about the truth or falsity of a proposition with A and B as terms. Thus, e.g., “all men are mortal” and “Some trees are evergreen” are indifferent, and so are any two propositions with the same four terms.)




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Immediate inference.

In immediate inference we draw a conclusion from a single given proposition or “premise”, as contrasted with mediate inference in which a conclusion is drawn from a number of premises taken together. We have seen that from any universal proposition its subaltern may be inferred, and thus subalternation may be described as a type of immediate inference. But we may also see that A proposition implies any equivalent of its subaltern, that XaY implies YiX, and that the two E propositions, XeY and YeX, are equivalent or imply one another. The inferring, from a given proposition, of a proposition with the same terms in the opposite order is conversion.

The possibility of conversion depends on the “convertiblity” of terms, on the fact that any subject can be a predicate and any predicate a subject. But it is not possible to convert all propositions; the O proposition has no converse, the truth of XoY being compatible with the truth and with the falsity of any proposition with Y as subject and X as predicate. And, whereas the converse of XeY is equivalent to it, each being the converse of the other, and the same is time of XiY, the converse of XaY is YiX, which is equivalent only to the subaltern of XaY and does not imply XaY itself. We could, from XeY, infer YoX, but, since we always can infer YeX and then, if we so desire, draw the further inference YoX by subalternation, it is customary to call YeX the converse of XeY. Thus we have: XaY—converse YiX (which has as its converse XiY); XeY—converse YeX (which has as its converse XeY); XiY—converse YiX (which has its converse XiY); XoY—no converse. (When XoY is true, X may include Y in which case YaX and YiX are true, or


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X may intersect with Y in which case YiX and YoX are true, or X may exclude Y in which case YeX and YoX are true.)

Obversion is the second main type of immediate inference. Conversion, as we have noted, is bound up with the fact that any term is general, has instances as well as characters, and thus with the question of quantity. Obversion is bound up with the question of quality and may be said to bring out the negative aspect of any affirmative proposition and the affirmative aspect of any negative. It does this by means of the conception of “opposites”. The opposite of a term X is expressed by the term non-X (and symbolised by (non-X)); and two opposites, X and (non-X) (e.g., mortals and non-mortals), are such that nothing can be both and anything must be one or other.

The obverse of a proposition with X as subject and Y as predicate is a proposition which it implies, having X as subject and non-Y as predicate, the universal inference being preferred (as in the case of the conversion of E) to the particular which could be got from it by subalternation. On that understanding we can see not merely that every proposition has an obverse but that a proposition and its obverse are equivalent, being obverses of one another. Thus we have: XaY—obverse Xe(non-Y) (which has as its obverse XaY); XeY—obverse Xa(non-Y); XiY—obverse Xo(non-Y); XoY—obverse Xi(non-Y).

By using the processes of obversion and conversion alternately (either beginning with obversion or beginning with conversion) we can obtain a number of inferences from each of the four forms of propositions, as set out in the following table:-


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Proposition  XaY  XeY  XiY  XoY 
Obverse  Xe(non-Y)  Xa(non-Y)  Xo(non-Y)  Xi(non-Y) 
Partial Contrapositive  (non-Y)eX  (non-Y)iX  —  (non-Y)iX 
Contrapositive  (non-Y)a(non-X)  ((non-Y)o(non-X))  —  (non-Y)o(non-X) 
Inverse  (non-X)i(non-Y)  (non-X)o(non-Y)  —  — 
Partial Inverse  ((non-X)oY)  (non-X)iY  —  — 
Obverted Converse  Yo(non-X)  Ya(non-X)  Yo(non-X)  — 
Converse  YiX  YeX  YiX  — 
Proposition  XaY  XeY  XiY  XoY 

The inferences are to be understood as proceeding downwards from the original proposition, when the first step is obversion, and upwards from the original, when the first step is conversion; and the brackets indicate where the chain of inferences, in either direction, from the universal propositions stops—in each case (and so also with inferences from the particular propositions) with the impossibility of converting an O proposition. This is important, as the fact (e.g.) that (non-X)oY and Yo(non-X) can both be inferred from XaY might lead to the belief that they can be inferred from one another—a belief which would involve taking an O proposition to have an O converse.

The fact that O has no converse may be connected with the question of distribution. It may be assumed that an affirmative connection between two terms cannot be inferred from a negative one; but, if we took XoY to have a negative converse, then we should be inferring a proposition in which X is distributed (one applying to “any X we like to take”) from a proposition in which it is undistributed.

But now a question may be raised regarding the partial inverse of the A proposition. How can we infer (non-X)oY, in which Y is distributed,


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from XaY, in which it is not? The answer is that, in using obversion, in taking Y to have an opposite, we have taken it to be entirely excluded from something and thus have implicitly distributed it. If, given XaY, it were possible for (non-X)oY to be false and thus (non-X)aY to be true, the position would be that everything, whether X or not X, would be Y, and thus Y would have no opposite. The recognition of obversion, then, supplements the contention that all propositions have real terms by the contention that all real terms have real opposites; and this can be supported by saying that, just as there is no issue unless we are dealing with real things, so there is none unless these things exclude other things, unless in affirming something we are negating some other possibility.

The most important of the inferences made by combining conversion and obversion is contraposition, where as before (though this applies only to the contraposition of A) we take as the contrapositive the most that can be inferred, with subject (non-Y) and predicate (non-X), from a proposition with subject X and predicate Y. On this understanding A and its contrapositive (and similarly O and its contrapositive) are equivalent and are contrapositives of one another; E, on the other hand, has a contrapositive which is in subaltern relation to it (being equivalent to its subaltern, O). XaY and its contrapositive (non-Y)a(non-X) can be expressed as “When X is present, Y is present “and” When Y is about, X is absent”; and the equivalence of there (as will be seen later) is important in other connections. It is also important to note that “When X is present, Y is present” is not equivalent to (in particular, does not imply) “When X is absent, Y is about.” The belief that it is amounts to the fallacy of inferring YaX from XaY (e.g., “all mortals are men” from “all men are mortals”)—a fallacy, or fallacious argument,


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being the inferring from a given premise (or premises) of a conclusion which does not follow from it (or them); in other words, which is not necessarily true when it is (or they are) true.

The recognition of equivalences enables us to find the logical relation between any two propositions with two of the four terms, X, Y, (non-X) and (non-Y), as terms (it being understood that no proposition has two opposites, e.g., X and (non-X), as its terms). Thus, given the propositions (non-X)e(non-Y) and XaY, we can say that the former is equivalent to its converse (non-Y)e(non-X) and the latter to its contrapositive (non-Y)a(non-X), and, these being contraries, the original propositions are seen to be in contrary relation (they cannot both be true and may both be false).

Using subalternation as well as conversion and obversion, we find that eleven propositions may be inferred from a universal proposition—three universals, each equivalent to the original, and eight particulars, each in subaltern relation to it, and divisible into two groups of four equivalent propositions, any member of one group being indifferent to any member of the other. From a particular proposition, on the other hand, only three propositions can be inferred, each equivalent to the original.

Syllogism.

The commonest and the most important type of mediate inference is syllogism, which can be described as the inferring of a connection between two terms from the connection of each of them with a third term or as the elimination of a term common to two propositions so as to arrive at a proposition containing the other two terms. An outstanding case of this elimination is the attribution of any part of the intension of a term to any part of its extension, the attribution of any of its characters to any of its instances.


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An example would be

All men are mortals, All mathematicians are men. Therefore: All mathematicians are mortals. (BaC, AaB, AaC).

This argument is in syllogistic form because (a) it contains three propositions (two premises and a conclusion), (b) it has three terms, one common to the two premises, one common to the first premise and the conclusion, and one common to the second premise and the conclusion. The term common to the premises is called the middle term, the predicate of the conclusion is called the major term and the premise in which it appears the major premise, the subject of the conclusion is called the minor term and the premise in which it appears the minor premise. The use of these expressions may be traced to the fact that, in the main type of syllogism (as in the above example), the middle term is frequently included in the major term and the minor term in the middle term. But there is nothing in a syllogism with A propositions to say that this must be so; B and C could be co-extensive (and similarly with A and B) without affecting the force of the argument. This is another illustration of the need for keeping issues clear, for distinguishing between what is said and what is not said, even if it may be also known. Again, it is customary in setting out a syllogism to put down the major premise first, but the inference does not depend on the order of the premises and, whichever is written first, the major premise is still the one which contains the predicate of the conclusion. The customary order is adopted to facilitate recognition of the form of the argument and, in connection with that, of its validity or invalidity (whether the conclusion drawn follows from the given premises or not);


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and it should be noted that, in order to bring out the syllogistic character of an argument, it may be necessary not only to put colloquial statements into logical form but to use immediate inference (particularly, obversion) so as to have only three terms.

The rules of syllogism are certain general conditions of validity. The requirements of three propositions and three terms are sometimes mistakenly called “rules”; they merely indicate whether an argument is a syllogism or not, but it is quite possible for an argument which is not a syllogism to be valid. In determining, then, whether what is a syllogism is valid or not, we employ the following general considerations or rule:-

In a valid syllogism

I. The middle term must be distributed once at least (or if the middle term is undistributed in one premise, it is distributed in the other).

II. If the major or minor term is undistributed in its premise, it is undistributed in the conclusion (or a term distributed in the conclusion is distributed in its premise).

III. One premise at least is affirmative (or if one premise is negative, the other is affirmative).

IV. (a) If one premise is negative, the conclusion is negative; (b) If the conclusion is negative, one premise is negative (or if both premises are affirmative, the conclusion is affirmative).

There are thus two types of valid syllogism—those in which all three propositions are affirmative, and those in which one premise is affirmative and one negative and the conclusion is negative.

The first two rules are concerned with the distribution of terms and the last two with the quality of propositions. Considerations of


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the quantity of propositions in a valid syllogism are sometimes given as “rules” (e.g., in a valid syllogism one premise at least is universal—or “from two particulars nothing can be inferred”), but they are not independent rules; they can be derived from the four rules given above, while none of the four rules can be derived from them. It is better, then, to restrict the expression “rules” to the four independent conditions of validity.

It is important to distinguish the question of validity from the question of truth. When we say that the syllogism BaC, AaC ? AaB is invalid (pointing out that it breaks rule I because the middle term C is undistributed in both premises), we do not mean that, when the premises are given as true, the conclusion cannot be true (must be false); we merely mean that it need not be true, that the premises do not show it to be true. We can, of course, find examples of the falsity of such a conclusion while the premises are true (e.g., all horses are mortal, all dogs are mortal, therefore, all dogs are horses), but we can also find examples of the truth of such a conclusion (e.g., all men are mortal, all mathematicians are mortal, therefore, all mathematicians are men). The question, then, in the examination of a syllogism by reference to the rules, is not whether the conclusion is true but whether it is validly inferred (or follows) from the given premises.

When Rule I is broken, the syllogism is said to commit the fallacy of undistributed middle; when Rule II is broken, the fallacy is one of illicit process—either illicit process of the major term (“illicit major”) or illicit process of the minor term (“illicit minor”). Breach of Rule III could be referred to as “the fallacy of two negatives”, but if this expression is used, or if it


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is said that “from two negatives nothing can be inferred”, it has to be remembered (a) that this applies only when we have syllogistic form (three terms) and that two negative statements may imply (with the aid of obversion) propositions which can be the premises of a valid syllogism, (b) that, even though from two negative propositions with a common term no proposition with the other two terms can be inferred, it may still be possible from the two data to draw some conclusion; e.g., BeC, BeA imply no proposition with terms A and C (they are compatible with any relation whatever between these terms), but BeA does imply Ba(non-A), and BeC, Ba(non-A) imply syllogistically (non-A)oC. There is no special name for breaches of the fourth rule.

Syllogisms, valid or invalid, may be in any one of four figures, distinguished by the position of the middle term in the two premises. In the first figure, the middle term is subject of the major premise and predicate of the minor premise; in the second figure, it is predicate of both premises; in the third figure, it is subject of both premises; in the fourth figure, it is predicate of the major premise and subject of the minor premise. Thus the order of the terms (M being the middle, X the minor, and Y the major term) in each of the four figures is as follows:

image

According to the form (quality and quantity) of the propositions we have various moods in each figure; thus, “a syllogism of mood EIO in the third figure” means a syllogism of the form. MeY, MiX ? XoY. It is understood that the propositions are referred


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to in the order major premise, minor premise, conclusion; this is important since there are valid and invalid moods in each figure, and whereas EIO, in the above example, is valid, IEO would be invalid. EIO is in fact valid in any figure, and IEO is invalid in any figure; but there are moods which are valid in some figures and not in others (AAA, e.g., is valid only in the first figure), and it is better, therefore, not to take mood apart from figure—to speak only of moods in a particular figure.

If we include “subaltern moods” (in which a particular conclusion is drawn, though the corresponding universal could be drawn) there are six valid moods in each figure:-

First. AAA, AAI, EAE, EAO, AII, EIO.

Second. EAE, EAO, AEE, AEO, EIO, AOO.

Third. AAI, IAI, AII, EAO, OAO, EIO.

Fourth. AAI, AEE, AEO, IAI, EAO, EIO.

It will be noted (and can be easily proved from the rules of syllogism) that all valid syllogisms in the second figure have negative conclusions and all valid syllogisms in the third figure have particular conclusions. ‘Memorising the above list of valid moods has some advantages but should not replace the habit of examining any given argument with reference to the rules. Being found in a list of valid arguments is not a reason for validity.’

The first figure may be called the “most natural” one in that the subject of the conclusion is also subject in its premise and the predicate of the conclusion predicate in its premise, and in that as mentioned above, it exhibits the bringing together of the extension and intension of a term. The view that there is something specially


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clear and persuasive about first figure syllogisms lies at the basis of the theory of reduction, i.e., the demonstration, by means of a first figure syllogism, supplemented in many cases by immediate inference, that the conclusion of a valid syllogism in some other figure really does follow from the premises. In “direct reduction” the given premises are shown to imply first figure premises, and the first figure conclusion is shown to imply the required conclusion. In “indirect reduction” it is shown by a first figure syllogism (with or without the addition of immediate inference) that the premises and the contradictory of the conclusion of the given syllogism in another figure cannot all be true.

(Examples (a) Direct reduction of AEE in the second figure. Its premises YaM, XeM imply, by conversion of the given minor, MeX and YaM and these, in the first figure, imply YeX, which in turn implies by conversion XeY, the required conclusion. Thus, by conversion and a first figure syllogism, it has been shown that YaM and XeM imply XeY, or that AEE in the second figure is valid. (b)Indirect reduction of OAO in the third figure. The syllogism is MoY, MaX ? XoY. The contradictory of the conclusion is XaY. XaY and MaX, as first figure premises, imply MaY: i.e., imply that MoY is not true. Thus it has been shown, by means of a first figure syllogism, that XaY, MaX and MoY cannot all be true, hence that, if MoY and MaX are true, XaY is false, i.e. XoY is true; it has been shown, that is to say, that MoY and MaX imply XoY, or that OAO in the third figure is valid.)

But what is shown in these and similar cases is that certain arguments in figures other than the first really are valid, that the conclusion in each case really follows from the premises,


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and so the possibility of reduction to the first figure is no reason for dispensing with other figures. We can, in fact, recognise special functions of the second and third figures, at least. The second figure may be called the figure of contrast; in EAE and AEE we find that two classes are exclusive, by finding a property which is present throughout one of them and is entirely absent from the other. The third figure may be called the figure of illustration; in AAI we find that two characters are compatible by finding each of them in the same subject.

The theory of syllogism enables us to clear up the question of distribution. The formula given earlier was that, in stating a proposition in which X is distributed, we are saying something of any X we like to take (any part of the extension of X); this amounts to saying that we can, without loss of truth, substitute for X in the given proposition any part of the extension of X. But this substitution is simply syllogism. To say that X is distributed in XaY amounts to saying that, given ZaX (Z as a part of X's extension), we can infer that ZaY—in other words, the “distribution” of X in XaY means that a syllogism of the form XaY, ZaX ? ZaY (AAA in the first figure) is valid. Similarly, to say that X is undistributed in XiY means that a syllogism of the form XiY, ZaX ? ZaY (IAA in the first figure) is invalid. Considerations of the same kind apply to the other instances of distribution and non-distribution of terms in the four forms of propositions.

This explanation seems to involve us in circularity of


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reasoning as between distribution and validity—particularly, in the treatment of lack of distribution as an explanation of invalidity and of invalidity as an explanation of lack of distribution. The fact remains that the notion of “distribution” is one of the legitimacy of certain substitutions and so of the validity of certain syllogistic forms. Circularity is avoided, then, by recognising that we are directly aware of validity and invalidity, of implication and non-implication as relations between propositions of certain forms. We can, as a result of such direct knowledge, form a table of distributions and non-distributions, and we can use these as handy criteria for detecting validity and invalidity (as in the first two rules of syllogism) without taking them to tell us what validity and invalidity are.

More generally, with regard to the four rules of syllogism, we can say not only that any syllogism which breaks one of them is invalid but that any syllogism which does not break any of them is valid. But this does not mean that keeping the rules is a reason for, or justification of, validity; it means that, having found certain characters of valid and invalid syllogisms respectively, we can use these as reminders, as means of rapid decision on validity. We could not have learned to do so, however, if we had not first recognised validity and invalidity, or, more exactly, implication and non-implication as actual relations among propositions. The theory of logic, of course, attempts to systematise such initial findings; but it is a vital point in method that it does not impose forms upon the facts but finds them in the facts.




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Complex terms.

Another type of mediate inference can be approached by reference to the “illustrative” arguments of the third figure. Instead of drawing the syllogistic conclusion XiY from the premises MaY, MaX, we could draw the conjunctive conclusion MaXY: i.e., not merely that X and Y can be combined but that they are combined or “conjoined” in M. This is inference in that the term XY (things which are at once X and Y) does not occur in either premise; but it differs from syllogism in that we can infer the premises again from the conclusion—as we may put it, the conclusion in this valid conjunctive argument is equivalent to the two premises taken together, whereas the conclusion in a valid syllogism is in subaltern relation to the two premises taken together. (There are other types of conjunctive argument in which this equivalence of premises and conclusion does not hold.)

The argument MaY, MaX ? MaXY can be described as combining two parts of the intension of M, and the conjunctive term XY can be called a combination of intensions or “intensive combination.” It is a putting together of predicates, forming a complex predicate out of two simple ones (simple in the sense of being conveyed by a single expression), though once we have recognised the term XY we can, of course, also use it as a subject. This type of combination can be indicated by the use of “and”, though for brevity, and in accordance with ordinary usage, this is commonly omitted. (“All men are rational animals” conveys the same as “All men are rational and animals.”)

Now it is possible in a similar way to combine two


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parts of the extension of a term; we can argue MaY, LaY ? (L or M)aY (as we may put it, anything which is either L or M is Y). The term (L or M) is a disjunctive term or an “extensive combination”, and the argument is a disjunctive argument, the premises of which, as in the previous case, can be inferred from the conclusion. In this case we have put together subjects, formed a complex subject out of two “simple” ones, though, as before, the term (L or M) can be used as a predicate. (Cf. “All adults are men or women”.)

An important point concerning such terms is that the opposite of a conjunctive term is a disjunctive term, and the opposite of a disjunctive term is a conjunctive term. Thus the opposite of XY is ((non-X) or (non-Y)); the opposite of being both X and Y is being either not X or not Y. And the opposite of (X or Y) is (non-X)(non-Y); the opposite of being either X or Y is being both not X and not Y (which is the same as being neither X nor Y).

The recognition of such terms raises the question of reality or existence. It is quite possible that X and Y should exist and yet that XY should not exist (namely, when XeY). For example, there are men and there are horses, but there are no men-horses; “men-horses”, then, is a meaningless expression—we can conjoin the words “men” and “horses”, but there is no sort of thing which that verbal conjunction signifies. But since, as we saw in connection with obversion, any real term has a real opposite, we have to treat the formal “opposite” of any non-existent conjunction as also meaningless; thus “non-men or non-horses” has to be taken as meaningless. The point is that, for an assertion to be significant, it must distinguish certain sorts of things from certain other sorts of


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things; but, if we say of a subject X that it is either not a man or not a horse, we are not distinguishing it from anything else. Such meaninglessness is as far as possible avoided if, instead of forming conjunctive and disjunctive terms at random, we continue ourselves to such combinations as are arrived at by valid argument from premises we believe to be true. But, since our beliefs are sometimes mistaken, we may mistakenly believe that XiY and thus that “XY” is a real term. The general position is that, if XeY, “XY” is not a real term, and, if (non-X)e(non-Y) (if there is nothing that is neither X nor Y), “X or Y” is not a real term.

Definition and division.

In seeking a definition of a term, we try to find a conjunctive term which is co-extensive with it, and, in seeking a division of a term, we try to find a disjunctive term which is co-extensive with it; but there are additional points to be considered. A class is not considered to be properly “divided”, or broken up into smaller classes, unless these smaller classes exclude one another, i.e., unless they are independent parts of the extension of the original term. Using the terminology of genus and species, we can say that a genus is divided into various species which exclude one another and together make up the genus; and this means (taking the simplest case of the division of a genus A into species B and C) that the formal requirements of division are Aa (B or C), BaA, CaA, BeC. Thus BeC is the requirement additional to that of the coextension of A with the disjunctive term (B or C).

Similarly, we may take it that the characters, or parts of the intension, of X by which it is defined must be independent, i.e., if we take


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the simplest case, definition by two characters Y and Z, that neither is part of the intension of the other. On that understanding, the formal requirements of definition are XaY, XaZ, YZaX, YoZ, ZoY. This means that Y and Z intersect (they cannot be exclusive if XaY and XaZ), and that X is their common part. If Y were part of the intension of Z (ZaY), then, given XaZ and YZaX, it would be unnecessary to say that XaY, and Z by itself would be co-extensive with X. Thus YoZ and ZoY are the requirements additional to that of the co-extension of X with the conjunctive term YZ.

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