1 REASONING SYLLOGISM DISTRIBUTION OF THE TERMS The word "Distrlbution" is meant to characterise the ways in which terrns can occur in Categorical Propositions. A Proposition distributes a terrn if it refers to all members of the class designated by the term. In other words, if a terrri refers to all the members of the class for which it stands, it is said to be distributed term. Propositions A-Proposition (Untversal Affirmative) E-Proposition (Universal Negative) I-Proposition (Particular Affirmative) O-Proposition (Particular Negative) Distrfbution of Term Subject Predicate Let us consider how they are distributed or not distributed. A-Proposition : All aples are fruits. It is clear that all apples are included in the category of fruits. Graphic representation of this Proposition will be : Clearly, the first circle refers to all the men and the second to all the women. In other words, the two circles have nothing in common with each other. We cannot say that "one man is woman" or "one woman is man". Both refer to all the members of the class, man or woman, and are independent. Therefore, both the terms are distributed. I-Proposition : Some girls are dancers. Graphic representation of this Proposition : The two circles intersect each other, and the shaded portion stands for the given Proposition. The Proposition refers to the segments of the two circles and not the whole of the circles. Therefore, neither of the terms is distributed. O-Proposition : Some girls are not dancers. Clearly, the Subjective term refers to "some" only and the Predicative tenn refers to all the dancers. The graphic representation of this Proposition will be : In this Proposition, the Subjective term (Apples) refers to all apples but the Predicative term does not refer to aü kinds of fruits; it refers to only one category of fruits, i.e., apples. E-Proposition : No man is woman. In this Proposition the Subjective term refers to all the men and the Predicative term refers to all the women. Graphic representation of this Proposition will be : The segment representing "some girls" has nothing in contrnon with the circle representing "dancers". The term "dancers" stands for all the dáncers. Thereforé, in such a Proportion only Predicative term is distributed. SYLLOGISM AND INFERENCE An argument is any group of Propositions of which one is claimed to follow from the others, which are regarded as providing support or grounds for the truth of that one. An argument, in reasoning, is not a mere collection of PropositiOns; it has a structure. In defining this structure, the terms "Premise" and "Conclusión" are usually used. The concltision of an argument is the Proposition that is affirmed on the basís of the other Propositions of the argument, and these other Propositions, which are affirmed (or assumed) as providing support or reasons for accepting the conclusion, are the premises of that argument. Thus, a premise can be defined as "a statement or an idea on which reasoning is based."
2 The simplest kind of argument consists of just one premise and a conclusion that is claimed to follow from it, or to be implied by it. For example, Premise : All girls are dancers. Valid Conclusion : Some girls are dancers. A complete Syllogism (or Categorical Argument) consists of three Prepositions and three terms. The three Propositions are known as Major Premise, MinorPremise and Inference or Conclusion respectively. The three terms are known as MajorTerm, Minor Term and Middle Term respectively. Major Term : The Predicate of the Inference or Conclusion.lt is denoted by Capital letter P. Minor Term :The Subject of the Inference or Conclusion. It is denoted by Capital letter S. Middle Term : It is the term common to both the Premises and is denoted by Capital letter M. The middle term does not occur in the Inference or Concluion. Example Premises : (1) AIl games are outdoor. M (2) All outdoor are indoor. M Conclusion : All games are indoor. S P Major Premise : (a) The Major Premise is that in which Major Term (P) is present, or (b) the Major Premise is that in which the Middle Term (M) is the Subject. In the above example second Premise is Major. Minor Premise : (a) The Minor Premise is that in which Minor Term (S) is present, or (b) the Minor Premise is that in which the Middle Term (M) is the Predicate. In the above example, the first Premise is Minor. REASONING OR INFERENCE The term Inference refers to the process by which one Preposition is arrived at and afiirmed on the basis of one or rnore other Propositíons accepted as the starting point of the process. To determine whether an Inference i's correct, we examine those Propositions that are the initial and end points of that process and the relationships between them. Thus, Inference or Reasoning is the process of passing from one or more Propositions to another, which is justífied by them, and the product is called an Inference (Conclusion). Arguments are traditionally divided into two types : Deductive and Inductive. Every argument involves the ciaim that its Premises provide some grounds for the truth of its Conclusion, but only a Deductive argument involves the claim that its Prémises provide conclusive grounds for its Conclusion. When the reasoning in a Deductive argument is correct, we call that argument valid, when the reasoning of a Deducüve argument is incorrect, we call that argument invalid. In every Deductive argument, either the Premises succeed in providing conclusive grounds for the truth of' the Conclusion or they do not succeed. Therefore, every Deductive argument is either valid or invalid; If a Deductive argument is not valid, ít must be invalid; if it is not invalid, it must be valid. In Inductive argument its Premises do not give conclusive grounds for the truth of its Conclusion, but only they do provide some support for the Conclusion. Therefore, Inductive arguments cannot be valid or invalid in the sense in which these terms are applied to D'eductive arguments. Thus, a Deductive argument is one whose Conclusion is claimed to follow from its Premises with absolute necessity, this necessity not being a matter of degree and not depending in any way on whatever else may be the case; in sharp contrast, an Inductive argument is one whose Conclusíon is claimed to follow from its Premises only with probability, this probability being a matter of degree and dependent upon what else may be the Case. In Deductive argument, the Conclusion cannot be more general than the Premise or Premises whíle in Inductive argument, the Conclusion must be more general than the Premises. Deductive Inferences have been divided into two types : Immediate and Mediate. In the Immediate argument the Conclusion is drawn from only one given Proposition. For example : Premise : All men are mortal. From this Proposition a Conclusion can be derived that "Some men are mortal". In Mediate argument, Conclusion follows from more than one Proppsition. Where there are only two Propositions and the Conclusion follow from them jointly, the form of Mediate argument is, called Syllogism. IMMEDIATE INFERENCES Kinds qf Immediate Inference By an immediate inferenceis meant whatever conclusion may be drawn from a single proposition, as distinguished from what may only be inferred from two or more propositions jointly. The problem of the determination of the various types of immediate inference falls into two parts. In the first part are considered the various infer.ences which may be drawn from a given proposition in terms, or in respect, of another proposition having the same subject and the same predicate as the given proposition, but differing from it in respect of quality, or of quantity. This part is known as the doctrine of the opposition of propositions. The second part deals with inferences which may be drawn from a given proposition involving certain other subjects and predicates than those of the given proposition. This part is known as the doctrine of educttons.
3 The Laws of Contradiction and Excluded Middle The assumptions in quesüons are included among the so-called Laws of Thought, and are known as the IJiw qf Contradiction and the Ixiw ofexclu.dedmiddle. According to the Law of Contradiction the same predicate cannot be both affirmed and denied of precisely the same subject S (the same S) cannot both he P and not he P. According to the Law of Excluded Middle, given predicate must either be affirmed or denied of a given subject S rnust either he P or not be P, it cannot be neither, just as it cannot be both. These are fundamental assumptions on which all consistent thinking res.ts; ail apparent exceptions rest on misunderstandings, cr on quibbles. With the aid of these Laws oi' Thought we may now consider the opposition of propositions. The Formal Oppasition of Categorical Propositions We are concerned here with the relations between propositions having the same subject and predicate, and differing only in form. that is, in quality or in quantíty. Now there are only four such propositional forms SaP, SiP, SeP, SoP and we have to determine the relation of each to the others. SaP: According to it Pis affirmed of every S without exception. Therefore if SaP is true SiP must be true ; if it were not, that is, jfit were possible not to affirm P of one or more S's, it would be impossible to affirm P of every S, that is, SaP could not be true. Again, SaP implies the falsity, or rejection. of SeP, for if SeP could be true at the same time as SaP then the same subject, each S, would at once he and noí be P; and this would be a violation of the Law of Contradietion. Thereibre, SaPirnplies the falsity of SeP. Similarly, SaP implies the falsity of SoP. For if both could be true together, then some S's would both be P (because SoP) and not be P (because SoP) ; and this is excluded by the Law of Contraction. Thus SaP implies StP, but excludes SeP and SoP. Sip : SaP cannot be tnferred from SiP, for the inference must not distribute a term (S in this case) not distributed in the premise ; but, of course, SaP, though not inferable from SiP may be true at the same time. Again, SiP excludes SeP, for both could be true, then some S's would both be P (because SiP) and not be P (because SeP); and this would violate the Law of Contradiction. But SiP neíther implies nor excludes SoP. It does not imply SoP because when SiP is true SaP may also be true, in which case SoP could not be true. It does not exclude SoP, because the "some S's" which are asserted to be P may be different S's from those which are asserted not to be P, and this would not involve a violation of the Law of Contradiction. They are, therefore, simply compatible ; neither inferable one from the other, nor exclusive one of the other. SeP rthis is inconsrstent with both SaP and SiP. On the other hand, it implies SoP for the same reason that SaP implies SiP. SoP: Thts is inconsistent with SaP, is compatible with SiP, and fis implied by SeP. Suppose SaP is not true. This means that it is incortect to affirm P of évery S in other words, there are at least some S's (one Or more) of which one should not say that it is P. But, according to the Law of Excluded Middle, anything must either be P or not be P. Consequently, of those S's (one or more) of which it is incorrect to assert that they are P, one must assert that they are not P, that is, SoP. Thus the falsity, or rejection, of SaP implies the truth, or íhe acceptance of SoP. On the other hand, SiP and SeP may either of them be true (not both, of course) when SaP is false, one simply cannot tell. Suppose Sip is false. This means that it is incorrect to say of even one S that it is P. Consequently, according to the Law of Excluded Mrddle, it is right to say of every S that it ts not P, or SeP. Thus is the falsity, or rejection, of SiP involves the truth, or acceptance, of SeP, and therefore also of SoP, which is implied by SeP. Obviousiy the lalsity, or rejection, oi' SíP implies a jbrtiori the falsity, or rejection, of SaP - if it is incorrect to say of even one S that it is P, it must be even more incorrect to say of every S that it isp. Suppose SeP is untrue. This means that it is incorrect to assert of every S that it is not P, or, in other words, that there is at least one S of which it is incorrect to assert that it is notp. líso, then, by the Law of Excluded Míddle, it is correct to assert of at leasi that S that it is P, or StP. Thus the f'alsity. or rejection, of SeP implies the truth, or the acceptance, ol' SiP. On the óther hand, it carries no implication w'ith regard to SoPor SaPeither of which (though not both) may be (rue, or not if SeP is not true). Suppose SoP is false. This means that it is incorrect to assert of even one S that it is not P. Therefore, by the Law of Excluded Middle, it rs correct to assert of every Sthat it is P SaP. Thus the rejectron, or falsity, of SoP implies the acceptance, or truth, of SaP, and therefore of SiP. it also implies the rejection, or falsity, of SeP what cannot be asserted of amj S can certainly not be asserted of every S. It may be helpful to sum up the foregoing results in the following table : TABLE OF RELATIONS OR OPPOSITIONS Given SaP SíP SeP SoP SoPtrue true false false SiP true not knówn faise not known SeP true false. false true SoPtrue faise not known not known SaP í'alse not: known nót known true SiPfalse false true true SeP false not known true not known SoP false true true false
4 Some of the above relations between propositions having the same subject and predicate have received special names. They are summarised in the following diagram, which is known as The Square of Oppositíon. The following points should be specially noted: Contraries : SaP and SeP are extreme opposites, and do not between them exhaust all possibilities. They cannot both be true; but they may both be false, namely, when both SiP and SoP are true. Singular propositions and particular propositions have no formal contraries, only contradictories, Sub-contraries : The reiationship between subcontraries must be carefully distinguished from that between contraries, as they are the precise reverse of each other. Of SiP and SoP. one must be true, and both may be true or both cannot be false, and neither need be. Contradictori.es : Of the two propositions SaP and SoP both cannot be true, but one of them must be true. In other words, they are mutually exclusive (without being extreme opposites) and collectively exhaustive. The same holds good of SeP and SiP. EDUCTIONS A proposition having S for its subject and P for its predicate may imply a proposition having P for its subject'and S for its predicate, or propositions containing thé contradictories of S and P. Such propositions are called eductions. There are two principal types of eduction, and five derivative forms.the principal types are known as the converse and the obverse; the other forms are obtained (when they can be obtained) by combining, or repeatlng alternately, the steps by which the converse and the obverse are obtained. Contradictory Terms and their Symbols Any pair of terms, by means of which a class of objects is divided into two mütually exclusive and collectively exhaustive classes, are called contradictory terms. Of a pair of contradictory terms one is positive, namely, the one which indicates the presence of the. characteristic (or group of characteristics) in which one is interested at the moment, while the other is
5 Another point to be noted is that if the obverse of a given proposition is itself obverted we simpiy get back to the original proposition. So that no new result can be obtained by merely repeating the process of obversion. Conversion The converse of a given proposition is a proposition rmplied by it, but having its subject for predicate, and its predicate for subject. In other words, the terms of the given proposition and those of its converse (if it has a converse) are related as S Pto P S. every proposition, has a converse. SeP : This may be said to mean that the whole class S is outside of, or different from, the whole class P. If we express precisely the same relation from the point of view of P, instead of from the standpoint of S, we get PeS. Thus SeP = PeS. Again, SiPmeans that some S's are included in the class P, and this means that they are identical wifh some P's. What we really have, then, are objects which are both S and P, ánd whích can therefore be described indifferently as some S's which are P, or as some P's which are S. Thus SiP = PiS. It will thus be seen that E and / propositions can be converted simply one may just transpose their terms without changing in any other way the character of these propositions. It is different with A and O propositions. SaP means that the whole class S is included in the class P, that is, is identical with some indefinite part of the class P. The two classes may occasionally coincide, as for instance in the statement AU. equilateral triangles are equiangular triangles, but théy may not, as, for example, in the statement All rectangles are parallelograms. What we always have when SaP is true is a class of objects which are both S and Pwhich consists of the whole class S, but may not consistof all P's, though it must include at least some P's. P ís undistributed here. Therefore, in the absence of additional evidence, P may not be distributed in the conclusion, that is, we must not make SaP imply PaS, but only PiS. Thus SaP -> PiS. This is called conversion by limitation, that is, by limiting, or restricting, the (universal) quarrtity of the original proposition in contrast with simple conversion, where the given proposition and the converse have the same quantity. Lastly, SoP has no converse at all. To convert SoP into PoS would be to distribute, in the inference, a term (S) undistributed in the premise, and that is not permissible. The fact is that PaS may be true at the same time as SoP. For example, Some rectangles are not squares, yet Aü squares are rectangles, or Some Europeans are not Swedes, but All Swedes are Europeans. Hence SoP cannot imply PoS, which is the contradictory of PaS. Thus SoP has no converse. It should be noted that when theconverse of a given propositiori, is itself converted then in the case of E and I propositions we simply come back to the original pf oposition, while in the case of A propositions even that much is not achieved, for we get SiP instead of SaP. Therefore, in the case of conversion, as in the case of obversion, no new result can be obtained by merely repeating the process. The only way of obtaining new eductions is by applying the processes of obversion and conversion alternately, each to the result of the other. That, indeed, is the way in which the other eductions are obtained. Table of Principal Eductions Original S P Obverse S-P, Converse P S SaP SeP, PiS SiP SoP, PiS SeP SaP, PeS SoP SiP, None CONCEIVABLE EDUCTIONS All propositions imply ah obverse the terms of which are related to the terms of the original as S P, to S P, and that some propositions have a conveise, the terms of which are related to those of the original as P S to S P. Let us now consider in a purely abstract manner what other implications are conceivable involving the terms of a given proposition and their contradictories. The terms in question will be four in number, namely S, P, S,, P,, of which S and P represent the subject and predicate of the given prop osition. Let us omit merely tautological statements, such as S is S, or S is not S, and self-contradictory statements like S is not S, or S is Sr We then left with the following conceivable combinations of terms for Concervable additional education, namely, P S,, Pj S, P S, S, P, S Pr If we add to these the orig inal S P, the obverse S P, and the converse P S, we obtain th'e fcllowing table of conceivable combinations of terms of propositions in relation to any given proposition. 1. S P original proposition. obverse. converse. obverted converse. contrapositive. obverted contrapositive. inverse. obverted inverse. Of these combinations it will be seen that Nos. 4, 6 and 8 are each related to its immediately preceding combination (Nos. 3, 5, and 7 respectively) ih exactly the same way as the obverse (No. 2) is related to the original proposition (No. 1). They are accordingly called each the obverse of the preceding form (obverted con" verse, etc). If the converse can be obtained, then there is no difficulty in getting its obverted form, since every proposition has an obverse. Símilarly, if and when the contrapositive and inverse forms (Nos. 5 and 7) can be obtained, there will be no difficulty in determining their obverted forms (Nos. 6 and 8).
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