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The Logician © Avi Sion All rights reserved

A FORTIORI LOGIC © Avi Sion, 2013. All rights reserved.
CHAPTER 3  STILL MORE FORMALITIES
1. Understanding the laws of thoughtMany people regard Aristotle’s three ‘laws of thought’ – the laws of identity, of noncontradiction and of the excluded middle[1] – as rigid prejudices. They think these are just conventions, that some moronic old fellow called Aristotle had the bad grace to impose on the rest of us, and that we can just chuck ’em out at will. In each of my past works, I have tried to explain why these are fundamental human insights that cannot under any pretext be discarded. I would like to add a few more explanations in the present work. The laws of thought must not be thought of as mechanical rules, but as repeated insights of our intelligence. Every ‘application’ of these laws in a new context demands a smart new insight from us. We must in each new context reaffirm these laws, and use them creatively to deal with the complexities of the case at hand. In a fortiori logic, where new forms are encountered, and new problems need solutions, we can expect our intelligence and creativity to be called upon. We have already come across many contexts where subtlety was required. The distinction between a proposition like ‘X is Y’ and ‘X is R enough to be Y’ was one such context. Another was our development of a distinction between absolute terms (R and notR) and relative terms (R and notR). The laws of thought are ever present in logical discourse, but they must always be understood and adapted in ways that are appropriate to the context at hand – so they are not mechanical laws, but ‘smart laws’. The laws of thought have to repeatedly be adapted to the increasing complexity of discourse. Originally, no doubt, Aristotle thought of the laws with reference to indefinite propositions, saying that ‘A is B’ and ‘A is not B’ were incompatible (law of noncontradiction) and exhaustive (law of the excluded middle). In this simplest of contexts, these laws implied only two alternatives. However, when Aristotle considered quantified propositions, ‘All A are B’ and ‘Some A are B and some A are not B’ and ‘No A is B’ – he realized that the application in this new context of the very same laws implied three alternatives. From this example, we see that the subtleties of each situation must be taken into consideration to properly ‘apply’ the laws. They are not really ‘applied’; they are intelligently formulated anew as befits the propositional forms under consideration. We could say that the disjunction “Either ‘A is B’ or ‘A is not B’” refers to an individual subject A, whereas the disjunction “Either ‘All A are B’ or ‘Some A are B and some A are not B’ or ‘No A is B’” refers to a set of things labeled A. But then the question arises: what do we mean when we say that an individual A ‘is B’? Do we mean that A is ‘entirely B’, ‘partly B and partly not B’? Obviously, the mutually exclusive and exhaustive alternatives here would be: “Either ‘A is wholly B’ or ‘A is partly B and partly not B’ or ‘A is not at all B’.” It seems obvious that in most cases ‘A is B’ only intends ‘A is partly B and partly not B’ – for if ‘A is wholly B’, i.e. ‘A is nothing but B’ were intended, why would we bother verbally distinguishing A from B? Well, such tautologies do occur in practice, since we may first think of something as A and then of it as B, and belatedly realize that the two names in fact refer to one and the same thing. But generally we consider that only B is ‘wholly B’, so that if something labeled A is said to have some property labeled B, A may be assumed to be intended as ‘only partly B’. To give a concrete example: my teacup is white. This is true, even though my teacup is not only colored white, but also has such and such a shape and is made of such and such a material and is usually used to drink tea. Thus, though being this teacup intersects with being white, it does not follow that the identity of this individual teacup is entirely revealed by its white color (which, moreover, could be changed). With regard to classes, even though we may choose to define the class of all A by the attribute B, because B is constant, universal and exclusive to A, it does not follow that A is thenceforth limited to B. B remains one attribute among the many attributes that are observed to occur in things labeled A. Indeed, the class A may have other attributes that are constant, universal and exclusive to it (say C, D, etc.), and yet B alone serves as the definition, perhaps because it intuitively seems most relevant. Thus, to define concept A by predicate B is not intended to limit A to B. If A were indeed limited to B, we would not name them differently. These thoughts give rise to the logical distinction between ‘difference’ and ‘contradiction’, which calls forth some further use of ad hoc intelligence. When we say that ‘A and B are different’, we mean that these labels refer to two distinct phenomena. We mean that to be A is not the same as to be B, i.e. that Bness is different from Aness. It does not follow from this that No A is B. That is to say, even though A is not the same thing as B, it is conceivable that some or all things that are A may yet be B in some way. To say the latter involves no contradiction, note well. Therefore, the laws of noncontradiction and of the excluded middle cannot in this issue be applied naïvely, but only with due regard for the subtleties involved. We must realize that ‘difference’ is not the same as ‘contradiction’. Difference refers to a distinction, whereas contradiction refers to an opposition. Two propositions, say X and Y, may have different forms and yet imply each other. It is also possible, of course, that two propositions may be both different and contradictory. Another subtlety in the application of the laws of thought is the consideration of tense. Just as ‘A is B’ and ‘A is notB’ are compatible if they tacitly refer to different places, e.g. if they mean ‘A is B here’ and ‘A is notB there’, so they are compatible if they tacitly refer to different times, e.g. if they mean ‘A is B now’ and ‘A is notB then’. Thus, if a proposition is in the past tense and its negation is in the present or future tense, there is no contradiction between them and no exclusion of further alternatives. Likewise, if the two propositions are true at different moments of the past or at different moments of the future, they are logically compatible and inexhaustive. These matters are further complicated when we take into consideration the various modalities (necessity, actuality, possibility), and still further complicated when we take into consideration the various modes of modality (natural, extensional, logical). I have dealt with these issues in great detail in past works and need not repeat myself here. In the light of considerations of the categories and types of modality, we learn to distinguish factual propositions from epistemic propositions, which qualify our knowledge of fact. In this context, for instances, ‘A is B’ and ‘A seems not provable to be B’, or even ‘A is B’ and ‘A seems provable not to be B’, might be both true. One of the questions Aristotle made a great effort to answer, and had some difficulty doing, was how to interpret the disjunction: “Either there will be a sea battle tomorrow or there will not be a sea battle tomorrow”[2]. But the solution to the problem is simple enough: if we can truly predict today what will (or will not) happen tomorrow, it implies that tomorrow is already determined at this earlier point in time and that we are able to know the fact; thus, in cases where the fact is not already determined (so that we cannot predict it no matter what), or in cases where it is already inevitable but we have no way to predict the fact, the disjunction obviously cannot be bipolar, and this in no way contravenes the laws of thought. Nothing in the laws of thought allows us to foretell whether or not indeterminism is possible in this world. As a matter of fact, either now there will be the sea battle tomorrow or now there won’t be one or the issue is still undetermined (three alternatives). As regards our knowledge of it, either now there will be the battle tomorrow and we know it, or now there won’t be and we know it, or now there will be and we don’t know it, or now there won’t be and we don’t know it, or it is still undetermined and so we cannot yet know which way it will go (five alternatives). We could partially formalize this matter by making a distinction between affirming that some event definitely, inevitably ‘will’ happen, and affirming only that it just possibly or even very likely ‘will’; the former is intended in deterministic contexts, whereas the latter is meant when human volition is involved or eventually when natural spontaneity is involved. These alternatives can of course be further multiplied, e.g. by being more specific as regards the predicted time and place tomorrow. What all this teaches us is that propositions like ‘A is B’ and ‘A is not B’ may contain many tacit elements, which when made explicit may render them compatible and inexhaustive. The existence of more than two alternatives is not evidence against the laws of thought. The laws of thought must always be adapted to the particulars of the case under consideration. Moreover, human insight is required to properly formalize material relations, in a way that keeps our reading in accord with the laws of thought. This is not a mechanical matter and not everyone has the necessary skill. Another illustration of the need for intelligence and creativity when ‘applying’ the laws of thought is the handling of double paradoxes. A proposition that implies its contradictory is characterized as paradoxical. This is a logical possibility, in that there is a quick way out of such single paradox – we can say that the proposition that implies its contradictory is false, because it leads to a contradiction in knowledge, whereas the proposition that is implied by its contradictory is true, because it does not lead to a contradiction in knowledge. A double paradox, on the other hand, is a logical impossibility; it is something unacceptable to logic, because in such event the proposition and its contradictory both lead to contradiction, and there is no apparent way out of the difficulty. The known double paradoxes are not immediately apparent, and not immediately resolvable. Insights are needed to realize each unsettling paradox, and further reflections and insights to put our minds at rest in relation to it. Such paradoxes are, of course, never real, but always illusory. Double paradox is very often simply caused by equivocation, i.e. using the same word in two partly or wholly different senses. The way to avoid equivocation is to practice precision and clarity. Consider, for instance, the word “things.” In its primary sense, it refers to objects of thought which are thought to exist; but in its expanded sense, it refers to any objects of thought, including those which are not thought to exist. We need both senses of the term, but clearly the first sense is a species and the second sense is a genus. Thus, when we say “nonthings are things” we are not committing a contradiction, because the word “things” means one thing (the narrow sense) in the subject and something else (the wider sense) in the predicate. The narrow sense allows of a contradictory term “nonthings;” but the wider sense is exceptional, in that it does not allow of a contradictory term – in this sense, everything is a “thing” and nothing is a “nonthing,” i.e. there is no “nonthing.” The same can be said regarding the word “existents.” In its primary sense, it refers to actually existing things, as against nonexisting things; but in its enlarged sense, it includes nonexisting things (i.e. things not existing in the primary sense, but only thought by someone to exist) and it has no contradictory. Such very large terms are, of course, exceptional; the problems they involve do not concern most other terms. Of famous double paradoxes, we can perhaps cite the Barber paradox as one due to equivocation[3]. Many of the famous double paradoxes have more complex causes. See for examples my latest analyses, in Appendix 7.4 of the Liar paradox, and in Appendix 7.5 of the Russell paradox. Such paradoxes often require a lot of ingenuity and logical skill to resolve.
2. QuantificationMany people, even trained logicians among them, find the field of a fortiori logic difficult because of the variety of issues of quantity in the terms. We have already dealt with the issue of proportionality, and the differences between purely a fortiori argument, pro rata argument and a crescendo argument. Here, we will deal with three distinctions between terms: that between constants and variables; that between individuals and classes; and that between distributive and collective terms. Terms may be constants or variables We have earlier presented and validated various moods of a fortiori argument, without specifying whether the terms they involve are constants or variables. The reason is simply that they may be either. Although initially all the terms involved might be thought of as constants, we eventually realize that some or all of them may equally well be variables, provided the patterns of interrelationship between the variables continue to fit in to the formulae required for valid a fortiori argument. That is, provided each value of one variable is related to the corresponding value of the other variable in accord with the said formulae; which simply means that the various values of variables are effectively alternative constants. This should be obvious. But let us consider an example the following positive subjectal argument:
Here, P, Q, R, and S may all at first be assumed to have individual values, i.e. to be constants. But it is conceivable that P and Q may have different values of R at different times or places or in different circumstances or in different instances. That changes nothing in our argument, provided the relation ‘Rp > Rq’ remains unaffected for every pair of values of the variables Rp and Rq. If when variable Rp has value Rp1 and variable Rq has value Rq1, ‘Rp1 > Rq1’, and if the same is true for every other pair of values, like Rp2 and Rq2, Rp3 and Rq3, etc., then the a fortiori argument holds. It is also conceivable that Rp is constant while Rq is a variable, or vice versa, provided ‘Rp > Rq’ remains true. In all such cases, though the a fortiori argument seems like one statement, it may also be viewed as a summary of many similar statements[4]. Again, the predication ‘Q is R’ in the minor premise may intend the term R (which here more precisely means Rq) as a constant or as a variable. In the latter case, so long as every value of Rq – namely, Rq1, Rq2, Rq3, etc. – is sufficient to imply Q to be S (whether S is here a constant, or itself also a variable), the minor premise as a whole remains true. The conclusion then (given also the major premise, of course) naturally follows, whether Rp is a constant or a variable (with values Rp1, Rp2, etc.). Note here again that it is possible for Rq to be a variable while Rp is a constant, or vice versa, or they might both be constants or both variables. The important thing to note here is that whatever value S has in the minor premise, it must be repeated in the conclusion. Where S is variable in the former, it is identically variable in the latter. The a fortiori argument as such cannot change the value of S in the transition from the minor premise to the conclusion; only the very same value of S can be inferred a fortiori. This issue should not be confused with that of ‘proportionality’ of conclusion, which we dealt with in the preceding chapter (2.22.3). As we saw there, the value of S in the conclusion may differ from that in the minor premise, if and only if we have an additional premise capable of justifying such change. Terms may be individuals or classes We have so far treated the major and minor terms (P and Q) of subjectal a fortiori arguments as indivisible units. Obviously, P and Q may be named or pointedto individuals; e.g. Tom is richer than Harry, or this man is richer than that man. Also, obviously, P and Q may be whole classes; e.g. gold is worth more than silver, meaning any amount of gold and any equal amount of silver; or again, humans are more intellectual than other animals, meaning all humans are more intellectual than all other animals. The latter universal major premise could conceivably give rise to an a fortiori argument in which the minor premise is particular (or even singular) and the conclusion is general, note well. This means the following positive and negative forms are valid:
That is, given the major premise “All P are more R than all Q,” it follows that “if there is any Q that is R enough to be S, then all P are so too;” and inversely, “if there is any P that is R not enough to be S, then no Q can be so.” Note well, “all P” here refers to each and every P, and “all Q” to any and every Q; consequently, the minor premises might be particular, i.e. refer to some Q or P, or even to just one specified or unspecified Q or P. This is obvious enough. If the major premise is entirely particular, we can still draw a particular conclusion provided the minor premise is general. That is to say, the following positive and negative forms are also valid:
These moods are valid because there is still in them a guarantee of overlap between the instances referred to in the major and minor premises. Note that if we could pinpoint instances, we could treat particulars as generalities; i.e. the following would be valid moods:
When we have a mix of quantities in the major premise, we can for the same reason in certain cases draw a valid conclusion as follows:
This should cover all cases of subjectal argument. Moods not above mentioned as valid are intended as invalid. All this can be formally demonstrated in the usual manner. Note well that the quantities intended in the major premises are not intended as “one for one” correspondences between P and Q. That is, when we say “All (some) P are more R than all (or some) Q,” we do not specifically mean that “for each instance of P referred to, there is a corresponding instance of Q such that this P is more R than that Q” (more on this topic presently); we are dealing with the classes in bulk (though “one for one” may be applicable in specific cases). As regards predicatal a fortiori argument, the situation is very different since major and minor terms, P and Q, are predicates and not (as above) subjects, and therefore cannot properly be quantified. On the other hand, we can here quantify the subsidiary term, S, which is here a subject and not (as above) a predicate. The rule in this context is simple enough: the quantity in the conclusion is the same as that in the minor premise. If the minor premise (whether positive or negative) addresses some specified, some unspecified or all instances S, then so will the conclusion do. Again, this is obvious enough:
So much for copulative argumentation. Regarding implicational a fortiori argument, the theses involved may in principle contain any quantity or mix of quantities, without affecting the given implications; so there is nothing much to say about it. That is to say, though it is of course conceivable that in some cases the implications originally depend on the quantities involved, it remains true that once we are given the implications as premises the quantities in the theses concerned become irrelevant to the drawing of a conclusion from them. In such cases, quantities count as content, not as form, as regards the a fortiori process as such.[5] Correspondences between terms. A special case of a fortiori argument that ought to be mentioned is when the terms involved are tied together by some correspondence. For example, the argument “all parents (P) are older (R) than their children (Q); therefore, if the children are old enough to vote, so are their parents” – it is clear that the major and minor terms refer to parents and their respective children, and not just to any children of any parents. With this in mind, we can construct the following valid moods in general terms:
In the above two subjectal moods, the correspondence is between P and Q. There could also be correspondence between P and Q and the subsidiary term (S), as in the following two predicatal moods.
Sometimes, additionally or alternatively, the middle term (R) is tied in this sense. For example, we may say that Noah was more righteous in his generation than many an absolutely more righteous man was righteous in his own respective generation. Valid moods can be constructed with such a variable middle term if this is done carefully – avoiding the fallacy of two middle terms by making sure the effective middle term throughout the argument is a generality. Clearly, in all such cases, the qualification “its corresponding” must be considered as part of the terms it qualifies, even though it is for emphasis stated outside them. Needless to say, correspondences between terms are special cases; usually the terms involved are not tied together in this manner.
Terms may be distributive or collective What has been said above about quantification concerns distributive terms – i.e. cases where the quantity all or some or whatever refers to the instances intended each one singly. The situation is very different where collective terms are intended, i.e. where the quantity is used to signify that a number of specified instances together make up a unit. The latter situation can cause havoc in a fortiori argument, if misunderstood. Consider the following examples. With terms intended distributively, a fortiori argument is always possible: e.g. “If (any of) these five men are rich enough to pay for this object, then this one (of them) is rich enough.” However, when the intent is collective, we cannot readily infer a fortiori. Given that five designated men together can lift a certain weight, it obviously does not follow that three of them are strong enough to do so; or: given that three designated men together can go through a certain door, it obviously does not follow that five of them are thin enough to do so. Thus, note, we cannot formulate an a fortiori argument in either direction – i.e. neither from five to three nor from three to five. Clearly, what is involved here is an issue of upper and lower limits, i.e. of maxima and minima. A quantity can be enough, or too much or too little. We can formalize our above examples as follows. Suppose ‘X’ is a designated group of individuals, ‘n’ refers to the number of them involved, and ‘Y’ is a predicate concerning them collectively. Given the lower limit ‘Less than nX together cannot be Y’, it follows that ‘nX together are Y’ does not imply ‘less than nX together are Y’. Similarly, given the upper limit ‘More that nX together cannot be Y’, it follows that ‘nX together are Y’ does not imply ‘more than nX together are Y’. In some cases, both limits are set – neither more nor less than a stipulated quantity is allowed. Moreover, note that collective terms are very specific in their intents: while these five strong men are able to jointly lift said weight, five (or even more) other (weaker) men may not be able to do so; or again, while these three thin men are able to jointly go through said door, three (or even less) other (fatter) men may not be able to do so. More formally put, given a limit for ‘nX1’, i.e. a number n of designated individuals labeled X1, it does not follow that the same limit obtains for ‘nX2’, i.e. a number n of other designated individuals labeled X2. Collective terms can be very tricky in yet other ways. For instance, these five men may be able to lift a certain weight together, if they all have a handle to take hold of it, but not if they have a hard time grabbing it. Or these three men may be able to go through this door at once, if they are stacked one on top of the other, but not if they try to simply walk through it simultaneously. Thus, a collective term is not fully defined by numbers (like n) and designated individuals (like these men), but may involve more complex conditions. Note this well. Also to keep in mind, a collective may have very different properties than its components; and conversely, its components may have very different properties than a collective. If we assume that what is true of some or all of its components is true of a collective, we commit the ‘fallacy of composition’. If we assume that what is true of a collective is true of some or all of its components, we commit the ‘fallacy of division’.
As regards application of the concept of existential import to the commensurative and suffective propositions constituting a fortiori argument, my position is simply that the existence of subjects is generally assumed. See Appendix 7.2 for a fuller presentation of this position.
3. A fortiori through inductionAs we have shown, a fortiori argument is essentially a deductive argument, one that can readily be formalized and validated. There is no denying that. Yet many commentators persist in erroneously regarding it as an essentially inductive argument, no doubt simply because they have not perceived its deductive form. We shall now examine the various ways induction may be involved in the formulation and justification of an a fortiori argument. Induction in general. First, we must distinguish formal induction from material (or informal) induction. An inductive argument can be considered as part of formal logic if the relation between its premise(s) and conclusion can be expressed in formal terms, i.e. with reference to symbols acting as placeholders for any material terms or propositions. If we cannot treat the inductive argument in purely abstract ways, it constitutes a material (or informal) one. Many people think that only deduction can be treated formally, but the truth is that much inductive reasoning can be formulated and justified in formal terms. There are several types of formal induction. a. The most important type of formal induction is factorial induction. It is the most important type, because it is the most distinct from deduction and the most reliable. I have treated this topic in great detail in my work Future Logic (part VI). Factorial induction may be described in the following general terms: given a premise p, or a set of premises p1 and p2 and…, then: if we have a single necessary implication c, then it is the deductive conclusion; whereas, if we have two or more logically possible implications c1 or c2 or…, and for some reason one of them (say, c1) is logically preferable to the other(s), then it is the adopted inductive conclusion. The latter conclusion may of course change as new data is discovered which changes the premises; or c1 may be found empirically untrue, in which case the next most probable inductive conclusion (say, c2) is selected. The conclusions c1, c2, c3, etc. are called the factors of the premise p, or of the set of premises p1, p2, p3, etc. The most probable conclusion is called the strongest factor. To understand this process, let us look at the simplest example: generalization of a particular proposition. A deductive argument is distinguished by having a single conclusion regarding certain terms (say, X and Y) from the given premise(s); for example, that ‘All X are Y’ (symbol A) implies ‘Some X are Y’ (symbol I) is a deductive argument (an eduction). On the other hand, an inductive argument has in principle two or more conclusions; for example, ‘Some X are Y’ (I) implies either ‘All X are Y’ (A) or ‘Some X are Y and some X are not Y’ (IO) is (part of) an inductive argument[6]. But that is not all. The argument ‘I implies either A or IO’ gives us a choice of inductive conclusions, but it does not give us any advice as to which of the two conclusions to opt for (or at best, it gives us a 5050 probability for each conclusion, by virtue of symmetry). However, through the inductive act of generalization, we are able to prefer the conclusion A to the conclusion IO, by arguing that whereas A has the same positive polarity as given in the premise, IO involves an additional negative polarity not at all found in the premise. Thus, A requires less assumption than IO. For, though it is true that both A and IO claim quantitatively new information regarding the whole class, nevertheless IO introduces a claim to negative polarity not at all found in the premise I, while A remains entirely consistent in respect of polarity with the premise I. Thus, given I, alternative A is more probable than IO, because the latter is more speculative than the former.[7] Thus, though our inductive argument starts by yielding a disjunction of conclusions, what makes it truly useful is that we are able to formally (before reference to any content) select one conclusion in preference to the other. It is the latter feature that allows us to refer to the argument as specifically inductive and distinguish it from deductive argument. An inductive argument allows deductively for two or more conclusions, but additionally (in most cases) provides us some formal reason(s) for preferring one of these conclusions over the others. Such induction is thus not a matter of guesswork or of purely material considerations, but is part of ‘formal logic’ in its own right. What we have described above is called generalization from I to A. This process depends for its validity, note well, on two premises. First, that I is true; this might be determined empirically (i.e. by observation through the senses, as regards material phenomena, or by observation through the proverbial “mind’s eye,” as regards mental phenomena) or by deduction from previous inductions or deductions. Second, that there is no known evidence for claiming that O is true. Given these two premises together, and not just one of them alone (as some people erroneously think), we can inductively conclude that A is true. It is obvious that if I concerns positive phenomena and is truly empirically established, it is very unlikely to ever turn out to be untrue. However, since I is usually made up of concepts and not just percepts, it is not inconceivable that it be later found inaccurate and should be abandoned; though, to repeat, that is relatively rare. On the other hand, the nonknowledge of O is not a definitive denial, and may more readily be conceived as being overturned. If upon further scrutiny we discover that in fact ‘Some X are not Y’, we must imperatively correct our previous judgment, and conclude with IO instead of A. This is called particularization. This is also a valid process within formal inductive logic. Particularization does not invalidate generalization, note well, but complements it by ensuring that all judgments are subject to correction if the need arise. One more thing needs clarifying here: what constitutes inconclusiveness. If we lack, and cannot currently infer, any information regarding the relation of X and Y, so that the disjunction ‘either A or IO or E’ is true, we have effectively no conclusion from any known premises. If now, whether by observation or inference, we discover that I is true, this eliminates the E alternative of that disjunction, and we are left with the narrower disjunction ‘either A or IO’. This is, as we have seen, a conclusion of sorts[8]. Thus, conclusiveness is distinguishable from inconclusiveness with reference to the range of alternatives implied. If the range is unlimited, there is no conclusion; if it is partly limited, we have a set of inductive conclusions; if it is limited to one conclusion, we have either a deduction or a preferred induction. Another approach to factorial induction is with reference to individual cases. Starting with any particular X, we know from the laws of thought that it is either Y or notY. Having found that some other X are Y, and not having found any other X that are notY, we can predict with considerable certainty that this here X is Y. This is another way of presenting the above detailed generalization from I to A. Similarly, having found that some other X are notY, and not having found any other X that are Y, we can predict with considerable certainty that this here X is notY. This is generalization from O to E. If on the other hand, from the very beginning we find that some X are Y and some X are not Y, we obviously cannot generalize either to A or to E. If we have already generalized to one of the latter universal forms, we must retract and admit IO instead. In that case, what can we predict concerning an individual case of X, about which we have no information as yet as to whether it is Y or notY? The way to resolve this issue is to compare the frequencies of cases of X that are known to be Y and cases of X that are known to be notY. If it looks as if the frequencies are about the same on both sides of the contingency, the probability that this here X is Y and the probability that it is notY are even, and no inductive conclusion can be drawn; since the factors have the same weight, they remain formally indistinguishable. But if one side is more frequent than the other, we can identify it as the strongest factor. That is, we can conclude: ‘this X is most probably Y and less probably notY’. In some contexts, though not all, the probabilities involved can be precisely quantified. The process just described may be described as statistical induction, because it relies on approximate or precise enumeration of known cases. If most (i.e. more than half of the) known instances of X are Y, then this unknown instance of X may be assumed to be Y. If most known instances of X are notY, then this unknown instance of X may be assumed to be notY. If the known instances of X are evenly distributed, with as many Y as notY, then this X being Y and this X being notY are equally probable, which means nothing definite can be said about the relation of X and Y or notY. Probabilities are of course subject to change as more instances of X are otherwise identified (empirically or by other arguments) to be Y or to be notY, so that our preferred conclusion may need updating. In this light, generalization may be viewed as the special case where all known instances have the same polarity, so that we may assume the universal proposition A or (as the case may be) E to be true. We can also view the statistical treatment of contingencies, i.e. the movement from ‘Most known instances’ to ‘Most instances’ as a sort of generalization, in that we reiterate information available for known instances to unknown instances. b. The second most important type of formal induction is adduction, also known as ‘the scientific method’. This form of reasoning is an offshoot of the logic of hypotheticals, in contrast to the preceding form, which is an offshoot of the logic of categoricals. The following are the two valid moods of apodosis, or hypothetical deduction:
These moods are respectively called ‘affirming the antecedent’ (modus ponens) and ‘denying the consequent’ (modus tollens). These moods are used, in adduction, to connect new theories to given data or to make new predictions from old theories and, respectively, to exclude new theories unconnected to the data or reject old theories with false predictions. They are valid because the major premise, ‘If P, then Q’ (i.e. if the antecedent P is true, then the consequent Q is true), means that ‘the theses P and notQ cannot both be true’. From which it follows that if P is true, notQ must be false; and if notQ is true, P must be false. The following the two moods are, on the other hand, invalid:
Given the major premise, we cannot infer that P is true from the fact that Q is true, or that Q is false from the fact that P is false. These moods are invalid as apodoses, i.e. as deductive arguments. But they still have some utility as inductive arguments. We can say regarding the positive mood that the evident truth of Q in the context of ‘If P, then Q’ gives some probability to the hypothesis that P is true; for it could be that Q is true because P is true. Likewise, we can say regarding the negative mood that the evident falsehood of P in the context of ‘If P, then Q’ gives some improbability to the hypothesis that Q is true; for it could be that P is false because Q is false. Such arguments are of course very tenuous taken individually. But if we have many, and still more, cases of evidence like Q confirming hypothesis P, thesis P becomes more and more credible. Likewise, if we have many, and still more, cases of evidence like notP undermining hypothesis Q, thesis Q becomes more and more incredible. Thus, these forms of argument are used in adduction, respectively, to strengthen or weaken existing theories. Strengthening and weakening do not, of course, mean proving or disproving; they just refer to cumulative credibility. All such cumulative credibility can fall apart instantly, if even a single instance is found that belies the thesis concerned. That is, if P is repeatedly confirmed through evidence like Q, and an instance of Q is found which implies notP, the hypothesis that P is true must be rejected. Likewise, if Q is repeatedly undermined through evidence like notP, and an instance of notP is found which implies Q, the hypothesis that Q is false must be rejected. How definitive such theory ‘rejection’ is depends on how empirical the reason for it is. If the reason is truly empirical, the rejection is definitive; whereas, if the reason is itself merely theoretical, the rejection is more mooted. Such thoughts can be compared to generalization and particularization. When we repeatedly find evidence that confirms our hypothesis, and find no evidence to the contrary, we generalize and assume all future evidence will go the same way. When and if we eventually encounter evidence to the contrary, we particularize that generality, and adopt a more contingent stance, if not the very opposite idea. Here again, given a contingency, we may still for statistical reasons prefer a thesis over its negation, or we may find the evidence evenly distributed. Adduction also, like generalizationparticularization, uses disjunctive argument. There may be, and usually there are, two or more hypotheses that could explain the accumulated evidence. In such case, we compare the probabilities for each of the alternative hypotheses, and we opt for the strongest. Other considerations may affect this judgment, such as the simplicity or complexity of a hypothesis; such considerations may be viewed as adding or subtracting credibility to the hypothesis, i.e. as increasing or diminishing its net probability. Thus, the disjunction of possible explanations of varying probability may be said to be a list of the factors of the data to be explained, ordered according to their relative strengths. So adduction may be viewed as a special case of factorial induction. It is also true that we may regard generalization and particularization as special applications of theory confirmation and elimination, i.e. of adduction. When we generalize from I to A, say, we are effectively regarding the instances subsumed by I as confirmations of the hypothesis A. When we particularize, we reject hypothesis A, and opt for the more empirical thesis IO. We then look towards the relative frequencies of the two polarities, and formulate a new hypothesis that ‘Most X are Y’ or that ‘Most X are not Y’ or that ‘As many X are Y as are notY’, as the case may be. This in turn might be modified, as new data comes in. Thus, adduction and generalizationparticularization are very closely related reasoning processes. In this context, quite parenthetically, I would like to say a word or two about the term ‘abduction’, which many people confuse with and prefer to the term ‘adduction’. The term ‘abduction’ refers to the assumption of a hypothesis that explains available evidence[9]. From this definition it is clear that the term refers to a positive argument: If P, then Q; and Q; therefore, P. There is notably no mention here of the possible negative aspect, i.e. the possibility that hypothesis P might entail some false prediction, such as Q2, so that it is eliminated by the argument: If P, then Q2; and notQ2; therefore, notP. Nor is the competitive aspect brought out, i.e. that P may be supplanted by some other hypothesis P2 which is found more probable. Abduction is thus very naïve guesswork. Adduction, on the other hand, refers to the full range of the scientific method: repeatedly testing a hypothesis by means of new evidence, which may turn out to be counterevidence, and comparing competing hypothesis, to opt for the most likely, as above described. Thus, the two terms should not be confused. c. The third most important type of formal induction is induction based on deduction. What this refers to is the following: suppose we have a duly validated deductive argument, say ‘P and Q implies R’. Then we can logically rely on the arguments: ‘If P and probably Q, then equally probably R’, or ‘If probably P and Q, then equally probably R’, or again ‘If probably P and probably Q, then probably R’. In these arguments, the degrees of probability of the premises are passed on to the conclusion, by multiplication. Thus, if one of the premises is 100% probable and the other 50%, the conclusion is 50% probable; whereas if both premises are 50% probable, the conclusion is only 25% probable. This is reasoning mathematically[10]. The important point to note is the deductive underpinning of such induction. The inductive form apes a deductive form. If the deductive form intended is valid, then the inductive form is a credible induction. But if the deductive form intended is not valid, then the inductive form is not a credible induction. The credibility of the induction is based on the credibility of the deduction it imitates. Thus, to give an example: given that ‘All Y are Z and this X is Y, therefore this X is Z’ is deductively valid, we can refer to a parallel inductive argument with, say, ‘this X is probably Y’ as the minor premise and ‘this X is probably Z’ as the conclusion. But, given the same premises, we cannot conclude ‘this X is probably not Z’ because such inductive conclusion has no deductive underpinning. It may well be true for other reasons that ‘this X is probably not Z’ – but this is not an inductive conclusion from the said premises. Thus, an ‘induction based on deduction’ is very different from an unsupported statement of logical possibility. It is a type of formal induction, which relies for the credibility of its conclusion on the conclusion having a form that is validated by deduction. The relation between premises and conclusion is thus warranted. The implication involved is not merely probable but certain. The process remains certified, even if the premises and therefore the conclusion are only probable. In truth, most arguments we encounter in practice are of this sort, since most of our material knowledge is somewhat open to doubt. Our deductions naturally pass the doubtfulness of the premises onto the conclusion. But the process of deduction is unaffected: once shown logically necessary, it is reliable, no matter what the reliability of the propositions involved. d. The last and most unreliable type of ‘induction’ is precisely inductive inference that relies on a doubtful deductive process. Here, the basis is only that ‘P and Q probably implies R’ (rather than ‘P and Q implies R’) – so that, whether the premises are certain or only probable, the conclusion is at best only probable, if at all credible. This may be called induction in only a very loose sense of the term, since by definition we have no formal justification for drawing the putative conclusion from the given premises. Does this happen? Yes, it does. And people indulge in it because it seems to them ‘better than nothing’. Someone may have information that suggests he might be able to formulate a certain deductive argument, but as yet he does not have enough information to actually formulate that argument – so he refers to a merely ‘probable implication’. There is some truth in that since, indeed, the argument might eventually crystalize; but until it does, the expectation that it might must be regarded with the utmost skepticism. A ‘probable implication’ is, strictly speaking, no implication at all. Thus, though we have premises, we have no real process through which to elicit a conclusion from them. The conclusion’s ‘probability’ is thus not much greater than the ‘probability’ of anything we imagine offhand. This is comparable to having a few pieces of a jigsaw puzzle and trying to guess what the picture it comes from might be. It is in such contexts that the word ‘abduction’ is most appropriately used. And it is in such contexts that the skills of statisticians are called for, to find ways to select the most credible hypothesis from miniscule indices. The validity of such ‘inference’ is obviously very relative; in absolute terms, it is of doubtful worth. It should be stated that the basis of all inference is analogy and disanalogy, although of course there are more sophisticated and reliable inferences and more simplistic and unreliable ones. Our deductive and inductive capacities all depend on, among other cognitive processes, the cognition of similarities and differences. If appearances did not seem ‘similar’ in various respects and ‘different’ in others, we could not perceive or conceive any continuity in identity in any individual object across time, or be able to class it with some other objects and differentiate it from yet other objects. Everything would seem completely different or completely the same. We could not claim any individual subjects to think or talk about, nor any abstractions to predicate of them. We would have no particular propositions to generalize, and all the more no general propositions. Nothing could be related in any way, positively or negatively, to anything else. There could be no analogy and no disanalogy. Abstract theories could not be formed or tested, since nothing could be claimed repetitive and distinguishable. In short, any attempt to deny human ability to cognize similarities and differences, and thence deductive and inductive processes, is a selfdefeating proposition. Induction in a fortiori logic. Now, let us illustrate the above four types of induction with reference to a fortiori argument. Consider an a fortiori argument, say the positive subjectal mood:
Put it this form, the argument takes for granted that the premises are true, so that the truth of the conclusion follows necessarily. However, in practice, more often than not, there is some degree of uncertainty in the premises, which is transmitted to the conclusion. Thus, insofar as the conclusion has some material uncertainty, it may be said to be somewhat ‘inductive’. But this induction is quite significant and reliable insofar as the deductive process above described is certain, note well. The following is a general statement of the corresponding inductive argument:
The validity of this argument as induction is entirely based on the validity of the deductive argument it is modeled on. It is not an arbitrary construct designed to give an illusion of inference – it is inference. The probability of the conclusion (z%) is presumably a product of the probabilities of the premises (x% and y%), which may each range from > 0% to 100%; thus, we can say: z/100 = x/100 * y/100. When x% and y% both equal 100%, so does z. If x and/or y = 0% (i.e. if a premise is false), z = 0% (which means not that the conclusion is proved false, but that it cannot be determined from these premises). We may be able to pinpoint more precisely where the uncertainties in the premises and conclusion lie, by looking at the components that make up each proposition. Thus, as we have seen, the major premise “P is more R than Q” is composed of three propositions, to wit: P is Rp, Q is Rq, and Rp > Rq. If P and Q are individuals, we may be assuming by generalization that they are all the time respectively Rp and Rq; or that the R value of P (Rp) is always greater than the R value of Q (Rq). If P and Q are classes, we may have generalized their stated relation from some instances to all instances. Similarly with regard to the minor premise: as we have seen, it is composed of four propositions, viz. Q is Rq, Whatever is at least Rs is S, Whatever is not at least Rs is not S, and Rq ≥ Rs. Here again, generalization may be involved in the claim that Q is Rq, whether Q is an individual or a class; furthermore, the two clauses relating Rs to S positively and negatively, and the clause comparing Rq and Rs all assume generalities. Thus, beneath the surface discourse, we very probably rely on a number of inductions, which may have proceeded directly by generalization from experience, or indirectly by deduction from such generalities, or via adductive reasoning, or by means of induction based on deduction, or even by means of induction not based on deduction. We can very well illustrate ‘induction not based on deduction’ with reference to a fortiori argument as follows. Instead of the above positive subjectal forms, someone might propose the following arguments as a deductive a fortiori (on the left) and the corresponding inductive a fortiori (on the right):
Notice that the clause “R enough to be” is missing in both the minor premise and conclusion of these two arguments. This admittedly often occurs in practice, although the missing clause may usually be taken as tacitly intended. If it is not tacitly intended, but omitted out of ignorance of its being a requisite for validity, or as a sophistic attempt to mislead, the argument is inevitably deductively invalid – that is to say, there is no way for us to formally prove that the putative conclusion follows from the given premises. The process might superficially seem somewhat credible, because its minor premise and conclusion partly resemble the minor premise and conclusion of valid a fortiori argument. But this is of course an illusion; in fact, the conclusion is not implied by the given premises. If the alleged deductive underpinning of the inductive argument is invalid, then the inductive argument is of course all the more so. Nevertheless, some people look upon such arguments as inchoate a fortiori, attempts at argument which might eventually crystalize into a fortiori form, at least inductive and then maybe even deductive. But we must keep in mind that the term ‘inductive’ here has its most disreputable sense. It does not refer to any actual inference, but only to an imaginary ‘possible’ inference. The excuse often given in such circumstances is that it is ‘material’ or ‘informal’ inference. But what does this mean? The suggestion is that we are able to intuit logical relations ad hoc, without reference to logical principles. I agree we are able to do so; indeed, I believe all general logical principles are derived from particular logical intuitions. However, most appeals to ‘material implication’ are in fact admissions of the speaker’s inability to elucidate the ‘formal implication’ underlying it – i.e. they are excuses for ignorance or laziness. For someone else, more skilled in logic and informed regarding the subjectmatter, may very often be able to state the formal sources of the material intuition – or alternatively to demonstrate it to be invalid. While a fortiori argument is often best characterized as inductive rather than deductive, for the above stated reasons, a crescendo argument is very often best so characterized, because of the additional problems involved in affirming proportionality, and all the more so in affirming some quantitatively specific proportionality. As we have seen, an a crescendo argument consists of an a fortiori argument combined with a pro rata argument, the two together yielding a ‘proportional’ conclusion. For instance, the positive subjectal form is:
As can be seen, the third premise, viz. “S varies in proportion to R” is rather vague, and can only justify the vague conclusion that P is not merely R enough to be S, but even R enough to more than S. Furthermore, the conclusion is at best probable, since the additional premise about proportionality is, of course, normally known by induction, and so is only probably true (say, to degree g%) rather than absolutely certain. Thus, here our formula for calculating the probability of the conclusion would be z/100 = x/100 * y/100 * g/100, where symbols x, y and z have the meanings already above assigned them. The inductive status of a crescendo argument needs to be further emphasized when the additional premise is not merely that “S varies in proportion to R,” but more specifically that S varies in proportion to R “in accord with such and such formula, say S = f(R),” when the formula concerned is natural (rather than conventional). The formula allows us to calculate a precise value for Sp (i.e. S in relation to P) given a precise value for Sq (S in relation to Q). Such a formula is of course conceivable, and we are often able to produce one. But of course, such a formula is not easy to establish with certainty (unless merely conventional, as happens – e.g. a price list for products of different sizes). Very often, scientific experiments are necessary. We extrapolate from a number of correspondences between the values of the variables concerned. We propose a summary formula that allows us to predict untested values from the pattern suggested by past events. Thus, the formula constitutes a scientific hypothesis, which is relied on until and unless we come across some serious hitch in its application. Alternatively, the formula might be deduced from more general principles, which have earned our respect over time. But even then, it is still inductive, insofar as the general principles it is based on are themselves products of induction. Thus, in any event, the mathematical formula referred to must be admitted to be probable rather than certain. Thus, we must remain aware that a crescendo argument is even more likely, than purely a fortiori argument, to deserve to be viewed as inductive. Needless to say, the same can be said for all forms of a crescendo argument, besides the example given above. Nevertheless, in practice, many a crescendo arguments may be viewed as effectively deductive, because the proportionality claimed seems pretty obvious and straightforward – all the more so if the putative conclusion is vague rather than precise. Their validity is so probable that we may take it for granted. But very often, in everyday discourse, we come across a crescendo claims that are nowhere substantiated, and even that would be hard to substantiate if we tried. It is best to say in such cases that the speaker is proposing an inductive a crescendo argument, meaning that he is not so much deducing the conclusion by a crescendo from established premises but proposing his conclusion in the framework of a possible a crescendo development. That is, he is not so much inferring the putative conclusion as proposing it, although his proposition is not entirely arbitrary but comes embedded within a larger context.
4. Antithetical itemsa. We saw earlier that arguments with antithetical middle items, such as: “If Q, which is not R, is S, then, all the more, P, which is R, is S,” or “If S is P, though P requires R, then all the more S is Q, which does not require R,” or their negative equivalents, or their implicational equivalents, all of which often occur in practice[11], can readily be assimilated into standard forms by viewing their middle items, R and notR, as inhabiting a common continuum, which is labeled R (or notR, as convenient). In this perspective, the absolute terms R and notR are both seamlessly included in a more expansive relative term R (or notR), and a standard form of a fortiori argument (or similarly, of a crescendo argument) can be formulated instead of the more awkward (i.e. difficult to formally validate) formula commonly given. In this way, the standard forms are brought to bear to validate the said nonstandard forms, which (to repeat) are often used in common discourse. However, it should be made clear that the standard forms we thus construct are based on the information provided in the said nonstandard forms. That is to say, it is thanks to the information in the latter that we can put together the major premise of the former, which makes possible inference of the conclusion from the minor premise. In subjectal argument, given that P is R and Q is not R, we can say obviously that P (for which R > 0, i.e. a positive quantity of R) is more R than Q (for which R ≤ 0, i.e. a zero or negative quantity of R) is. Again, in predicatal argument, given that R is required to be P and R is not required to be Q, we can say obviously that More R is required to be P (for which R > 0) than to be Q (for which R ≤ 0). Thus, there are forms of a fortiori argument (and similarly, a crescendo argument) which explicitly or implicitly involve a middle item R with contradictory or contrary values in relation to the major and minor terms. Such arguments can be formally validated through standardization, by replacing the absolute items R and notR with an allencompassing but relative middle item R (or notR, as convenient), which ranges in value from any negative or zero absolute value (= notR) to any positive absolute value (= R). Note well, the validity of such arguments is not affected by the occurrence of antithetical middle items. Let us now pursue this theme of antithetical terms or theses further, and investigate its application to major and minor items and/or to the subsidiary items. b. A fortiori argument (whether pure or a crescendo) is sometimes developed in relation to antithetical major and minor items. Antithetical means contrary or contradictory. In copulative reasoning, the positive subjectal form of such argument is as follows:
Clearly, X and notX are here both R, with different values of R, the value of the first being superior to that of the second. We can obviously construct a similar negative subjectal argument. We can likewise formulate a positive predicatal argument, as follows:
And we can obviously construct a similar negative predicatal argument. Needless to say, in all these argument forms, X and notX could just as well switch roles and be Q and P, respectively. The fact that the major and minor terms (whether X and notX, or notX and X) are contradictory does not affect the validity of the argument, since it has standard form. It is just a special application of a fortiori, occasionally found in practice[12]. Similar comments can be made with respect to implicational reasoning. The important thing to note here is that the validity of the standard forms of argument is not affected by their involving antithetical major and minor items, for the simple reason that in each mood the major premise (presumably given, explicitly or implicitly) formally allows us to draw the conclusion from the minor premise (and third premise, if any). We do not have to prove anything new in relation to the antitheses – all the information we need is already given in the premises. c. A crescendo argument (unlike purely a fortiori argument) sometimes concerns antithetical subsidiary items. Antithetical means contrary or contradictory. In copulative reasoning, the positive subjectal form of such argument is as follows:
Clearly, notY and Y are here two different values within a common continuum called S, one being labeled S1 and the other S2. Assuming direct proportionality of S to R, then S1 < S2, since the argument is from minor to major. Thus, the negative notY is the lesser value S1, corresponding to S ≤ 0, and the positive Y is the greater value S2, i.e. S > 0. In cases of inverse proportionality of S to R, the positive Y would be S1 and the negative notY would be S2. We could of course switch the roles of Y and notY (i.e. reason from Y to notY while S is directly proportional to R, or reason from notY to Y while S is inversely proportional to R) provided the meanings of these terms were such that S1 < S2. I have chosen the above order of presentation as more natural, taking Y as something more positively S and notY as something more negatively S. But these are just symbols, and the reverse order may be more accurate in some cases. We can obviously construct similar negative subjectal arguments. We can likewise formulate a positive predicatal argument, as follows:
Here again, Y and notY are two different values within the continuum S, one being labeled S1 and the other S2. Assuming direct proportionality of R to S, then S1 > S2, since the argument is from major to minor. Thus, the positive Y is the greater value S1, i.e. S > 0. the negative notY is the lesser value S2, corresponding to S = 0. In cases of inverse proportionality of R to S, the negative notY would be S1 and the positive Y would be S2. The reverse order of Y and notY is here again conceivable, needless to say. And we can obviously construct similar negative predicatal arguments. Similar comments can be made with respect to implicational reasoning. Now, the big question here is: are these arguments formally valid? That is to say, does the putative conclusion of each follow from the given premises? The answer is obviously: no – although it may be that with additional information these conclusions might well be logically justified. The answer is obviously ‘no’, because the third premise, which tells us about the proportionality of S to R or of R to S, only allows us a vague conclusion, namely that the subsidiary term in the conclusion is more than the subsidiary term in the minor premise, in the case of minortomajor arguments (namely, the positive subjectal and negative predicatal moods), and that the subsidiary term in the conclusion is less than the subsidiary term in the minor premise, in the case of majortominor arguments (namely, the positive predicatal and negative subjectal moods). If the given subsidiary term is the negative notY and the conclusion is supposed to be more than notY, it does not necessarily follow that the concluding subsidiary term is the positive Y. For if in a given case notY happens to refer to a specific negative value in the continuum S (i.e. S < 0) which happens to be well below zero, then ‘more than notY’ cannot be assumed to indeed refer to Y (i.e. S > 0), for it might conceivably still refer to another specific negative value in the continuum S (i.e. S ≤ 0), which is closer to zero but still not above zero. Likewise, if the given subsidiary term is the positive Y and the conclusion is supposed to be less than Y, it does not necessarily follow that the concluding subsidiary term is the negative notY. For if in a given case Y happens to refer to a specific positive value in the continuum S (i.e. S > 0) which happens to be well above zero, then ‘less than Y’ cannot be assumed to indeed refer to notY (i.e. S ≤ 0), for it might conceivably still refer to another specific positive value in the continuum S (i.e. S > 0), which is closer to zero but still above zero. It is only when Y and notY do not admit of degrees, i.e. when things are either precisely Y or precisely notY (S being binary, either = 1 or = 0), than we can take ‘more than notY’ to mean ‘Y’ and ‘less than Y’ to mean ‘notY’. This happens, but is not a general truth. Therefore, though the moods above described with antithetical subsidiary terms are in exceptional cases indeed valid, they are not universally valid, so we must be careful when we reason in this way that we do not rush to judgment, but ponder the matter carefully. To clarify this further: if we say that Q or P “is R enough to be” Y (or notY), do we mean every value of Y (or notY), or some value of Y (or notY)? Obviously, the latter. Therefore, even if our explicit reference to Y (or notY) is generic, it is tacitly specific – and we cannot tell from such a vague statement what the precise specification is. The only exception to this rule would be when we know for a fact that ‘Y’ has only one possible value (say, 1) and ‘notY’ has only one possible value (say, 0) – i.e. when they form a binary pair. In cases where ‘Y’ and ‘notY’ refer to ranges of values of S, i.e. to S > 0 and S ≤ 0 respectively, we would need a precise mathematical formula, viz. S = f(R) or R = f(S), as the case may be, to draw the desired conclusion (i.e. one with an antithetical subsidiary term). That is, to know that ‘more than notY’ here means ‘Y’ we must be able to calculate by how many degrees of S the inferred ‘more than notY’ is more than the given ‘notY’. Or, to know that ‘less than Y’ here means ‘notY’ we must be able to calculate by how many degrees of S the inferred ‘less than Y’ is less than the given ‘Y’. Note that the problem here is not specific to antithetical terms, but applies to any specific terms. Given only a vague premise about proportionality, i.e. “S varies in proportion to R” or “R varies in proportion to S” (as the case may be), we can only come to a vague conclusion. Even if the minor premise contains a very specific subsidiary term, the a crescendo conclusion cannot produce an equally specific subsidiary term, but only a term that is vaguely ‘more than’ or ‘less than’ the one given in the premise. An antithetical subsidiary term being a specific term, this rule applies equally to it. Of course, if we can amplify our premise about proportionality with a formula that allows us to precisely calculate the concluding subsidiary term from the given subsidiary term, the problem dissolves and we can produce an exact conclusion – which means, in some cases, an antithetical subsidiary term. Though such a formula is conceivable and often available in scientific discourse, it is rarely available in everyday discourse. In the latter, we usually have no practical means of producing such a formula, and must rely more on intuitive understanding to decide whether the putative conclusion, involving an antithetical subsidiary term, seems reasonable or not. Thus, what shall we declare at the last? Shall we say such arguments are valid or invalid? I would suggest we declare them in principle invalid, while admitting that they are in practice often accepted as reasonable[13]. It is clear that to draw the desired conclusions we strictly need more precise information than the vague proportionality given in the above premises. However, being human, we function in practice in a more permissive mode, and grant many inferences on the basis of mere intuitions of proportionality. In such cases, if the inference of a positive from a negative or the inference of a negative from a positive seems reasonable to everyone, we might accept the argument as close enough to valid. Obviously, conclusions based on such relatively lenient standards are inductive rather than deductive. They are probable rather than certain, accepted as true until and unless some objection to them be found. The logician’s role is not to inhibit human knowledge if it does not meet strict ideal standards, but to help practitioners approach these standards as much as possible in each given context. We have to remain lucid at all times, and be aware of our limitations – but this should not stop us from proceeding apace with our pursuit of knowledge, which is a necessity for us as living beings. On this basis, while I have above demonstrated that deduction of an antithetical subsidiary term on the basis of a vague premise of proportionality is strictly speaking invalid, I would recommend that we in practice, in most cases, adopt a lenient attitude towards such inference. Its result must, of course, be recognized as inductive rather than deductive. If the argument is based on public information, it is in principle subject to revision when and if new information arises. However, very often we have no means of scientific verification, so there is no likelihood that we shall change our mind that way. In such cases, we rely on a primary insight. A new such insight might conceivably arise in a wider context of knowledge and replace the former. In that event, of course, we would naturally revise our a crescendo argument. Thus, even an argument based on mere insight can in principle be reviewed. So it is fair to call such arguments inductive, and view them as more than just speculative pronouncements. d. Let us now consider a crescendo arguments (not purely a fortiori arguments, note) which involve both antithetical major and minor items and antithetical subsidiary items, in tandem. Such arguments may be characterized as a contrario a crescendo arguments, because the polarities of both the terms in minor premise are inverted in the conclusion, i.e. the subject and predicate in the former are negated in the latter. Such arguments do occur in common discourse, although very rarely. For instance, R. Hananiah ben Gamaliel says, in Mishna Makkoth 3:15: “If in one transgression a transgressor forfeits his soul, how much more should one who performs one precept have his soul granted him!” Here, ‘transgression’ is replaced by ‘performing precepts’ and ‘forfeiting of one’s soul’ is replaced by ‘being granted one’s soul’. If vice causes death, then virtue causes life. Sound reasonable. But is it? Does the first sentence logically imply the second? The question posed is whether such arguments are valid. Consider for instances the following most typical forms (the first positive subjectal and the second positive predicatal):
What distinguishes such arguments is that they compound the feature of antithetical major and minor terms and that of antithetical subsidiary terms. The expression a contrario is appropriately used in relation to such a crescendo arguments, because the movement of thought involved in them is obviously that of inversion. In our two samples, this means: If notX, then notY; therefore, if X, then Y. If Y, then X; therefore, if notY, then notX. However, we know that inversion is not a valid process of eduction (i.e. immediate inference), even if many people wrongly imagine it to be. That is to say, the proposition ‘if notX, then notY’ does not formally imply the proposition ‘if X, then Y’; these propositions are admittedly sometimes true together, but just as often one is true and the other false. This can be proven very precisely[14]. Similarly, of course, for the other pair, ‘if Y, then X’ and ‘if notY, then notX’. Admittedly, someone who formulates an argument like those above described is not relying on mere immediate inference, but on a complex a crescendo argument. Well, we have already in (b) above examined arguments with antithetical major and minor terms and found them formally valid, since the major premise guarantees the process. We have also in (c) above examined arguments with antithetical subsidiary terms and found them in principle invalid without more precise information regarding the proportionality, although often in practice accepted as effectively valid merely on the basis of intuitive understanding. On the basis of these earlier findings, we should be able to readily determine the validity or invalidity of the more complicated a contrario forms. As regards the minor and major terms, they present no problem of validity here again, since the major premise explicitly refers to them. As regards the subsidiary terms, the movement of thought is strictly invalid if we lack a precise formula, although reasonable enough in most cases commonly encountered. However, we cannot resolve the issue of a contrario argument simply by this extrapolation, because here the problem of antithetical items is compounded. That is, in a contrario a crescendo argument we are dealing with two changes of polarity in tandem, from negatives to positives (in our first sample, from notX to X as well as from notY to Y), or from positives to negatives (in our second sample, from Y to notY as well as from X to notX). Thus, here the difficulty is much greater: we need a formula that can express the parallelism between the X and Y values, not merely in a general way, but in a way so precise that it can pinpoint for us that the positives (X and Y) will occur together and the negatives (notX and notY) will occur together[15]. This is surgical precision that cannot simply be assumed offhand. Of course, it may happen that the terms behave in such an orderly manner, in lockstep, as tied couples. But how would we express the underlying determinism in a further premise? Perhaps in a statement like “Change from notX to X and change from notY to Y are linked” (for the first mood) or “Change from Y to notY and change from X to notX are linked” (for the second mood). But how would we know this? Presumably by observing the concomitant variation of the two terms. But then, if we already know that X is coupled with Y or that notY is coupled with notX, what need have we of the proposed a crescendo argument? Does its conclusion teach us anything more? Conversely, if we do feel the a crescendo argument to be useful, is it not because we do not have the said information on coupling, and seek to obtain it by deduction? I submit that the only way we could express the premise about coupling would be to adduce the conclusion! That is to say, the needed information is so precise that it cannot be specified indirectly. It can only be stated. But if it is stated, this is tantamount to granting the putative conclusion as a premise. We would be begging the question, engaging in circular argument. I admit I could be wrong in this reasoning, but I think it is best to adopt it, until if ever someone proposes some credible additional premise that would formally guarantee the conclusion without repeating it. In sum, I prefer as a precaution to declare the proposed argument invalid. This does not mean that, in the above described a contrario arguments, the minor premise and conclusion cannot be true together, but it does mean that we cannot deduce the latter from the former, even granting the said major premise and third premise. There is a nonsequitur, but no antinomy.[16] Thus, argument with antithetical minor and major items and antithetical subsidiary items is best avoided and regarded as rhetorical. Such argument appears reasonable to some people because they see it as mere inversion; but as we have seen, mere inversion is not valid inference, anyway. All the more is it not valid in the more complicated context of a crescendo argument.
5. TraductionsIn the course of interpreting numerous Mishnaic and Talmudic a fortiori arguments, I noticed that there were often two or more ways a given argument could be interpreted to the same effect. I therefore resolved to try and justify such correspondences in formal terms, to facilitate such reinterpretation in the future. I have called the establishment of such rewriting of arguments in alternate forms – ‘traductions’ (which in French means ‘translations’). Traduction should not be identified with reduction, the validation of arguments, since both the source and target arguments here are already known to be valid forms. The purpose of traduction is more material – it is simply deriving from an argument, of one form, another argument, of a different form; that is, it is merely a different verbal expression of the same thought. The value of studying such changes of wording is that it helps one find the interpretation of a given material argument that is formally closest to the given speech or text, rather than imposing a more common form on it. In this way one demonstrates full understanding of the original movement of thought. More specifically, the utility of traduction is that it allows us to pass from a positive to a negative form, or vice versa; or from a ‘minor to major’ to a ‘major to minor’ form, or vice versa; or from a subjectal (or antecedental) to a predicatal (or consequental) form, or vice versa – ideally, without loss of information. Correlations between copulative and implicational forms (some of which were dealt with earlier) can also of course be used for purposes of traduction. As we shall see, some traductions are of logical significance, others are more akin to linguistic manipulations. We shall deal with copulative arguments, and assume offhand that the same can be done with implicational ones (the reader is invited to verify this, as an exercise). Regarding the former, knowing there are four moods of primary copulative a fortiori argument, we shall need to investigate the following possible traductions: the uniformly subjectal or predicatal, and mixtures of subjectal and predicatal. That is, from +s to –s, and vice versa; from +p to –p, and vice versa; from +p to +s and –s, and from –p to +s and –s; and finally from +s to +p and –p, and from –s to +p and –p.
Let us first consider the uniform traductions. The first type is the purely subjectal one: (1a) from +s to –s:
This is a logical process and one not hard to prove. The two major premises imply each other, given that their middle terms R1, R2 are relative. The minor premises likewise imply each other because they both imply that Q is S, and both the ranges R1 and R2 are fully inclusive, though in opposite directions. Similarly, the conclusions imply each other, and that P is S. Thus, albeit apparent differences in middle terms and in the polarities of their minor premises and conclusions and their subsidiary terms, these two a fortiori arguments are formally equivalent. Compare for example the following two arguments: “given P is longer than Q, and Q is long enough to be S, then P is long enough to be S” and “given Q is shorter than P, and Q is not short enough to be notS, then P is not short enough to be notS.” Clearly, if Q is not short enough to be notS, then it must be long enough to be S; and if Q is long enough to be S, then it cannot be short enough to be notS; and the conclusions follow. Note well that both arguments are subjectal, and one is positive and goes from minor to major and the other is negative and goes from major to minor. Keep in mind, also, the special case where R1 is some concept R, and R2 its antithesis notR; this sometimes occurs. Needless to say, if we substituted notS for S in the first argument, and put S instead of notS in the second, we would have another pair of equivalent arguments. A likewise easily proved corollary of the above traduction is (1b) from –s to +s:
The second sort of uniform traduction is the purely predicatal one: (2a) from +p to –p:
This is also a logical process and one easy enough to prove. The two major premises imply each other, given that their middle terms R1, R2 are relative. The minor premises likewise imply each other because they both imply that S is P, and both the ranges R1 and R2 are fully inclusive, though in opposite directions. Similarly, the conclusions imply each other, and that S is Q. Thus, albeit apparent differences in middle terms and in the polarities of their minor premises and conclusions and their major and minor terms, these two a fortiori arguments are formally equivalent. Compare for example the following two arguments: “given more strength is required to be P than to be Q, and S is strong enough to be P, then S is strong enough to be Q” and “given more weakness is required to be Q than to be P, and S is weak not enough to be notP, then S is weak not enough to be notQ.” Clearly, if S is not weak enough to be notP, then it must be strong enough to be P; and if S is strong enough to be P, then it cannot be weak enough to be notP; and the conclusions follow. Note well that both arguments are predicatal, and one is positive and goes from major to minor and the other is negative and goes from minor to major. Needless to say, we could equally well have made the argument about R1 negative and that about R2 positive[17]. Keep in mind, also, the special case where R1 is some concept R, and R2 its antithesis notR; this sometimes occurs. A likewise easily proved corollary of the above traduction is (2b) from –p to +p:
More complex are the mixed traductions, aimed at correlation of predicatal and subjectal forms of a fortiori argument. It is not easy, if not impossible, to effect such changes of form, because in subjectal arguments the minor premises and conclusions have P or Q as subjects and S as predicate, whereas in predicatal ones they have S as subject and P or Q as predicates, and we cannot simply convert one form to the other. However, such changes of form can be effected in a more convoluted manner, by constructing new items from the given elements of the argument. Some such processes are of logical interest. But in many cases the change is rather artificial, in the sense that the underlying logical form of the argument is not really changed but only superficially made to appear to have been changed; that is, though the explicit wording looks different, the implicit thought is unchanged. Although such processing is thus a bit make believe from a logical point of view, it is still useful in that we can by this means verbally reproduce someone’s actual thought process.[18]
The processes that go from predicatal arguments to subjectal ones are based on a fusion of the given middle term R with the relational concept of its being (or not being) required for some result, yielding either the new positive middle term “demanding of R” or the negative relative term “undemanding of R” as appropriate. We have in all four such traductions to consider. (3a and 3b) From +p to –s and +s: starting with the following positive predicatal (major to minor) argument:
…we can construct the following two subjectal ones, whose equivalence is evidenced by their having the same net implications, viz. that S is P and Q:
(3c and 3d) From –p to +s and –s: similarly, starting with the following negative predicatal (minor to major) argument:
…we can construct the following two subjectal ones, whose equivalence is evidenced by their having the same net implications, viz. that S is notQ and notP:
Note that the move from +p to –s feels more natural than that from +p to +s, because in the former case, even though the minor premise and conclusion change polarity, the polarity of the middle term and the movement from major to minor remain the same. For the same reasons, the move from –p to +s feels more natural than that from –p to –s. This is why I have listed the traductions in that order. The reason I have characterized such traductions as artificial, i.e. as more verbal than logical, is partly of course due to their having recourse to a new middle term, viz. “demanding of R” (or “undemanding of R”), which has a formal element hidden in it, viz. the fact of requirement (or its lack) that is in fact indicative of predicatal argument. But there is a second, ultimately more important reason: it is that the subsidiary item is not really the same in minor premise and conclusion. Although the original subsidiary term S is still present and stands in the foreground at an explicit level in the derived arguments, at a more implicit level we have to specify its being the subject of the subject of the proposition. That is to say, though we say “for S” or “to prevent S,” by the term “S” here we really mean the proposition “S is P” or “S is Q” (as appropriate). As we all know by now, an a fortiori argument is formally invalid if the subsidiary item is not exactly the same in minor premise and conclusion. So we must regard such traductions as being, strictly speaking, misleading. But as is evident the derived arguments do carry some conviction! Why so? The reason they do so is that they are only apparently subjectal, at the surface level of their wording. In fact they are, at a deeper level, as regards their logical form, still very much predicatal, since the idea of requirement is inherent in their middle terms, viz. “demanding of R” (or “undemanding of R”). We could perhaps remedy the said fault, of the derived arguments having tacitly unequal subsidiary terms, by resorting to a more abstract subsidiary item. That is to say, instead of specifically saying (or even thinking) “S is P” or “S is Q” – we would state more vaguely “the subject (of the whole suffective proposition concerned, i.e. P or Q as the case may be) is predicated of S.” This would indeed considerably reinforce the subjectal appearance of the derived arguments. But I maintain that these arguments would not be fully understood and believed if we did not have in mind the underlying predicatal discourse. Consider, for instance, the first of the listed four traductions, viz. “from +p to –s.” Note that both the arguments involved are from major to minor, and that both imply that S is P and Q. The middle term of the first is R, but the middle term of the second is “demanding of R.” The major and minor terms P and Q remain the same, but the given subsidiary term S is amplified by a P or Q predicate (as appropriate) in the derived argument (though this may not be stated out loud, and only tacitly intended). Clearly, although the first argument is predicatal and positive, while the second is subjectal and negative, they tell us exactly the same thing. The latter argument is just as predicatal in essence as the former; such traduction is just a change of wording. But, to repeat, it is still useful sometimes. For example, consider the positive predicatal argument: “given that more money (R) is needed to buy a car (P) than to buy a bicycle (Q), it follows that if $1000 (S) is enough money (R) for a car (P), then it (S) is enough for a bicycle (Q).” This can be restated in the following negative subjectal form: “given that a car purchase (P) calls for more funds (new R) than a bicycle purchase (Q) does, it follows that if a car purchase (P) calls for funds (new R) not large enough that $1000 cannot effect it (S, in relation to P), then a bicycle purchase calls for funds not large enough that $1000 cannot effect it (S, in relation to Q).” We can similarly explain and exemplify the other three traductions; the reader should perhaps do that as an exercise.
The processes that go from subjectal arguments to predicatal ones are based on making a very abstract subsidiary term, “the subject concerned,” out of the subjects, P and Q; and fabricating two new major and minor items, “P is S” and “Q is S” (or their negations), out of the original major, minor and subsidiary terms. The middle term used here is R (or any relative of it, such as its negation[19]). We have in all four such traductions to consider. From +s to –p and +p: starting with the following positive subjectal (minor to major) argument:
…we can construct the following two predicatal ones, whose equivalence is evidenced by their having the same net implications, viz. that Q and P are S:
From –s to +p and –p: starting with the following negative subjectal (major to minor) argument:
…we can construct the following two predicatal ones, whose equivalence is evidenced by their having the same net implications, viz. that P and Q are notS:
Note that the move from +s to –p feels more natural than that from +s to +p, because in the former case, even though the minor premise and conclusion change polarity, the polarity of the middle term and the movement from major to minor remain the same. For the same reasons, the move from –s to +p feels more natural than that from –s to –p. This is why I have listed the traductions in that order. We can strongly criticize this set of four traductions as we did the preceding set, and even more so. The major premises of the derived arguments are credible enough, merely regrouping information already present in the original arguments. The problem lies rather in the new minor premises and conclusions. What is problematic in the latter is not the middle term R (or its relative, notR) or the predicated items, viz. “P is S” and “Q is S” (or their negations), which are clearly intended in the given arguments – the problem lies in the new subsidiary term. This very abstract term, viz. “the subject concerned,” is intended as a single substitute for the two original subjects, P and Q. We need a single term in this role, because an a fortiori argument cannot have more than one subsidiary. If there are two subsidiaries, the argument becomes invalid. So a vague term is introduced, “the subject concerned,” which tacitly refers to a term (P or Q) which is present elsewhere in the same suffective proposition, namely within a new predicated item, viz. “P is S” and “Q is S” (or their negations), as the case may be. This verbal artifice allows us to make predicatal arguments out of subjectal ones. But of course, it is a bit of a sleight of hand, because we cannot really understand the abstract term without mentally referring to the P or Q it stands for. Therefore, the tacitly intended P or Q remains the effective subject of the proposition concerned, even if we have hidden it away. So we must admit either that the argument is fallacious (having two subsidiary terms) or that it is not what it seems. That is to say, in the latter case, though we have reformulated the given subjectal argument in such a way that it now looks like a predicatal argument, the thought process involved is still really subjectal. Verbally, on the surface of thought, the argument may seem predicatal, but logically, in the depth of thought, it is quite subjectal. Although such rewording is theoretically a dead end, it can still as earlier indicated be useful for the practical purpose of interpretation. To analyze one of the listed four traductions, consider for instance the first, viz. “from +s to –p.” Note that both the arguments involved are from minor to major, and that both imply that P and Q are S. The middle term of the first is R, but the middle term of the second is “R in the subject.” The new subsidiary term is the very abstract “there,” and the new minor and major items are “Q is not S” and “P is not S,” which contain the original minor and major terms (Q and P), respectively, to which the original subsidiary term (S) is negatively predicated. Since the new middle term is not enough to entail these negative items, it follows that Q and P are both S. For example, consider the positive subjectal argument: “given that selling a car (P) generates more income (R) than selling a bicycle (Q) does, it follows that if selling a bicycle (Q) generates income (R) enough to buy a new suit (S), then selling a car (Q) generates income (R) enough to buy a new suit (S).” This can be restated in the following negative subjectal form: “given that more income generation is required for [a car sale to enable a suit purchase], than for [a bicycle sale to do so], it follows that if the subject concerned (here, sale of a bicycle) generated income not enough for [a bicycle sale not to enable purchase of a new suit], then the subject concerned (here, sale of a car) generated income not enough for [a car sale not to enable purchase of a new suit].” We can similarly explain and exemplify the other three traductions; the reader should perhaps do that as an exercise.
To conclude this topic: uniform traductions (purely subjectal or purely predicatal ones) may be qualified as logical processes, whereas mixed traductions (from predicatal forms to subjectal ones, or vice versa) are rather verbal than truly logical. Mongrel arguments. In this context, somewhat incidentally, I would like to draw attention to a mistake I have often found myself making when attempting to interpret examples of a fortiori argument in formal terms, i.e. when trying to fit them into some standard form. What happens is that we formulate an argument of mixed form; that is, mixing a subjectal major premise with a minor premise and a conclusion of predicatal form, or mixing a predicatal major premise with a minor premise and a conclusion of subjectal form. This produces arguments like the following positive copulative moods:
Such mixtures are best described as mongrels. The problem with them is not so much the order of the major and minor terms in the minor premise and conclusion, i.e. whether P is inferred from Q or Q is inferred from P; for the order could be changed. The problem is that the major and minor terms are subjects in the major premise and then predicates in the next two propositions, or predicates in the major premise and then subjects in the next two propositions. This is a problem, because it makes validation of these arguments impossible. Moreover, such arguments feel unnatural and unconvincing. Note that the major premise of the first argument may also be stated as “More R is involved in being P than in being Q,” which gives it a more predicatal air. Similarly, the major premise of the second argument may also be stated as “(to be) P requires more R than (to be) Q does,” which gives it a more subjectal air. But such verbal reconstructions do not affect the essence of the matter. The conceptual difference between subjectal and predicatal argument is clearcut, and the two forms should not be confused or mixed. As well, very often when one does this, the terms, though closely related, are not quite the same in the major premise on the one hand and in the minor premise and conclusion on the other hand. One should always make sure the terms are identical.
[1] Aristotle states the laws of noncontradiction and of the excluded middle in his Metaphysics, B, 2 (996^{b}2630), Γ, 3 (1005^{b}1923), Γ, 7 (1011^{b}2324). Metaph. Γ, 7 (1011^{b}2627) may be viewed as one statement by Aristotle of the law of identity: "It is false to say of that which is that it is not or of that which is not that it is, and it is true to say of that which is that it is or of that which is not that it is not." These references are found in the Kneales, p. 46 (although they interpret the latter statement as somewhat defining truth and falsehood, rather than as expressing the law of identity). [2] See De Interpretatione, 9 (19^{a}30). [3] I deal with this one in my Future Logic, chapter 32.3. [4] This implies, incidentally, that generalization may have been used – i.e. there may be an element of prediction of future instances based on past instances. [5] We could perhaps, however, conceive of implicational equivalents of the above copulative quantifications, with reference to the modalities of implication. In this perspective, “if X, necessarily Y” is equivalent to a positive general proposition, and “if X, possibly Y” to a positive particular proposition. Similarly, “if X, necessarily not Y” is equivalent to a negative general proposition, and “if X, possibly not Y” to a negative particular proposition. This idea needs further consideration, but I will not do that here. [6] Of course, I implies ‘either A or IO’ could be regarded as a deductive argument with a single disjunctive conclusion; so when we said that A implies I is a deductive argument because it has a single conclusion, we meant in this case that it has a single distinctively categorical conclusion. The inductive argument I implies either A or IO, by way of contrast, offers a choice of two possible categorical conclusions: either I implies A, or I implies IO. This is said in passing, being a minor issue. [7] Needless to say, we can similarly generalize from O to E, preferring the latter to IO. [8] Which conclusion can eventually be further narrowed down to A (by generalization) or to IO (by particularization). [9] See for instance the definition given in The Oxford Companion to Philosophy (Ed. Ted Honderich, Oxford UP, 1995). The term was introduced by Charles S. Pierce (1839–1914). [10] If we toss two coins, what are the chances of getting, say, two ‘heads’ at once? Obviously, 50% * 50% = 25%. But it could alternatively be argued, with syllogistic logic in mind, that the probability of the conclusion is 100% minus the sum of the probabilities of the premises, for the conclusion is only sure to occur in contexts where the premises are sure to overlap. In that view, if the probabilities of the premises are both 50%, or otherwise add up to 100% or less, the premises might never overlap since their contexts of occurrence may be at odds, and the probability of the conclusion (from that conjunction of premises) must be taken as 0%. This issue needs to be reflected on. [11] See for example Mishna Menahoth 8:5: “Just as the Menorah which is not to do with eating, requires ‘pure olive oil’, so meal offerings, which are to do with eating, is it not an inference that they should require ‘pure olive oil’?” [12] See for example Mishna Nedarim 10:7. “If he [a husband] can cancel vows [by his wife] which have already had [for a time, before he cancelled them] the force of a ‘prohibition’ [as any vow of his wife that he cancels], can he not also cancel vows which have not yet the force of a ‘prohibition’?” This is positive predicatal. [13] A possible illustration is 2 Samuel 12:18. “Behold: while the child was yet alive, we spoke unto him, and he hearkened not unto our voice; then how shall we tell him that the child is dead, so that he do himself some harm?” The relevant antithesis here is between ‘being distracted’ and ‘harming himself’. [14] See my work The Logic of Causation in this regard. [15] Or similarly in cases of inverse proportionality, that X and notY and notX and Y will be pairedoff. [16] The best way to demonstrate the truth of my contention that a contrario a crescendo argument is not necessarily valid and is therefore invalid would be to produce an example where the premises are obviously correct and the conclusion is obviously wrong. For the time being, I have not managed to do that; but admittedly I have not tried very hard. [17] But we cannot, of course, just change the polarities of the major and/or minor term(s) in the minor premise and/or conclusion without adjusting it/them in the major premise too. So this case differs from the preceding (subjectal argument), where we could just switch the polarity of the subsidiary term in the minor premises and conclusions without worrying about the major premise. [18] These are the best solutions I have found so far to the problems posed, at any rate. It is not excluded that other, even perhaps better, traductions be found in the future, by me or others. [19] I take the pair R and its negation notR here, but we could equally take two relatives R1 and R2. 