Original writings by Avi Sion on the theory and practice of inductive and deductive LOGIC  

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Avi Sion

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Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.




1.    Generalities.

2.    Valid/Invalid.

3.    Figures

4.    Moods.

5.    Psychology.


 1.      Generalities.


            We call inference the mental process of becoming aware of information implicit in given information, be it concrete or abstract. When we draw ideas from experience or generalities from particulars, we are involved in induction; otherwise, it is deduction. In any case, the original data is called the premises, and the logically derived proposition, the conclusion.

            When the conclusion is already known to us, and we are considering its validity in the context of other knowledge, we are said to argue. Furthermore, if the motive of our argument was to arrive at the conclusion for its own sake, we are said to be proving it; if on the other hand our motive was to show the contradictory or a contrary of our conclusion to be false, we are said to be engaged in a process of refutation.

            The difference in connotation between inference and argument is merely one of sequence: what was posited first, premise(s) or conclusion? The distinction between proof and refutation lies in our motive. But the logical form of all these processes is the same, so their names are used interchangeably here.

            The term deduction is sometimes used in a restricted sense which excludes eduction. Eduction has already been defined as eliciting information from one proposition (granting that the logical principles involved in this are not regarded as premises too). The deductive process which concerns us here, in contrast, is drawing information implicit in two or more propositions together, and not separately. P and Q are true, ergo R is true.

            This is called mediate inference, because it is found that the premises must have some factor in common, which serves as the medium of inference, making possible the eliciting of a conclusion. This might be thought of as 'conduction'. The technical name for it is syllogism, from Greek, the language of Aristotle.

            It can be shown that arguments involving more than two categorical propositions are reducible to a series of syllogisms and eductions. In this analysis, we will concentrate on categorical syllogism, that involving only categorical propositions. Argument involving noncategorical propositions will be dealt with later.


2.      Valid/Invalid.


            Now, an argument may be valid or invalid. The science of Logic shows that the validity of the method is independent of the truth or falsehood of the premises or conclusion. A formal argument only claims that if the premises are true, the conclusion must be true; if the conclusion is found false, then one or more of the premises must be false. It may happen that the premises are false, yet the conclusion is independently true; rejection of the premises does not necessarily put the conclusion in doubt. The validity or invalidity of an argument is a formal issue, irrespective of the content of the propositions involved.

            Logic analyses the variety of forms possible, and distinguishes the valid from the invalid, by reference to the Laws of Thought. The results are analyzed, in the search for general rules. Strictly speaking, only valid syllogisms are ultimately so called; invalid syllogisms are mere fallacies. But at the outset, Logic lists all possible combinations of propositions on an equal footing, to ensure the exhaustiveness of its treatment; then it finds out which are good and which bad.

            Its ultimate aim is of course to draw the maximum consequent information from any data. This allows us to correlate the different aspects of our experience, and improve our knowledge of the world. By comparing and connecting together all our beliefs, we can through logic discover inconsistencies, which cause us to reassess our assumptions at some level, and correct our data banks. In this way our beliefs are 'proved'; at least until there is good reason to think otherwise.

            Scientific proof always depends on the context of knowledge. It is always conceivable that some aspect of knowledge turns out to be open to doubt, even after seeming fundamental and unassailable for ages. For instance, certain axioms of Euclidean geometry. So proof never entirely frees a conclusion from review, given some new motive. Finding an inconsistency does not in itself guarantee that we will succeed in finding the source of the error, i.e. some false premise. In such cases, we register that there is some doubt yet to resolve, and either wait for new experience or search for an answer imaginatively.


3.      Figures.


            A syllogism, then, involves three propositions, two premises and a conclusion. These together involve three, and only three, terms. They are: the middle term, one common to both premises, but absent in the conclusion; the minor term, which is the subject of the conclusion, and present in one of the premises; and the major term, which is the predicate of the conclusion, and present in the other premise. The minor and major term are also called the extremes; the middle term acts as intermediary between them, to yield the conclusion. The premise involving the minor term is called the minor premise, that with the major term the major premise.

            The position of the middle term in the premises, that is, whether it is subject or predicate in each, determines what is called the 'figure' of the syllogism. (The colloquial expression for thought, 'to figure' or 'to figure out' may derive from this usage.) There are four possible figures of the syllogism. They are shown in the  following table, with S, M, P symbolizing the minor, middle and major terms, respectively:


Table 8.1         Figures of the Syllogism.







Major premise:





Minor premise:











            Note well the variety in the position of the terms. The order of the propositions in Logic is conventionally set as major-minor-conclusion, so that symbolic references can always be understood. But of course in actual thought any order of appearance may occur. Thus it is seen that syllogism is mediate inference; from their respective relationships to a middle term, a relationship may be found to follow between the extremes.

Each figure of the syllogism reflects a structure of our thinking. In practise, the Fourth figure is not regarded by many logicians as very significant. Aristotle, though aware of its existence, had this viewpoint. Galen, however, introduced it as a formal alternative for the sake of completeness.


4.      Moods.


            We previously identified six categorical forms, A, E. I, O, R and G, which can be involved in such syllogism. Each of the propositions in each figure might at first glance have any of these six forms. So there are 6X6X6 = 216 possibilities per group of proposition in each figure. Each of these combinations is called a mood of the syllogism. Altogether, in the four figures, there are 216X4 = 864 imaginable syllogistic forms. Each such form can be designated clearly by mentioning its figure and mood; for example, 'mood EAA in the first figure', or more briefly, '1/EAA'.

            Our task is differentiate the valid from the invalid, in this multiplicity of theoretical constructs. It will be seen that very few actually pass the test. The valid moods per figure should be justified, and the invalid ones shown wrong. This will enable us to know when a conclusion can be drawn from given premises, and when not.

            Note that each of the propositions may be positive (+) or negative (-), so that there are 2X2X2 = 8 possible combinations of polarity in each figure; they are: +++, ++-, +-+,+--, -++, -+-,  --+, ---. Likewise, as three quantities exist, viz. universal (u), particular (p), and singular (s), there are 3X3X3 = 27 possible combinations of quantity in each figure; which are: uuu, uup, uus, upu, upp, ups, puu, pup, pus, ppu, ppp, pps, and so on. It will be seen that many of these combinations are nonsensical, and rules concerning polarity and quantity can be formulated. Some rules are general to all figures, some are specific to each. In any case, the conclusion sought is always the maximal one; if a universal can be concluded, the subaltern conclusion is not of interest, though it follows a-fortiori.

            A more traditional way to express the task of logic with respect to syllogism is as follows. In each figure, which of the 6X6 = 36 combination(s) of premises yield a conclusion? Or which of the 2X2 = 4 combination(s) of polarity: ++, +-, -+, --? And which 3X3 = 9 combination(s) of quantity: uu, up, pu, pp, us, su, sp, ps, ss?


5.      Psychology.


            Some critics of Logic have accused it of puerility, arguing that the syllogism is too simple in form, and yields no new information, whereas actual thinking is somehow a more creative and complex process. But the 'event' of syllogistic reasoning is not as mechanical and automatic as it is made to appear on paper. Logic presents a static picture of what is psychologically a very dynamic and often difficult process.

            There is a mental effort in bringing together the concepts which form the separate propositions involved; this requires complex differential perceptions and insights. We also have to think of bringing together the propositions which constitute our premises; they are not always joined and compared automatically, sometimes a veritable inspiration is required to achieve this. And even then, actual drawing of the conclusion is not mechanically inevitable; honesty, will, and intelligence are needed.

            Thus, Logic merely establishes standards of proper reasoning, identifying common aspects of thought and justifying its sequences. But mentally, in practise, the processes are complexes of differentiation and integration. Sometimes such events are easy to produce, but often years of study and even genius are necessary to produce even a single result. Virtues such as open-mindedness, reality-orientation, perceptiveness, intuition, will-power are involved.



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