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The Laws of Thought

A Thematic Compilation by Avi Sion

27. Understanding the Laws of Thought


1.     Adapting the Laws of Thought


Many people regard Aristotle’s three ‘laws of thought’ – the laws of identity, of non-contradiction 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 non-contradiction) 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 B-ness is different from A-ness. 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 non-contradiction 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 not-B’ are compatible if they tacitly refer to different places, e.g. if they mean ‘A is B here’ and ‘A is not-B there’, so they are compatible if they tacitly refer to different times, e.g. if they mean ‘A is B now’ and ‘A is not-B 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 “non-things 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 “non-things;” 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 “non-thing,” i.e. there is no “non-thing.”

The same can be said regarding the word “existents.” In its primary sense, it refers to actually existing things, as against non-existing things; but in its enlarged sense, it includes non-existing 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 chapter 30, below] of the Liar and Russell paradoxes. Such paradoxes often require a lot of ingenuity and logical skill to resolve.


2.     Two More Laws of Thought[4]


The first three laws of thought, which were formulated by Aristotle, are that we admit facts as they are (the law of identity), in a consistent manner (the law of non-contradiction) and without leaving out relevant data pro or con (the law of the excluded middle). To complete these axioms of logic, and make them fully effective in practice, we must add two more. The fourth, which I have called the principle of induction; and a fifth, which I call the principle of deduction.

These five laws are nothing new, being used in practice by mankind since time immemorial; only our naming them in order to spotlight them and discuss them is a novelty. They qualify as ‘laws of thought’ because they are self-evident, and necessary to and implied in all rational thought.

The principle of deduction is a law of logic that no information may be claimed as a deductive conclusion which is not already given, explicitly or implicitly, verbally or tacitly, in the premise(s). The premises must obviously fully justify the conclusion, if it is to be characterized as deduced. This fundamental rule is true for all forms of deductive (as against inductive) arguments, which helps us avoid fallacious reasoning. It may be viewed as an aspect of the law of identity, since it enjoins us to acknowledge the information we have, as it is, without fanciful additions.

It may also be considered as the fifth law of thought, to underscore the contrast between it and the principle of induction, which is the fourth law of thought. The principle of induction may, in its most general form, be stated as: what in a given context of information appears to be true, may be taken to be effectively true, unless or until new information is found that puts in doubt the initial appearance. In the latter event, the changed context of information may generate a new appearance as to what is true; or it may result in some uncertainty until additional data comes into play.

Deduction must never be confused with induction. Although deduction is one of the tools of induction in a broad sense, it is a much more restrictive tool than others. Deduction refers specifically to inferences with 100% probability; whereas induction in a narrow sense refers to inferences with less than 100% probability.

Inductive reasoning is not subject to the same degree of restriction as deduction. Induction is precisely the effort to extrapolate from given information and predict things not deductively implied in it. In inductive reasoning, the conclusion can indeed contain more information than the premises make available; for instance, when we generalize from some cases to all cases, the conclusion is inductively valid provided and so long as no cases are found that belie it. In deductive reasoning, on the other hand, the conclusion must be formally implied by the given premise(s), and no extrapolation from the given data is logically permitted. In induction, the conclusion is tentative, subject to change if additional information is found, even if such new data does not contradict the initial premise(s)[5]. In deduction, on the other hand, the conclusion is sure and immutable, so long as no new data contradicts the initial premise(s).

As regards the terms, whereas in induction the conclusion may contain terms, denotations or connotations that are not manifest in the premise(s), in deduction the terms, denotations and connotations in the conclusion must be uniform with those given in the premise(s). If a term used in the conclusion of a deductive argument (such as syllogism or a fortiori) differs however slightly in meaning or in scope from its meaning or scope in a premise, the conclusion is deductively invalid. No equivocation or ambiguity is allowed. No creativity or extrapolation is allowed. If the terms are not exactly identical throughout the argument, it might still have some inductive value, but as regards its deductive value it has none.

Any deductive argument whose conclusion can be formally validated is necessarily in accord with the principle of deduction. In truth, there is no need to refer to the principle of deduction in order to validate the conclusion – the conclusion is validated by formal means, and the principle of deduction is just an ex post facto observation, a statement of something found in common to all valid arguments. Although useful as a philosophical abstraction and as a teaching tool, it is not necessary for validation purposes.

Nevertheless, if a conclusion was found not to be in accord with the principle of deduction, it could of course be forthwith declared invalid. For the principle of deduction is also reasonable by itself: we obviously cannot produce new information by purely rational means; we must needs get that information from somewhere else, either by deduction from some already established premise(s) or by induction from some empirical data or, perhaps, by more mystical means like revelation, prophecy or meditative insight. So obvious is this caveat that we do not really need to express it as a maxim, though there is no harm in doing so.

The principle of deduction is that the putative conclusion of any deductive argument whatsoever must in its entirety follow necessarily from (i.e. be logically implied by) the given premise(s), and therefore cannot contain any claims not supported in the said premise(s). If a putative conclusion contains additional information and yet seems true, that information must be proved or corroborated from some other deductive or inductive source(s). Inference in accord with this principle is truly deductive. Inference not in accord with this principle may still be inductively valid, but is certainly not deductively valid.

In truth, the principle of deduction is a redundancy. That the conclusion cannot go beyond what is given in the premises is true of all deductive argument, without any need to state it as a special principle; it is the very definition of deduction, as against induction or fallacious thought, and so the subtext of any deductive act. Clearly, the principle of deduction is not an artificial, arbitrary or conventional limitation, but a natural, rational one.


Drawn from A Fortiori Logic (2013), Chapters 3.1, 7.2.



[1]           Aristotle states the laws of non-contradiction and of the excluded middle in his Metaphysics, B, 2 (996b26-30), Γ, 3 (1005b19-23), Γ, 7 (1011b23-24). Metaph. Γ, 7 (1011b26-27) 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 (19a30).

[3]           I deal with this one in my Future Logic, chapter 32.3.

[4]           This essay was originally written for A Fortiori Logic, and may still be found there in a scattered way, notably in chapter 7.2.

[5]           For example, having generalized from “some X are Y” to “all X are Y” – if it is thereafter discovered that “some X are not Y,” the premise “some X are Y” is not contradicted, but the conclusion “all X are Y” is indeed contradicted and must be abandoned.

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