8. Geometrical logic

It is worth briefly investigating reasoning with propositions we might call ‘geometrical’[1], which compare the relative positions in space of two geometrical items (points, lines, surfaces or volumes) X and Y. This refers principally to the following set of forms, which are commonly used in discourse:

‘X is in Y’, ‘X is out of Y’, or ‘X is partly in and partly out of Y’.

These forms are implicit in Euler diagrams, which are often used by logicians to clarify and resolve syllogistic issues. For example, the syllogism ‘X is Y and Y is Z, therefore X is Z’ is read as ‘X is in Y and Y is in Z, therefore X is in Z’. In such cases, predication is interpreted as subsumption, or membership in a class, and the geometrical analogy is then obvious.

Deductive features. With regard to their logical oppositions, the said three forms are evidently contrary to each other. The ‘in’ form is here intended to mean ‘wholly inside’; the ‘out of’ form, ‘wholly outside’; and the ‘partly inside or outside’ form is intended to cover cases in between. It follows that if we use the indefinite form ‘X is (at least) partly in Y’, we mean the disjunction ‘X is either wholly in or only partly in Y’; and similarly for ‘X is (at least) partly out of Y’.

Thus, ‘not (wholly) in’ here would mean ‘either only partly in or wholly out of’; ‘not (wholly) out of’ would mean ‘only partly out of or wholly in’; and ‘not partly in and not partly out of’ would be understood as ‘either wholly in or wholly out of’. It follows from these oppositions that we can educe the negative forms ‘X is not outside Y’ and ‘X is not partly in and not partly out of Y’ from the positive form ‘X is inside Y’; and similarly in the other cases.

Note that such propositions cannot be ‘permuted’ at will. That is to say, just because we have verbalized the relation concerned with two or more words like ‘is in’ or ‘is out of’, it does not follow that we can freely separate these words from each other, treating one as the effective copula and the rest as part of the predicate. For instance, we treat ‘X is in {Y}’ as logically equivalent in all respects to ‘X is {in Y}’ – for to do so may lead to errors of reasoning[2].

As regards syllogism involving these six forms (the three positives and three negatives), we would simply refer back to Aristotelian methods and findings – i.e. consider all possible moods within three (or four) figures, and determine which are valid and which are not. I won’t bother doing this here systematically, but leave the job to the reader as an exercise.

Suffices here to give just a couple of examples. The example given earlier, viz. ‘X is in Y and Y is in Z, therefore X is in Z’, is the most obvious case, suggesting a circle X, within a larger (or equal) circle Y, within a larger (or equal) circle Z. A more interesting example would be the following:

These arguments can be illustrated as in the above diagram (where X3 represents the first syllogism’s minor term, while X2 and X1 the two conceivable values of the minor term in the second syllogism). Note that while such an Euler diagram traditionally presents the intersecting domains as circles, we should not take this literally – they might have any shape or even be physically scattered, so long as the relevant intersection(s) apply.[3]

The account here given of geometrical propositions and arguments does not, of course, cover the whole field of geometrical logic. Other commonly used forms may be mentioned, many of which are compounds involving the above forms. We may, for instance mention the forms ‘X is close to Y’ or ‘ X is far from Y’ (both of which imply X is outside Y). A special case of adjacency would be: ‘X is contiguous to Y’ (meaning the boundaries of X and Y are in contact at some point(s) of their boundaries). Such propositions may of course appear in combination with the others in mixed-form syllogisms.

Comparative propositions, like ‘X is bigger than Y’, ‘X is smaller than Y’, ‘X is equal in size to Y’, etc. are also to be classed as geometrical forms. So are propositions signifying sequence, like ‘X is before Y’, ‘X is after Y’, ‘X is simultaneous to Y’, etc. Other common forms: ‘next to’, ‘on top of’, ‘under’, ‘east of’, ‘west of’, etc. indicate relative directions. The point made here is that the whole field of geometry (as a branch of mathematics and eventually of physics[4]) has a parallel in formal logic, which focuses on the specifically discursive aspect of geometrical thought – and this may be called geometrical logic.

Inductive aspects. When the conclusion of an argument is uncertain, in the sense that we have a disjunction of two (or more) possible categorical conclusions as our valid formal inference (as in the second example of geometrical syllogism, illustrated above), the syllogism is still quite informative, in that while there is indeed more than one conceivable result, many other theses are thereby formally excluded (e.g. the conclusion ‘X is wholly in Z’ [as shown for X3] is excluded from the premises ‘X is only partly in Y and Y is wholly in Z’), i.e. rendered logically inconceivable.

Another point worth making with regard to such alternative conclusions is that they may not have the same degree of probability. If the situation in our above second example is exactly as described by our Euler diagram (although, to repeat, this illustration is only one possible visual interpretation), then the outcome labeled X2 would seem more probable than the one labeled X1. Clearly, there are many more places around the circumference of Y where X2 might lie; in comparison, X1 has a very limited scope. In that case, we could say that the conclusion of the premises ‘X is only partly in Y and Y is wholly in Z’ is most probably ‘X (i.e. X2) is wholly in Z’ and less probably ‘X (i.e. X1) is only partly in Z’).

Note this last comment well, because it relates to the interface between deduction and induction. One way to define the distinction between these two types of inference is to regard single conclusions as deductive and multiple possible conclusions (especially when their relative probabilities have been worked out) as inductive arguments; in that perspective, deduction is the limiting case of induction, when the probability of a certain conclusion is one hundred percent. We could alternatively view such disjunctive formal conclusions to some syllogisms as being as ‘deductive’ as categorical conclusions; but in that case, the expression ‘deductive’ simply corresponds to what we mean by ‘inference’.

These two viewpoints can be reconciled if we understand the difference between focusing on the inferred disjunctive proposition as a whole, which is forcefully ‘deduced’, and focusing on the individual disjuncts composing it, which may eventually be variously ‘induced’ in accord with of their relative probabilities (which, note well, require further argumentation to establish).

When the disjuncts are ordered by their respective probabilities, it means that the most probable disjunct is our first choice as conclusion. If this choice turns out to be belied by other considerations (i.e. by further experience or other, more reliable conceptual inferences), then we opt for the second most probable disjunction. If the latter is also eliminated, we go for the third, and so forth, till (if ever) we are left with only one option. This is of course the process of adduction – where, faced with more than one solution to a problem, we opt for the most credible solution, but may gradually be driven (by the concrete evidence or abstract issues) to prefer initially less obvious solutions.[5]

But note too that in some cases even the least probable option may eventually be found (empirically or otherwise) wanting! In such a case, we would have to backtrack through our chain of reasoning to find out exactly which assumption we made earlier needs to be revised so as to recover a logical situation. For it is logically unacceptable that all the valid alternative inferences from true premises be found false. If the consequent of an antecedent is certainly false, the antecedent cannot be entirely true but must contain some error.

Note here that, in view of the possibility of erroneous premises in deduction (whether the conclusions ultimately be found true or false), deduction is much more tentative than it seems at first sight. In that sense, deduction ought to be viewed as one tool in the toolbox of induction (together with observation, generalization and particularization, adduction, and so forth). Even the results of direct applications of the laws of thought are ultimately inductive, in the sense that the empirical or conceptual data the laws are applied to are products of prior induction.[6]

Since deduction is impossible without some given information from which further information might be deduced, all knowledge is ultimately inductive. All knowledge requires some sort of experiential input (whether sensory, mental or intuitive) from somewhere or other to take shape. Deductive logic is simply the ordering of such knowledge with reference to the laws of thought (identity, non-contradiction and excluding the middle).[7] These comments are in no way intended to devaluate deduction. We can point out, conversely, that induction beyond plain observation is impossible without some deduction, and that the moment we begin to analyze and synthesize purely empirical data, we are engaged in deductive acts.

Our interpretations or explanations of given data may variously be referred to as inductive or deductive conclusions, according to where we put the emphasis. If we want to stress the tenuousness of the result, we call it inductive; if we want to underline the rigor of our reasoning, we call it deductive. A merely ‘most probable’ conclusion is still deductive, in the sense that it is the best possible hypothesis in the context of knowledge available. This means: given the premises available, we can indeed deduce that conclusion; but if we were (or are later) given additional (or modified) premises, another conclusion might be deduced.

[1] I had the idea of developing this topic following a debate with Plamen Gradinarov in one of his Internet sites relating to Indian logic.

[2] See my work Future Logic, chapter 45. Note that we sometimes do verbally permute geometrical propositions. For example, ‘the worm went underground’ fuses the words ‘under’ and ‘the ground’ as if they formed a predicate, although the spatial relation of ‘under’ is formally more precisely tied of the copula ‘went’. This is more obvious if we contrast ‘the worm is aboveground’ – if the expressions ‘under’ and ‘above’ were indeed inextricably tied to the word ‘ground’, we would not realize that they are two relations the same worm might have to the same ground. Or indeed we might forget the third possible such relation, viz. ‘partly above, partly below’ the ground. In any case, we would miss out on syllogistic reasoning specific to geometrical propositions – for example: the premises ‘X is {in Y} and Y is {in Z}’ would yield no conclusion, since they lack a common middle term (i.e. ‘Y’ is not logically identical with ‘in Y’).

[3] With regard to methodology, it should be stressed here that logical issues cannot be credibly settled by only proposing concrete examples (as some amateur logicians are wont to do). Particular premises may seem to yield a categorical conclusion, because that ‘conclusion’ happens to be true in that particular case (independently of the reasoning process), although in fact – when we consider the issue more formally – there may be two or more possible conclusions. For instance, “the box is on the truck and the truck is in the garage, therefore the box is in the garage” might superficially seem correct (because in some cases it happens to be so) – but in fact the correct conclusion is “the box is at least partly in the garage, but it may be partly out” (for the box may be much longer than the truck). On the other hand, of course, a single concrete case may on occasion suffice to invalidate a proposed form.

[4] Geometry deals with mathematical abstractions, which do not necessarily have obvious material expression. For example, if we say ‘the planet Earth is in the Solar System’, we do not mean that the Solar System has a visible physical boundary. We would rather think of its boundary as being the outer limits of the gravitational pull of the Sun and the planets and other bodies close to it (versus the pull of other eventual bodies in the same galaxy) – beyond which a body would travel off unhindered. That effective boundary may of course be variable as conditions change.

[5] To give a specific traditional example: suppose I see something that resembles phenomena I have in the past labeled “smoke”. I cannot immediately call it smoke without risking error. Before I apply the same label to my new visual experience, I have to diligently ensure it fits in the conditions of applicability previously established (or I may have to adapt those conditions). Thus, I would want to sniff and find out if what I have just seen not only looks like smoke but smells like smoke. If I cannot be sure that the smell came from the same source as the sight, I may have to test the matter further by looking for an underlying fire. Better still (since smoke is not always accompanied by visible flames or tangible heat), I may chemically analyze the phenomenon. If it turns out to consist of H₂O, for instance, I would conclude it to be not smoke but mist. If on the other hand the chemical composition is found to be consistent with the composition previously established for smoke, I can at last with reasonable probability conclude that what I saw was smoke (unless or until some further objection is proposed). This is the process of adduction in observation, classification and naming: gradually eliminating alternative interpretations of an initial observation by means of additional observations and arguments; narrowing down the possibilities until we can attain reasonable certainty.

[6] I am here referring to arguments showing up a self-contradiction in some idea or thesis; for example: it is self-contradictory to say “all knowledge is false”. The point made here is that, in the latter example, we depend on understanding what is meant by “all”, by “knowledge”, by “is” and by “false” before we can realize that saying so is itself a claim (to an item of knowledge that is true) and therefore is self-contradictory. Indeed, even the concept of self-contradiction has to be understood. It follows that no act of reasoning, however primary, is ever deductively an island unto itself; there’s always some element of induction beneath the surface.

[7] Granting this reflection, it is easy to see the foolishness of Kant’s “analytic/synthetic dichotomy”; and similarly, of the work of logicians who assume there is such a thing as “purely deductive logical systems”. Such philosophers and logicians do not stop to ask how they managed to obtain their knowledge apparently out of nowhere. There is a failure of self-criticism on their part; they assume their insights to be irreducible primaries, as if they have been granted an epistemological privilege.