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

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RUMINATIONS

Sundry notes and essays on Logic

© Avi Sion, 2005. All rights reserved.

 

Part I - Chapter 4

About Formal Logic

 

1.         Form and Content

2.         Singular Subject

3.         Special Forms

4.         Fuzzy Logic

5.         Added Determinants

6.         Relational Expressions

7.         Disjunction

8.         Material and Strict Implication

9.         Nesting of Hypotheticals

10.       Compound Theses

11.       Validation of Nesting

12.       Brackets in Logic

 

1.         Form and Content

The notions of form and content are simple enough, though the uninitiated must first have them explained. ‘Form’ and ‘content’ are relative terms used in different contexts within formal logic. The basic idea is that of container and contained.

In one sense, a word is a form, and the word’s meaning – the real or imagined things it refers us to (i.e. that we intend when we use the word) – is the content. Thus, the personal name ‘Joe’ refers to an individual man we know by that name; the common name ‘man’ refers to an open-ended group of individuals like Joe, Jim, Nathalie and others.

We can also call any abstraction (or concept) a form and its (perceptual or intuitive) concretes the content. In this perspective, our concept of Joe is a collection of an immense number of sights, sounds, etc., across time and in various circumstances, that we have found fit to unify under this one idea. Similarly, in the case of general concepts like that of man: they refer to a presumed unity in the midst of large ongoing collections of material, imaginative and introspective data about Joe, Jim, etc.

Till here, we are in the everyday practice of logic. We enter the more abstract field of formal logic, the moment we posit a symbol like ‘X’ or ‘Y’ to stand in for any term like ‘Joe’ or ‘man’. At this stage, we formalize propositions.

For example, categorical propositions are written: ‘X is Y’, where X is the subject, Y is the predicate, and ‘is’ is the copula, i.e. the intended relation between X and Y. Note that ‘is’ is a form in the preceding sense; it is a bit less abstract than ‘X’ or ‘Y’. Note also that the relation we signify by ‘is’ does not exist apart from its particular terms (X and Y, in this case); it refers to the cement between them, which is also part of them. It is only cognitively that a distinction between these items exists; at a concrete level, they are inseparable.

Besides ‘is’ there are many other categorical relations, including copulas like ‘becomes’, ‘makes’ and so forth. Moreover, categorical propositions have other features not so far mentioned that are on a similar level of abstraction as the copula, such as the quantity (all, some, this one) and the modality (must be, can be, actually is).

We could therefore further formalize categorical forms, by means of symbols like (say) “Q(X)MR(Y)”. This concoction is of little use, however, because little can be said about categorical propositions viewed so generally. More interesting is to realize that there are other propositional forms, like the comparative (e.g. “X is more Y than Z”) or the hypothetical (e.g. “if p, then q”), for instances.

At this stage, logicians propose a general symbol, say: “P” (or “Q” or whatever), to stand for just any sort of proposition. This artifice of formalization has been found very useful. For instance, the p and q in “if p then q” are such general symbols. They allow us to study what the form “if-then” entails, without having to specify what kind of proposition p or q might represent in a particular case. We can study the “if-then” relation as such, in a very general way.

Formalization, then, is just a way to freely study the logical properties of different propositional forms, without regard to their content. A ‘form’ is simply a shorthand expression for any number of particular propositions, or ‘contents’. What we say about the form applies to all the contents. The wider the form, the broader the range of possible behavior, and the less rules there are for it. If any content is specified, or a more specific form is considered, the behavior pattern becomes more narrow, and the rules more restrictive.

 

2.         Singular Subject

A singular subject is usually identified:

·         indicatively (as “this X”), or

·         by name or pronoun (e.g. “John” or “he”).

But there are also unidentified singulars, which may be:

·         categorical (as in “someone stole my car”), or

·         conditional (as in “whoever it was, I’ll find him”).

The same distinctions apply to plural subjects, groups: “these X” is indicative; “The Brothers Karamazov” are named, “they” is the corresponding pronoun; “some people” and “whoever they be” are the unidentified group of individuals.

The singular subject “This X…”, which is sometimes read as “This thing, which is an X,…”, may be further analyzed as follows:

1.      First, I say “this” (demonstratively pointing to something, or in some other way directing attention to it), to produce in the auditor an awareness of a specific object, i.e. the minimum knowledge of it given within the perceptual glance or remembrance produced.

2.      Secondly, I categorize the object as “X”, i.e. classify it (occasionally, exceptionally, if the sentence is intended to convene on a word or transmit existing language, I just name it). This adds to the auditor’s knowledge, making him aware of the character X apparent in the object previously indicated (or, in the case of mere naming, aware of the name).

3.      In a third stage, I may propose a predicate, e.g. “This X is Y”, thereby more precisely categorizing the object and increasing knowledge of it. Such predication may be based on some experience or on a rational inference (such as syllogism).

The point being made here is that propositions are not ready-made, static wholes (as some modern logicians seem to regard them), but thoughts that are gradually built up in the mind and comprehended stage by stage. Subsumption is not a mere abstract relation, but signals a mental process of becoming aware of and assimilating information.

Likewise, a sentence is not mentally or orally formed in one shot, but gradually emerges. Our first try might just be an approximation of what we mean, expressing a thought that as yet remains ambiguous and uncertain in some respects. Then, reflecting on what we have just said, we might be moved to attempt a clearer, more precise expression of our insight or observation. Or we might discover and repair logical or grammatical faults in our previous statement. Sometimes, all such refinement is achieved on the first try; other times, we need successive tries to obtain a satisfactory result.

Conversely, when a logician seeks to logically evaluate some discourse, he has to do a considerable amount of preliminary linguistic analysis[1], to properly and fully interpret what is being said, before issues of formal logic come into play. This background processing, be it conscious or subconscious, is often insufficiently stressed by logicians.

 

3.         Special Forms

The form “It happens that X” or “There is X” is used to refer to events without pressing them into a strict subject-predicate form. For example, “It is raining” could have been stated as “Water drops are falling from clouds in the sky” – but we colloquially prefer the brief and phenomenal description, acknowledging or indicating the fact that there is rain in the field of vision (out there somewhere), without specifying whence this rain is making its appearance (the sky, clouds).

In such cases, we may retain the copula “is” to signify the presence of something, but we do not intend thereby to force the event described into a standard “S is P” format. The “it” in “it is raining” is not intended as an authentic subject. The latter (the sky, clouds) remains tacit. The form used is only superficially like the standard form.

Contrary to the certain critics of ‘Aristotelian’ logic, we do not hold that “S is P” is the only form of proposition. There are other categorical forms, as well non-categorical forms. The job of logicians is precisely to notice and investigate an ever-widening circle of forms and arguments. Nevertheless, the “S is P” form is basic to rational knowledge. Propositions like “it is raining” do not belie this standard, but require reference to it to be fully clarified.

The question to ask in differentiating a form is – is its logical behavior different? If the eductions and syllogistic inferences from it are different from the standard form, then the deviant form should certainly be given special treatment. If its inferences are essentially the same, then logicians need not give it more attention than necessary.

Logicians have tended to regard propositions as built up from their parts (terms, copulae and operators), but the process is in truth more inductive: we first look upon the whole event (e.g. a woman smoking), and out of many such events, by comparison and contrast, we mentally isolate the parts (woman, is, smoking). Only after this, can we name the parts and put the words together in a sentence.

All propositions are concepts, like the terms, copulae and operators (e.g. if-then, either-or, etc.) that constitute them. If they have (established, existing) referents, they are “true”; if not, not. Likewise, all concepts are implicitly propositions, since they affirm their referents (tentatively or definitely to exist). Forms like “There are…” or “It is…” reflect this technical equivalence between whole and partial concepts.

Note that such forms can also be modal: “There might be…”, “There are sometimes…”, “There is necessarily…”, etc.

 

4.         Fuzzy Logic

One function of “fuzzy logic” is to process concepts whose referents are not clearly definable.

Logicians in their theories should be careful to reflect the varieties of human thought processes, and not try to put their square pegs in round holes, e.g. by demanding that all subjects be defined by a universal predicate.

Normal concepts are defined by a common and distinctive character, and are therefore mutually exclusive: anything with some X in it is an X; and anything without any X in it is a non-X. However, some concepts refer to the predominance of some character Y, without insisting on the total absence of its contraries.

For example, the term “Indo-European language” refers to words and grammatical forms that are mostly Indo-European in origin, although some other roots and constructions (e.g. Semitic) may admittedly be found in it. Or again, the Bible’s “historical books” are not only historical, but contain some legal and other material.

The “fuzziness” of such concepts is not due to their having not yet attained (inductively) an optimum clarity of definition – it is inherent to their subject matter. It is not a conceptual flaw, but a reflection of the mixed state of the things referred to.

How can such concepts be logically processed according to Aristotelian syllogism? An example is the third figure mood:

 

Most Q are R, and

Most Q are P

Therefore, some P are R.

 

This argument is strictly valid, since the middle terms (“most Q”) of the two premises overlap. If the effective middle term is a “fuzzy concept”, the premises would both apparently refer to “all Q”, but in fact be based on the form “each Q is mostly Y, though partly non-Y”.

One should of course be careful in such contexts not to commit the fallacy of Four Terms. This formal fallacy is very common in practice, usually by way of ambiguity – the middle term verbally seems (or is made to seem) the same in both premises, but in fact does not refer to the same cases, so that any inference linking the major and minor terms through it is invalid.

 

5.         Added Determinants

Certain ‘arguments’ remain informal, because they cannot be formally validated. They are intended more rhetorically or poetically, than strictly logically. They are commonly used because of their usefulness in discourse: they make a point with a punch.

These include processes that have been labeled ‘added determinants’ (or ‘complex conception’), although I would hesitate to consider such processes to be sufficiently uniform to be clearly categorized. Rather, I would speak of a loose collection of diverse forms: an open-ended catchall for leftovers.

A couple of examples should suggest the degree of variety.

(a) “Power corrupts – absolute power corrupts absolutely”. A statement like this is not meant as an argument; rather, the first general statement is augmented by the second one, which quantifies it and specifies the extreme degree of it.

By syllogism, we can infer from “power corrupts” that “absolute power corrupts” – but not that the latter “corrupts absolutely”, which is an additional observation.

It could also be argued that “power corrupts” refers to small quantities of power, so that we may a fortiori infer that larger quantities of power corrupt even more. Note however that such inference would not be in accord with the sufficiency (“dayo[2]) principle.

In sum, the statement “power corrupts” does not by itself reveal whether there is concomitant variation and proportionality between power and corruption, and so cannot formally imply that “absolute power corrupts absolutely”.

(b) “All love is wonder; if we justly do account her wonderful, why not lovely too?” Here, a John Donne intent on seduction argues that given ‘love implies wonder’ it follows that ‘if the woman is lovely, she is wonderful’ (actually, he reverses the conclusion, but let us ignore this poetic license).

Such an eduction is of course formally open to debate. The premise seems to be a psychological statement, that being in love gives rise to an experience of wonder; whereas the putative inference, characterizes the female object of his attentions as lovely and wonderful. The process is made to seem like an application of a generality to a particular case. The terms are admittedly not unrelated, in that the poet’s psychological condition affects his perception of the woman. Still, the shift from his rather subjective assessments to quasi-objective characterizations is not strict logic.

Note the grammatical differences between the above two examples. In (a), a more or less common determinant (absoluteness) is added, as an adjective and an adverb respectively, to the initial noun and verb. In (b), two abstract nouns are turned into adjectives relative to some added third noun (the woman); in this case, the addition is a ‘determined’ rather than a ‘determining’ factor.

Many more examples can be adduced, to show that it is best not to quickly generalize. Each example encountered should be analyzed individually, to understand both its power of conviction and its hidden sophistries.

 

6.         Relational Expressions

When considering a propositional form commonly used in our thought and discourse, we should identify its minimum meaning, the most widely applicable interpretation.

Thus, the form “Unless P, Q” means that if P is absent (or false), Q is surely present (or true), though it does not really tell us whether in the presence of P, Q is necessarily or just possibly absent. Thus, this form implies “if not P, then Q” – and “if P, not-then Q”, though possibly (in some cases) also “if P, then not Q”.

The form “Though A is B, C is D” may be variously interpreted. This could just be intended as a statement of contrast, drawing attention to the divergent attributes (respectively B, D) of the subjects (A, C). Or it could be a statement intended to undermine or incite rejection of some theory of consequence – i.e. perhaps someone thought that “if A is B, then C is not D”, and here we are told that this alleged connection is in fact absent. In either case, note, the underlying relation (between the theses ‘A is B’ and ‘C is D’) is conjunctive rather than conditional.

The form “As a B, A is C” may be expanded to the syllogism: “A is B and all B are C, therefore A is C”. Notice the underlying general statement involved (tacit major premise). Incidentally, this form is often used as a means of ego construction: we (A) identify with a certain denomination (B), and on this basis attach to a certain behavior pattern (C); for example, “as your father, I advise you not to do this!”

More generally, a statement of the form “Since P, Q” is not simply a proposition, but an abridged argument. It apparently intends the apodosis: “If P, then Q; and P; therefore, Q”. Usually in practice, it means somewhat more, in that there may be a tacit major or minor premise R that we take as granted and understood. In such cases, we would render it as: “If (R +) P, then Q; and (R +) P; therefore, Q” (the argument is traditionally then referred to as an ‘enthymeme’).

Note also: very often in practice, the relationship between the antecedent and consequent in “Since P, then Q” is not mechanical, but volitional. For example, “Since you did this, I will do that!” In such cases, though the underlying conditional proposition has natural modality, and the consequent does not automatically follow the antecedent – i.e. the “then” is a bit of an overstatement.

Such overstatement of connection is common in discourse, even in purely mechanical contexts, to repeat. As long as some conditions remain tacit, the “then” involved is not to be taken literally. For example, in “if the machine has this extra gadget, it functions continuously”, it is clearly intended that all the other parts of the machine, in addition to the gadget mentioned, also play a role in producing the movement described.[3]

Another form worth mentioning is “It (P) is as if Q”. Here, ‘it’ (P) refers to some event, condition, result, or connection, and ‘Q’ to another; and ‘is as if’ indicates that if the two are compared they will be found similar to some degree. The degree of resemblance might be qualified by adding “a bit” or “much” to “as if”.

 

7.         Disjunction

With regard to disjunction, the following insights are worth adding to my past comments, because I have found many people to be confused by the varieties of senses the operator ‘or’ may have in ordinary discourse.

The expression “P or Q or …” is very vague; it only informs us that some manner of ‘disjunction’ is involved, but does not tell us what form it has. The operator ‘or’ is thus, in formal logic, to be understood very broadly.

This indefinite sense is somewhat narrowed down by making the distinction between ‘exclusive’ disjunction, for which the form “P or else Q …” may be agreed, and ‘inclusive’ disjunction, for which the form “P and/or Q…” may be agreed.

Thus, “P or Q…” may be taken to formally mean: “P or else Q…” and/or “P and/or Q…” is/are true. The exclusive and inclusive forms of disjunctions are thus more specific and explicit; and each of them implies the more generic and indefinite form.

If only two items (P, Q) are involved, exclusive disjunction just means “if P, then not Q” (and vice versa), whereas inclusive disjunction just means “if not P, then Q” (and vice versa). Thus, the first refers to the logical relation of incompatibility, while the second refers to exhaustiveness.

Moreover, exclusive and inclusive disjunctive propositions, though not as indefinite as generic disjunction, are themselves vague or open forms. The form “if P, then not Q” leaves unanswered the question as to whether not-P implies or does not imply Q; and likewise, the form “if not P, then Q” leaves unanswered the question as to whether P implies or does not imply not-Q.

If the two forms are combined, as is formally possible, they together imply P and Q in contradiction; if P and Q are incompatible but not exhaustive, they are contrary; if they are exhaustive but not incompatible, they are subcontrary. In common discourse, contradictories are placed in the form “either P or Q”; contrariety is expressed through the form “P or Q or neither”; and subcontrariety, through the form “P or Q or both”.

Therefore, we could say that “P or else Q” means: either “either P or Q” or “P or Q or neither”; and likewise, “P and/or Q” means: either “either P or Q” or “P or Q or both”. Thus, the indefinite form “P or Q” can also be read as: “P and Q are either contradictory or contrary or subcontrary”.

Note that our choice of the words “P and/or Q” to express the generic relation “if not P, then Q” is clearly not very appropriate, suggesting that P and Q are compatible, whereas they need not be so. The term ‘inclusive’ disjunction suffers from the same imperfection, seeming limited to subcontrariety. Since that terminology is too well established to be changed, we must simply ignore these misleading verbal connotations.

Thus, to summarize, disjunction may be considered as a generic relation between two terms or theses. This relation may be specified as exclusive or inclusive (or both); or even more precisely as contradictory, contrary or subcontrary. Contradiction occurs when both exclusive and inclusive disjunction are applicable.

All this can be compared to saying of two items (P, Q) that they are “related by implication”. This does not tell us whether “P implies Q” or “P is implied by Q” or both. If both directions of implication are true, P and Q are mutual implicants; if only one is true, then either “P subalternates Q” or “P is subalternated by Q”.

If more than two terms are involved (P, Q, R…), the formulas are more complex. Namely, in exclusion: if any one item is true, all the others are false; in inclusion: if all but one item are false, the remaining one is true. Note that, in the former case, no two items are compatible; whereas, in the latter case, the exhaustiveness concerns the complete set of items, but if we take any two of them at random, it does not have to apply.

These two relations between three or more items may, as with two items, occur in combination or separately. In such cases, distinguishing between ‘or else’ and ‘and/or’ becomes impractical, and the best course is to use ‘or’ and verbally define the intended set of relations. Note that matters may be further complicated in some cases because some of the items in the set have special relations that the others lack – i.e. we may intend mixed-form disjunctions. In such situations, explicit clarifications as to what we mean are all the more necessary.

We should keep in mind that much of the terminology in this field was invented by logicians; it is not a product of popular discourse. The word ‘disjunction’, etymologically connoting negation of conjunction (i.e. separation), first appeared in the 14th Cent. The conceptual distinction between ‘exclusive’ and ‘inclusive’ disjunction was made much later, and these terms were apparently coined only in 1942 (according to the Merriam-Webster Collegiate Dictionary).

The clear distinction between contradiction, contrariety and subcontrariety is, however, ancient, dating back at least to Aristotle, if not earlier. The concept of incompatibility is doubtless earlier than those of contradiction or contrariety; though these three terms may originally all have had the same meaning. The concept of exhaustiveness, being more subtle, probably arose later; and that of subcontrariety no doubt much later.

However, the word ‘or’ was not invented by logic theorists, but is found (in some form or other) in common discourse since way back. Certainly, the underlying notion must be very ancient. With regard to its verbal expression, I am not so sure, having noticed that discourse in the Talmud often struggles with this. For instance, it says (Sabbath ב"פ): “doubting sunset, doubting not sunset, [don’t do so and so]” (where we might have said: “if in doubt as to whether or not the Sun has set, don’t…”). The items are there listed, but their relation of disjunction is left tacit, as if there was no word for it (though words existed long before in Biblical Hebrew).

The many modern variants of the word ‘or’ – phrases such as ‘or else’, ‘and/or’, and others[4] – are also apparently natural linguistic developments, although evidently much more recent. They presumably arose as more or less deliberate attempts, within some ordinary discourse, to remove some of the ambiguity in the word ‘or’. Finally, of course, some logician came along and conventionally ‘froze’ the predominant meaning of each variant, so as to facilitate formal treatment.

Let us now examine the probable development of the notion of ‘or’. In English, the word is etymologically related to the word ‘other’ – suggesting that the second item listed is somehow ‘other than’ the first item listed. Now, ‘other than’ could be interpreted as ‘opposed to’ (suggesting exclusive disjunction) or as ‘different from’ (suggesting inclusive disjunction).

It might be thought that the first interpretation most accurately reflects the original meaning of ‘or’; some dictionaries seem to claim this. But in my opinion, both interpretations were vaguely intended from the start; for there is a common notion underlying the two.

The ‘or’ within exclusive disjunction means ‘not together’. Here, “P or Q” means P to the exclusion of Q, i.e. P only, P alone, whence P without Q (or vice versa, provided P and Q are not both true). The ‘or’ within inclusive disjunction means ‘not same’. The latter is softer: it allows that P may occur without Q, but does not insist on it (or vice versa, provided P and Q are not both false). The two forms are thus analogous in some respect, and the difference between them may be viewed as one of degree.

The disjuncts (P, Q) are rightly labeled ‘alternatives’, to indicate the essential fact of their being considered ‘in succession’. In exclusive disjunction, the alternatives displace and replace each other, whereas in inclusive disjunction, they do not necessarily do so. In the latter, the items are merely listed as individual possibilities, without prejudice as to whether they have to be separate or may eventually not be so.

We very often need to draw up a list of possibilities, without at the outset deciding whether all the alternatives are mutually incompatible, or even knowing full well that some or all of the alternatives may occur together. Sometimes, in writing, we simply use a comma instead of the word ‘or’ in such lists, so as to just avoid this issue of relation between the disjuncts. Because the practice of simple listing has obviously always existed in discourse, it cannot convincingly be argued that exclusive disjunction antedates inclusive disjunction.

We must thus suppose that a broad sense of the word ‘or’, which leaves open the issue of whether the disjunction is exclusive or inclusive or a mix of the two, has always existed (in some form or other). It follows that all senses of the word ‘or’ are equally legitimate in discourse, but we must remain aware as to how it may be intended.

The speaker or writer should opt for clarity; and the hearer or reader should carefully weigh the word in each context. In practice, sometimes, we make no verbal distinction between the disparate senses of ‘or’, letting context determine intent. In case of doubt, only the minimal, most indefinite sense may be assumed – i.e. the sense that is neutral with regard to the exclusive/inclusive distinction, i.e. the common property of all disjunctions.

Note that some people tend to use the unqualified form “P or Q” for exclusive disjunction, and get more explicit in cases of inclusive disjunction; while some people do the exact opposite. Different people behave differently, and even the same person at different times; so, no hard and fast rule can be handed down.

It is, note well, always possible to say exactly what we mean when we wish to, or when (as in formal contexts) we must. We need only declare our preferred language, and that becomes our convention in subsequent discourse.

With regard to the two constructs “P or Q or neither” and “P or Q or both”, the following may be added. Here, “P” means “P alone, i.e. without Q”; “Q” means “Q alone, i.e. without P”; and “both” means “P and Q”, whereas “neither” means “not-P and not-Q”. Thus, each form clearly lists all the alternative events acceptable to it, leaving out the defining unacceptable alternative – viz. “P and Q” in exclusive disjunction, and “not-P and not-Q” in inclusive disjunction.

The ‘or’ operator throughout these two forms is therefore the same: it refers implicitly to exclusive disjunction. The final disjunct ‘or neither’ or ‘or both’ serves to declare the disjunction not only exclusive, but also exhaustive. Note that we may construct similar forms with more than two disjuncts, of course (using as our last disjunct ‘or none of them’ or ‘or all of them’).

The vague “P or Q” form is often intended as an abbreviated version of these explicit forms. That is, when we use it we may be tacitly thinking and implying ‘or neither’ or ‘or both’, as the case may be, but we omit to say that explicitly out of laziness or the desire to be brief. More often than not, we leave the matter open, simply because it is not very relevant to our present discursive needs. Very often, too, as already pointed out, we have not yet determined the interrelations between the theses.

Lastly, note the function of the word ‘either’ at the beginning of a disjunction, be it exclusive or inclusive. This word serves to signal that the set of (two or more) alternatives listed is exhaustive, i.e. that the list is complete and there are no more alternatives to consider. Thus, in the case of “either P or Q”, the intent is that “P without Q” and “Q without P” are the two only acceptable outcomes.

Similarly in cases with more than two alternatives, i.e. “either P or Q or R or…”: all possibilities are declared foreseen. If the multiple disjunction is meant exclusively, the final outcome will consist in affirmation of only one of the alternatives and denial of all the others. If the multiple disjunction is meant inclusively, the final outcome will consist in affirmation of at least one of the alternatives, though possibly more or even all of them.

The word ‘either’ delimits a list. A list without it (i.e. just “P or Q or…”) is normally considered open – i.e. it may be incomplete: we may have intentionally or unintentionally ignored some other alternative(s).

It is a redundancy to add the word ‘either’ in front of a disjunction ending in the words ‘or neither’, ‘or both’, or the like – since, as we have seen, these words already signal exhaustiveness. The word ‘neither’, by the way, simply means ‘not either’ – i.e. it indicates that there are indeed other alternatives than those listed. Thus, in “P or Q or neither”, the ‘neither’ refers directly to “not-P and not-Q” (i.e. “neither P nor Q”), but also less directly to unstated alternatives “R or S… etc.”

Finally, it should be kept in mind that there are different modes of disjunction. In addition to the logical mode, there are the natural mode and the extensional mode, as well as the spatial and temporal modes. These are often mixed and undefined in ordinary discourse. For example, ‘or both’ or ‘or neither’ may be intended as a statement of fact (de re modality) or as something logically conceivable given our ignorance of the facts (de dicta modality). Failure to take such ambiguities into account can lead to some quite fallacious interpretations!

 

8.         Material and Strict Implication

Material and strict implication exhibit significant formal differences in behavior. This can be made manifest as follows.

In the case of material implication, which refers to inactualities of truth, the hypothetical “if P then Q” means “(P and not Q) are not both true”. Here, the reverse is also valid; i.e. “The conjunction of P and not Q is not true” formally implies “P (materially) implies Q”.

In the case of strict implication, which refers to impossibilities of truth, the hypothetical “if P then Q” means “(P and not Q) cannot be true together”. Here too, the reverse is also valid; i.e. “The conjunction of P and not Q cannot be true” formally implies “P (strictly) implies Q”.

The relations involved are parallel. However, when we mix the two categories of modality, the result is significantly different. Since impossibility implies inactuality, but inactuality does not imply impossibility, it follows that “P and not Q are incompatible” implies but is not implied by “P and not Q are not jointly true”.

Thus, whereas in material implication “if P then Q” is fully equivalent to “(P and not Q) are not both true”, i.e. these two forms mutually imply each other – in the case of strict implication, “if P then Q” implies but is not implied by “(P and not Q) are not both true”, i.e. the implication between these two forms is unidirectional. Knowing actual negation of conjunction does not justify assuming strict implication. [5]

It should be added that these reflections provide us with an unbeatable argument in favor of strict implication, against the advocates of material implication. If we ask: what is the “formal implication” or “implication between forms” that we refer to in this very discussion (and indeed in all discussions of formal logic - or of mathematics, for that matter)? Is it material or strict? The answer has to be: strict implication.

When we say: “(P materially implies Q) implies and is implied by (P and not-Q are not both true)”, the implications between the bracketed items are strict implications, even though the implication within the first item is material. Formal implication is logical necessity; i.e. it is applicable under all possible conditions, whatever the content of the forms involved. Therefore, strict implication is more important to logic than material implication.

Does material implication then have any place in natural discourse, or is it artificial? I believe it still does have a place, due to the fact that all implication is denial of conjunction. When we know that, say, “P and not-Q are not both true”, we may indeed turn it around in our minds, thinking “well, that means if P is true, Q is not false, etc.” This shows that material implication is useful to the understanding, helping us mull over certain indefinite statements.

Note well, then, strict implication is essential to logic, and cannot be ignored or discarded in favor of material implication, as some logicians (and mathematicians) think, even though the latter has some utility.

Additionally, an oft-ignored advantage of strict over material implication is the negative form it provides us. If we understand “if P, then Q” in the strict sense, then its contradictory is the negative form “if P, not-then Q” (or “If P, it does not follow that Q”), meaning: “(P and not-Q) is a possible conjunction”; whereas, in the material sense, its contradictory would simply be the actual conjunction “P and not-Q”.

Clearly, the negation of strict implication gives discourse an important formal tool. We can, for instance, use it to point out the common fallacy of confusing a non-sequitur demonstration with a disproof. If we show that a conclusion does not follow from certain premises (“if P, not-then Q”), it does not mean we have disproved the conclusion (“if P, then not Q”).

With regard to logic history, I would like to here correct a suggestion I made in [the original version of] Future Logic, that the Megarian Philo’s view of implication may have not corresponded to our modern concept of material implication. The following quotation from the Encyclopaedia Britannica (2004) convinced me:

“Diodorus also proposed an interpretation of conditional propositions. He held that the proposition “If p, then q” is true if and only if it neither is nor ever was possible for the antecedent p to be true and the consequent q to be false simultaneously. Given Diodorus' notion of possibility, this means that a true conditional is one that at no time (past, present, or future) has a true antecedent and a false consequent. Thus, for Diodorus a conditional does not change its truth value; if it is ever true, it is always true. But Philo of Megara had a different interpretation. For him, a conditional is true if and only if it does not now have a true antecedent and a false consequent. This is exactly the modern notion of material implication. In Philo's view, unlike Diodorus', conditionals may change their truth value over time.”

Following this reading, we can safely assert that strict implication was first elucidated by Diodorus Cronus (also a Megarian, d. circa 307 BCE)[6]. Note that Philo was a student of Diodorus.

One last note on this: material implication is a logical (i.e. “de dicta”) relation – and is not to be confused with any of the “de re” types of conditioning, i.e. with natural, temporal, extensional or personal conditionals. Some logicians are led into this confusion by the name “material” implication and its implied contrast to “formal” implication. But the truth is that so-called material implication is a subcategory of logical relations, just one that is weaker than strict implication.

 

9.         Nesting of Hypotheticals

Concerning nesting of hypothetical propositions: the nested form “if p then (if q then r)” may be considered as equivalent to (implying and being implied by) the form with a compound antecedent “if (p and q) then r”[7]. From which it follows that it also means: “if q then (if p then r)”. And since the nested clause can be contraposed to “if p then (if not r then not q)”, we can further educe: “if (p and not r) then not q” and “if not r then (if p then not q). Or again: “if q then (if not r then not p)”; whence: “if (q and not r) then not p” and “if not r then (if q then not p)”.

No matter which of these forms we choose to use in our discourse, they all mean the same thing, namely “(p + q + not r) is an impossible conjunction”, all seven other combinations of p, q, r, and/or their negations, being left open, i.e. remaining logically possible in the stated context. This is the underlying ‘matrix’ of meaning, which remains constant for the form concerned, however complicated the way we express it. If in a given context, additional forms are specified as true, one or more of these combinations left open is declared impossible, and the range of logical possibilities becomes narrower.

The laws of thought teach us that there are only eight ways p, q, r, and/or their negations, can combine together. They cannot all be false: one of them must be true; no ninth way is ever logically possible (law of the excluded middle). Furthermore, if one of these combinations is true, all seven others are false; i.e. no more than one of them can be true in any given case or context (law of non-contradiction).

Conjunctions as such are compound categorical propositions. Hypothetical (if-then) propositions, on the other hand, are defined by general[8] negations of such conjunctions. Whereas (positive) conjunctions are directly about the truth or falsehood of the combined propositions, the general negations of conjunctions (i.e. hypotheticals) are about the logical impossibility of specified combinations – i.e. they determine truth and falsehood more vaguely.

When we say that a conjunction of propositions is logically “possible”, we mean that, as far as we know, or can logically predict in the given context of knowledge – that conjunction may yet turn out to be true, i.e. that form may well be realized in the case concerned. A combination is logical “impossible”, on the other hand, if no matter what its content or eventual changes in our knowledge context, we can predict with certainty that it will never be found true.

If all eight conjunctions (of our three items) are still possible, it is as if nothing has been said (since this is logically given universally). We begin to say something significant, when we narrow down the possibilities by declaring (for whatever reasons) one or more of the conjunctions impossible. The more combinations are negated, the more specific our statement. If it turns out eventually that seven conjunctions have thus been eliminated (for various reasons), the leftover conjunction has got to be “true”. Of course, this implied truth is contextual, i.e. it remains dependent on the correctness of all experience and reasoning that led up to it; but granting the latter, it is true. [9]

 

10.       Compound Theses

The logic of hypotheticals with compound theses. Nesting may be viewed as anticipation of the consequences of gradual realization of a compound antecedent. This gives us, for an antecedent compounding two propositions:

·         “(p and q) implies r” implies and is implied by “p implies (q implies r)”.

The same can be done, by successive application of the preceding argument, with compounds involving any number of elements (as with the example with three below):

·         “(p and q and r and…) implies s” implies and is implied by “p implies (q implies (r implies… s))”.

Having considered the logic of conjunctions in the antecedent of hypotheticals, let us, in passing, also mention the corresponding logic for their consequents. Note specifically the following useful arguments, which are easy to validate (by referring to the underlying conjunctions, as usual):

·         “p implies q” implies and is implied by “p implies (p and q)”[10]. This may be labeled ‘adding the antecedent to the consequent’, or in the reverse direction ‘subtracting it’.

·         “p implies q” and “p implies r” together imply and are implied by “p implies (q and r)”. This may be labeled ‘adding together consequents of the same antecedent’, and in the reverse direction ‘splitting them apart’.

These arguments may be compounded with the following hypothetical syllogism:

·         “p implies q” and “q implies r” together imply (though are not implied by) “p implies r”

… to yield the following two derivative arguments:

·          “p implies q” and “q implies r” together imply (though are not implied by) “p implies (q and r)”.

·          “p implies q” and “q implies r” together imply (though are not implied by) “p implies (p and q and r)”.

The above mentioned process of ‘adding together consequents of the same antecedent’ may be viewed as a special case of the following process, of ‘merging hypotheticals’, i.e. compounding both their antecedents and their consequents:

·         “p implies r” and “q implies s” together imply (though are not implied by) “(p + q) implies (r + s)”.

This process would seem valid only on the proviso that p and q are compatible, granting that if the antecedents are not compatible, they couldn’t occur in conjunction, and so the shown conclusion would not be possible. Or perhaps we should without fear say it is valid unconditionally, since the conclusion does not in fact affirm the antecedent (p + q), and denial of that antecedent would not logically imply (r + s) to be impossible.

The reverse process of ‘splitting’ is anyway not conditional, but it concerns only the consequent, not the antecedent, note well:

·         “(p + q) implies (r + s)” implies “(p + q) implies r” and “(p + q) implies s”, and is implied by them together.

Note well: we cannot here reverse the previous merger, and conclude “p implies r” and “q implies s”. This would be an ‘illicit splitting of the antecedent’. Beware also, therefore, of the following common fallacious argument (which could be classed as an apodosis):

 

If (p + q), then (r + s)

but: if p, then r

therefore: if q, then s.

 

The erroneous tendency here is to mentally ‘subduct’[11] both p and r, leaving q and s. But if we first split the major premise into two, we see that the minor premise eliminates one hypothetical, leaving us with the conclusion “if (p + q), then s”, or in nested form “if p, then (if q, then s)”. Note well, the latter, correct conclusion is in fact an eduction from the major: we have no need of the minor to infer it. Notice too, the precondition “if p” remains operative in it, until and unless “p” is categorically affirmed; and even then, “if q, then s” should be kept in mind as a mere contextual truth, since strictly speaking there are no ‘actual hypotheticals’. If the “if q, then s” conclusion does sometimes seem true in practice, it is no doubt because we tacitly regard the precondition “p” as already satisfied in the case at hand.[12]

Finally, note: it might be worthwhile looking for similar processes with respect to disjunctive propositions.

The logic of nesting and compound theses is considered as having been founded by the Stoic logician, Chrysippus of Soli (Greek, 280-206 BCE).[13]

 

11.       Validation of Nesting

Let us look a bit more in detail at the issue of validation in the logic of nesting.

We may refer to the eduction from “if (p and q) then r” to “if p then (if q then r)” as the production of a nest (or nesting); and to the reverse immediate inference from “if p then (if q then r)” to “if (p and q) then r” as the removal of a nest (or ‘unnesting’). How are these two processes validated? For a start, they make sense from a common sense viewpoint….

Nesting can be understood as follows. Knowing that a set of conditions (p and q) implies a certain conclusion (r), and knowing that some of these conditions (p) are already satisfied, we can predict that when the remaining conditions (q) are also satisfied, the conclusion (r) will indeed follow[14]. The reverse process of ‘unnesting’ can be understood as follows. Knowing that under a certain condition (p) a further condition (q) implies a certain conclusion (r), we can predict that when both conditions (p and q) are satisfied, the conclusion (r) will indeed follow.

On these grounds (exposition), it is reasonable to consider the forms “if p then (if q then r)” and “if (p and q) then r” as equivalent, i.e. that each implies and is implied by the other. We may also argue that both forms have the same underlying meaning, namely that “the conjunction of p, q and not r is an impossible one”.

However, if we analyze matters more precisely some doubt might be justified….

The form with a compound antecedent means “(p and q) and not r” is impossible, i.e. the bracketed conjunction of p and q is impossible in the context of not r, which is clearly equivalent to “(p + q + not r) is an impossible conjunction”; whereas the nested form means “p and ‘not (if q then r) is impossible”, which means “the conjunction of ‘p’ with ‘(q and not r) is not impossible’ is impossible”. The latter form is less clear, because it could apparently be interpreted in two ways: either as meaning that “p” is incompatible with the possibility of “q and not r”; or as meaning that “p” is incompatible with the actualization of the latter possibility (viz. when “q” and “not r” are both true).

We know from tropology (the theory of modality) that the necessary implies (but is not implied by) the actual, which in turn implies (but is not implied by) the possible; and similarly the impossible implies (but is not implied by) the inactual, which in turn implies (but is not implied by) the unnecessary. On such grounds, it could be objected that these interpretations are not equivalent.

However, one could reply that “p is incompatible with the possibility of (q and not r)” at least implies “p is incompatible with the “actuality of (q and not r)”, though the reverse may not hold. In that case, the nested form would be admitted to at least imply (though perhaps not be implied by) the unnested form; i.e. unnesting would be validated, but not nesting.

But I would be inclined to dismiss such objection altogether, and insist that “p is incompatible with the possibility of (q and not r)” is only superficially about conjunction with a possibility and ultimately is only concerned with conjunction of actuals (i.e. p, q and not r). In this view, the meaning of hypothetical propositions, however intricately constructed, is always the impossibility of one or more of the underlying actual conjunctions and the leftover possibility of at least one such actual conjunction[15].

Admittedly, some flavor of doubt remains, and some people will surely subscribe to the dissident view. But it occurs to me that we do have a reliable technical means to settle the issue once and for all – viz. the advanced methods of matricial analysis I have developed in phase II of The Logic of Causation (microanalysis). I shall have to eventually look into this matter in that context (and might conceivably find that my intuitive assumptions here are simplistic).

Note these techniques will also make possible the clear interpretation of intricate forms involving negative hypotheticals – such as “if (p and q) not-then r”, or such as “if p then (if q not-then r)”[16]. The mental acrobatics involved in the comprehension of such forms are daunting, and there is an obvious need for more objective and mechanical methodology. I look forward to developing software for this purpose.

 

12.       Brackets in Logic

In my past treatment of logical compositions[17], I did not fully deal with the issue of whether brackets in logic transmit polarity as they do in mathematics. The answer to that question is: not always – i.e. the analogy between symbolic logic and algebra should not be pushed too far or blindly applied.

Our analogy begins by labeling the affirmation of a thesis as positive polarity, and its negation as negative polarity. Thus, “P” may be written “+ P” and “not P” must be written “– P”. Then we ask whether “– (– P)” equals “+ P”? The answer is yes, this mathematical formula applies, since “not (not P)” means the same as “P”.

The brackets seem to also transmit polarity in the following case:

“– (P v Q)” = “– P – Q”

since “not (P or Q)” means “not (not (not-P and notQ)”, whence “not-P and not-Q”. This suggests further analogy between logic and mathematics, albeit in somewhat forced fashion.

However, the analogy breaks down entirely in view of the following invalid case:

“– (P – Q)” = “– P + Q”

This process would be fallacious, since “not (P and not-Q)” is not logically equivalent to “not-P and Q”, but also allows for the alternatives “P and Q” and “P and not-Q”.

Thus, we should avoid attempting to make parallels between logic and mathematics; it is artificial and misleading.[18]

 


 

[1]           Such as that taught by J. Searle.

[2]           See Judaic Logic, chapter 4.3.

[3]           Generally, in causation (ignoring natural spontaneity) in contrast to volition, sudden motion cannot emerge from static conditions – a trigger is needed. Thus, as I mention in Volition and Allied Causal Concepts (chapter 8.1) causation of motion refers to the transition from “if x, then y” to “if not x, then not y” or vice versa, rather than to a state x (or non-x) completely causing a movement y (or non-y). Although some if–then statements seem to suggest otherwise, it is only because they refer to partial causation, i.e. they conceal tacit factors.

[4]           Note also the forms “Only P or only Q”, “P alone or Q alone”, “P or alternatively Q”, “P, respectively Q”, “P or even Q”; and there are probably many more. The meaning may not always precisely or only correspond to the ones considered here. For instance, in “I could use a hammer or even a stone for this job”, the hammer is my first choice and the stone is rather a last resort, and I would not use both. Note how, although usually indifferent, in some cases, the order of listing of the alternatives (P, Q…) is relevant, signifying an order of preference.

[5]           This is why I insisted, in the original version of Future Logic (chapter 24.3), that the truth-table relative to implication is only an effect, not a cause.

[6]           For comparison, Aristotle died 322 BCE.

[7]           The first form means “p and (q and not r) is impossible”; the bracketed conjunction of q and not r is impossible in the context of p, which is the same as “the conjunction of all three items is impossible”. The second form means “(p and q) and not r” is impossible, which is equivalent to “(p + q + not r) is an impossible conjunction”.

[8]           I say “general negations” to stress that we are here dealing with strict implication. We do not just deny the actual truth of a combination, but its logical possibility ever. In this framework, the negation of an “if – then” form is not a conjunction, but an “if – not-then – “ form.

[9]           It is interesting to analyze the specific case of nesting: “if p then (if p then q)”. Clearly, it is equivalent to “if (p + p) then q”, which just means “if p then q” or “(p + not-q) is impossible”. An Internet correspondent, David Brittan, asked me the question: how to interpret such a form when its consequent is contraposed: “if p then (if not-q then not-p)”? This eduction would seem to suggest the possibility of contradiction – i.e. the coexistence of p and not-p, at least in the context of not-q, which might be taken to imply that not-q is impossible!

But the answer is simply: if we rewrite it as “if (p + not-q) then not-p”, it becomes clear that this form is not per se illogical – it is merely paradoxical, telling us that “if not-q then (if p then not-p)”. The consequent of this hypothetical proposition, viz. “if p then not-p”, is logically quite viable; it just implies “not-p” categorically. Thus, the overall conclusion is still “if not-q then not-p” (which is merely the contraposite of our initial conclusion “if p then q”). Note well that the inference is not “if not-q then (p and not-p)” – if that had been the case, then indeed “not-q” would have been logically impossible (as my correspondent feared).

[10]          One might add: since “p implies p”, but that premise being universally true need not be mentioned. Note also that if the theses p, q are not synchronous in the premise, they are of course not synchronous in the consequent of the conclusion. This is important when dealing with natural conditioning: it would be fallacious to ignore the original temporal difference (if any) and regard the theses in the conjunction “p and q” as simultaneous. Similarly in other cases, needless to say.

[11]          I take this term from J. S. Mill’s method of residues (see the 2005 revised version of my essay on his methods – in Part II, chapter 1).

[12]          Note that I mention this form of argument, as being common in rabbinic reasoning, in Judaic Logic, chapter 9.1 (p. 116). Of course, the conclusion I give there is only valid provided the antecedent’s conjuncts left out in it are tacitly considered categorically true.

[13]          As I mention in Future Logic, chapter 63.2. But I do not know on what evidence this claim is based. How many of the theorems here listed were known to him, or to anyone since, I also do not know.

[14]          It might be objected: but what if (as occurs in some cases) the first conditions (p) are sufficient without the others (q) to imply the conclusion (r); i.e. what if q is redundant? This refers to a situation where “if (p and q) then r” and “if p then r” are both true, and the question asked is: is the inference “if p then (if q then r)” still valid? The answer would be: yes, in such a situation, in the context of “p”, both “r” and “if q then r” would follow, and these two propositions are quite compatible; if “q” also happened to be true, then “if q then r” would simply reconfirm “r”. Indeed, given “if p then r” and that “(p and q) is possible”, it follows that “if (p and q) then r”, because the given is that “p implies r under all conditions”; note well however that “p” and “q” must be known to be compatible, before making such an inference.

[15]          In this view, the ‘matrix’ of any form refers to all logically possible combinations of the items concerned (in the case of three items – as here – there are 2*2*2 = 8 formally conceivable combinations), labeling some as ‘impossible’ and leaving the others as ‘possible’. The mode of modality intended by the word possibility here may admittedly vary slightly: sometimes it means ‘formal logical possibility’, in other cases it means ‘possibility by virtue of ignorance’; but such distinction is academic, the effect on discourse being the same.

[16]          I would also like to investigate conjunctions of hypotheticals. For instance, what is the conclusion given the two premises: “if p then (if q then r)” and “if not p then not (if q then r)” ?

[17]          Future Logic, chapter 28.

[18]          Note also, in passing: a logical disjunction is sometimes in the sciences replaced by an average. For instance, if we know the value of some physical variable is “either 0 or 1”, we may suppose, granting equal probabilities for both outcomes, that on average its value is ½. Such reasoning is partly logical and partly mathematical. The logical part is the (presumably exclusive and exhaustive) disjunction, and the awareness that one of the disjuncts will ultimately turn out to be true and the other(s) false; logic also admits of the existence of probabilities, ranging from 100% on one side or the other, or somewhere in between. However, the task of calculating probability belongs to mathematics. Additionally, by the way, physical science is involved here: in gathering the relevant empirical data, and also (at least in quantum mechanics) in the discussion as to whether the probabilities are factual or epistemic.

 

 

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