CHAPTER 27. INTRICATE LOGIC.
People think that logic is a linear enterprise, antithetical to the curvatures of poetic knowledge. But, viewed holistically, knowledge is not essentially a mechanical activity and product, but more akin to a living organism.
Just as any living organism functions on many levels, from the physical-chemical, through the biochemical and cellular, to the gross level of our sensory perceptions, and beyond that, as an intricate part of the natural environment as a whole, through the intellectual and spiritual dimension, the whole being sustained by the Creator — so knowledge ought to be viewed.
Logic is the way we establish the chemical bonds between the different data elements of our knowledge. These bonds vary in kind and effect, and can occur cooperatively in any number and complication of combinations. The result is a network, from the microscopic level of precise logical relations, to the less-magnified level of clusters of information, to the organic whole, to the cultural context.
This knowledge network is not stationary, but like an organism, pulses and glows with life, growing, ordering, clarifying, strengthening. This life has a mechanical level, a vegetative level, and a conscious and volitional level, which is animal and human, and therefore spiritual.
So viewed, logic and the poetic side of us are not in conflict, but in easy, friendly, fruitful togetherness. A balanced, healthy mind, requires some degree of rigor in observation and thought, and also some degree of freedom to move, some room to maneuver. Because knowledge is always in flux, and there is always some inconsistency involved.
One has to be able to flow with the tides of information, the momentary waves, and even the momentary storms, and remain patient and awake at one’s center. Logic maps for us the wide terrain of the mind, improving our research skills. Thus, ultimately, logic is an aspect of wisdom, knowing to navigate smoothly in the changing sea of information.
Logic teaches us to clarify information, by engineering tools for this purpose. Especially multiple-theses, mixed-form, modal logic provides us with ways to express ideas precisely, and thus construct and check them more rigorously. Let us look at some of the possible intricacies of logical relations between items of knowledge.
Let us now broaden our understanding of conjunctive logic, in different directions. Note that we refer to any specialized field of logic as ‘a logic’.
a. Multiple-Theses Logic. We have talked of conjunction with reference to two theses, because the logic of conjunctions of more than two theses is derivable from it.
Thus, we may inspect the theses of a proposition of the form ‘P and Q’, and find that, say, Q is itself composed of two theses ‘Q1 and Q2’; from this we conclude that ‘P and Q1 and Q2’ is also true. Likewise, though with diminishing statistical probability, any of the theses P, Q1, Q2, may in turn be found subdividable. Thus, conjunctives may have any number of theses.
It follows that a conjunctive clause within a conjunction, is equivalent to a larger conjunctive proposition, so that we need not think in terms of clauses. This process may be viewed, in analogy to mathematics, as ‘addition of conjuncts’; or we may refer to it as ‘logical composition’, the formation of composites out of elements or other components. For example:
‘P and {Q and R}’ is identical to ‘P and Q and R’.
A corollary of this is that we can isolate part of a conjunction as a clause, at will. All that should be obvious, since ‘P and Q..’. simply informs us that the theses are individually, as well as together, true. Since the order of the theses is irrelevant in the case of two-theses conjunction, meaning ‘P and Q’ equals ‘Q and P’, it can likewise be shown that order does not affect the logical relation of any number of conjuncts.
This begins for us the topic of multiple-theses logic.
b. Matrix Logic. It is well, as we shall see, to think of multiple conjunctions as forming a continuum. The number of conjuncts (ands) is one less than the number of theses. Here, a single thesis is the limiting case of the continuum, a conjunction without conjunct, as it were. Clearly, the more theses are conjoined, the more overall information we have. Thus:
P (one thesis)
P and Q (two theses)
P and Q and R (three theses)
P and Q and R and… (and so on).
The logic of nonconjunction should follow, though it is more complex. Thus, the negative conjunction ‘not-{P and Q}’, where the theses are entirely problematic, signifies that any of the positive conjunctions ‘P and nonQ’, ‘nonP and Q’, or ‘nonP and nonQ’ might be true, since they are formally the only conceivable alternatives to the negated one.
We can therefore think of negative conjunctions with reference to positive ones entirely. The existence of a negative is expressed only through positives; negation is a lesser, derivative expression of existence. It is useful therefore, to view negative conjunctions as equivalent to ‘matrixes’ of positive alternatives, as follows:
Table 27.1 The Matrixes of Negative Conjunctions.
not-{P+Q} | not-{P+nonQ} | not-{nonP+Q} | not-{nonP+nonQ} |
P+nonQ | P+Q | P+Q | P+Q |
nonP+Q | nonP+Q | P+nonQ | P+nonQ |
nonP+nonQ | nonP+nonQ | nonP+nonQ | nonP+Q |
We may call each of the alternatives in a matrix, a ‘root’ conjunction of the nonconjunction. Such matrixes are very useful in clarifying the logic of negative conjunction, since we need only find the common ground of the positive alternatives (the roots) in each matrix, to know the properties of the corresponding negative. Thus, for instance, we can here too prove that the order of the theses is irrelevant, since all the alternatives of the matrix can be reversed.
It can thus be shown that clauses may be inserted or removed arbitrarily, with negative conjunctions as well. We can accordingly develop a logic of negations of multiple conjunctions, again thinking of all such conjunctions as forming a continuum. The difference here being that the more alternatives there are the wider, vaguer, and weaker, is their overall negation. That is, the more theses are involved in a negative conjunction, the less information we have; negation of one thesis being the most definite, limiting case. Thus, we have an upside-down continuum:
not-{P and Q and R and…} (and so on).
not-{P and Q and R} (three theses)
not-{P and Q} (two theses)
not-{P}…or P (one thesis)
The one-thesis case may be P, as well as nonP, if we understand these negations of conjunctions as effectively disjunctions, meaning ‘P or Q or..’., for then P is one of the ways the disjunction can be resolved (since Q or R may be negated instead).
In this way, this here continuum of negative conjunctions, can be attached to the previously described positive conjunctions continuum, resulting in a larger continuum, stretching from the negative forms with the most theses, through the central one-thesis case (the ‘P’ common to both positive and negative conjunction), up to the positive forms with the most theses.
The negative, say left, side is a virtual kind of knowledge, getting ever vaguer, a storehouse of possibilities. The positive, say right, side is a growing categorical knowledge, ever more precise. As we move from left to right, our knowledge becomes more specific; we have more information, a higher, wider, deeper view.
It is interesting to note, in passing, that in Hebrew the word for ‘and’ is ‘oo’ (spelt, vav; also pronounced as ‘ve’), and the word for ‘or’ is ‘o’ (spelt, alef-vav). This similarity confirms that the notions conjunction and disjunction are intuitively conceived as continuous, different degrees of the same thing.
Each of the multiple-theses negative conjunctions may be dissected into a matrix of positive conjunctions, the alternatives to the one negated. Negation of one thesis, P, leaves us with only one alternative, nonP. Effectively, every theses should be viewed as including the denial of its contradictory; P, say, may be taken as implying ‘P and not-{nonP}’; nonP likewise becomes ‘nonP and not-{P}’.
Negation of two theses, as we saw above, leaves us with three alternatives. Since three theses and their negations are combinable in eight ways, negation of a conjunction of three theses, leaves us with seven positive alternatives out of the eight:
P + Q + R | nonP + Q + R |
P + Q + nonR | nonP + Q + nonR |
P + nonQ + R | nonP + nonQ + R |
P + nonQ + nonR | nonP + nonQ + nonR |
Beyond that, the general formula is clearly, for n theses, there are: two to the nth power combinations, and therefore that number minus one positive alternatives to the negation of any root. For instance, for 5 theses, there are 2X2X2X2X2 = 32 possible combinations.
Accordingly, the positive conjunction of two (or more) negative conjunctive clauses, may be also expressed by reference to the leftover positive combinations. We can thus develop a general logic of conjunction; that is, any complex of positive and negative conjunctions can be interpreted in positive terms. I will not go into such detail here, however.
c. Modal Logic. The above concerns factual conjunction; modal conjunction has yet to be considered. To say that a conjunction is logically necessary means that it holds, no matter what the surrounding conditions. In contrast, a logically contingent conjunction depends for its eventual realization on certain conditions. If those conditions are unspecified, we have a nonhypothetical modal proposition; if sufficient conditions are specified, we have a precise hypothetical relation. All this applies to positive and negative conjunctions.
It follows that modal conjunctions can always be understood in terms of factual ones, whether the latter are framed by conditions, specified or unspecified, or unconditional. In conjoining modal conjunctions, we must however be careful, and consider whether the conditions under which each clause is realized are compatible with the conditions for the other clause(s) to become factual.
Consider, for instance, the following complex: {P and Q} is possible and {Q and R} is possible. Does it follow that: {P and Q and R} is possible? The answer is clearly, No! It is conceivable that, though these possibilities are compatible as modal propositions, they are incompatible in their factual embodiments. That is, it may be that: {P and Q and R} is impossible, and the given possibilities can only be realized separately, through {P and Q and nonR} or {nonP and Q and R}, respectively.
In this way, by focusing on the underlying factual conjunctions, we can develop a detailed logic of modal conjunction. In formal logic, using variables for terms or propositions, whatever is conceivable is logically possible. But in practise, when dealing with specific terms and specific relations, we must be careful to distinguish between problemacy and logical contingency.
In the above example, for instance, if the given two ‘possibilities’ are mere problemacies, then any combination is conceivable; and we can say (also problematically) that {P and Q and R} might well be true. But if the premises are logical possibilities, we cannot conclude that the {P and Q and R} conjunction is also logically possible.
d. Thus, a complete logic of conjunction, whether positive or negative, factual or modal, evolves entirely from the logic of positive conjunctions.
Since hypothetical and disjunctive propositions are in turn defined with reference to conjunctions, the logic of all mixtures of logical relations is likewise reducible to the logic of positive conjunctions.
Any statement, whatever its mix of logical relations — of whatever modalities and polarities — can thus be analyzed through matrixes, and compared to any other statement similarly analyzed.
a. The form of argument. We present an argument by listing its premises and conclusions as follows. There are of course arguments with one premise (eductions), and arguments with more than two premises (as in sorites), and some with more than one conclusion, but the typical unit of deduction is two premises, one conclusion.
P,
and Q,
therefore R.
A valid categorical, Aristotelean syllogism, for instance, may be regarded as establishing a hypothetical link between premises and conclusion, by way of the common terms in these propositions, in specific figures and with precise polarity, quantity and modality specifications.
Thus, although we cannot say generally of any group of propositions that P and Q imply R, we do know that under specific conditions (where for instance P means ‘X is Y’, Q means ‘Y is Z’, and R means ‘X is Z’), such a bond can be established, for all cases of that form. Thus, categorical syllogism may be viewed as one condition under which the form ‘if P and Q, then R’ may be viewed as universally true.
Now, this can be interpreted as a hypothetical proposition with a conjunctive antecedent: (a) ‘If {P and Q}, then R’. Alternatively, we tend to interpret it as a hypothetical proposition with a hypothetical clause as its consequent: (b) ‘if P, then {if Q, then R}’, meaning that, under the condition P, Q implies R. The former states that ‘{P and Q and nonR} is impossible’, whereas the latter states that ‘{P and possibly[Q and nonR]} is impossible’.
At first sight, the two statements may seem significantly different, yet if we analyze them with reference to the underlying positive conjunctions, it is seen that they make identical allowances. The form ‘{P and Q and nonR} is impossible’ obviously allows for seven alternative positive conjunctions. The form ‘{P and possibly[Q and nonR]} is impossible’ allows for:
(i) ‘P and impossibly{Q and nonR}’, implying the factual ‘P and not-{Q and nonR}’, which might be realized as ‘P and {Q and R}’, ‘P and {nonQ and R}’, or ‘P and {nonQ and nonR}’;
(ii) ‘nonP and possibly{Q and nonR}’, which grants the realizability of ‘nonP and {Q and nonR}’;
(iii) ‘nonP and impossibly{Q and nonR}’ implying the factual ‘nonP and not-{Q and nonR}’, which might be realized as ‘nonP and {Q and R}’, ‘nonP and {nonQ and R}’, or ‘nonP and {nonQ and nonR}’.
Clearly, here again all seven alternatives to ‘P and {Q and nonR}’ are eventually permitted. Thus, the two expressions compared are equal: they have the same root conjunctions. This is an important finding for hypothetical logic.
The allowances in all cases are of course problemacies. In purely formal contexts, these problemacies do ordinarily signify that there are unspecified contents fitting the various alternatives. But in contexts of specified content, these problemacies should not be taken as formally logical possibilities, since some of the alternatives may well be excluded by additional statements.
b. Nesting. The definition of hypotheticals accurately reflects our formation of such thoughts. Assuming the antecedent clause allows us to hold it mentally in place, so that we can be free to deal with other matters, namely the relative status of the subsequent clause. This process may be called ‘nesting’, or ‘framing’. It is similar to the technique of control in the experimental sciences, where, while keeping all other things equal, we observe the effects on our subject, of a precise change in the single remaining factor.
In the case of two theses, appropriately related, we frame the one by means of the other, in a simple hypothetical proposition, ‘If P, then Q’. In the case of three theses, we can say ‘If P, then {if Q, then R}’, meaning that P is a context or framework for Q implying R. Likewise, for four theses, ‘If P and Q and R, then S’ can be reformed as ‘If P, then {if Q, then [if R, then S]}’.
We can in this manner nest any number of hypotheticals within each other. In practise, much of the framework is often left tacit, note. Such multiple-theses hypotheticals serve to express partial or conditional antecedence. They may be viewed as forming a continuum, ranging from a single, unconditional thesis, to one framed by more and more difficult demands.
The value of such successive framing by hypotheticals can be seen in analysis of the process of reductio ad absurdum used in validation of syllogisms. To prove that ‘If {P and Q}, then R’; we infer ‘If P, then {if Q, then R}’ by framing; then we contrapose the inner hypothetical to obtain ‘If P, then {if nonR, then nonQ}’; then we remove the frame to obtain ‘If {P and nonR}, then nonQ’; thus showing that denial of the conclusion leads to denial of a premise.
c. Mixed-Form Logic. Just as the antecedent of a hypothetical may be composite, so may the consequent be, as in ‘if P, then {Q and R}’; this is equivalent to the conjunction of ‘if P, then Q’ and ‘if P, then R’. Just as the consequent may be hypothetical, so may the antecedent be, as in ‘if {if P, then Q}, then R’; this is not equivalent to ‘if {P and Q}, then R’, note well.
We can also use the disjunctive format in complicated propositions, which present alternative antecedents and/or consequents. For example, ‘If {P or Q}, then R’ (which is ordinarily taken to imply ‘if P, then R’ and ‘if Q, then R’); or again, ‘if P, then {Q or R}’, which, though not incompatible with ‘if P, then Q’ or ‘if P, then R’, does not imply them. Those methods are used to find alternate conclusions from weaker premises (as seen in transitive syllogism), or weaker conclusions from alternate premises (as we shall see with ‘double syllogism’).
More broadly still, any kind of conjunction, hypothetical, or disjunction, positive or negative, may be involved with any other(s), in countless, intricate relations. Of course, it is wise not to get too carried away, it must be possible for the mind to unravel the meaning with relative ease. Going into the mechanics of all these relations in detail is beyond the scope of this book, but it can be expected to be an interesting field.
a. Multiple Disjunctions. Disjunctions may involve more than two alternatives, as in ‘P or Q or R or..’.. We tend to use the general operator ‘or’, rather than the more specific ‘and/or’, ‘or else’, and ‘either-or’, because with three or more alternatives, disjunction has more nuances in meaning. Indeed, we need not specify any disjunctive operator at all, but could just list the theses under consideration (P, Q, R, etc.) and verbally specify their collective relations (as explained below).
Usually, of course, the inclusive form ‘P and/or Q and/or R and/or…’ may be supposed to mean ‘at least one of P, Q, R, etc. must be true’ (leaving open whether each of the others can or must be true or false). Similarly, the exclusive form ‘P or else Q or else R or else…’ may be supposed to mean ‘all but one of P, Q, R, etc. must be false’ (implying only one can be true, but leaving open whether it can be false or must be true; note too that any pair of theses are incompatible). If both these disjunctions are affirmed, the two or more theses involved may be said to be both exhaustive and incompatible.
More generally conceived, a multiple disjunction depends for its definition on how many theses, out of the total number listed, must be true, and/or how many must be false. These components specify the degrees of exhaustiveness and/or incompatibility of the alternatives. In some cases they are independent variables, in others, they affect each other, according to the total number of theses available.Strictly, we should specify the definitions of our disjunction parenthetically; though in practise they are often left unsaid, when we do not know them precisely, or when we consider them as obvious in the context. Note well that the definitions do not tell us exactly which of the theses are true and which false; they only tell us that some stated number are this or that.
With two theses, as already seen, ‘one must be true’ signifies that both cannot be false, ‘one must be false’ signifies that both cannot be true, and those two specifications may occur without each other or together.
With three theses, the specifications ‘one must be true’, ‘two must be true’, ‘one must be false’, ‘two must be false’, can be combined every which way, except for ‘two must be true and two must be false’ together, normally (though in abnormal logic, this is not excluded).
With four (or more) theses, likewise, we can specify that one to three (or more) of the theses must be true and/or that one to three (or more) of the theses must be false, though the total number of theses so specified should not normally exceed the total number of theses available.
Whatever the number of theses, it is clear that the more of them are specified as having to be true or false, the firmer the implied bond between them. For instance, ‘two must be true’ is a more forceful relation than ‘one must be true’. The more definite the bond, the more restrictive the relation, but also the more informative.
In the maximalist case, where we are given that all the theses must be true, or all must be false, or exactly which are true and which false is specified, we are left with no degree of freedom, and no ignorance. In the minimalist case, where any number may be true or false, in any combination, there is no link between the theses, and all the issues remain unresolved. The various degrees of disjunction lie in between these extremes.
Inversely, the greater the total number of theses listed in a disjunction, the looser the bond implied by the ‘or’ operators in it. For instance, ‘one thesis must be true’ represents a weaker relation with reference to a total of three or four theses than to a total of two theses. The more alternatives are available, the more of them we have to eventually eliminate to arrive at categorical knowledge, therefore the less we know so far.
Thus, the operator ‘or’ has many gradations of meaning, depending on various factors. However, we can think of all disjunctives as aligned in a continuum, ranging from one to any number of theses in toto, and from one to any number among them specified as having to be true or false. In some cases the inclusive and exclusive specifications diverge, in some cases they converge. Ultimately, all disjunctives are part of the same continuum as conjunctives.
b. Matrixes. It is best, when faced with such multiplicity of alternatives, to think in terms of the underlying possible outcomes of positive conjunction. For example, ‘one of {P or Q or R} must be true, and two must be false’ may be interpreted as ‘{P and nonQ and nonR}, {nonP and Q and nonR}, and {nonP and nonQ and R}, are possible (that is, at least problematic) conjunctions of the given theses’.
This format is least ambiguous, because we may on formal grounds understand the disjunction of the factual conjunctions listed to be formally of the ‘one must be true and all the others must be false’ degree, without having to say so, no matter what the original number of theses. We earlier referred to this as matrix logic.
Note that any of the underlying positive conjunctions involving a negative thesis, may themselves conceal an internal disjunction. For negation is often a shorthand expression of a number of positive alternatives; thus, nonP might mean ‘P1 or P2’, if it so happens that P, P1, and P3 are exhaustive. This is applicable even to elements, and all the more so to compounds and all composites.
Thus, we may find disjunctions within disjunctions within disjunctions; these may be referred to as different levels of disjunction. This phenomenon is interesting, because it illustrates the complexities of stratification which occur among propositions. There is an enormous wealth of possible relations among propositions.
Disjunctions may also may be expressed in hypothetical form, and vice versa. For instance, ‘P or Q or R’ (as defined in the above example) can be reformulated as: ‘If nonP and nonQ, then R, and if nonP and nonR, then Q, and if nonQ and nonR, then P’ (the ‘one thesis is true’ component), and ‘If P, then nonQ and nonR, and if Q, then nonP and nonR, and if R, then nonP and nonQ’ (the ‘two theses are false’ component). But such formulas can get pretty intricate and confusing. This is what justifies disjunction as a valuable form in itself.
But it follows anyway that the laws of intricate logic for hypotheticals may be used to obtain analogous laws for disjunctives; and vice versa. Thus, for instance, the case of a disjunctive proposition with a disjunctive clause as one of its theses, corresponds to the case of premise nesting we encountered in an earlier section.
We found that a modal conjunction within a larger modal conjunction, is equivalent to a factual clause. That is, since ‘nonP and {nonQ and nonR} is impossible’, and ‘nonP and possibly{nonQ and nonR} is impossible’, yield the same matrix of seven alternative conjunctions, they have the same logical properties. It follows that the corresponding disjunctives ‘P or Q or R’ and ‘P or {Q or R}’, intended in the ‘one thesis must be true’ sense, are equivalent.
A disjunction may be taken as a gross unit, as well as with reference to the alternatives it lists. We may focus on the whole or the parts, and determine the one or the others as our clause(s).
Such intricacies will not be covered in any great detail in this work, though interesting. All this is part of a yet broader field of research. The nesting case concerns a possible conjunction within impossible conjunction. But other combinations of ‘modality within modality’ can also be worked out.
c. Another direction of development for disjunctive logic, is the introduction of modalities of disjunction. The concepts of connection and basis are applicable to disjunction. Purely connective disjunction has entirely problematic bases; if the base of each thesis is specified, whether as logical contingency (normally) or as incontingency (abnormally), special logics may apply.
The ‘connection’ of the disjunction is the impossibility of the conjunction(s) which are excluded from the underlying matrix. Here, the law of contradiction is that at least one of all the possible conjunctions in a matrix, for the given number of theses, must be true; the law of the excluded middle is that all but one of them must be false. Thus, connection is inherently incontingent.
One could argue that, since we can place a disjunction as the consequent of a hypothetical statement, we can think of conditional levels of disjunction as well. In that event, the connection may be logically contingent, valid in some specific (though not always specified) context(s). It follows that we can also think of a factual level of disjunction (loosely speaking), signifying that it is operative in the presently held context.
A more modal logic of disjunction may accordingly be developed, and here again basis may come into play. Possible disjunction implies that the disjunction is consequent to certain conditions, and therefore can be made factual by revealing the implicit antecedent. Problematic or logically possible disjunctives, underlie hypothetical propositions with a disjunction as antecedent or consequent. Disjunctions may of course also appear within larger disjunctions.
However, factual (contextual) versus incontingent (unconditional) disjunction, may be compared to material versus strict implication. So these concepts may be used to some extent, if we remain conscious of their main pitfall — namely, the difficulty of pin-pointing precisely just which parts of the overall context frame our propositions, making up our effective so-called ‘context’. In practise, we wordlessly ‘know’ the intended context, but in formal work this vague knowledge is not a useable capital.
In conclusion, the concept of modality provides us with a means of clarifying thoughts to a much greater degree than purely factual logic, giving us a new/improved tool of analysis of data. I leave it to you, to explore this field more thoroughly; this may be compared to presenting you with an object for inspection under your own microscope, using the techniques developed in this treatise.