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The Logician © Avi Sion All rights reserved

FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
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
physicalchemical, 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
lessmagnified 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 multipletheses, mixedform, 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.
MultipleTheses
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 twotheses 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 multipletheses 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.
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
upsidedown 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 onethesis 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 onethesis 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, alefvav). This similarity
confirms that the notions conjunction and disjunction are intuitively conceived
as continuous, different degrees of the same thing.
Each of the multipletheses 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:
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
multipletheses 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.
MixedForm
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 'eitheror', because with three or more
alternatives, disjunction has more nuances in meaning. 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 pinpointing precisely just which parts of
the overall context frame our propositions, making up our effective socalled
'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.
