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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 23.
CONJUNCTION.
In this chapter, we begin to analyze the various ways two or more
propositions, or sets of propositions, of any kind, may be correlated. A
proposition so considered, in relation to other propositions, is called a thesis;
we symbolize theses by using letters such as P, Q, R,
. The negation of any
thesis is called its antithesis;
that is its exact logical contradictory: the antithesis of thesis 'P' is 'nonP',
and vice versa.
The primary form of correlation is conjunction; this is expressed by
means of the operator 'and', or its negation. On a factual level, the
conjunction (or positive conjunction) of two theses typically takes the form 'P
and Q'. The contradictory of 'P and Q' would be 'Not-{P
and Q}', where the 'not' negates the 'and'; this may be called
nonconjunction (or negative conjunction). In the context of conjunction, a
thesis may be called, more specifically, a conjunct.
Most simply, the theses are categorical propositions of any form, so that
their conjunction may be viewed as a compound categorical. But, by extension, a
thesis may itself be a conjunction of two or more categorical propositions; or
it may consist of any other, more complex, kind of proposition, or any mix of
various kinds of propositions conjoined together. Thus, a thesis may ultimately
be a whole, intricate theory.
Logical conjunction of two theses simply affirms both of them as true,
implying that they are true separately as well as together. Thus, 'P and Q' (or
'P with Q') may be read as '{P and Q} is true', implying '{P is true} and {Q is
true}'. The 'is true' segment may be left tacit or made explicit, as with
categorical affirmations.
The contradictory form, 'not-{P and Q}' simply denies that the two theses
are both true, without asserting that
they are both false. All it tells us
is that at least one of the two theses is false, without excluding that an
unspecified one of them be true, nor excluding that both be false. Thus, 'not-{P
and Q}' may be read as '{P and Q} is false', which does not imply that '{P is
false} and {Q is false}'.
Thus, whereas the 'and' relation is fully assertoric, with regard to the
parts as well as the whole, the 'not-and' relation is much more indefinite. It
gives us limited information: it is assertoric with regard to the whole, but
leaves the parts problematic. This problemacy should not even be interpreted as
a logical contingency: not only do we not know of each thesis in isolation
whether it is true or false, we do not even know whether it is contingent or
incontingent. Keep that well in mind.
By definition, P and Q' and 'Q and P' are equivalent: the relation is
reversible; also, 'P and P' is equivalent to 'P' alone: repetition of a thesis
does not affect it. Likewise, by definition, 'not-{P and Q}' and 'not-{Q and P}'
are equivalent: the relation is reversible; note however that 'not-{P and P}' is
equivalent to 'not-{P}' alone, since 'P and P' means 'P'.
The three forms 'P and nonQ' (or, 'P without Q'), 'nonP and Q' (or 'Q
without P'), 'nonP and nonQ' (or 'neither P nor Q'), are derivative forms of
positive conjunction, obtained by substituting antitheses for theses in the
original formula. Likewise, the three forms 'not-{P and nonQ}', 'not-{nonP and
Q}', 'not-{nonP and nonQ}', are derivative forms of negative conjunction,
obtained by substituting antitheses for theses in the original formula. We thus
have a grand total of eight forms.
Note the we have used the word 'conjunction' in two senses. In a wider
sense, it includes both the positive and negative forms. In a narrower sense, it
includes only the former, the latter being called 'nonconjunction'. Note that a
positive conjunction is denied by negating any one, or any set, or all, of its
parts, which means that one of the remaining alternative positive conjunctions
must be true; thus, nonconjunction may be viewed as an abridged reference to the
outstanding conjunctions.
Conjunction may of course involve more than two theses, as in 'P and Q
and R and..'., signifying that they are all true individually as well as
collectively. Conjoining an additional thesis to a conjunction of two or more
other theses, just results in a conjunction of all the theses, in a normal
string: '{P and Q} and {R}' simply means 'P and Q and R'. Knowledge as a whole
may be viewed as a conjunction of all the propositions in our minds.
Nonconjunction of more than two theses, as in 'not-{P and Q and R
and
}' accordingly signifies that the theses are not all
true, without implying any further information concerning each thesis alone. Any
combination of theses and antitheses other than the one denied, would be
acceptable. We shall develop the theory of conjunction with reference to
two-theses forms, but the results can be extended with appropriate carefulness
to forms with more than two theses.
The following table lists the various forms of conjunction (or positive
conjunction), and shows the truths (T)
and falsehoods (F) of theses and antitheses they imply. We see that, in contrast,
nonconjunctions (or negative conjunctions) leave the individual theses and
antitheses problematic (?): their
information is purely collective. I have labeled these forms K1-K4
and H1-H4, as shown, for convenience. Table
23.1 Truth-Table
for Factual Conjunctions.
The four positive conjunctions exhaust the possible ways two theses and
their antitheses may be positively conjoined, and are mutually exclusive. That
is, one of them must be true, and three of them must be false. If any
one is true, the other three must be false; but if one of them is false, the
status of each the others is undetermined. Thus, the oppositional relation of
any pair of positive conjunctions is contrariety.
The oppositions of the four negative versions relative to each other is: three
of them must be true, and one of them must be false. If one of them false,
the other three must be true; but if one of them is true, it is uncertain what
the status of each of the others is. This follows from the interrelations of the
positive versions. Thus, the oppositional relation of any pair of negative
conjunctions is subcontrariety.
The opposition of any pair of positive and negative conjunctions, other
than a pair of formal contradictories, is therefore subalternation. Proof:
consider any positive conjunction, its truth implies the three others to be
false, and therefore implies their contradictories to be true; on the other
hand, its falsehood does not have further implications.
Thus, we could present the eight forms of conjunction in a cube of
opposition, with the four positive forms in the upper corners and the four
negative forms in the lower corners. The top plane involves contrariety, the
bottom plane involves subcontrariety, the diagonals through the cube involve
contradiction, and the four remaining faces involve subalternation in a downward
direction.
The eight factual forms of conjunction are the singular level of logical
modality. Let us now investigate the corresponding plural levels of logical
modality. Each of the factual conjunctions has a possible equivalent below it
and a necessary equivalent above it. Thus, we have to consider 2X8 = 16 modal
conjunctions, in addition to the 8 factual ones. They are (always referring to
logical modality, needless to repeat): Table
23.2 List
of Modal Conjunctions.
The factuals fit in between these two levels of modality, of course; they
are less than necessary, but more than possible.
Now, just as the factual positives implied that their respective theses
are not only collectively true, but individually true so the necessary
positives imply that their theses are each (as well as all) necessary, and the
possible positives imply that their theses are each (as well as all) possible.
However, in the latter case, it does not follow that the antitheses are equally
possible, note well.
In contrast, none of the negatives tell us anything about the logical
modalities of their respective theses. In all cases, the statuses of the
individual theses are left entirely problematic; all we have is collective
information. Not only are we left in the dark as to whether any thesis is true
or false, but there is no specification as to whether it is necessary or
possible or unnecessary or impossible.
Thus, for examples. 'P and Q are necessary' implies that P is necessary
(and nonP is impossible); and likewise for Q. 'P and Q are possible' implies
that P is possible, not impossible (and nonP is unnecessary, not necessary)
but without excluding that P be necessary or contingent: both are acceptable;
and likewise for Q.
'P and Q are impossible' (meaning: 'not-{P and Q} is necessary') does not
imply that P and Q are each impossible, but is equally compatible with each of
them being contingent or necessary except that in the latter case, if one
theses is necessary, the other would needs be impossible, to satisfy the overall
requirement of the form. 'P and Q are unnecessary' (meaning: 'not-{P and Q} is
possible') allows for each of the theses to be necessary, contingent or
impossible provided they are not both necessary at once.
Similarly, for the remaining forms. Thus, we see that each form delimits
some collective property of the theses, in some cases implying some individual
properties; but in most cases, the form leaves some open questions, some areas
of doubt, which would require additional statement(s) to specify in full.
Only the necessary positives fully define the factual and modal status of
the theses (they are equally necessary). The factual positives establish the
factuality and possibility of the theses, but leave their exact modal status
(necessary or contingent) undetermined. The possible positives establish the
possibility of the theses, but leave their factual and exact modal status
untold.
The negatives are even less committed with regard to their theses. It
is very significant to note that although a negative conjunction makes mention
of a proposition as one of its theses, it does not thereby imply it as even
logically possible. One might think that the mere mention of a proposition
is always an admission of its possible truth; but here we learn that such
assumption is unjustified.
The value of such indeterminacy is that it allows us to verbally capture
just precisely those relational details which are of interest to us, without
being forced to know more than we do at that point in time. If we could only
make statements where every issue is already resolved, we would be left wordless
until we had all the requisite details.
Be careful not to confuse problemacy and logical contingency. A
proposition may be so problematic, that we do not even know whether it is
logically contingent or incontingent, let alone whether it is true or false; or
it may be only problematic to the extent that, though we know it to be
contingent, we do not know whether this contingency is realized as truth or
falsehood on the factual level.
The following table lists the various forms of modal conjunction, and
shows the necessity (N),
impossibility (M), possibility (P),
unnecessity (U), or problemacy (?),
of individual theses and antitheses, implied by each modality (cum polarity) of
conjunction, in accordance with our previous comments. Note the labels assigned,
namely K1-K4, H1-H4,
with suffix n or p,
as the case may be, for convenience. Table
23.3 Truth-Table
for Modal Conjunctions.
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