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The Logician © Avi Sion All rights reserved

FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 26.
DISJUNCTION.
One way to introduce the topic of 'disjunction', is to view it in
contradistinction to 'subjunction'. According to this approach, we may divide
hypotheticals into two groups, with reference to the emphasis they put on their
theses and antitheses. 3.
Broadening the Perspective.
Hypotheticals which relate two theses as such, or two antitheses as such,
may be called 'subjunctive'. The reason these two sets are grouped into one
class becomes clearer when their definitions are considered.
The primary form of subjunction is 'If P, then Q', which tells us that
'{P and nonQ} is logically impossible' (H2n).
This is known as implication. Its negation is 'if P, notthen Q', meaning '{P
and nonQ} is possible' (K2p).
The other form of subjunction, 'If nonP, then nonQ', tells us that '{nonP
and Q} is logically impossible' (H3n),
and so is equivalent to the statement 'If Q, then P', which has a similar
meaning to 'If P, then Q', but in the antiparallel direction. This could
therefore be called reverse implication. The corresponding negative form is 'if
nonP, notthen nonQ', meaning '{nonP and Q} is possible' (K3p).
We may view implication and its reverse as forms of subjunction, and
their contradictories as forms of nonsubjunction. Or we may conventionally
broaden the sense of the word subjunction, and speak of positive and negative
subjunction, respectively.
Now, taken individually, these various logical relations are indefinite.
Hypotheticals are elementary forms, capable of various combinations, called
compounds, which define relationships more definitely. The forms are
intentionally left open, to allow expression of the maximum number of
combinations using a minimum number of building blocks. These effects have
already been encountered in the context of opposition theory, and will only be
briefly reviewed here for the sake of thoroughness.
Implication and its reverse are oppositionally neutral to each other
(likewise, therefore, their contradictories). They are therefore capable of four
combinations: they may be both true, or one true and the other false, or both
false. The hypotheticals conjoined in such combinations are called
complementary, in that they together serve to define the relationship between
the theses in both directions.
In such case as 'If P, then Q' and 'If nonP, then nonQ' are both true,
the resulting relation is one of mutual or reciprocal implication of P and Q (or
nonP and nonQ). This may be called implicance, and viewed as asserting the
logical equivalence of these two theses (or of their antitheses).
In such case as 'If P, then Q' and 'If nonP, notthen nonQ' are both
true, P is said to subalternate Q; in such case as 'If P, notthen Q' and 'If
nonP, then nonQ' are both true, P is said to be subalternated by Q. Thus,
subalternation, in contrast to implicance, is oneway subjunction, and not
reversible.
In such case as 'If P, notthen Q' and 'If nonP, notthen nonQ' are both
true, we are left with a relation which might be called 'unsubjunction'. This is
not a fully defining combination, unlike the preceding three compounds, in that
it allows the possibility of disjunction.
In contrast, we call 'disjunctive' those hypotheticals which relate a
thesis with an antithesis, or an antithesis with a thesis. We usually express
such relationship by means of the word 'or'. Rephrasing a hypothetical in
disjunctive form allows us to conceal the negative polarity of the antitheses
involved, so that the statement is made purely in terms of theses. The two
theses are known as the 'alternatives' (or disjuncts).
Two essential manners of disjunction may be distinguished. As usual in logic, we
must adopt some clearcut differences in terminology to facilitate treatment;
but, although the underlying distinctions of meaning are indeed intended in
practise, they are not always verbalized so exclusively.
(i) 'P and/or Q' (or 'P or also Q') signifying simply 'If nonP, then Q'
(or 'if nonQ, then P'), in other words, '{nonP and nonQ} is logically
impossible' (H4n). This is known as
inclusive disjunction, and expresses the exhaustiveness of P and Q: one of them must be true. This is the
more commonly intended sense of 'P or Q'; it stresses the theses (P, Q), rather
than the 'or' operator.
The negation of this form 'not{P and/or Q}' (which could be written 'P
not{and/or} Q') means 'If nonP, notthen Q' (or 'if nonQ, notthen P'); in
other words '{nonP and nonQ} is not logically impossible' (K4p).
This of course signifies inexhaustiveness.
(ii) 'P or else Q' (or 'P otherwise Q') signifying simply 'If P, then
nonQ' (or 'if Q, then nonP'); in other words, '{P and Q} is logically
impossible' (H1n), suggesting a
difference. This is known as exclusive disjunction, and expresses the incompatibility
of P and Q: one of them must be false. This is a rarer sense of 'P or Q'; it
stresses the separation of the theses (P, Q), the 'or' operator.
The negation of this form 'not{P or else Q}' (which could be written 'P
not{orelse} Q') means 'If P, notthen nonQ' (or 'if Q, notthen nonP'); in
other words, '{P and Q} is not logically impossible' (K1p).
This of course signifies compatibility.
We may view exhaustiveness and incompatibility as forms of disjunction,
and their contradictories as forms of nondisjunction. Or we may conventionally
broaden the sense of the word disjunction, and speak of positive and negative
disjunction, respectively.
Note, sometimes when we say 'P and/or Q', we intend to admit of only two
alternatives, 'P and Q' or 'nonP and Q', in advance excluding or not meaning to
include 'P and nonQ', as well as 'nonP and nonQ'. Sometimes, this is what we
intend when we say 'P or else Q', for that matter; meaning, 'at least Q, whether
or not P'. Likewise, 'P or also Q' may be intended to mean: 'P and Q' or 'P and
nonQ'; that is, 'at least P, possibly without Q but also possibly with it'.
Sometimes, 'P or Q' is understood to mean 'P and nonQ' or 'P and Q'.
Such implications are often obvious to us by virtue of the subject
involved; the subjectcontent is well known to everyone to exclude certain
alternatives, so that these exclusions are virtually formal. The logic of such
forms can easily be derived from the logic of the forms here considered, so they
will be ignored.
The recasting of a hypothetical form into disjunctive form, or vice
versa, may be called 'transformation'. This may be viewed as a form of
inference, or of elucidation, insofar as the mind may favor such process to more
fully understand the relationship under consideration.
Note that disjunctives, like hypotheticals, may each be dissected into
their implicit connection and basis. The general case comprises only the
'connective' (a modal conjunction) for its definition, whereas normal and
abnormal disjunctions specify the logical modalities of the theses in various
ways. Many processes are only valid for contingencybased disjunctions.
Needless to say, the theses of disjunctions may be any kind or complex of
proposition(s): categoricals, conjunctives, hypotheticals, or also disjunctive
clauses. The logic involved becomes progressively more intricate and
complicated, accordingly. Some such logical 'compositions' will be analyzed in
the next two chapters.
Each of the forms of disjunction is, we note, nondirectional, unlike the
forms of subjunction. By reference to their definitions, it is easy to see that:
'If P, then nonQ' is equivalent to 'If Q, then nonP'; 'If P, notthen nonQ' is
equivalent to 'If Q, notthen nonP'; 'If nonP, then Q' is equivalent to 'If
nonQ, then P'; and 'If nonP, notthen Q' is equivalent to 'If nonQ, notthen P'.
These equations have already been encountered under the heading of
contraposition.
The forms of elementary disjunction are complementary; any pair of them,
other than contradictories of course, may be used in conjunction to define a
compound relationship, as follows. Note that each of these relations is
reversible.
Contradiction combines 'If P, then nonQ' and 'If nonP, then Q'. We could
assign to the disjunctive form 'Either P or Q' this specific meaning, comprising
both incompatibility and exhaustiveness of P and Q. The proposition 'Either nonP
or nonQ' is equivalent, note well.
Contrariety combines 'If P, then nonQ' and 'If nonP, notthen Q'. Thus,
contrariety means incompatibility without exhaustiveness.
Subcontrariety combines 'If nonP, then Q' and 'If P, notthen nonQ'.
Thus, subcontrariety means exhaustiveness without incompatibility.
'Undisjunction' might be used to label the combination of 'If P, notthen
nonQ' and 'If nonP, notthen Q', which means inexhaustive and compatible. This
is not a fully defining combination, unlike the preceding three compounds, in
that it allows the possibility of subjunction.
The oppositions of all forms of subjunction and disjunction, elementary
or compound, to each other, and the eductions feasible from each of them, are
all easily inferred from the findings for the corresponding hypotheticals. I
will not list them all, to avoid repetition, but a couple are worth
highlighting.
Thus, note that 'P and/or Q' and 'nonP or else nonQ' are equivalent, and
likewise, 'P or else Q' and 'nonP and/or nonQ' are equivalent. Also, 'either P
or Q' and 'either nonP or nonQ' are identical. 3.
Broadening the Perspective.
a.
Interface
of Subjunction and Disjunction.
Since the conjunctive roots of subjunctions and disjunctions, namely H2n,
H3n, and H4n, H1n, are neutral to
each other, they are in principle combinable together. However, normally,
subjunctions and disjunctions are contrary to each other and not combinable;
this applies to formal logic, where the theses and antitheses are all granted
the status of logical contingency, as in the theory of opposition. This further
justifies their division into two classes.
In contrast, nonsubjunctions and nondisjunctions, namely K2p,
K3p, and K4p, K1p, are generally
combinable, since they are compatible both in absolute terms (neutral) and in
formal situations (subcontrary).
In opposition theory (ch. 6), we identified seven fully defining logical
relations. The six main ones — implicance, subalternating,
beingsubalternated, contradiction, contrariety, and subcontrariety — have
been reviewed in the previous sections of the present chapter. The remaining one
was, you will recall called 'unconnectedness' or 'neutrality', in formal logic
discussions. This may be defined as a combination of 'unsubjunction' and
'undisjunction'. Although each, taken alone, is still an indefinite compound,
taken together they form a fully defining and reversible relationship.
In formal logic contexts, these 7 fully defining compounds are all
mutually exclusive and constitute an exhaustive list of possibilities; if any
one holds, the other six are out, and if any six are rejected, the remaining one
must stand. The negation of any one of them means one or more of its constituent
hypotheticals is false, without specification as to which one(s); so we must be
careful not to make errors here.
In particular, note that the expression 'neither P nor Q' is normally
equivalent to 'both nonP and nonQ', and should not be thought to be the logical
negation of 'either — or —' in the above suggested sense, though it is
sometimes so intended.
Beyond these definitions, we will not further discuss compound forms, so
as not to complicate matters further. The inferences possible from them are all
implicit in those concerning the constituent elementary forms, and can easily be
derived.
b.
Vague
Disjunctions.
The
important thing is not to confuse the elementary forms with their compounds, and
to be aware of the reducibility of compound forms to their elementary positive
and negative constituent hypotheticals. Especially, disjunctive propositions are
in practise often notoriously ambiguous.
Sometimes, when we say, 'P and/or Q' we only intend 'if nonP, then Q',
sometimes an additional 'if P, notthen nonQ' is sousentendu.
The elementary case merely forbids 'nonP and nonQ', without specifically
allowing or forbidding 'P and Q', whereas the compound case specifically allows
for the latter. Similarly, mutadis mutandis, with regard to 'P or else Q'.
The difficulty is due to the previously mentioned inductive rule for weak
relations in logical modality: here, there is little distinction between the
'open' and the 'possible'. Ultimately, a conjunction which is neither
specifically allowed nor specifically forbidden, is effectively allowed. The
difference is merely one of degree. If the open turns out to be impossible, it
is just eliminated from the list of alternatives as a matter of course, without
affecting the overall truth of the disjunctive proposition.
In practise, we often use a vague form of disjunction, 'P or Q', which
might mean anything from an elementary inclusive or exclusive disjunction, to a
compound like subcontrariety, contrariety, or even contradiction. It is thus
relatively uninformative; nevertheless, it shows why we can class all these
relations under the common heading of disjunctions.
The forms 'P and/or Q' and 'P or else Q' and 'either P or Q' all suggest
that 'P or Q', though for different reasons. The form 'P or Q' in its broadest
sense recognizes at least 'P and nonQ' or 'Q and nonP' as conceivable outcomes,
without telling us at the outset whether 'P and Q' or 'nonP and nonQ' are
allowed or forbidden, though it is understood that at least one of them (if not
both) is forbidden.
The implicit questions are left open, unless the relation is further
specified by 'and' or 'else' or 'either', in which case the additional allowance
is made more firm (given a greater degree of eventuality) by what is
specifically forbidden. If both the open questions are answered negatively, then
'or' means 'eitheror'.
The vague form 'P or Q' may thus be defined by the disjunction of all the
clearer forms of disjunction. The following table shows the common ground
between these forms. Note that the 'allowances' here should be interpreted
minimally, as problemacies, though they are often in practise meant to be
logical possibilities in the stricter sense. Table
26.1 Common
Ground of Disjunctions.
The negation of 'P or Q' may be stated as 'not{P or Q}' (or 'P notor
Q'). What we mean by that of course depends on what we intend by 'P or Q'.
c.
Involving
Antitheses.
We presented subjunction and disjunction as subdivisions of
hypotheticals. But unlike subjunction, disjunction involves a distinct set of
operators, 'or' and its derivatives. So disjunction deserves to be viewed as a
logical relation in its own right. We can see from its name that we intend this
relation as conceptually opposed to conjunction.
What this means is that, in addition to 'P or Q', we should consider 'P
or nonQ', 'nonP or Q', 'nonP or nonQ'. Similarly for the less vague operators
'and/or', 'orelse', and their compounds, including 'eitheror': we can insert
one or both antitheses, in place of the original theses, to obtain other forms,
as we did for hypotheticals. And of course, all these have contradictories.
It is very easy to determine the conjunctive definition for each form,
and then compare it to all the others. Since each operator gives rise to four
impossible conjunctions and four possible ones, and these eight conjunctions are
ubiquitous, there is bound to be a corresponding number of equations.
I will not go into this domain in any detail, so as not to expand this
treatise unnecessarily. The reader is invited to explore it for him or her self.
