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, not-then 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 anti-parallel direction. This could therefore be called reverse implication. The corresponding negative form is ‘if nonP, not-then 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, not-then nonQ’ are both true, P is said to subalternate Q; in such case as ‘If P, not-then Q’ and ‘If nonP, then nonQ’ are both true, P is said to be subalternated by Q. Thus, subalternation, in contrast to implicance, is one-way subjunction, and not reversible.
In such case as ‘If P, not-then Q’ and ‘If nonP, not-then 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 clear-cut 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, not-then Q’ (or ‘if nonQ, not-then 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-{or-else} Q’) means ‘If P, not-then nonQ’ (or ‘if Q, not-then 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 subject-content 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 contingency-based 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, not-then nonQ’ is equivalent to ‘If Q, not-then nonP’; ‘If nonP, then Q’ is equivalent to ‘If nonQ, then P’; and ‘If nonP, not-then Q’ is equivalent to ‘If nonQ, not-then 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, not-then Q’. Thus, contrariety means incompatibility without exhaustiveness.
Subcontrariety combines ‘If nonP, then Q’ and ‘If P, not-then nonQ’. Thus, subcontrariety means exhaustiveness without incompatibility.
‘Undisjunction’ might be used to label the combination of ‘If P, not-then nonQ’ and ‘If nonP, not-then 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, being-subalternated, 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, not-then nonQ’ is sous-entendu. 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 ‘either-or’.
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.
Conjunction | P+Q | P+nonQ | nonP+Q | nonP+nonQ |
P and/or Q | allow | allow | allow | forbid |
P or-else Q | forbid | allow | allow | allow |
Either P or Q | forbid | allow | allow | forbid |
The negation of ‘P or Q’ may be stated as ‘not-{P or Q}’ (or ‘P not-or 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’, ‘or-else’, and their compounds, including ‘either-or’: 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.