FUTURE LOGIC

CHAPTER 23.CONJUNCTION.

1.Factual Forms.

2.Oppositions of Factuals.

3.Modal Forms.

4.Oppositions of Modals.

1.Factual Forms.

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 athesis; we symbolize theses by using letters such as P, Q, R,…. The negation of any thesis is called itsantithesis; 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 arebothtrue, without asserting that they arebothfalse. 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 notalltrue, 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.

2.Oppositions of Factuals.

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 formsK1–K4andH1–H4, as shown, for convenience.

Table 23.1Truth-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.

3.Modal Forms.

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 23.2List 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.

4.Oppositions of Modals.

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, namelyK1–K4,H1–H4, with suffixnorp, as the case may be, for convenience.

Table 23.3Truth-Table for Modal Conjunctions.

Since the categories of logical modality are by definition distinguished with reference to a quantity of contexts, the oppositions of the various modalities of conjunction among themselves, can be deduced from the oppositions between the corresponding factual conjunctions, given in an earlier section of this chapter, and the general doctrine of ‘quantification of oppositions’, which we worked out in an earlier chapter (14.1) with reference to categoricals.

Thus, since the formsK1andH1are contradictory, the oppositions betweenK1n,K1,K1p,H1n,H1,H1p, are the same of those between the categoricalsA,R,I,E,G,O. Likewise for similar sets.

Since the formsK1,K2,K3,K4, are contrary to each other, it follows that: the forms which subalternate these factuals,K1n,K2n,K3n,K4n, are contrary to each other, and to them; and the forms which these factuals in turn subalternate,K1p,K2p,K3p,K4p, are neutral to each other, and to them.

Since the formsH1,H2,H3,H4, are subcontrary to each other, it follows that: the forms which subalternate these factuals,H1n,H2n,H3n,H4n, are neutral to each other, and to them; and the forms which these factuals in turn subalternate,H1p,H2p,H3p,H4p, are subcontrary to each other, and to them.

Since the formK1subalternates the formsH2,H3,H4, it follows that:K1nsubalternatesH2n,H3n,H4n, and thereforeH2,H3,H4, andH2p,H3p,H4p; butK1is neutral toH2n,H3n,H4n, though it subalternatesH2p,H3p,H4p; andK1pis neutral toH2,H3,H4, though it subalternatesH2p,H3p,H4p. Likewise, for similar sets.