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FUTURE LOGIC

© Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.

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CHAPTER 38. TEMPORAL CONDITIONALS.

1. Structure and Properties.

2. Relationships to Naturals.

3. Mixed Modality Arguments.

1. Structure and Properties.

a. Structure. The forms of conditional proposition of temporal modality, are very similar to those of natural modality. I will therefore analyze them only very briefly. They are presented below without quantifier, but of course should be used with a singular or plural quantifier.

When S is P, it is always Q

When S is P, it is never Q

S is P and Q

S is P and not Q

When S is P, it is sometimes Q

When S is P, it is sometimes not Q

(The symbolic notation for temporal conditionals could be similar to that used for naturals, except with the suffixes c, t instead of n, p; m and a are of course identical.)

Temporal conditional propositions have structures and properties very similar to their natural analogues. There is no need, therefore, to reiterate everything here, since only the modal type differs, while the categories of modality involved remain unchanged.

Temporal conditionals signify that at all, this given, or some time(s), within the bounds of any, the indicated, or certain S being P, it/each is also Q (or: nonQ), as the case may be. (Similarly, it goes without saying, with a negative antecedent, nonP.)

Here, ‘when’ means ‘at such times as‘. The actuals (momentaries) exist ‘at the time tacitly or explicitly under consideration’, the modals (constants or temporaries) concern a plurality of (unspecified) times.

The antecedent and consequent events are actualities. The modal basis of their relationship is the temporal possibility: ‘this/those S is/are sometimes both P and Q (or: nonQ)’. The connection between them is expressed by a temporal modifier placed in the consequent; for constants, it is ‘this/those S is/are never both P and nonQ (or: Q)’, for temporaries, it is identical with the basis. The quantifier specifies the instances of S concerned.

The order of sequence of the events, though often left unsaid, should be understood. Each has a relative duration, as well as location in time. Expressions like ‘while’, ‘at the same time as’, ‘before’, ‘thereafter’, ‘whenever’, are used to specify such details.

b. Properties. With regard to opposition, constant conditionals (like ‘Whenever S is P, it is Q’) do not formally imply the corresponding momentaries (‘S is now P and Q’, for example), although both the former and the latter do imply temporaries (their common basis, ‘S is sometimes P and Q’, here).

A constant like ‘When this S is P, it is always Q’, is contradicted by denial of either its basis or connection; that is, by saying ‘This S is never P’ or, ‘This S is sometimes both P and nonQ’. A temporary like ‘When this S is P, it is sometimes Q’, is contradicted by denying the base of either or both events; that is, by saying ‘This S is never both P and Q’.

Other oppositional relations follow from these automatically, and the same may be repeated for negative events. Momentaries are identical to, and behave like, actuals, of course.

The processes of translation, eduction, apodosis, syllogism, production, and dilemma, likewise all follow the same patterns for temporals as for naturals.

Temporal disjunction is also very similar to natural disjunction, and its logic can be derived from that of temporal subjunction.

2. Relationships to Naturals.

Although temporal and natural conditionals have analogous structure and properties, each within its own system, the continuity between the two systems is here somewhat more broken than it was in the context of categoricals.

In conditionals, natural necessity does not imply constancy. Compare, for instance, ‘When this S is P, it must be Q’ and ‘When this S is P, it is always Q’. Although the natural connection ‘This S cannot be P and nonQ’ implies the temporal connection ‘This S is never P and nonQ’ — the natural basis ‘this S can be P and Q’ does not imply (but is implied by) the temporal basis ‘this S is sometimes P and Q’.

Since the higher connection is coupled with an inferior basis, while the lower connection is coupled with a superior basis, the ‘must’ conditional as a whole is unable to subalternate the ‘always’ version. This is easy to understand, if we remember that even within natural conditioning, ‘must be’ does not imply ‘is’; it follows that ‘must be’ cannot imply ‘is always’, which is essentially a subcategory of ‘is’ (though it too does not imply ‘is’, as already mentioned).

This breach in modal continuity, in the context of conditionals, further justifies our regarding natural and temporal modal categories, as belonging to distinct systems of modality. In categorical relationships, these two types of modality differ merely in the frame of reference of their definitions (circumstances or times); but a more marked divergence between them takes shape when they are applied to conditioning.

For similar reasons, natural necessity does not even imply temporariness. On the other hand, temporariness does imply potentiality, since, for instance, ‘When this S is P, it is sometimes Q’ implies ‘When this S is P, it can be Q’. Here, the categorical continuity is still operative.

Also, the actualities for both types coincide: ‘in the present circumstances’ and ‘at the present time’ mean the same thing. ‘Circumstances’ refers to the existential layout of the world, how all the substantial causes are positioned in the dimensions of space; while ‘time’ focuses on the positioning of these various circumstances along the dimension of time; at any given present, these two aspects of a single happening are bound to correspond, like two sides of the same coin.

These first principles allow us to work out the valid processes which correlate natural and temporal conditionals in detail.

3. Mixed Modality Arguments.

I will not explore deductive arguments which mix natural and temporal modalities, in any great detail, but only enough to make the reader aware of their existence.

In syllogism, we should note valid arguments such as the following (which follow from 1/naa by exposition):

1/ncc

When this S is M, it must be Q (or: cannot be Q)

When this S is P, it is always M

so, When this S is P, it is always Q (or: is never Q).

1/ntt

When this S is M, it must be Q (or: cannot be Q)

When this S is P, it is sometimes M

so, When this S is P, it is sometimes Q (or: nonQ).

However, an argument like the following would be invalid, because there is no guarantee that the circumstances for this S to be P are compatible with those for it to be Q (or, nonQ, as the case may be).

1/cnp

When this S is M, it is always Q (or: is never Q)

When this S is P, it must be M

so, When this S is P, it can be Q (or: nonQ).

This mode is invalid, note well. Although 1/ccc, 1/cmm and 1/ctt are valid, the temporal conditionals c, m, or t are not subalterns of the natural conditional n.

In production, modes of mixed modal type are subalterns of modes of uniform type, in accordance with the rules of categorical syllogism. This may result in compound conclusions, as in the following case:

All P must be Q (implying, is always P)

This S is sometimes P (implying, can be P)

therefore, When this S is P, it must be Q (1/npn)

and, When this S is P, it is always Q (1/ctc)

(likewise with a negative major term.)

In apodosis, mixed-type ‘modus ponens‘, like the following ones in ncc or ntt, are valid (since they can be reduced to a number of naa arguments):

When this S is P, it must be Q (or: nonQ)

and This S is sometimes, or always, P

hence, This S is sometimes or always Q (or: nonQ).

And also, note well, mixed-type ‘modus tollens‘, like the following ones in ncc or ntt, are valid (since they can be reduced to a number of naa arguments):

When this S is P, it must be Q (or: nonQ)

and This S is sometimes not, or never, Q (or: nonQ)

hence, This S is sometimes not, or never, P.

This result is interesting, if we remember that the arguments below are not valid, since they involve inconsistent premises (the minor contradicts a base of the major):

When this S is P, it must be Q (or: nonQ)

and This S cannot be Q (or: nonQ)

hence, This S cannot be P.

When this S is P, it always be Q (or: nonQ)

and This S is never Q (or: nonQ)

hence, This S is never P.

Additionally, note, a constant major premise coupled with a naturally necessary minor premise, yield a conclusion, granting that for categoricals n implies c. Thus, cnc is valid, as a subaltern of ccc. But since ccc is invalid in cases of denial of the consequent, cnc only applies to cases of affirmation of the antecedent:

When this S is P, it is always Q (or: is never Q)

and This S must be P (implying, is always P)

hence, This S is always Q (or: is never Q).

We can similarly investigate disjunctive arguments of mixed modal type, and dilemma.

2016-08-20T07:17:48+00:00