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The Logician © Avi Sion All rights reserved |
FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 38.
TEMPORAL CONDITIONALS.
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.
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.
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. |
