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The Logician © Avi Sion All rights reserved |
FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 21.
LOGICAL MODALITY.
I do not claim that my theory of logical modality as it stands solves all
issues, but I think you will find it very productive, an impressive integrative
force.
The concepts of 'logical modality' enable us to predict systematically
all the ways credibility may arise in knowledge over the long-term. Credibility
itself is not a type of modality, but the ground and outcome of logical
modality. We shall immediately define the primary categories of logical
modality, and thereafter discuss their development, their significance, and
their justification:
Truth
is the character of a proposition which seems more convincing than its negation,
in a given context of knowledge. In the case of any proposition implied by its
own negation, its credibility is extreme.
Falsehood
is the character of a proposition which seems less convincing than its negation,
in a given context of knowledge. In the case of any proposition implying its own
negation, its incredibility is extreme.
A proposition is 'problematic',
with regard to its truth or falsehood, if it seems to carry neither more nor
less conviction than its negation, in the given context of knowledge. This is
indicated by such expressions as 'might or not be' or 'perhaps is and perhaps is
not'.
In practical terms, the degree
of credibility, whether high, low, or median, of a proposition is a
measure of the amount of evidence or counterevidence put forward on its behalf
or against it. This refers to the weighting of information by confirmation or
undermining, which topic will be dealt with more fully under the heading of
adduction.
By (logical) context is meant, the
accumulated experiences and conceptual insights of the knower (a person or
society) at the time concerned.
The context-specific concepts of logical modality are built on the
awareness that: at every stage of knowledge, some things somehow seem 'true',
other things somehow seem 'false', yet others seem 'problematic'; and that these
attributes often vary with the growth of experience and reasoning.
These observations suggest that, although every appearance is accompanied
by some such characterization, the characterization is not in all cases firmly
attached to the object, but is often a function of the experience and reasoning
which have preceded them.
The concepts are thus formed, to begin with, only in recognition that
such events occur, and that they are distinguishable by our consciousness, and
that they each display such and such properties. Then we say: 'Let us call this
truth or falsehood or problemacy, as the case may be….'
It must be stressed that underlying the foregoing definitions of truth,
falsehood, and problemacy, is the assumption that a sincere effort of awareness
took place. It is difficult to insert such technical specifications in our
definitions explicitly, without engaging in circularity, but there is no doubt
that the definitions would lose all their value and significance without this
tacit understanding.
A true or false proposition is called 'assertoric',
because it makes a definite claim. A problematic proposition is not assertoric:
it presents an appearance with equal tendency in both directions, and therefore
devoid of tendency; it calls upon us to consider a hypothesis.
Problemacy signifies a suspension of judgment. It does not signify the
existence of 'real' indeterminacy, but only recognizes the appearance of
indeterminacy in contexts less than complete. In reality, we believe, every
issue is settled, once the event takes place; in omniscience, there would
accordingly be no problemacy — it only arises in more limited viewpoints.
Problemacy has no equivalent outside logical modality; being freely open
to change as knowledge evolves, there is no error in saying that any proposition
we choose to formulate is at first encounter problematic.
Note that meaningful, precise, and clear, propositions may be true, false
or problematic. Meaningless propositions are classified as false. Vague or
obscure propositions, as at best problematic, if not false.
Factual assertorics of less than extreme credibility and problematics,
give a semblance of co-presence or co-absence of opposites. The laws of
contradiction and of the excluded middle are our reminders that that impression
is transient; ultimately, everything is either totally credible or completely
incredible. In other words, so long as we make no attempt to at once apply both
truth and falsehood, or both untruth and unfalsehood, no law is broken; but as
soon as we lay claim to more than the propositions suggest, we err.
For this reason, we can effectively discard nonextreme assertions and
problems, and say of any proposition: it cannot be both true and false, and
cannot be neither true nor false. There is ultimately no mixing or in-between of
these attributes; our goal is to arrive to the extremes, not to linger on
intermediate stages. There would be no point in constructing a logical system
with reference to the finer gradations of credibility: it would be immobile.
Truth and falsehood are the categories of logical modality with
a single, given context as their frame of reference.
Truth is a category of logical modality lying between logical necessity
and possibility. Falsehood is the exact contradictory of truth, lying between
logical impossibility and unnecessity. Truth is fact and falsehood is fiction,
ideally. So we may call them the 'factual'
level of logical modality; in analogy to the actual level of natural or temporal
modality, or the singular level of extensional modality; but this is only an
analogy, not an equation.
The categories of logical modality referring to a
plurality of unspecified contexts:
Logical necessity characterizes a
proposition which is true in every context, and in that sense is true
irrespective of any given context.
Logical impossibility
characterizes a proposition which is false in every context, and in that sense
is false irrespective of any given context.
Logical contingency characterizes
a proposition which has neither the attribute of necessity nor that of
impossibility, as they are above defined, so that it is true in some contexts
and false in others.
Logical incontingency is the
negation of contingency, the common attribute of necessary and impossible
propositions. Logical possibility is
the negation of impossibility, the common attribute of necessary and contingent
propositions: truth in some contexts. Logical unnecessity is the negation of necessity, the common attribute of
impossible and contingent propositions: falsehood in some contexts.
With regard to corresponding concepts of logical probability or improbability.
We can say that, in this system, truth or falsehood correspond to mere
incidence or nonincidence; necessity or impossibility signify the extremes
(100%) of probability or improbability, and contingency concerns intermediate
degrees (less than 100%) of these. Thus, to be consistent, we must define the
logically probable as what would be true in most contexts (or false in a
minority of contexts), and the logically improbable as what would be true in few
contexts (or false in a majority of contexts).
These concepts would then enable us to specify our breadth of vision —
effectively, how many eventual changes of context we have taken into
consideration in making a prediction. The practical feasibility of this, with
some precision, and the relation of logical probability and credibility, will be
explored when we deal with adduction.
Thus, in summary, logical
modality may be defined as a qualification of propositions as such,
informing us as to whether each is true or false, in this (i.e. a given)
context, only some (unspecified) contexts, or all contexts, or somewhere in
between these main categories.
Here again, it must be emphasized that 'is true' (meaning, seems more
convincing than not) and 'is false' (seems less convincing than its
contradictory), depend for their plausibility on our having sought out and
scrutinized the available information with integrity. This issue is discussed in
more detail in the next chapter.
I want to emphasize here that the
concepts of logical modality, as here defined, are prior to concepts of logical
relation, like implication, which (as we shall see) they are used to define.
The former are built on the vague, notion of a proposition being
variously credible 'in' some context(s). Although this 'in' suggests that a kind of
causality is taking place, it is not yet at the stage where specific relations
like implication may be discussed. There is only a mental image of items
'pushing' others into existence; a very sensory notion.
Likewise, our first encounter with 'credibility' is very intuitive,
something intrinsic to our every consciousness. The later systematic
understanding of credibility, with reference to adduction, is merely a report on
when it occurs, not a substitute for that primitive, inner notion.
It is interesting that, in Hebrew, the word for 'with' is 'im'
(spelt ayin-mem), and that for 'if' is 'im'
(spelt aleph-mem). In that language, if I am not mistaken, when verbal roots are
that close, it signifies that the thoughts underlying them are also close. I
wonder if the English words 'in' and 'if' have similar origins, rather than
those most philologists assume.
Incidentally, also similar in Hebrew, are the words 'az'
(spelt alef-zayin), meaning 'then' in time or logic, and 'oz' (spelt ayin-zayin), meaning 'strength'. This confirms what I
said above, that the notion of logical causality is rooted in an intuitive
analogy to physical force.
Various analogies and contrasts between the singular and plural
modalities are worthy of note. The former measure credibilities in any one
context. The latter take a broader perspective, and compare credibilities in a
variety of contexts. Thus, true, false, and problematic are comparable to
necessary, impossible, and contingent — but they are not identical.
Contingent truth and falsehood are contextual, whereas necessity and
impossibility (incontingent truth and falsehood) effectively transcend context.
What holds in every context, holds no matter what the context, whereas the
contextual is tied to context and in principle liable to revision (though that
may never happen).
Note that it is the realization
of contingency as truth or falsehood, which is relative to context, but the
contingency in itself is no less absolute (with respect to context) than
necessity or impossibility.
A careful distinction must be made between the truth, falsehood, or
problemacy, of a proposition whose logical necessity, contingency, or
impossibility is unspecified — and the truth, falsehood, or problemacy, of any
proposed modal specification for that proposition. Failure to distinguish
between these perspectives can be very confusing.
A proposition may be problematic to the extent that, not only do we not
know whether it is true or false, but we do not even know whether it is
logically necessary, contingent, or impossible.
Less extremely, we may know the proposition to be true or false (and
thus, possible or unnecessary), yet not know whether it is logically necessary,
contingent (possible and unnecessary),
or impossible. In such case, the singular modality (the proposition per se) is
assertoric, but the plural modality is still to some extent problematic.
If a proposition is known to be logically necessary or impossible, then
it is assertoric with regard to both its plural modality (the incontingency) and
to its singular modality (accordingly, true or false).
If a proposition is known to be logically contingent, it is assertoric
with respect to its plural modality (the contingency). We may additionally know
that the proposition per se is true or false, in which case it is also
assertoric with respect to its singular modality. Or we may still be at a loss
as to whether it is true or false, so that it is problematic with respect to its
singular modality.
In any case, here again, problemacy does not signify real indeterminacy,
but merely absence of sufficient knowledge, remember.
Our definitions make clear that problemacy should not be confused with
logical contingency. A proposition may be definitely true or false, and so
unproblematic, and still contingent; and a problematic proposition may after
serious consideration be found to be necessary or impossible, whereas a properly
contingent proposition should not thus change status.
Yet problemacy and contingency have marked technical analogies, which
allow us to treat any problematic proposition (and therefore any proposition
whatever, at first encounter) as effectively
contingent in logical properties. Logic repeatedly makes use of this valuable
principle. As will be seen, if the proposition is not indeed contingent, it will
be automatically revealed so eventually through dilemmatic argument, so that no
permanent damage ensues from our assumption.
Note that the definitions of the logical modalities are very similar to
those of extensional, natural and temporal modalities. There is a marked
quantitative analogy (this, some, all), so that we can refer to them as
'categories of modality'; and there is a broad qualitative analogy (inclusion or
exclusion in a wider perspective), yet with enough difference that we can refer
to them as distinct 'types of modality'.
Logical modality puts more emphasis on epistemology than ontology, in
comparison to the other types. It primarily qualifies knowledge, rather than the
objects of knowledge. Whereas natural modality refers to the objective
circumstantial environment of events, temporal modality to surrounding times,
and extensional modality to cognate instances — logical modality looks at the
informational setting.
With regard to technical properties, logical modality is often similar to
the other types, but some notable differences also occur, as we shall see as we
go along.
The many-contexts concepts of logical modality are formed by reference to
the awareness that there are items of knowledge which somehow would seem to be
true or false no matter what developments in knowledge may conceivably take
shape, while others seem somehow more dependent on empirical evidence for their
acceptance or rejection. The former are often called 'a priori' or 'apodictic',
and the latter 'a posteriori'.
At first sight, apodictic statements present a difficulty. They seem
inaccessible to anyone with less than total knowledge. Only the fully omniscient
could know what is necessary or impossible in the widest context. A normally
limited mind like ours cannot have foreknowledge of any final verities. Indeed,
even if we ever reached omniscience, how could we be sure we have reached it?
However, these skeptical arguments can be rebutted on several grounds. To
begin with, they are self-defeating in that they themselves claim knowledge
about the capabilities of omniscience, and they do so in no uncertain terms:
therefore, they are intrinsically conceptually flawed. Logically, then, it is
conceivable for a limited mind to acquire apodictic knowledge, somehow.
Secondly, it is noteworthy that our minds, though admittedly less than
omniscient, are not rigidly limited in their powers of imagination. We are able
to construct innumerable hypotheses even with a limited amount of factual data
to play with. Thus, we are never limited to one context, the present one, but
can manipulate ideas which go beyond it. Of course, this does not mean that our
imagination is able to foresee all contexts. The more factual data we have to
feed on, the more our imagination can stretch out — but we never have all the
seeds.
Thirdly, the skeptical arguments misconstrue the issues. We defined the
necessary as true, and the impossible as false — 'in every context'. We did
not say, the necessary is what is true, and the impossible is what is false —
'to the omniscient'. Our definition does not exclude that the quality of
necessity or impossibility be given as such within any single context, as an inherent component of
the appearance. It does not logically mean that we have to foretell what goes on
in other contexts besides our own.
And indeed, we find within common knowledge many instances of manifest
necessity or impossibility, without need of further investigations. Such events
constitute the experiential basis for these concepts.
The primary examples of this are Aristotle's laws of thought. They strike
us as intrinsically overwhelming, as in themselves capable of overriding any
other consideration of knowledge. We can only ever deny them reflectively, by
obscuring their impact; but the moment we encounter them plainly, their
practical force is felt. When we are face to face with a specific contradiction,
we see that it is nonsense and that something, somewhere must be amiss. That is
why the laws of identity, of contradiction, and the excluded middle are
naturally adopted as the axioms of logical science.
But other examples abound. More generally, as we shall see, a proposition
is self-evident, if it is implied by its own negation, or implied by any
contradictories; and a proposition is self-contradictory, if its affirmation
implies its own negation, or implies any contradictories. It will be shown that
a self-evident proposition displays the consequent property of being implied by
any conceivable proposition, and a self-contradictory proposition that of
implying any conceivable proposition. 'Any' here means 'every' — so that these
are cases of logical necessity or impossibility.
This may occur formally, for all propositions of a certain kind whatever values be
assigned to their variables. Indeed, the science of logic itself may be viewed
as a record of all such occurrences. Or it may occur contentually (or 'materially'), in the sense: not for all
propositions of a certain kind, but only with certain specific contents. Note
that this distinction is somewhat relative, depending on what we hold fixed and
what we allow to vary.
Another way apodictic knowledge (or, for that matter, any knowledge)
might conceivably be made available to a limited mind is through revelation,
a communication from an omniscient mind. This is the logical premise of
religion. Faith might be defined as
the conviction that the information does indeed come from an infallible source,
G-d. This topic is too vast to be discussed in this treatise, but I merely
wanted to indicate the entry point.
Now, if logical necessity or impossibility are somehow given as
components of the appearance of things in any context of knowledge, what is
their difference from (contingent) truth or falsehood, which are also given?
Theoretically, once a proposition has been seriously scrutinized and
found not to be necessary or impossible, it henceforth remains permanently
contingent — just as once a proposition is seen to be necessary or impossible,
its status is thenceforth established. In practise a mistake might conceivably
be made, but this does not affect the principle.
The essence of necessity or impossibility is their property of
self-evidence or self-contradiction; it is not their permanence, which is only
incidental. Contingent truths or falsehoods may also be permanent; a proposition
may happen to remain true or false without change as knowledge evolves, and yet
never lose its contingent status. That some contingent truths or falsehoods do
change over time, is irrelevant. Even in a total knowledge context, truths or
falsehoods may be characterized as contingent.
Thus, we do not regard an obvious empirical truth like 'it is now
raining', or a well-established law of nature like 'the amount of matter and
energy in the universe are constant', as logically necessary, even though we
believe them to happen to be fixed truths (each in its own way), because they do
not seem self-evident; they are both therefore intrinsically logically
contingent. The raw, factual finality of the former or the natural necessity of
the latter do not affect their common logical status.
On this basis, we can also say that logical contingency is conceptually
distinct from problemacy. In omniscience, problemacy disappears, but not logical
contingency. The latter remains as a further qualification of certain truths and
falsehoods, distinguishing them from logical necessities and impossibilities,
respectively. It follows that contingency as such is not a lower status than
necessity or impossibility.
Lastly, note, a necessity or impossibility may be immediately apparent to
anyone, or we may need to go though a long or complicated reasoning process to
make it apparent. But in either case, the sense of obviousness is given within
the appearance itself, so that the ease or difficulty with which we were brought
to the insight are irrelevant to its finality.
It is hard to distinguish a priori and a posteriori knowledge by
reference to the concepts of reason and experience. The former is indeed more
purely analytical, but it cannot occur without the minimum of experience on the
basis of which the concepts involved are meaningful and clear. Likewise, the
latter is indeed more likely to be affected by changes in experience, but its
conceptualization and logical evaluation involve a great deal of rational
activity. |
