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The Logician © Avi Sion All rights reserved

FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 24.
HYPOTHETICAL PROPOSITIONS. 3.
Strict or Material Implication.
We saw in the previous chapter that two or more propositions may be
correlated in various ways, with reference to conjunctions (involving the
operator 'and') of various polarities and logical modalities.
Implicit in certain conjunctive forms are relationships of
'conditioning'; they signify a certain amount of interdependence between the
truths and/or falsehoods of the theses involved. These relationships are
definable entirely with reference to modal conjunction, so that we may fairly
view all forms of conjunction, and all forms which may be derived from them, as
one large family of propositions called 'conditionals'.
However, in a narrower sense, and usually, we restrict the name
conditional to the derivative forms which employ operators like 'if'. The
remaining derivative forms, which employ operators like 'or', are called
disjunctive.
These issues of terminology are of course of minor import. What counts is
that conjunctive, conditional, and disjunctive propositions are ultimately all
different ways of saying the same things, as far as logic is concerned.
Nevertheless, because each of these formats reflects a quite distinct turn of
thought, they are worthy of separate analyses.
We are in this part of our study concerned with conditioning in the
framework of logical modality. But as we shall see eventually, each other type
of modality also gives rise to a distinct type of conditioning.
Logical conditionals are more commonly known as 'hypothetical'
propositions — this more easily distinguishes them from nonlogical (not
meaning illogical) conditionals, meaning natural, temporal or extensional
conditionals, which may therefore simply be called 'conditionals', in a narrower
sense.
Hypothetical propositions are essentially concerned with the logical
relations between propositions, or sets of propositions. This area of Logic is
therefore quite important, as it constitutes a selfanalysis of the science, to
a great extent — the 'logic of logic'. But it is also a specific
investigation, like any other area of Logic, for the purposes of everyday
reasoning.
The sequence in hypotheticals, the ordering of their theses, is what we
call 'logical'. It is not essentially temporal, though the mental sequence is of
course temporal, one thought preceding the next — we can be aware of only so
much at a time, beyond that we function linearly, in trains of thoughts. Some
thoughts are linked into chains by precise relational expressions, but their
sequence should not be viewed as to do with natural causation between mental
phenomena per se. Thought processes are sometimes apparently involuntary, but
for the most part there plainly seems to be a volitional element involved;
indeed, if thought was automatic, there would be no call for logic.
Logical sequence has rather to do with conceptual breadth. The wider
proposition is viewed as including, or implying, its consequences, in a timeless
manner. The exclusive proposition 'Only if P, then Q', though formally identical
to the reciprocal relation 'If P, then Q, and if Q, then P', suggests that P and
Q are not logically quite interchangeable, but that P has a certain conceptual
primacy over Q, that their order matters. The suggested order is not merely in
the time of arrival of the thoughts about P and Q, but more deeply concerns the
hierarchy of their factual contents.
a.
The paradigmatic form of hypothetical proposition is 'If P, then Q', where P and Q are any theses. The former, P, is
known as the antecedent, and the latter, Q, as the consequent. The relation
between them is minimally defined by saying that the conjunction of P and nonQ is impossible.
This means that the affirmation of P and the denial of Q are
incompatible; given that P is true, Q cannot be false, and it follows that Q
must also be true. We can also say: P implies Q.
Note the correspondence of this proposition to the negative modal
conjunction labeled H2n in the previous chapter; as we saw, this leaves the individual
theses P and Q entirely problematic at the outset: they need not even be
logically possible. Note well also that the unmentioned conjunctions 'P and Q',
'nonP and nonQ', and 'nonP and Q' are all left equally problematic; one should
not surmise, from the allusion to P being followed by Q, that the conjunction of
P and Q is given as logically possible.
The expression 'if' normally suggests that the truth of the antecedent
'P', and thereby of its consequent 'Q', are not established yet; they are still
in doubt. Note that the 'if' effectively colors both the theses.
The expression 'then' (which in practise is often left out, but tacitly
understood) informs us that, in the event that the truth of the antecedent is
established, the truth of the consequent will logically follow. The form 'if P,
then Q' does not specify whether P is likewise implied by Q, or not; it takes an
additional statement to express a reverse relation.
A note on terminology: officially, in logical science, the whole relation
'if P, then Q' is called a hypothetical proposition, in the sense that it
includes one thesis in another. The proposition as a whole is assertoric, not
problematic (unless we specify uncertainty about it, of course); it is the two
theses in it which are normally problematic. But colloquially, we understand the
expression 'hypothetical' as signifying problemacy, so confusion is possible.
Etymologically, the word 'hypothesis' could suggest a thesis which is
placed under another, and so might be applied to the consequent; here, the sense
is that it is 'conditioned' upon the truth of the antecedent (which, however, is
normally in turn conditioned by other theses). But, again in practise, we often
look upon the antecedent as the 'hypothesis', because it is qualified by an 'if'
and underlies the other thesis; here the sense is that our thesis is placed
before the consequent (which, however, is normally more or less equally 'iffy').
Be all that as it may, logical science has frozen the various expressions
in the special senses described.
b.
The contradictory of the 'if P, then Q' form is 'If
P, notthen Q'. This merely informs that the conjunction of P and nonQ is not impossible. It tells us that:
if P is true, it does not follow that Q is true; Q may or not be true for all we
know, given only that P is true. We can also say, P does not imply Q.
Note the correspondence to the positive modal conjunction labeled K2p
in the previous chapter; as we saw, this implies that P is logically possible
and Q is logically unnecessary, though both individual theses are of course left
problematic with regard to their factual status. One should not surmise, from
the allusion to Q rather than nonQ, that Q is given as logically possible. Note
well also that the unmentioned conjunctions 'P and Q', 'nonP and nonQ', and
'nonP and Q' are all left equally problematic.
It is not excluded that P and Q have some other positive relation; for
instances, that P together with some additional conditions imply Q, or that Q
implies P. It is also conceivable that P is not only compatible with the
negation of Q, but implies it; or at the other extreme, that P and Q are totally
unrelated to each other. In any case, here again, the theses P, Q are normally
problematic, though the proposition as a whole is assertoric.
The name 'hypothetical' may be retained for such negative forms insofar
as the prefix 'if' is equally involved; likewise, the name 'antecedent' for P
remains correct; but for Q, the name 'inconsequent' would be more accurate here.
For, whereas the positive form 'If P, then Q' suggests that Q is a logical
consequence of hypothesizing P, the negative form 'If P, notthen Q' denies such
connection (for this reason it is called the 'nonsequitur' form, the Latin for
'it does not follow'). We may use the word 'subsequent' (without chronological
connotations) to mean 'consequent or inconsequent'; or we may simply use the
word 'consequent' in an expanded sense.
The form 'if P, notthen Q' should not be confused with the form 'If P,
then nonQ', which means that the conjunction of P and Q are impossible;
sometimes we say the latter with the intent to mean the former. There is a world
of difference between 'P does not imply Q' and 'P implies nonQ'. To make matters
worse, we sometimes leave out the 'then', and just say 'if P, not Q', which can
be interpreted either way.
It is important to note that we
commonly assume that 'if P, notthen Q' is true, whenever we have searched and
found no reason to think that 'if P, then Q' is true. This is effectively an
inductive principle for negative hypotheticals: strong relations like 'if P,
then Q' require specific proof, whereas weak relations like 'if P, notthen Q'
may usually be taken for granted, so long as their contradictory has eluded us.
c.
The following table clarifies the relations between the antecedent and
consequent and their antitheses, in positive and negative hypotheticals. It
shows what follows as true (T), false
(F), or undetermined (?),
from the truth of any of them. Note well that the table is an outcome of the
hypothetical relations, but does not constitute their definition. Table 24.1 TruthTable for Hypotheticals.
d.
Hypotheticals are not only used in everyday reasoning, but also to
develop logical theory; they express the formal connections between theses. The
hypothetical relations validated by formal logic are not defined by mere denial
of the occurrence of this or that conjunction in a specific instance, but by
claiming the logical impossibility of it with any content.
We use them to indicate the oppositional relations between any
propositional forms, or the inferences which can be drawn from one, or a
conjunction of two or more, propositional forms. Premises are antecedents, valid
conclusions are consequents; an argument is valid if the premises imply the
conclusion, invalid if they do not. Likewise, when we speak of assumptions and
predictions, we refer to such logical relations.
The psychology of assumption consists in mentally imagining as true a
proposition not yet so established, or even which is already known false. In the
latter case, we phrase our hypothetical as 'If this had been true, that would
have been true'. Because of logic's ability to deal with form irrespective of
content, even untrue contents may be considered and analyzed.
As will be seen, hypothetical relations are established through a process
of 'production'. Most, if not all, of the logical relations we intuit in
everyday reasoning processes are in fact expressions of formal connections. 3.
Strict or Material Implication.
Note well that the definitions of both the positive and negative
hypothetical forms involve two essential factors. First, they refer to a
conjunction of two theses, symbolized by 'P' and 'nonQ' (meaning, the negation
of Q). Secondly, hypotheticals are essentially modal propositions; they refer to
the logical impossibility or possibility of such a conjunction.
Many logicians have defined the 'if P, then Q' form as identical with the
negative conjunction 'not{P and nonQ}'. They have called this 'material'
implication to distinguish it from the above 'strict' implication. The
suggestion being that implication is a relation which ranges from singular
contextuality or actuality (material), to all contexts or necessity (strict).
It is true that we often for practical purposes, intend an implicative
statement as merely applicable to the present context. However, since the
'present context' is notoriously difficult to identify precisely, this is a
practise which cannot be subjected to formal treatment. Two propositions cannot
be compared or combined, if it is unclear what parts of the everchanging
context they depend on. The unstated conditions may be different enough that
their fluxes are not in harmony.
My position is therefore that the idea of 'material implication' is
mistaken. There is no such thing as nonmodal implication, in the sense they
intended. All implication is inherently modal, 'strict'. The realization of
implication is not a more restrictive implication, but simply a factual
conjunction or nonconjunction.
One mere denial of the bracketed conjunction is not implication: such
definition only seems to work because it conceals a repetitive denial, coming
into force whenever we bring the definition to mind.
The reason why the error arose, is because negative conjunction, even on
a factual level, is intrinsically indefinite. When we say 'not{P and nonQ}', we
think: 'well, if P, then not nonQ, and if nonQ, then not P'. However, these
seemingly implicit hypotheticals are not themselves assertoric: they are
preconditioned by a tacit 'if not{P and nonQ}, then: if P, then not nonQ, and if nonQ, then
not P'. There is a hidden nesting involved. The consequent hypothetical
proposition is in fact quite modal; it only appears nonmodal, because the
antecedent nonconjunction is taken for granted.
I very much doubt that the form 'not{P and nonQ} ever occurs in
practise, except insofar as it is logically implied by a factual conjunction
like 'nonP and nonQ' or 'P and Q' or 'nonP and Q', or by the modal form 'if P,
then Q' (in the sense of '{P and nonQ} is impossible'). For example, even though
the conjunction '{chickens have teeth} and {squares are round}' is indeed false,
we do not interpret this to mean that these two happenstances are at all linked;
the proposition as a whole can only be constructed as
a result of our foreknowledge (in this case) that both clauses are
separately false, and would not be otherwise arrived at.
This misconception has caused the logicians in question to ignore the
contradictory 'if P, notthen Q' form altogether, since that would be equivalent
to the positive conjunction 'P and nonQ', according to that theory. Yet, we
commonly reason in such terms, saying 'it does not follow that' or 'it does not
imply that', without intending to affirm the theses categorically thereby [as in
negation of conjunction].
The antecedent does not merely happen to precede the subsequent, as that
theory suggests. In the 'if P, then Q' case, the consequent follows it as a
logical necessity; it means effectively, 'if P, necessarily Q'. In the 'if P,
notthen Q' case, the inconsequent is denied such necessary subsequence, without
affirming or denying that it may possibly happen to be conjoined; it means
effectively 'if P, possibly not Q'. If we compare the truthtables of ‘P strictly implies Q’ and ‘P materially implies Q’, we may be misled by the identity of the positive side (see the ‘if P, then Q’ half of table 24.1). But when we look at the negative side (i.e. the denial of ‘if P, then Q’), the difference between the two cases is glaring (for strict implication, see the second half of table 24.1; and for material implication, see row ‘K2’ of table 23.1). That is to say, though strict and material implication seem to have the same truthtable, their negations have very different truthtables, so their logical behaviors will be different. Moreover, the former is permanent (i.e. true for all time if true), whereas the latter (except when it is true by implication from the former) is temporary (i.e. true for a limited time if true). Of course, we can invent any forms we please; but logical theory should reflect practice, and not be allowed to degenerate into an arbitrary game. What the proponents of material implication were looking for, the seed of truth they were trying to express, was, I suggest, the analogues of implication found in other types of modality — the natural, temporal or extensional. I will discuss these in detail later, and the truth of this statement will become more apparent then.
For all these reasons, I have not followed suit. I ignore socalled
material implication (though not factual negative conjunction, of course), and
limit hypotheticals to strict implication.
Now, the forms 'If P is true, then (or notthen) Q is true' are
paradigms. If we substitute in place of P and/or Q, their respective
contradictories, that is, the antitheses nonP (P is false) and/or nonQ (Q is
false), we obtain the following full list of eight possible relations. The
symmetries involved ensure the completeness of our list of hypotheticals. Each
hypothetical is defined by a modal conjunction, as shown, on the basis of our
original definitions. Table
24.2 List
of Hypotheticals and their Definitions.
a.
As earlier decided, hypotheticals with the 'if, then' operator, which
posit a consequence, are classified as 'positive'; these are fully defined by
reference to the logical impossibility of a conjunction. Hypotheticals with the
'if, notthen' operator, which negate a consequence, are classified as
'negative'; these are fully defined by the logical possibility of a conjunction.
The unmentioned conjunctions in each case are of undetermined status; this means
problematic, and should not be taken to mean logically contingent.
The oppositions between hypotheticals and factual conjunctives follow
accordingly. Given the truth of a positive hypothetical, it follows that the
conjunction which it by definition denies as possible is false,; and vice versa:
so these are contraries. Given the falsehood of a negative hypothetical, then the
negation of the conjunction which it by definition admits as possible is
true; so these are subcontraries. With regard to all other factual conjunctions, hypotheticals are
neutral.
b.
There is another respect in which polar expressions might be applied to
hypotheticals. We will reserve the labels 'affirmative'
and 'negatory'
for this new division; here, unlike with categoricals, the terms must not be
confused.
Thus, 'If P, notthen not Q', involving a double negation, is essentially
as positive as 'If P, then Q' towards the subsequent Q; these forms, and their
equivalents with nonP as antecedent, will therefore be classified as
affirmative. Whereas 'If P, then not Q' or 'If P, notthen Q', which involve
only one negation, effectively negate the subsequent thesis Q; so that they, and
likewise the corresponding forms with nonP as antecedent will be said to be
negatory hypotheticals. Such polarity considerations, also, as will be seen,
clarify the basis of validity of certain hypothetical syllogisms.
c.
Although the hypotheticals included in our initial list of forms are all
tenable and useful, half of them are somewhat artificial as they stand.
Forms involving a thesis 'P' as antecedent can be regarded as 'perfect'
in comparison to those involving an antithesis 'not P' as antecedent, labeled 'imperfect', whether the
forms are positive or negative, and whether the consequent or inconsequent is
'Q' or 'not Q'.
These characterizations are relative, and not of great importance, but
they are useful. The significance of this division of hypotheticals will become
more apparent in due course, when we deal with hypothetical inference. But the
following are some explanations.
Those with the antecedent P are most 'true to form' and express a normal
'movement of thought', and may therefore be called perfect, whether P be in
itself a thesis with an affirmative or negative content. But those with the
antecedent notP, qua antithesis (and not because it may present a negative content),
are not as such representative of a natural way of thinking. If notP is taken up
as a thesis in itself (be it intrinsically affirmative or negative in form or
content), rather than by virtue of its being the antithesis of P, the form is
quite normally hypothetical, proceeding from a posited antecedent, which may happen to be of negative polarity, to
some consequences or inconsequences. But if the focus or stress is on the antiP
aspect of our 'nonP', the form is relatively artificial, and so 'imperfect'.
d.
The use of 'substitution', putting an antithesis in place of a thesis, or
vice versa, is a theoretical device of the science of formal logic, rather than
a process in the practical art of logic. The science of logic is built as a
conceptual algebra, with 'variables' open to any content, related by selected
'constants'. In categoricals, the variables are terms, the constants, the
copula, the polarity, the quantity, and so on. In hypotheticals, the variables
are propositions, the constants, the relational factors peculiar to them.
But the use of substitution, in the sense of putting specific 'values' in
the place of logic's variables, is a practical, rather than theoretical,
process, and should be counted as a form, or at least stage, of inference. Here,
the thinker is applying logical principles to a given situation, appealing to
generally established processes to justify a particular act of thought. Such
movement from knowledge of logical science to practical application, is in
itself a reasoning process.
