FUTURE LOGIC
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.
Proposition 
Given 
P 
Q 
NonP 
NonQ 
If P, then Q 






P 
T 
T 
F 
F 

Q 
? 
T 
? 
F 

NonP 
F 
? 
T 
? 

NonQ 
F 
F 
T 
T 
If P, notthen Q 






P 
T 
? 
F 
? 

Q 
? 
T 
? 
F 

NonP 
F 
? 
T 
? 

NonQ 
? 
F 
? 
T 
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.
Form 
Equivalent Modal Conjunction 
Symb. 
If P, then Q 
{P and nonQ} is impossible 
H2n 
If P, notthen Q 
{P and nonQ} is possible 
K2p 
If P, then nonQ 
{P and Q} is impossible 
H1n 
If P, notthen nonQ 
{P and Q} is possible 
K1p 
If nonP, then Q 
{nonP and nonQ} is impossible 
H4n 
If nonP, notthen Q 
{nonP and nonQ} is possible 
K4p 
If nonP, then nonQ 
{nonP and Q} is impossible 
H3n 
If nonP, notthen nonQ 
{nonP and Q} is possible 
K3p 
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.