W. E. Johnson, Logic: Part I (1921)

CHAPTER III
COMPOUND PROPOSITIONS

§ 1. Having examined the proposition in its more philosophical aspects, and in particular from the point of view of its analysis, the present chapter will be mainly devoted to a strictly formal account of the proposition, and will be entirely concerned with the synthesis of propositions considered apart from their analysis. The chapter is intended to supply a general introduction to the fundamental principles of Forrrml Logic; and formulae will first be laid down without any attempt at criticism or justification -- which will be reserved for subsequent discussion. For this purpose we begin by considering the different ways in which a new proposition may be constructed out of one or more given propositions.

In the first place, given a single proposition, we may construct its negative -- expressed by the prefix not -- not-p being taken as equivalent to p-false. Next, we consider the construction of a proposition out of two or more given propositions. The proposition thus constructed will be called compound, and the component propositions out of which it is constructed, may be called simple, relatively to the compound, although they need not be in any absolute sense simple. The prefix not may be attached not only to any simple proposition, but also to a compound proposition, any of whose components again may be negative.

The different forms that may be assumed by compound propositions are indicated by different conjunctions. A proposition in whose construction the only formal elements involved are negation and the logical conjunctions is called a Conjunctional Function of its component propositions. The term conjunctional function must be understood to include functions in which negation or any one or more of the logical conjunctions is absent. We have to point out that the compound proposition is to be regarded, not as a mere plurality of propositions, but as a single proposition, of which truth or falsity can be significantly predicated irrespectively of the truth or falsity of any of its several components. Furthermore, the meaning of each of the component propositions must be understood to be assignable irrespectively of the compound into which it enters, so that the meaning which it is understood to convey when considered in isolation is unaffected by the mode in which it is combined with other propositions.

§ 2. We will proceed to enumerate the several modes of logical conjunction by which a compound proposition may be constructed out of two component propositions, say p and q. Of all such modes of conjunction, the most fundamental is that expressed by the word and: this mode will be called par excellence conjunctive, and the components thus joined will be called conjuncts. Thus the compound propositions --

  1. 'p and q,'
  2. 'p and not-q,'
  3. 'not-p and q,'
  4. 'not-p and not-q,'
are the conjunctive functions of the conjuncts p, q; p, not-q; not-p, q; not-p, not-q; respectively. There are thus four distinct conjunctive forms of proposition involving the two propositions p, q, taken positively or negatively.

The significance of the conjunctional and will be best understood in the first instance, by contrasting it with the enumerative and. For example, we use the merely enumerative and when we speak of constructing any compound proposition out of the components p and q. Here we are not specifying any mode in which p and q are to be combined so as to constitute one form of unity rather than another; we are treating the components (so to speak) severally, not combinatorially. In other words, the enumeration -- p and q -- yields two propositions, the enumeration -- p and q and r -- yields three propositions, etc.; but the conjunctive 'p and q,' or the conjunctive 'p and q and r' etc., yields one proposition. Again, of the enumerated propositions -- p and q and r and... -- some may be true and others false; but the conjunctive proposition 'p and q and r and...' must be either definitively true or definitively false. Thus in conjoining two or more propositions we are realising, not merely the force of each considered separately, but their joint force. The difference is conclusively proved from the consideration that we may infer from the conjunctive proposition 'p and q' a set of propositions none of which could be inferred from p alone or q alone. The same holds, of course, where three or more conjuncts are involved: thus, with p, q, r, as components, seven distinct groups1 of propositions are generated: viz. the three groups consisting of propositions implied by p, by q, by r respectively; the three groups consisting of propositions implied by 'p and q,' by 'p and r,' by 'q and r' respectively; and lastly, the group consisting of propositions implied by p and q and r.'

§ 3. In our first presentation of formal principles we shall introduce certain familiarly understood notions, such as equivalence, inference etc., without any attempt at showing how some of them might be defined in terms of others. The same plan will be adopted in regard to the question of the demonstrability of the formal principles themselves; these will be put forward as familiarly acceptable, without any attempt at showing how some of them might be proved by means of others. Ultimately, certain notions must be taken as intelligible without definition, and certain propositions must be taken as assertible without demonstration. All other notions (intrinsically logical) will have to be defined as dependent upon those that have been put forward without definition; and all other propositions (intrinsically logical) will have to be demonstrated as dependent upon those that have been put forward without demonstration. But we shall not, in our first outline, raise the question of the dependence or independence of the notions and propositions laid down.

Thus the formal law which holds of Negation is called the Law of Double Negation: viz.

not-not-p ≡ p.

§ 4. We now lay down the formal laws which hold of compound propositions constructed by means of the conjunction and. They are as follows:

Laws of Conjunctive Propositions
1. The Reiterative Law:
p and p ≡ p.
2. The Commutative Law:
p and q ≡ q and p.
3. The Associative Law:
(p and q) and r ≡ p and (q and r).
Here the notion of ejmivalence (expressed by the shorthand symbol ≡) is taken as ultimate and therefore as not requiring to be defined. These laws and similar formal principles are apt to be condemned as trivial. Their significance will be best appreciated by reverting to the distinction between the mental acts of assertion and progression in thought on the one hand, and the propositions to which thought is directed on the other. Thus the laws above formulated indicate, in general, equivalence as regards the propositions asserted, in spite of variations in the modes in which they come before thought. Thus the content of what is asserted is not affected, firstly, by any re-assertion; nor, secondly, by any different order amongst assertions; nor, thirdly, by any different grouping of the assertions.

§ 5. Having considered the Conjunctive form of proposition, we turn next to the consideration of the remaining fundamental conjunctional forms. These will be classed under the one head Composite for reasons which will be apparent later. So far, compound propositions have been divided into the two species Conjunctive and Composite, and we shall now proceed to subdivide the latter into four sub-species, each of which has its appropriate conjunctional expression, viz.:

(1) The Direct-Implicative function of p, q: -- If p then q.
(2) The Counter-Implicative function of p, q: -- If q then p.
(3) The Disjunctive function of p, q: -- Not-both p and q.
(4) The Alternative function of p, q: -- Either p or q.

In the implicative function 'If p then q,' p is implicans2 and q implicate; in the counter-implicative function 'If q then p,' p is implicate and q is implicans2; in the disjunctive function 'Not-both p and q,' p and q are disjuncts; and in the alternative function 'Either p or q,' p and q are alternants. These four functions of p, q, are distinct and independent of one another. The technical names that have been chosen are obviously in accordance with ordinary linguistic usage. The implicative and counter-implicative functions are said to be Complementary to one another, as also the disjunctive and alternative functions. Each of the four other pairs, viz. (1) and (3); (1) and (4); (2) and (3); (2) and (4) may be called a pair of Supplementary propositions. These names are conveniently retained in the memory by help of

The Square of Independence

Now when we bring into antithesis the four conjunctive functions:

(1) p and not-q;
(2) not-p and q;
(3) p and q;
(4) not-p and not-q;

with the four composite functions:

(1) if p then q;
(2) if q then p;
(3) not-both p and q;
(4) either p or q;

we shall find that each of the composite propositions is equivalent to the negation of the corresponding conjunctive. This is directly seen in the case (3) of the conjunctive and the disjunctive functions of p, q. Thus, 'Not-both p and q' is the direct negative of 'Both p and q.' Again, for case (4) 'Either p or q' is the obvious negative of 'Neither p nor q.' The relations of negation for all cases may be derived by first systematically tabulating the equivalences which hold amongst the composite functions, as below: abbreviating not-p and not-q into the forms ~p and ~q respectively.

Table of Equivalences of the Composite Functions
Implicative
Form 		Counter-
Implicative Form 		Disjunctive
Form 		Alternative
Form
1. If p then q = If ~q then ~p = Not both p and ~q = Either ~p or q
2. If ~p then ~q = If q then p = Not both ~p and q = Either p or ~q
3. If p then ~q = If q then ~p = Not both p and q = Either ~p or q
4. If ~p then q = If ~q then p = Not both ~p and ~q = Either p or q

In the above table it will be observed:

(a) That each composite function can be expressed in four equivalent forms: thus, any two propositions in the same row are equivalent, while any two propositions in different rows are distinct and independent.

(b) That the propositions represented along the principal diagonal are expressed in terms of the positive components p, q; being in fact identical respectively with the implicative, the counter-implicative, the disjunctive and the alternative functions of p, q.

(c) That all the remaining propositions are expressed as functions of p and not-q, or of not-p and q, or of not-p and not-q.

We may translate the equivalences tabulated above in the form of equivalences of functions, thus:

1. The implicative function of p, q; the counter-implicative function of not-p, not-q; the disjunctive function of p, not-q; and the alternative function of not-p, q, are all equivalent to one another. Again,

2. The counter-implicative function of p, q; the implicative function of not-p, not-q; the disjunctive function of not-p, q; and the alternative function of p, not-q, are all equivalent to one another. Again,

3. The disjunctive function of p, q; the alternative function of not-p, not-q; the implicative function of p, not-q; and the counter-implicative function of not-p, q, are all equivalent to one another. Again,

4. The alternative function of p, q; the disjunctive function of not-p, not-q; the implicative function of not-p, q; and the counter-implicative function of p, not-q, are all equivalent to one another.

Since the force of each of the four composite functions of p, q can be represented by using either the Implicative or the Counter-implicative or the Disjunctive or the Alternative form, the classification of the four functions under one head Composite is justified. And since each Composite function is equivalent to a certain Disjunctive proposition, it is also equivalent to the negation of the corresponding Conjunctive proposition. Thus :

1. The implicative 'If p then q' negates the Conjunctive 'p and ~q.'
2. The counter-implicative 'If q then p' negates the Conjunctive '~p and q.'
3. The disjunctive 'Not both p and q' negates the Conjunctive 'p and q.'
4. The alternative 'Either p or q' negates the Conjunctive '~p and ~q.'

Thus inasmuch as no composite function is equivalent to any conjunctive function, we have justified our division of compound propositions into the two fundamentally opposed species Conjunctive and Composite.

The distinction and relation between these composite forms of proposition may be further brought out by tabulating the inferences in which a simple conclusion is drawn from the conjunction of a composite with a simple premiss. In traditional logic, the Latin verbs ponere (to lay down or assert) and tollere (to raise up or deny) have been used in describing these different modes of argument. The gerund ponendo (by affirming) or tollendo (by denying) indicates the nature of the (simple) premiss that occurs; while the participle ponens (affirming) or tollens (denying) indicates the nature of the (simple) conclusion that occurs: the validity or invalidity of the argument depending on the nature of the composite premiss. There are therefore four modes to be considered corresponding to the four varieties of the composite proposition, thus:

Table of Valid Modes
ModusForm of Composite Premiss
1. Ponendo Ponens:If p then q; but p; ∴ q | The Implicative.
2. Tollendo Tollens:If q then p; but ~p; ∴ ~q| The Counter-Implicative.
3. Ponendo Tollens:Not both p and q; but p; ∴ ~q| The Disjunctive.
4. Tollendo Ponens:Either p or q; but ~p; ∴ q| The Alternative.

The customary fallacies in inferences of this type may be exhibited as due to the confusion between a composite proposition and its complementary:

Table of Invalid Modes
Modus Form of Composite Premiss
1. Ponendo Ponens:If q then p; but p; ∴ q| The Counter-Implicative.
2. Tollendo Tollens:If p then q; but p; ∴ ~q| The Implicative.
3. Ponendo Tollens:Either p or q; but p; ∴ ~q| The Alternative.
4. Tollendo Ponens:Not both p and q; but ~p; ∴ q | The Disjunctive.

The rules for correct inference from the above table of valid modes may be thus stated:

1. From an implicative, combined with the affirmation of its implicans, we may infer the affirmation of its implicate.

2. From an implicative, combined with the denial of its implicate, we may infer the denial of its implicans.

3. From a disjunctive, combined with the affirmation of one of its disjuncts, we may infer the denial of the other disjunct.

4. From an alternative, combined with the denial of one of its alternants, we may infer the affirmation of the other alternant.

§ 6. We ought here to refer to an historic controversy as regards the interpretation of the conjunction 'or.' It has been held by one party of logicians that what I have called the Alternative form of proposition, viz., that expressed by either-or, should be interpreted so as to include what I have called the Disjunctive, viz., that expressed by not-both. This view has undoubtedly been (perhaps unwittingly) fostered by the almost universal misemployment of the term Disjunctive to stand for what ought to be called Alternative. This prevalent confusion in terminology has led to a real blunder committed by logicians. The blunder consists in the fallacious use of the Ponendo Tollens as exhibited in the table above given. Consider the argument: 'A will be either first or second'; 'It is found that A is second'; therefore 'A is not first.' Here the conclusion is represented as following from the promising qualifications of the candidate A, whereas it really follows from the premiss 'A cannot be both first and second.' In fact, the Alternative proposition which is put as premiss is absolutely irrelevant to the conclusion, which would be equally correctly inferred whether the alternative predication were false or true.

It remains then to consider whether the logician can properly impose the one interpretation of the alternative form of proposition rather than the other. The reply here, as in other similar cases, is that, in the matter of verbal interpretation, the logician can impose legislation -- not upon others -- but only upon himself. However, where any form of verbal expression is admittedly ambiguous, it is better to adopt the interpretation which gives the smaller rather than the greater force to a form of proposition, since otherwise there is danger of attaching to the judgment an item of significance beyond that intended by the asserter. This principle of interpretation has the further advantage that it compels the speaker when necessary to state unmistakeably and explicitly what may have been implicitly and perhaps confusedly present in his mind. I have therefore adopted as my interpretation of the form Either-or that smaller import according to which it does not include Not-both. Those logicians who have insisted on what is called the 'exclusive' interpretation of the alternative form of proposition (i.e. the interpretation according to which Either-or includes Not-both) seem sometimes to have been guilty of a confusion between what a proposition asserts, and what may happen to be known independently of the proposition. Thus it may very well be the case that the alternants in an alternative proposition are almost always 'exclusive' to one another; but this, so far from proving that the alternative proposition affirms this exclusiveness, rather suggests that the exclusiveness is a fact commonly known independently of the special information supplied by the alternative proposition itself.

In this connection, the significance of the term complementary which I have applied to the implicative and counter-implicative as well as to the disjunctive and alternative, may be brought out. Propositions are appropriately called complementary when a special importance attaches to their conjoint assertion3. Thus it may be regarded as an ideal of science to establish a pair of propositions in which the implicans of the one is the implicate of the other; and again to establish a number of propositions which are mutually co-disjunct and collectively co-alternate. The term complementary is especially applicable where propositions are conjoined in either of these ways, because separately the propositions represent the fact partially, and taken together they represent the same fact with relative completeness.

We next consider the inferences that can be drawn from the conjunction of two supplementary propositions. These may be tabulated in two forms, the first of which brings out the fundamental notion of the Dilemma; and the second that of the Reductio ad Impossibile.

First Table for the Conjunction of Supplementaries
The Dilemma

(1) 'If p then q' and (4) 'If ~p then q': therefore, q.
(3) 'If p then ~q' and (2) 'If ~p then ~q': therefore, ~q.
(2) 'If p then q' and (4) 'If ~q then p': therefore, p.
(3) 'If p then q' and (1) 'If ~q then ~p': therefore, ~p.

The above table illustrates the following principle:

The conjunction of two implicatives, containing a common implicate but contradictory implicants, yields the affirmation of the simple proposition standing as common implicate. Or otherwise:

Any proposed proposition must be true when its truth would be implied both by the supposition of the truth and by the supposition of the falsity of some other proposition.

Second Table for the Conjunction of Supplementaries
The Reductio ad Impossibile

(1) 'If ~q then ~p' and (4) 'If ~q then p': therefore, q.
(3) 'If q then ~p' and (2) 'If q then p': therefore, ~q.
(2) 'If ~p then ~q' and (4) 'If ~p then q': therefore, p.
(3) 'If p then ~q' and (1) 'If p then q': therefore, p.

This second table illustrates the following principle:

The conjunction of two implicatives, containing a common implicans but contradictory implicates, yields the denial of the simple proposition standing as common implicans. Or otherwise:

Any proposed proposition must be false when the supposition of its truth would imply (by one line of argument) the truth and (by another line of argument) the falsity of some other proposition.

§ 7. In tabulating the formulae for Composite propositions as above I have merely systematised (with slight extensions and modifications of terminology) what has been long taught in traditional logic; and it is only in these later days that criticisms have been directed against the traditional formulae, especially on the ground that their uncritical acceptance has been found to lead to certain paradoxical consequences, which may be called the Paradoxes of Implication. In this connection it is (I think) desirable to explain what is meant by a paradox. When a thinker accepts step by step the principles or formulae propounded by the logician until a formula is reached which conflicts with his common-sense, then it is that he is confronted with a paradox. The paradox arises -- not from a merely blind submission to the authority of logic, or from any arbitrary or unusual use of terms on the logician's part -- but from the very nature of the case, as apprehended in the exercise of powers of reasoning with which everyone is endowed. In particular, the paradoxes of implication are not due to any unnatural use of the term implication, nor to the positing of any fundamental formula that appears otherwise than acceptable to common sense. It is the formulae that are derived -- by apparently unexceptionable means from apparently unexceptionable first principles -- that appear to be exceptionable.

Let us trace the steps by which we reach a typical paradox. Consider the alternative 'Not-p or q.' If this alternative were conjoined with the assertion 'p,' we should infer 'q.' Hence, 'Not-p or q' is equivalent to 'If p then q.' Similarly 'p or q' is equivalent to 'If not-p then q.' Now it is obvious that the less determinate statement 'p or q' could always be inferred from the more determinate statement 'p': e.g. from the relatively determinate statement 'A is a solicitor' we could infer 'A is a solicitor or a barrister' i.e. 'A is a lawyer.' Hence, whatever proposition q may stand for, we can infer 'p or q' from 'p'; or again, whatever p may stand for we can infer 'not-p or q' from q. Hence (i) given 'p' we may infer 'If not-p then q,' and (ii) given 'q' we may infer 'If p then q,' whatever propositions p and q may stand for. These two consequences of the uncritical acceptance of traditional formulae have been expressed thus: (i) A false proposition (e.g. not-p when p has been asserted) implies any proposition (e.g. q); (ii) A true proposition (e.g. q, when q has been asserted) is implied by any proposition (e.g. p). Thus '2 + 3 = 7' would imply that 'It will rain to-morrow'; and 'It will rain to-morrow' would imply that '2 + 3 = 5.' That these two implicative statements are technically correct is shown by translating them into their equivalent alternative forms, viz.: (i) 'Either 2 + 3 is unequal to 7 or it will rain to-morrow'; (ii) 'Either it will not rain to-morrow or 2 + 3 = 5.' We may certainly say that one or other of the two alternants in (i) as also in (ii) is true, the other being of course doubtful.

Taking 'If p then q' to stand for the paradoxically reached implicative in both cases, we have shown that (i) from the denial of p (the implicans), and (ii) from the affirmation of q (the implicate) we may pass to the assertion 'If p then q.' This is, of course, only another way of saying that the implicative 'If p then q' is equivalent to the alternative 'p false or q true.' Thus when we know that 'If p then q' is true, it follows that we know that 'either p is false or q is true'; but it does not follow that either 'we know that p is false' or 'we know that q is true.' The paradoxically reached implicative merely brings out the fact that this may be so in some cases: i.e. when asserting 'If p then q,' there are cases in which we know that 'p is false,' and there are cases in which we know that 'q is true.' But it is proper to enquire whether in actual language -- literary or colloquial -- the implicative form of proposition is ever introduced in this paradoxical manner. On the one hand, we find such expressions as: 'If that boy comes back, I'll eat my head'; 'If you jump over that hedge, I'll give you a thousand pounds'; 'If universal peace is to come tomorrow, the nature of mankind must be very different from what philosophers, scientists and historians have taken it to be'; etc., etc. Such phrases are always interpreted as expressing the speakers intention to deny the implicans; the reason being that the hearer is assumed to be ready to deny the implicate. Again, on the other hand, we find such forms as: 'If Shakespeare knew no Greek, he was not incapable of creating great tragedies.' 'If Britain is a tiny island, on the British Empire the sun never sets.' 'If Boswell was a fool, he wrote a work that will live longer than that of many a wiser man.' 'If Lloyd George has had none of the advantages of a public school education, it cannot be maintained that he is an unintelligent politician.' Such phrases are always interpreted as expressing the speaker's intention to affirm the implicate, the reason being that the hearer may be assumed to be willing to affirm the implicans.

Looking more closely into the matter we find that when a speaker adopts the implicative form to express his denial of the implicans, he tacitly expects his hearer to supplement his statement with a tollendo tollens; and when he adopts it to express his affirmation of the implicate, he expects the hearer to supplement it with a ponendo ponens. Furthermore, inasmuch as the alternative form of proposition requires to be supplemented by a tollendo (ponens) and the disjunctive by a ponendo (tollens), we find that an implicative intended to express the denial of its implicans is quite naturally expressed otherwise as an alternative: e.g. 'That boy won't come back or I'll eat my head,' to which the hearer is supposed to add 'But you won't eat your head'; therefore (I am to believe that) 'the boy won't come back' (tollendo ponens); and we find that an implicative intended to express the affirmation of its implicate is quite naturally expressed otherwise as a disjunctive: e.g. 'It cannot be held that Shakespeare both knew no Greek and was incapable of creating great tragedies,' to which the hearer is supposed to add 'But Shakespeare knew no Greek,' and therefore (I am to believe that) 'he was capable of creating great tragedies' (ponendo tollens). We have yet to explain how the appearance of paradox is to be removed in the general case of a composite being inferred from the denial of an implicans (or disjunct) or from the affirmation of an implicate (or alternant). Now the ordinary purpose to which an implicative (or, more generally, a composite) proposition is put is inference: so much so that most persons would hesitate to assert the relation expressed in a composite proposition unless they were prepared to use it for purposes of inference in one or other of the four modes, ponendo ponens, etc. In other words, Implication is naturally regarded as tantamount to Potential Inference. Now when (i) we have inferred 'If p then q' from the denial of 'p,' can we proceed from 'If p then q' conjoined with 'p' to infer 'q'? In this case we join the affirmation of 'p' with a premiss which has been inferred from the denial of p; and this involves Contradiction, so that such an inference is impossible. Again, when (ii) we have inferred 'If p then q' from the affirmation of 'q,' can we proceed from 'If p then q' conjoined with 'p' to infer 'q'? In this case we profess to infer 'q' by means of a premiss which was itself inferred from 'q'; and this involves Circularity, so that this inference again must be rejected. The solution of the paradox is therefore found in the consideration that though we may correctly infer an implicative from the denial of its implicans, or from the affirmation of its implicate, or a disjunctive from the denial of one of its disjuncts, or an alternative from the affirmation of one of its alternants, yet the implicative, disjunctive or alternative so reached cannot be applied for purposes of further inference without committing the logical fallacy either of contradiction or of circularity. Now it must be observed that the rhetorical or colloquial introduction of a paradoxical composite, which is meant to be interpreted as the simple affirmation or denial of one of its components, achieves its intention by introducing -- as the other component of the composite -- a proposition whose falsity or truth (as the case may be) is palpably obvious to the hearer. The hearer is then expected to supplement the composite by joining it with the obvious affirmation or denial of the added component, and thereby, in interpreting the intention of the speaker, to arrive at the proposition as conclusion which the speaker took as his first premiss. Accordingly the process of interpretation consists in taking the same propositions in the same mode and arrangement as would have entailed circularity if adopted by the speaker.

§ 8. The distinction between an implicative proposition that can and one that cannot be used for inferential purposes may now be further elucidated by reference to the distinction between Hypothesis and Assertion. In order that an implicative may be used for inference, both the implicans and the implicate must be entertained hypothetically. In the case of ponendo ponens the process of inference consists in passing to the assertion of the implicate by means of the assertion of the implicans, so that the propositions that were entertained hypothetically in the implicative, come to be adopted assertively in the process of inference. The same holds, mutatis mutandis, for the other modes. Now when we have inferred an implicative from the affirmation of its implicate or from the denial of its implicans -- as in the case of the implicative which appears paradoxical -- the two components of the implicative thus reached cannot both be regarded as having been entertained hypothetically; and hence the principle according to which inference is a process of passing from propositions entertained hypothetically to the same propositions taken assertorically, would be violated if we used the composite for inference. This consideration constitutes a further explanation of how the paradoxes in question are solved.

The above analysis may be symbolically represented by placing under the letter standing for a proposition the sign |- to stand for assertorically adopted and the sign H for hypothetically entertained.

Thus the fundamental formula for correct inference may be rendered:

From 'p would imply q' with p; we may infer q,
HH|-|-

where, in the implicative premiss, both implicans an implicate are entertained hypothetically.

Now the following inferences, which lead to paradoxical consequences, may be considered correct: i.e.

(a) From q, we may infer 'p would imply q.'
|-H|-
(b) From ~p, we may infer 'p would imply q.'
|-|-H

But the implicative conclusions here reached cannot be used for further inference: i.e.

(c) From 'p would imply q' with p; we cannot infer q.
H|-|-|-
(d) From 'p would imply q' with p; we cannot infer q.
|-H|-|-

For in (c) the implicate, and in (d) the implicans enters assertorically, and these inferences therefore contravene the above fundamental formula which requires that both implicate and implicans should enter hypothetically. Thus while admitting (a) that 'a true proposition would be implied by any proposition,' yet we cannot admit (c) that 'a true proposition can be inferred from any proposition.' Similarly, while admitting (b) that 'a false proposition would imply any proposition,' yet we cannot admit (d) that 'from a false proposition we can infer any proposition.' In fact, the attempted inference (c), where the conclusion has already been asserted, would entail circularity; and the attempted inference (d), where the premiss has already been denied, would involve contradiction.

Still maintaining the equivalence of the composite propositions expressible in the implicative, the counter-implicative, the alternative or the disjunctive form, each of these four forms will give rise to a like paradox. The following table gives all the cases in which we reach a Paradoxical Composite; that is, a Composite which cannot be used for inference, either in the modus ponendo ponens, tollendo tollens, ponendo tollens or tollendo ponens. The sign of assertion in each composite must be interpreted to mean asserted to be true when the term to which it is attached agrees with the premiss, and asserted to be false when it contradicts the premiss.

Table of Paradoxical Composites

(a) From q we may properly infer
|-
(1) ~p or q = If p then q = If ~q then ~p = Not both p and ~q
H|-H|-|-HH|-
or
(2) p or q = If ~p then q = If ~q then p = Not both ~p and ~q
H|-H|-|-HH|-
(b) From ~q we may properly infer
|-
(3) p or ~q = If ~p then ~q = If q then p = Not both ~p and q
H|-H|-|-HH|-
or
(4) ~p or ~q = If p then ~q = If q then ~p = Not both p and ~q
H|-H|-|-HH|-

The above composites can never be used for further inference. Thus:

in line (1), the attempted inference
'p ∴ q' would be circular and 'not-q ∴ not-p' would be contradictory;
in line (2), the attempted inference
'not-p ∴ q' would be circular and 'not-q ∴ p' would be contradictory;
in line (3), the attempted inference
'not-p ∴ not-q' would be circular and 'q ∴ p' would be contradictory;
in line (4), the attempted inference
'p ∴ not-q' would be circular and 'q ∴ not-p' would be contradictory;

The paradox of implication assumes many forms, some of which are not easily recognised as involving mere varieties of the same fundamental principle. But I believe that they can all be resolved by the consideration that we cannot without qualification apply a composite and (in particular) an implicative proposition to the further process of inference. Such application is possible only when the composite has been reached irrespectively of any assertion of the truth or falsity of its components. In other words, it is a necessary condition for further inference that the components of a composite should really have been entertained hypothetically when asserting that composite.

§ 9. The theory of compound propositions leads to a special development when in the conjunctives the components are taken -- not, as hitherto, assertorically -- but hypothetically as in the composites. The conjunctives will now be naturally expressed by such words as possible or compatible, while the composite forms which respectively contradict the conjunctives will be expressed by such words as necessary or impossible. If we select the negative form for these conjunctives, we should write as contradictory pairs:

Conjunctives (possible)
~a. p does not imply q
~b. p is not implied by q
~c. p is not co-disjunct to q
~d. p is not co-alternate to q 		|
|
|
|
|
Composites (necessary)
a. p implies q
b. p is implied by q
c. p is co-disjunct to q
d. p is co-alternate to q

Or otherwise, using the term 'possible' throughout, the four conjunctives will assume the form that the several conjunctions -- p~q, ~pq and ~p~q -- are respectively possible. Here the word possible is equivalent to being merely hypothetically entertained, so that the several conjunctives are now qualified in the same way as are the simple components themselves. Similarly the four corresponding composites may be expressed negatively by using the term 'impossible,' and will assume the form that the conjunctions p~q, ~pq, pq and ~p~q are respectively impossible, or (which means the same) that the disjunctions p~q, ~pq, pq and ~p~q are necessary. Now just as 'possible' here means merely 'hypothetically entertained,' so 'impossible' and 'necessary' mean respectively 'assertorically denied' and 'assertorically affirmed.'

The above scheme leads to the consideration of the determinate relations that could subsist of p to q when these eight propositions (conjunctives and composites) are combined in every possible way without contradiction. Prima facie there are 16 such combinations obtained by selecting a or ~a, b or ~b, c or ~c, d or ~d for one of the four constituent terms. Out of these 16 combinations, however, some will involve a conjunction of supplementaries (see tables on pp. 37, 38), which would entail the assertorical affirmation or denial of one of the components p or q, and consequently would not exhibit a relation of p to q. The combinations that, on this ground, must be disallowed are the following nine:

a~bc~d, ab~cd, ~ab~cd, ~abc~d; ~abcd, bacd, ~cabd, ~dabc; abcd.

(The combinations that remain to be admitted are therefore the following seven:

ab~cd, cd~ab; ab~c~d, b~a~c~d, c~d~a~b, d~c~a~b; ~a~b~cd.

In fact, under the imposed restriction, since a or b cannot be conjoined with c or d, it follows that we must always conjoin a with c and ~d; b with ~c and ~d; c with ~a and ~b; d with ~a and ~b. This being understood, the seven permissible combinations that remain are properly to be expressed in the more simple forms:

ab, cd; a~b, b~a, c~d, d~c; and ~a~b~c~d.

These will be represented (but re-arranged for purposes of symmetry) in the following table giving all the possible relations of any proposition p to any proposition q. The technical names which I propose to adopt for the several relations are printed in the second column of the table.

Table of possible relations of proposition p to proposition q.

1. (a,b): p implies and is implied by q. p is co-implicant to q.
2. (a, ~b): p implies but is not implied by q. p is super-implicant to q.
3. (b, ~a): p is implied by but does not imply q. p is sub-implicant to q.
4. (a, ~b, ~c, ~d): p is neither implicans nor implicate
nor co-disjunct nor co-alternate to q. 		p is independent of q.
5. (d, ~c): p is co-alternate but not co-disjunct to q. p is sub-opponent to q.
6. (c, ~d): p is co-disjunct but not co-alternate to q. p is super-opponent to q.
7. (c, d): p is co-disjunct and co-alternate to q. p is co-opponent to q.

Here the symmetry indicated by the prefixes, co-, super-, sub-, is brought out by reading downwards and upwards to the middle line representing independence. In this order the propositional forms range from the supreme degree of consistency to the supreme degree of opponency, as regards the relation of p to q. In traditional logic the seven forms of relation are known respectively by the names equipollent, superaltern, subaltern, independent, sub-contrary, contrary, contradictory. This latter terminology, however, is properly used to express the formal relations of implication and opposition, whereas the terminology which I have adopted will apply indifferently both for formal and for material relations.

Notes

1 The term group is here used in its precise mathematical significance.

2 The plural of implicans must be written: implicants.

3 Thus complementary propositions might be defined as those which are frequently confused in thought and frequently conjoined in fact.