Faithful and Fruitful Logic

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Faithful and Fruitful Logic

Appropriate for a conference relating philosophy and education, we seek ways more faithful than the truth-functional (TF) hook to understand and represent that ordinary-language conditional which we use in, e.g., modus ponens, and that conditional’s remote and counterfactual counterparts, and also the proper negations of all three. Such a logic might obviate the paradoxes caused by T-F representation, and be educationally fruitful. William and Martha Kneale and Gilbert Ryle assist us: "In the hypothetical case in which p, it is inferable, on the basis that p and at least in the given context, that q." "Inferable" is explained. This paraphrase is the foundation of the logic of hypothetical inferability ("HI logic"). It generates the negative but non-TF device "hib" (= "there is a hypothetical-inferability bar against the conjoint proposition that"), followed by a bracketed conjunction. This is an enriched negative: "hib (p . -q)" is stronger than "-(p . -q)," and "-hib" ("dash hib" = "there is no h-i bar...") offers us "-hib (p . -q)," weaker than "p . -q." Thus equipped, we can test deductive arguments by the CI ("Compatible-or-incompatible?") method explained, and explode paradoxes. The paraphrase, "hib," and the CI method are fruitful in training students to understand this conditional, and to demonstrate genuine validity or invalidity.

The logic generally taught to English-speaking students is symbolic logic. How faithful is it when employed as a representation of the connectives they use and will use in their ordinary conversation and in most of their intellectual activity, at least if they are not mathematicians? How fruitful for their education? Is there a logic more faithful and likely to be more fruitful? A conference inviting us to relate philosophy and education makes those questions especially opportune.


Reviewing Strawson’s Introduction to Logical Theory in Mind (1953), Quine admits that Strawson is "good on ‘É ’ and ‘if/then’" and "rightly observes the divergence between the two". But he left unchanged his handling of "the conditional" in subsequent editions of his textbooks. In the review he writes unconcernedly (as would be impossible for Ryle, Austin or Strawson) of the "Procrustean treatment of ordinary language at the hands of logicians", defending it by offering symbolic logic as the appropriate language for science, and suggesting that philosophy of science comes close to being "philosophy enough".

Ackermann, in the Preface to his Modern Deductive Logic, takes quite a different approach. He emphasises the "mathematical and scientific applications" of symbolic deductive logic, but says "one may well wonder" whether it has "enough philosophical value" to justify a major place in the philosophy curriculum.

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"Faithful and Fruitful Logic." 17 Dec 2017
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Unsatisfied by the intrinsic-interest defence, he says (using ‘logic’ to mean symbolic logic):

The importance of logic, and its justification in the philosophical curriculum, seems rather to appear in the realization that logic is often most relevant to philosophical interests precisely where it fails to provide an appropriate assessment of validity.

The question seems obvious: why not seek a non-Procrustean system of representation, one that is concerned to match at least certain uses of ordinary-language connectives, and see if it does provide "an appropriate assessment of validity"?

In the absence of such a system, can we expect most philosophy students to have much patience with logic? At present, departments are largely divided between those staff and students who "lap up" symbolic logic and are not greatly concerned to relate it to ordinary-language arguments, and those whose interests are more in ethics and kindred subjects, who, by and large, do no more symbolic logic than they have to and afterwards use it very little or not at all. Dorothy Sayers said in relation to Abelard’s teaching of logic in Paris that the students clasped logic to their arms like a bride. There is little such reaction now.


Let us examine an example from Ackermann (p.78). He imagines a professor saying that "to pass his course, one must either write a passing paper or a passing exam", and then that "it is not necessary to write a passing paper or a passing exam". How unsatisfactory (and in the second case confusing) is the use of English here. The two propositions which the professor asserts are (if we presume compatibility and note Ackermann’s symbolic treatment) (i) that if she is to pass the course, some particular student (let us call her Jo) will do so through writing a passing paper or by writing a passing exam, and (ii) (as suggested by the first) that it is both not the case that if Jo passes the course she writes a passing paper and not the case that if she passes the course she writes a passing exam.

Let us move to abstraction, keeping Ackermann’s letters but not symbolizing ‘If/(then)’ or ‘or’. The forms of the assertions are (i) ‘If C then P or E’; and (ii) ‘–(if C then P) . –(if C then E)’. Whenever someone performs the valuable service of stating that there are two alternative ways of gaining some hypothetical result (not gainable any other way), or two alternative outcomes of some hypothetical state of affairs, he or she will be committed to assertions of these two forms.

Ackermann gives the truth-functional formulae which are all that symbolic logic allows. Let us label them with ‘Tf’. Tf(i) is ‘C É (P v E)’ and Tf(ii), where I shall go on using a dot and dashes instead of his inverted V and tildas, ‘–(C É P) . –(C É E)’. From Tf(ii) we get Tf(iii), ‘C . –P . –E’, incompatible with Tf(i), as with (i). "What has gone wrong?" he asks. In his answer (p.245f) he says of the second abstraction that "it incorrectly suggests there is no connection between passing and a paper, and between passing and an exam. But this isn’t what the professor said. What he said is more appropriately rendered as ‘(C . –P) v (C . –E)’." That will not do at all. Symbolic forms do not "suggest" anything. Because of the meaning of the hook, Tf(ii) simply asserts both that C and –P and that C and –E. That is why it is replaceable by Tf(iii). But all three of Tf(ii), Ackermann’s preferred Tf alternative to it, and Tf(iii) suffer from two fatal defects. Unlike (i), Tf(i), and (ii), they assert that C and rule out the possibility that C . P . E.

We have an appalling situation: the orthodox system of logic cannot properly represent both members of a pair of statements of which the first is of the form ‘If p, (then) q or r’ and the second denies both ‘If p then q’ and ‘If p then r’. Clearly those denials do not commit their utterer to affirm that p, or to deny that q and/or to deny that r. The trouble is seen at its clearest in Tf(ii) and Tf(iii), for there we have the only type of negation of a conditional which truth-functional logic allows us. If the negation of ‘If p then q’ is not ‘p . –q’, then ‘If p then q’ is not properly represented by the familiar ‘p É q’.

The absurdity of supposing that ‘If p then q’ is properly negated by ‘p . –q’ comes out even more clearly in relation to Lemmon’s claim (Beginning Logic, p.61) that it is a logical truth that either if it is raining then it is snowing or if it is snowing then it is raining. Again the negation is crucial; deny the symbolic representation of that supposed logical truth and you doubly contradict yourself, with p . –q . q . –p. But will any educated person (not already imbued with symbolic logic) who is aware how few of the infinite range of imaginable conditionals are true accept that, as a matter of logic, at least one of any pair of statements representable by ‘If p then q’ and ‘If q then p’ must be true? No: for such persons will agree that conditionals are not properly asserted merely on the strength of knowledge, or of belief, that –p, or that q, nor properly denied by categorical statements of the form ‘p . –q’.

What will have to occur before logicians content with the symbolic representations accept that they are seriously inadequate? We may recall Kuhn’s analysis of how much is needed before a paradigm is dropped by scientists: it is not enough that paradoxes emerge, for a paradigm will be clung to in spite of them until a better one is there to replace it, one which will do what the earlier did and more, as well as resolving the paradoxes.


In the ‘Whenever ...’ sentence in the second paragraph of Sec. II I twice used the word ‘hypothetical’. If we attempt a general account of what people are doing when they begin a sentence with ‘If’, we have to say that they are showing that they are dealing with a hypothetical case, i.e., one not declared to be actual but merely envisaged or entertained. When we consider the Latin si, the Greek e i , the French si and au cas que, and the German wenn and im Falle daß, we get the same answer.

Limiting ourselves now to conditional (or, we may say, hypothetical) statements, we may ask what is being done by their maker. Here we must draw a major distinction, which, to my knowledge, has been previously brought out only by William and Martha Kneale (The Development of Logic, pp.135-8). Some utterances of the kind ‘If p (then) q’ are intended as no more than predictions (or guesses), and of those the Kneales rightly say that they are defective truth-functions, with no truth-value if it turns out to be the case that –p; others assert what the Kneales call necessary connexion, and are not rendered true by its being the case that –p, or that q (nor by its happening to be the case that p and q). We should note here that the conditionals in Ackermann’s example are obviously of the latter type: the professor is declaring what the rule is on how passes may be attained. Moreover, the vital moves for logic of modus ponendo ponens and modus tollendo tollens, and of hypothetical syllogism, require statements of the second type.

Thus we must ask: "For this logically crucial kind of conditional, what paraphrase can we give that will reveal its nature and yield a proper negation, so that paradoxical outcomes of the kind we have seen are obviated?" And then: "Can such a paraphrase yield a form of representation which will permit a reliable form of testing for validity?"

I shall give a summary answer here, because of the limitation of space. Here is the paraphrase:

In the hypothetical case in which p, it is inferable, on the basis that p and at least in the given context, that q.

Let us see how well that fits the ‘If ... (then) ...’ statements contemplated by Ackermann and by Lemmon. "In the hypothetical case that Jo passes the course, it is inferable ... that she has written either a passing paper or a passing exam. In that hypothetical case, however, it is not inferable ... that she has written a passing paper, nor is it inferable ... that she has written a passing exam." The paraphrases are clear and consistent; there is no implication that Jo does pass the course, nor that she does not write both a passing paper and a passing exam. We deny Lemmon’s absurdity by saying simply "In the hypothetical case in which it is raining, it is not inferable ... that it is snowing; and conversely."

What is meant by ‘inferable’? As with ‘reasonable’ and other such words, the word is normative. If you are entitled to infer that q in the case (actual or hypothetical) in which p, and to do so on the basis that p and at least in the given context, you are entitled, in that case in which p, to hold that you are in no weaker epistemological relation to the proposition that q than you are to the proposition that p. If you know the latter to be true, you also know the former to be true; and so also with belief. If, however, in the actual or hypothetical case in which p, I say hastily ‘so q’, and then or later admit that it is or was not inferable that q (but only, perhaps, reasonable to predict or guess that q), then I am in a situation similar to that in which I say concerning a particular proposition ‘I was wrong in claiming to know that was true; it could be no more than a matter of belief.’ (However, I do not deny that people will sometimes say ‘I wrongly inferred that ...’, where ‘infer’ is used as a synonym of ‘conclude’.) A point that David Lewis makes about claims of knowledge, in "Scorekeeping in a Language Game", applies here too: people will differ somewhat in the degree of stringency with which they think it right to use the verbs ‘know’ and ‘infer’. I am in complete agreement with Ryle in "‘If’, ‘So’, and ‘Because’": ‘If p (then) q’ (in the Kneales’ second use) is closely akin to ‘p, so q’, and no more a truth-functional statement than is ‘q because p’.

One great advantage of our paraphrase as an interpretation of the class of conditional statements crucial for logic is that it is equally applicable to open, remote and counterfactual conditionals. All we need to distinguish them are parentheses to follow ‘in which p’: ‘and it is here left open whether that case is actual’; ‘and that case is here presented as a remote possibility’; and ‘and that case is not actual’.

Our paraphrase reveals the nature and importance of a relation which we may call hypothetical inferability, which holds when the proposition that q is inferable in the hypothetical case that p, on the basis that p and at least in the given context. It is of course perfectly true of ‘If p (then) q’, and essential to its meaning, that it denies the conjunction ‘p . –q’. However, it does not merely deny it, as one would if one knew that –p or that q; it rules it out in a special way, by asserting hypothetical inferability. In consequence, as Stevenson saw in "If-iculties" (Philosophy of Science, 1970) in relation to scientific statements, ‘p É q’ is too weak as a representation of ‘If p (then) q’ when that is no mere guess or prediction; and ‘p . –q’ is too strong as a representation of the negation of ‘If p (then) q’. (The proper negation of ‘If you pass the course you write a passing paper’ is ‘You could pass the course without writing a passing paper.’)

Now let us use the device ‘hib’, which we may call an enriched negative, and give it the meaning ‘there is a hypothetical-inferability bar against the conjoint proposition that’. It can be followed only by a bracketed conjunction, with two or more members. There is such a bar against any conjunction if a true ‘If ... (then) ...’ statement would rule it out. The negation is ‘–hib’ ("dash hib"): ‘there is no hypothetical-inferability bar ...’.

The professor’s first statement may be represented by ‘hib (C . –P . –E)’ and his second by ‘–hib (C . –P) . –hib (C . –E)’. Lemmon’s paradox is obviated by ‘–hib (p . –q) . –hib (q . –p)’ (cf. ‘It could, so far as hypothetical inferability is concerned, be raining and not snowing; and similarly it could be snowing and not raining’).

I have developed the CI method (‘CI’ standing for ‘"Compatible-or-incompatible?"’) which permits the testing of deductive arguments whose components cannot be adequately represented truth-functionally. Consider the simplest case of hypothetical syllogism. Let us number the premises after a capital P, write ‘Con’ and ‘NCon’ for the conclusion and its negation, C for ‘Compatible with’ and I for ‘Incompatible with’. Let us also write S for ‘Subjunct’ for anything we can join on underneath because we could infer it if we supposed to be true both the NCon and a premise, or both that conjunction and a further premise, and so on (we can number subjuncts if we have more than one). Thus we have:

P1 hib (p . –q)
P2 hib (q . –r)
Con hib (p . –r)
NCon –hib (p . –r)
C P1
S –hib (p. q . –r)
I P2
Argument valid.

Now suppose a truth-functionalist argues, with reference perhaps to Ackermann’s example, that ‘If p then q or r’ is incompatible with ‘–(If p then q) . –(If p then r)’. A truth-tree will produce the answer that these are, if the hook is employed in their representation, incompatible. But what I call the vertical CI test in HI logic (the logic of hypothetical inferability) goes as follows:

P hib (p . –q. –r)
Con –[–hib (p . –q) . –hib (p . –r)]
NCon –hib (p . –q) . –hib (p . –r)
{S –hib (p . –q . r) . –hib (p . –r . q)}
Argument invalid.

I have put the subjunct in curly brackets, because once a sole or sole-remaining premise can be seen to be compatible with the NCon, it is optional whether one then adds a subjunct. However, the subjunct matches the conversational explanation ‘There’s nothing to stop Jo from passing the course and not writing a passing paper, provided she passes the exam; nor from (etc.)’, which would appropriately follow the NCon-giving remark ‘Let’s see if it is compatible with our premise that there’s no bar (as far as the rules go) to the possibility that Jo passes the course without writing a passing paper, nor to the possibility that (etc.)’.


The logic of hypothetical inferability will be more fruitful than a merely truth-functional one for students, whether they specialize in logic or not, both because it can obviate the paradoxes to which truth-functional representation of conditional statements has been found to give rise and because, as we have just seen, it will help a student to show in ordinary writing or conversation, beginning from the negation of the conclusion, that a deductive argument is valid or is invalid. What I call the conversational form of CI has been illustrated at the end of the previous section. I invite the reader to try ‘hib’-representation and the CI method, vertical and perhaps conversational, on numerous argument-forms, including those given by Stevenson and by Hunter (PASSV 1983) where the truth-functional representation brings out an invalid argument as valid. ‘Hib’-representation will suit not only arguments in propositional logic which contain open, remote or counterfactual conditionals (provided that continuity of context is preserved), but also those in predicate logic. Jo’s case could obviously be generalized thus: ‘(x) hib (Cx . –Px . –Ex)’.

If there is a system available which avoids paradoxes by its faithfulness to ordinary language and offers the prospect of a logic which can accurately test for validity and could fruitfully interpenetrate one’s reasoning in any field, then it is reasonable to expect that this system will eventually prevail. Then logic may once again be clasped like a bride, as students learn, whatever their other disciplines may be, to understand and appreciate the conditional which is crucial for reasoning in any field, to test deductive arguments (reliably) by the CI method in both the conversational and the vertical form, and to argue hypothetico-deductively.

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