Blanshard on Implication and Necessity #5

This post is less about what Blanshard says in his book than the earlier ones and my last in this series.

Blanshard's critique is excellent. Still, I wonder why he did not say something like: Why are two of the rows in the truth table, when p is F in post #1, even relevant? To wit, what logical truth is implied by '7 < 4' or 'the moon is made of cheese'?

Subsequent to the problems with "material implication" logic being noticed and acknowledged, relevance logic arose. It was proposed before Blanshard’s book was first published in 1939. However, it didn’t become prominent until the 1970s.

Blanshard on Implication and Necessity #4

Blanshard next summarizes his critique. But what does “necessity” mean?

“It means, replies the empiricist, only that certain parts have been presented together with such unfailing regularity that we have become unable to dissociate them. … Formalism we have found more plausible. It admits the element of necessity that empiricism denies; its peculiarity is that it confines the necessity within certain highly general forms. … As for symbolic logic, we found it [ ] less helpful than the older logic, primarily because with its decision to ignore intension, it had abandoned interest in necessity. Of its three principal ways of conceiving implication, material, formal, and strict, we recognized an advance over the others, but could find in none of them a definition that would cover, even approximately, the necessity actually used in inference and understanding. We are left with this conclusions; necessity is not a habit, induced in us by an inexplicable regularity of presentation. Necessity is not a form or skeleton which, while sustaining the fleshy matter of the world, is sharply distinct from it” (397-8).

Blanshard’s use of “empiricist” clearly includes Hume, Mill, and other later philosophers. He references Hume and Mill. He doesn’t reference Locke, and I believe it would be unfair to include John Locke, the leading empiricist, among those whom Blanshard critiques. Locke wrote about habit and ideas by association (in ECHU), but he is not mentioned in The Nature of Thought, Volume 2, nor did he reduce all inference to habit like Hume did.

Blanshard on Implication and Necessity #3

This post is about Mr. Blanshard’s critique of strict implication as posited by C. I. Lewis.

Professor Lewis “would agree that when p implies q materially or formally this gives no assurance that q is deducible from p, nor does it give us what we usually mean by implication. He believes that his own system of ‘strict implication’ gives us both. This relation he defines as follows (the symbol stands for ‘strictly implies’ and  for ‘possible’ or ‘self-consistent’): p  q. = .   (p  q); that is, ‘ “p strictly implies q” is to mean “It is false that it is possible that p should be true and q false” or “The statement ‘p is true and q false’ is not self-consistent.” When q is deducible from p, to say “p is true and q is false” is to assert, implicitly, a contradiction” (385).

Note: is not the symbol in Blanshard’s book, but I couldn’t find how to use his and is often used to mean imply.

“There is no doubt that this sense of ‘implies’ is far nearer to the ordinary meaning than the previous senses. It no longer asks us to say anything so alien to common usage as that every true proposition implies every other, or that every false proposition implies all conceivable propositions; it is far more critical and selective” (385)

“Take an instance. ‘If anything is red, then it is extended.’ This, I think, is a fair example of implication in an ordinary sense. Now when we say that anything’s being red implies that it is extended, it is our meaning this, that if we denied that it was extended we whould also have to deny that it was red? I do not think so. I agree, of course, that when p implies q, to deny q does commit us to denying p also; I agree that in such a case to affirm p and deny q would be inconsistent. … When I say that p implies q, I am saying that a certain relation holds between them. The inability to insert not-q consistently instead of q is not the same as that relation, but something that holds in virtue of it” (386-7).

Blanshard on Implication and Necessity #2

This post is about Mr. Blanshard’s critique of formal implication.

“We turn, then, to formal implication. Our hopes rise as we do, for Mr. [Bertrand] Russell describes it as ‘a much more familiar notion’ which as a rule is really in mind even when material implication is mentioned. But these hopes sink again as we learn what formal implication means. ‘Formal implication is a class of material implications; it asserts that in every case of a certain set of cases material implication holds. … In the statement ‘Socrates is a man implies Socrates is mortal’ we have the expression of a material implication. In the statement ‘If anything is a man then it is mortal’ we have the expression of a formal implication …”

“Now as an account of necessity do we find here any advance? Certainly not so far as concerns the items summarized. Each statement of  a implying  a,  b implying  b, etc., is merely a factual statement that  a and  a (the truth of  a and the falsity of  a), etc. do not occur together. And we have seen there is no necessity there. Does it appear then in review by which we take in all the items as a glance? No again. If the connection of p with q in some one case falls short of being necessary, that same connection does not become so merely holding in all cases.”

“We begin to see, then, in what such logic involves us. It cuts us off altogether from the knowledge of universal truth” (381-2).

“The ignoring of necessity on the part of what is offered as logic, where if anywhere one would expect to find necessary connection, is a legitimate ground of dissatisfaction with the newer logistic disciplines … Stripped of its symbolism and regarded in bare logical essentials, this is the well worn atomism of Hume and Mill. There are no necessary propositions, only statements of class inclusion. There are no necessary inferences; what look like these are statement of exceptionless conjunction” (383).

“When he [the formalist] says ‘triangles have internal angles equal to 180 degrees’, does he mean ‘no triangles do in fact lack this characteristic'? If this is all he means, he has no reason to be surprised if he finds a triangle tomorrow with half or twice that number; there never was any must in the case; the new fact is merely one to be noted, and added to his collection. … Indeed extensional logic has here reversed the true order of priority; it is only because we have a prior insight into the nature of the triangle and what this nature involves that we can be so sure about particular cases. When we say a implies b, we surely mean that a in virtue of being a rather than c or d, implies b; the implication is bound up with intension. And we are clear that in the intension or content upon which thought is directed, we find connections far more intimate than the de facto togetherness to which material and formal implication are both restricted.” (384-5).

Blanshard on Implication and Necessity #1

In The Nature of Thought, Volume 2, chapter XXIX, Brand Blanshard critiques modern logic. He addresses three versions of ‘p implies q’ – material, formal, and strict. This post will be about the first of these. If the reader gets the notion that Blanshard made a straw man, see Wikipedia’s article strict conditional.

Take two propositions at random, p and q. There are 4 possible combinations of their being true or false. What that means for ‘p implies q’ (p → q) in a modern logic truth table is as follows.

          p    q    p → q

          T    T       T

          F    T       T

          F    F       T

          T    F       F

“Now in the usage of the formal logicians one proposition is said to imply the other materially when any one of the first three possibilities holds” (375). That is, when p → q is T. If the two propositions are both true, then they imply each other, even if one of them is ‘snow is cold’ and the other ‘grass is green’. That’s because we could assign either to p and the other to q.

“If two propositions are both false, again they imply each other, regardless of what they assert; ‘Darwin discovered gravitation’ implies ‘Benedict Arnold wrote Jerusalem Delivered. Finally, if the first proposition is false and the second true, once more the first implies the second; ‘Darwin discovered gravitation’ implies that Roosevelt was re-elected in 1936. It will be noted that our false proposition about Darwin has been used twice and that it implies not only every false proposition that can be made, but all true propositions as well; ‘a false proposition implies all propositions’. In sum: p always implies q except when p is true and q is false.”

“What are we to say of implication so defined? Does it describe or define the element of necessity we are seeking? On the contrary, necessity does not enter into it at all.”

“We shall see this more clearly if we ask whether it provides a basis for inference. For it will be admitted that in inference, if anywhere, we are generally using necessity, and that any satisfactory account of necessity must accord with the use we make of it there. Now it is plain that in the actual work of inference we constantly succeed in passing from one proposition to another without knowing the second independently. What sort of relation must hold between them to make this possible? It must be one in which the second is a consequence of the first. Mr. [Bertrand] Russell writes, as we have seen, ‘in order that one proposition may be inferred from another, it is necessary that the two should have that relation which makes the one a consequence of the other’ (375-6).

The logician “would admit that in the proposition ‘snow is cold’ there is not the slightest hint that grass is green, but would add that that is quite irrelevant to whether one implies the other” (376).

“A logic whose propositions are bound together only by this sort of implication is a logic in which, strictly speaking, nothing follows from anything else” (377). “Material implication itself is not, we have seen, a necessary relation” (379).