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).