there is much to be said for the material conditional as a version of “if-then”, there is nothing to be said for it as a version of “implies”…
We can now say:
material conditional = ‘if-then’
material conditional≠ implies, or ‘If A, then AimpliesB’
Carnap [1934] makes this position clear by analysing English usage. He argues:
‘to imply’ = ‘to contain’ or ‘to involve’
Clearly this means that in English ‘implies’ is not unlike Kant’s claim that in a subject-predicate grammatical expression the subject-term’s concept ‘contains’ the predicate-term’s
concept. Or, more generally, we say that ‘A implied B’ because with this expression of ‘A’ we can be find, after analysis, the implied ‘B’. Thus when someone implies B with A, he
does not want to stop people concluding B. He simply does not state B. We can say that A ‘involves’ B, as Carnap does.
All this is in opposition, so Quine and Carnap think, to logical consequence:
logical consequence≠ ‘AimpliesB’
This, Quine argues, is what Russell called ‘implication’. It left
no place open for genuine deductive connections between sentences.
Although Quine rejects the linguistic notion ‘implies’, therefore ‘A implies B’, he still believes that deductive connections were still ‘between sentences’, not between abstract or
concrete objects. We must now ask if
p⊃q = p implies q
Of course we must now add:
deductive connection = logical consequence
Finally
implication relation≠ consequence relation
(Carnap, 1934)
But even if we study everyday English language, we can still clearly see a distinction between ‘implies’ and ‘was a consequence of’. We can say
i)John implied B by saying A.
But we cannot say:
ii)B is a consequence of what John said (A).
or
iii)John did not say B, but it is a consequence of what he said (i.e., A).
We usually take the word ‘consequence’ as a consequence relation between B and A. That is
iv)B is a consequence of A.
Thus consequence can be a causal word connection, as in:
v)The consequence (B) of John holding that meeting (A) is that there were riots on the streets (B).
Clearly when we say ‘John implied B by saying A’, this is not a causal connection of any kind. It is, in a Kantian way, an instance of the conceptual containment of B in A. Thus we
can say that the concept [person] is contained in the concept [philosopher]. Linguistically, we can have the following:
Black people are animals.
And we can then say that if a racist said it, it would imply that blacks are not persons. Thus:
the concept [non-person] = the concept [animal]
Even if the concept [non-person] is not identical or synonymous with [animal], we can still loosely claim that
He implied that blacks are not persons when he called them ‘animals’.
This is the logical case because, strictly speaking, not only blacks are animals, so too are whites. All human beings, and therefore all persons, are animals (if of a higher order then
many other animals).
This situation is complicated by the fact that Carnap continued to believe that
i)a material conditional = an implication
and did not believe that
ii)logical consequence = an implication
Thus we need to ask: What, exactly, is a logical consequence? For example, is
p⊃ q
a case of logical consequence, or q’s being a logical consequence of p, or is it an implicational conditional in that q is implied in p? Clearly, because of our prior look at the English
language, we will now say that it doesn’t seem right to say that ‘p implies q’ or that ‘q is implied by p’ or that ‘q is the implication of p’. Thus we can intuitively see Quine and
Carnap’s point before any logical distinction is made. A logical consequence relation is a relation of entailment, not implication. Thus in ‘p⊃ q’ we can say that ‘p entails q’ or that
‘q is an entailment of p’. So an implication is not the same as a logical consequence. But does the notation ‘p⊃ q’ represent an entailment relation? Is entailment expressed by
something stronger? The notation ‘p ↔q’ symbolises entailment. That is, ‘p entails q’; or that if ‘p iff q’. Thus:
i)p↔q = entailment relation
and
ii)iff = entailment relation
Notes and Further Reading
Carnap, R – (1934), The Logical Syntax of Language
