Carnap and Quine on Implication and Entailment

According to Quine, Bertrand Russell (as well as others) confused “if-then” with “implies”.

Quine said that

“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 A implies B”)

Rudolph Carnap makes this position clear by analysing English usage. He argues:

“to imply” = “to contain” or “to involve”

Clearly this means that in English ‘implies’ isn't that unlike Kant’s position that in an analytic subject-predicate expression the subject-term’s concept ‘contains’ the predicate-term’s concept. Or, more generally, we say that “A implied B” because in the expression of A we can find (as it were) - after analysis - the implied B. Thus when someone implies B with A, he doesn't want to stop people concluding B. He simply doesn't want to state B. Thus we can say that A ‘involves’ B, as Carnap does.

All this is in opposition, so Quine and Carnap thought, to logical consequence:

logical consequence ≠ A implies B

This, Quine argues, is what Russell called ‘implication’. It left

“no place open for genuine deductive connections between sentences”.

Although Quine rejected the linguistic notion ‘implies’ (i.e., “A implies B”), he still believed that deductive connections were still “between sentences”, not between abstract or concrete objects (i.e., propositions and suchlike).

We can now ask if

p ⊃ q = p implies q

According to Carnap and Quine, it doesn't. We can now add:

deductive connection = logical consequence

Finally

implication relation ≠ consequence relation

Even if we study everyday English language, we can still clearly see a distinction between the words ‘implies’ and “was a consequence of”. We can say

i) “John implied B by saying A.”

though we can't say:

ii) “B is a consequence of what John said [A].”

or

iii) “John didn't say B; though it's 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 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 isn't 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:

“Child-killers are animals.”

Thus if someone said the above, it would imply that child-killers aren't human beings. Thus:

the concept [non-human being] = the concept [animal]

Even if the concept [non-human being] isn't identical or even synonymous with [animal], we can still loosely claim that

“He implied that child-killers aren't human beings when he called them ‘animals’.”

This situation is complicated by the fact that Carnap continued to believe that

i) a material conditional = an implication

and didn't 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 (i.e., 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 can now say that it doesn’t seem right to say that “p implies q”, “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 one of implication. Thus in p ⊃ q we can say that “p entails q” or that “q is an entailment of p”. An implication isn't the same as a logical consequence. Though does the notation p ⊃ q represent an entailment relation? Actually, entailment is said to be expressed by something stronger.

The notation p ↔ q symbolises entailment. That is, “p entails q”; or “p iff q”. Thus:

i) p ↔ q = entailment relation

and

ii) iff (if and only if) = is part of an entailment relation

References

Carnap, R. (1934) The Logical Syntax of Language

Marcus, Ruth (1990) 'A Backward Look at Quine's Animadversions on Modalities', in Philosophy of Logic, edited by Dale Jacquette.
Quine, W.V. O. (1961) 'Reply to Professor Marcus'