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

Linguistically, we can have the following:

*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)

Marcus, Ruth (1990) 'A
Backward Look at Quine's Animadversions on Modalities', in *The Logical Syntax of Language**Philosophy of Logic*, edited by Dale Jacquette.

Quine, W.V. O. (1961) 'Reply to Professor Marcus'

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