Why apodosis and protasis should be preferred to 'consequent' and 'antecdent'

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I recall a lovely symposium by my PhD advisor, E. A. Rabossi. He was, usually, and fastidiously, making his philosophical point again a linguist of the Labovian school, Beatrice Lavandiera. Rabossi would say, "Unlike the linguists' preference, I will avoid 'protasis' and 'apodosis' and use, instead, 'antecedent' and 'consequent'.

But of course when it comes to the horseshoe

p ) q

it is best to AVOID such 'implicaturalish' talk!

Note that unlike /\ and \/, which are the two devices that Grice mentions just before the horseshoe, the horseshoe involves a sub-ordinate construction. We have

the paratactic:

p /\ q -- She is a blonde and speaks French.
p \/ q -- She is a blonde OR speaks (something like) French.

But:

p ) q -- She speaks French.
---------- "She speaks French?" Simpliciter?
-------------- No. Not simpliciter. "IF". "IF she can articulate
her thoughts in this or that a way, we can deem her
to be able to speak French.

"if p, q"

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To use J's example -- and Grice's --

"If he has measles, he's bound to have those spots (on his skin).

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Or

"if there is smoke, there is fire"

CONSEQUENT:

fire.

Actually, the consequent is the smoke.

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To use a similar example:

"If there is smoke, there is smoked salmon"


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Grice on 'consequence':

His point: 'consequentia' is a VERY Abstract notion. Not necessarily entrenched with 'causation'.

A figure having three sides can HARDLY be called 'the cause' of it being a triangle. (And there are other examples from the apriori analytic fields of logic and mathematics -- which would include pure geometry).

Grice:

It's a bit like 'meaning' -- quite a bit of an abstract notion, whatever its etymology (Etymologically, 'mean' is NOT an abstract notion -- it relates to mind and opinion).

Grice:

"the root-idea in the notion of meaning, which in one
form or adaptation or another would apply to [every case] is
that if x means that y, then this is equivalent to, or at
least contains as a part of what it means, the claim that
y is a CONSEQUENCE of x. That is, what [all] cases of
meaning have in common is that, on SOME interpretation
of the notion of consequence, y's being the case is
a consequence of x." (WoW, 'Meaning Revisited', p. 292)

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Next lesson: how cause gets into the picture.