Carnap and Grice on ⊃

Anyone familiar with the German edition of that seminal work in symbolic logic by R. Carnap, "The Logical Syntax of Language" will be aware that Carnap used some of Whitehead/Russell's signs:

Notably

⊃

--- (which will be taken up again by Grice in the first page to his "Logic and Conversation" (2nd William James lecture). In the case of Grice, it was his very campaign against Strawson.

--- A cursory look at the index to Carnap's book will show how the man was so interested in such abtract formal questions. He has symbols for almost everything: e.g. the horseshoe above _is_ used, but also another symbol that will stand for the horseshoe along with, say "v", i.e. disjunction. In this symbology, with which Griceans are familiar from the work of Gazdar, one can write

p * q

to stand for "any" connective between "p" and "q", and derive generalisations or provide generalisations out of it.

Carnap, for the record, also uses variables not just for 'predicates' (praedikat, in German), but for n-argument predicates, i.e. predicates proper and relations.

The work by A. Smeaton in translating this magnum opus was a thing to behold!

The horseshoe will feature large later in Carnap's idea of a "meaning postulate", which fascinated a few logicians (and indeed linguists) since the day it was published.

A "meaning postulate" (Griceans will be aware of this since Lakoff/Johnson provided analogues for Grice's maxims in terms of what they dubbed, confusingly to some, 'conversational postulates'!) has the form

(x) Px --> Kx

e.g. pirots karulise elatically.

For this to hold 'analytically', added to the semantics of a particular interpretation of the sub-semantic domain of the formal system, we stipulate

Px =df Kx

but not quite, for the above holds the equivalence, in terms of 'iff', whereas the meaning postulate holds only a one-way 'horseshoe' and it's not thus 'definitional'. But there IS an element of 'definition' in the meaning postulate, though (to consider).

Suppose we symbolise the 'elatically' bit as "elaticaser" "Ex". We now can build something like a meaning-posulate, in terms of predicates, alla

bachelors are unmarried male

(x) Bx =df Ux & Mx

pirots karulise elatically

(x) Px --> (Kx & Ex)

-- We now use the strict 'horseshoe' and get

(x) (Px ⊃ (Kx & Ex))

-- Talk of strict horseshoe can be fun, seeing that the whole point of C. I. Lewis's enterprise against the horseshoe was to 'postulate' a sort of strict implication (a relevant 'if') that would do double duty for all things Philo never wanted the 'if' to do double duty for!

Etc.