Grice on Λx and Vx

----- By J. L. S.


------- ANYONE WHO HAS SEEN THE VARIOUS mimeos of Grice's "Logic and Conversation" -- I only saw a few -- will realise that he wasn't wedded to one particular notation. When I learned logic I had to learn all the dialects, because my teacher was too lazy to provide the standard. Grice WoW:ii sticks with the Russellian (∀x) -- where "∀" stands for German "alle" and (∃x), where "∃" stands for German "existenz". But I'm using the above to mark the parallelism with the symbols that Grice does use for conjunction (Λ) and disjunction (V).

From wiki, 'existential quantification':

"Consider a formula that states that some natural number multiplied by itself is 25: "(0 x 0 = 25) v (1 x 1 = 25) v (2 x 2 = 25) v (3 x 3 = 25), and so on.". This is a disjunction. However, the "and so on" makes it impossible to integrate and to interpret as a disjunction proper. Instead, the statement is rephrased more formally as, "For some natural number n, n x n = 25".

Now from wiki, under 'universal quantification', again the parallelism, but this time with 'conjunction':

"Suppose it is given that "2 x 0 = 0 + 0" Λ "2 x 1 = 1 + 1" Λ "2 x 2 = 2 + 2", etc. This is a conjunction. However, the "etc." cannot be interpreted as a conjunction. Instead, the statement must be rephrased, "For all natural numbers n, 2·n = n + n". This is a single statement using universal quantification."

There are multiple, all fascinating problmes with these quantifiers. A tricky one has to do with the interpretation as extensional (substitutional) or not. Notably when it comes to 'all'. Linguists (like Horn) use 'thetic' for that kind of interpretation, which the nominalist in me finds not as cute as a mere substitutional account. Grice never considered, other than in "Vacuous Names" the problems with the interpretation models for quantifiers. But he proposes introduction and elimination rules for each, and consideres both existential instantiation, universal generalisation, and their counterparts.

But back to the prehistory. Frege, one reads from wiki, 'quantification" "would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas." "He did not devise an explicit notation for existential quantification, instead employing his equivalent of ~∀x~. It all went unremarked until Russell's 1903 Principles of Mathematics." It was however Giuseppe Peano's notation for (Ex) in 1897, that Grice uses -- that was adopted by the Principia Mathematica of Whitehead and Russell." (Ax) that Grice uses for 'all' was introduced by Gentzen in 1935.

The substitutional approach to the quantifiers amounts to reading the
universal and existential quantifier as (respectively) a conjunction and a disjunction of formulas in which constants replace the variables in the scope of the quantifier.

(Λx)Fx iff Fa Λ Fb Λ Fc Λ ...

where a,b,c are individual constants replacing all occurrences of x in Fx.

--- And wait till we start to get, perhaps, _English_ into the picture! Nall!