System CR and System GHP

By J. L. Speranza, of The Grice Club
for The Carnap Corner

The following is a running commentary on a 'semantics' for a formal system like System C-R (apres Rudolf Carnap, that is -- a revised System C) and System G-HP (a hopefully plausible rewrite of System G, apres H. P. Grice).

Our guiding tutor here will be B. Mates, Elementary Logic (that Grice quotes in the building of his system Q to honour Quine):

In System C and System G, "the formulae of the languages are assembled from atomic formulas using the logical connectives and the two quantifiers, (x) and (Ex)".

"To ascribe meaning to all sentences of a first-order language, the following information is needed: a domain of discourse D, usually required to be non-empty."

But vide Grice, "Vacuous Names" for a lifting of this requirement.

"An object carrying full information about the domain is known as a structure (of signature σ, or σ-structure, or L-structure), or as a "model"".

"As to how to interpret formulas of the form ∀ x φ(x) and ∃ x φ(x), the idea is to see the domain of discourse as forming the range for these quantifiers."

"The idea is that the sentence ∀ x φ(x) is true under an interpretation exactly when every substitution instance of φ(x), where x is replaced by some element of the domain, is satisfied."

"On the other hand, the formula ∃ x φ(x) is satisfied if there is at least one element d of the domain such that φ(d) is satisfied."

EXTENSIONALISM:

"Because the first-order interpretations described here are defined in set theory, they do not associate each predicate symbol with a property (or relation), but rather with the EXTENSION of that property (or relation)."

"Example of a first-order interpretation"

Domain: pirots a, b, c, karulising

Individual constants:

a: pirot called Augustus.

b: pirot called Basil.

c: pirot called Crispin.


Px: x is a pirot
Kx: x karulises.
x is perceiving/potching x'
x is perceiving/pocthing x' as another pirot/karuliser.

In the interpretation of G:

some statements are true, and some are false.

"A first-order interpretation is usually required to specify a nonempty set as the domain of discourse. However, empty relations do not cause any problem for first-order interpretations, because there is no similar notion of passing a relation symbol across a logical connective, enlarging its scope in the process. Finally, the identity relation (x = y) is often treated specially in first order logic and other predicate logics. The axioms related to equality are automatically satisfied by every normal model, and so they do not need to be explicitly included in first-order theories."

"There are reasons to restrict study of first-order logic to normal models. If non-normal models are considered, then every consistent theory has an infinite model; this affects the statements of results such as the Löwenheim–Skolem theorem, which are usually stated under the assumption that only normal models are considered."

Etc.

Refs.

Carnap, Logical Syntax of Language -- the seminal work of 1937 (German original 1934) that set many a trend in this area
Grice, in Festchrift for Quine.
Mates, Elementary Logic.

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