The System GHP: a hopefully plausible (highly powerful) version of Myro's System G -- "in gratitude to H. P. Grice for the general idea"

-- by J. L. Speranza
-- for the Grice Club

ELSEWHERE, I have referred to the System G_HP. It draws from pp. 125 of Grice's festschrift contribution for Quine -- and thus it is in actuality Grice's System "Q" -- but I, with Myro, I will proceed accordingly and change the relevant subscripts. While the formal presentation starts on p. 125, there are earlier commentaries. On p. 120, Grice writes he will be:

"present[ing] and discuss[ing] [...]

a first-order predicate calculus"

which he will call "System [G]" --. He credits Mates's Elementary Logic, and mentions suggestions by C. D. Parsons and G. Myro.

On p. 125 He describes the System [G] as a "natural deduction system".

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It comprises:

A. Glossary:

The glossary has four points.

1. DEFINITION OF SUBSCRIPT

The subscript device. If eta is a symbol of Q, eta_n denotes the result of attaching to eta a subscript denotating n.

2. RANGING

reading -- of constant: of phi(aj, ..., ak).
-- of variable: phi (omegaj, ..., omegak).

3. HIGHEST SUBSCRIPT

phi_[n] = "a formula, the highest subscript within which denotes n".

4. REPLACEMENT

phi(th2/th1) = "the result of replacing each occurrence of thi1 in phi by an occurrence of th2..."

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The section B Grice titles 'set of rules for [G]' and comprisis a few subsections. The first, 1, corresponds to the symbols.

1. Symbols. These includes 7 points
---1) constants of predicate:
---2) constants of individual
---3) variables of individual
---4) contants of 'devices': --- there are four listed in the system (p. 125): "~", "/\", "\/" and ")".
---5) two quantifiers, (/\x), (\/x) -- a quantifier being a quantificatication symbol -- /\, \/, followed by a variable. (Grice uses the inverted A and E). (p. 125).
---6) numerical subscripts (denoting natural numbers)
---7) propositional letters.

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The second section, 2, provides the 'FORMation rules', i.e. what constitutes a 'formula', or as they sometimes redundantly have it, a well formed formula (as if an ill-formed thing would be such!).

There are 7 items under this second section of B ("Set of rules for" G):

Section II then, Formulae

1. Any subscripted n-ary predicate constant
followed by n unsubscripted variables is a formula;
any subscripted propositional letter is a formula.

2. If phin is a formula, phi(alphan+m/ww) is a formula

3. If phn is a formula, (omegan+mphi(omega/alpha) is a formula.

4. If phn is a formula (Ewn+mphi(omega/alpha) is a formula.

5. NEGATION -- the squiggly.
If phn is a formula, ~_n+mphi is a formula.

6. THE THREE CONNECTIVES: the inverted wedge, the wedge, and the horseshoe:
If phn-m and psin-l are formulae, phi &n psi, phi Vn psi, phi ?n psi are formulae.

7. CLOSURE
phi is a formula only if it can be shown, by application of (1)-(6), that phi is a formula.


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The important next section of "B" ("Set of Rules for" G) Grice entitles 'inference-rules').

There are 16 of them:

(1) [Assumption]. Any formula may be assumed at any point.

--- Rules for the squiggly:

(2) reductio ad absurdum.
-- introduction of the squiggly.

(3) DNE double negation elimination
-- elimination of the squiggly

Rules for the connectives

CONJUNCTION (inverted wedge)

(4) introduction of '/\'

(5) elimination of '\/'

DISJUNCTION (wedge proper)

(6) introduction of '\/'

(7) elimination of '\/'

HORSESHOE

(8) CP, introduction of the horseshoe

(9) MPP modus ponendo ponens -- elimination of the horseshoe

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THE TWO QUANTIFIERS

Inverted A

(10) Inverted A, introduction

(11) Inverted A, elimination.

Inverted E

(12) Inverted E, introduction

(13) Inverted E, elimination

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TERM-SUBSCRIPT RULES:

(14) Introduction of DOMINANCE:

If (1) alpha dominates phi
---(2) phi, khi1, .... khil /- psi(aj, ... ak),
----
(3) phi,khi1, ... khil /- phi((aj+m/aj),...(ak+n/ak).

(15) elimination of dominance

(16) Isomorphism. If phi and psi are isomorphs, phi /- psi

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The next section Grice entitles 'existence' and it contains a few subsections. The first, A, read:

A. Closed formulae containing an individual constant alpha.

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It involves two subsections, i, and ii. The first subsection, i, gets divided into six sections.

i. If a dominates phi, for any interpretation Z, phi
will be true on Z only if a is non-vacuous. The six clauses provide a recursive specification:
1) If a dominates phi, phi is E-committal for a.
2) double negation (double squiggly)
3) conjunction (inverted wedge)
4) disjunction (wedge)
5) horseshoe
6) inverted A and inverted E-- quantifiers.

The second point, ii: a reading of 'a exists'

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The Section B, reads: "inverted E" quantified formulae.

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On p. 133, Grice starts the discussion of identity. He calls this system [S]', i.e. an extension. His point is to consider the second-order predicate calculus

(inverted AF)(Fa ) Fb) or (Fa )( Fb) as "a definitional substituend" for a=b.

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a) A new symbol then: "="
b) two new formation rules:
--- 1) omega1 =n omega2 is a formula
--- 2) If aj+k =jBj+l is a formula, xj+l =mbj+l is a formula.
c) two new inference rules
--- I) a weak identity law. gamma /- Awn+mwn-l
--- II) substitutivity on both strong and weak identity.
-------- aj =m Bk, phi /- phi (b/a).

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It's on p. 134 that he adds the "SEMANTICS FOR" his system.

----- This proceeds by steps. The first, he calls, A, and it's Interpretation. The B section provides for FIVE specifications:

First specification:

For phi atomic:

If Phi is atomic, phi is Corr(1) on Z iff (i) each constant of individual in phi has in Z a designatum, and (ii) the designata of the constants o findividual in phi, taken in the order in which the constants of individual which designate them occur in phi, form an ordered n-tuple which is in the E-set assgined in Z to the constant of predicate in phi.

SECOND SPECIFICATION.

For the squiggly,

-- (1) If phi = ~n phi, psi is corr(0) on Z.

For the inverted wedge

-- (2)

For the wedge proper

-- (3)

For the horseshoe

-- (4)

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THIRD SPECIFICATION

For psi(alpha) being a closed formula

FOURTH Specification

For a closed formulaeted not Corr(1) on Z.

FIFTH SPECFIFICATION (Grice, p. 138):
If phi = invertedA omega n psi, phi is corr(1) on Z iff phi(a'/w) is corr(1) on every efficiency-quota preserving i.c.-variant of Z.

SIXTH SPECIFICATION:
If phi = invertedE omega n phi, phi is corr(1) on Z iff psi(a'/w) is corr(1) on at least efficiency-quota preserving i.c.-variant of Z.

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---- Grice did refer to the appeal of first-order calculus on him at various places: notable in his reaction to the disputes between modernists and neo-traditionalists in Strand 6 of Way of Words, but also in his "Reply To Richards" which he felt like entitling, "Prejudices and predilections, which become the life and opinions of Paul Grice" by Paul Grice.

Some of the work was undertaken by G. Myro, whom Grice credits for 'countless illuminating suggestions' and 'notational' considerations, in "Vacuous Names". Myro died in 1987; Grice in 1988. Myro had occasion to present what he called the "System G" (now in honour of Grice) in the Grice festshrift ed. by R. Grandy & R. Warner, "Time and essence" and in papers published elsewhere -- He would circulate "A sketch of System G in gratitude to Grice for the general idea" and in fact it is to Grice that he dedicated his book, "Rudiments of Logic". Grice also quotes from C. Parsons, and G. Boolos, and is particularly grateful to Mates for personal discussion and for his Elementary Logic which Grice quotes at various points -- Mates, like Myro, were Grice's colleagues at Berkeley. Both, in fact, Grice knew since their Oxford days. Parsons and Boolos Grice had got familiar via his association with Harvard.

The "System" by Grice was presented in the Davidson/Hintikka, "Words and objections", originally. The emphasis on the subscript device he will later balance with the square bracket device.

In fact, the square bracket device predates the subscript device. Grice makes use of the square-brackets to indicate scope considerations in Lecture IV of the William James Lectures (1967) and notably in his 1970, "Presupposition and Conversational Implicature" (repr. in WoW). Thus, his general references, in his 'reaction' to modernism and neo-traditionalism, to 'scope' indicating devices (subscripts and square brackets) is best to regard as a nod to these alternate ways of 'supplementing' the modernist apparatus with what are notably syntactic methods of dealing with nonconventional implicatures.

The "Words and objections" paper has not been reprinted in full, but Ostertag had the good idea, or partially good idea, of reprinting the bit of it that deals with Descriptions in his collection, "Definite Descriptions". I regret the whole thing was not reprinted in full, since it is like having Grice's systematicity without the system -- a shame -- but blame it on MIT Press!