Five 'formal' properties of Grice's natural deduction system (from Grice, in Davidson/Hintikka, p. 121)

by J. L. Speranza
for the Grice Club.

After stating the two goals or objectives that Grice's natural deduction system is designed to meet, Grice lists five 'formal' properties that will bear on the system:

FIRST PROPERTY:

This is the natural inclination regarding UG and EG. i.e. UG and EG will 'hold without restriction', including with respect to 'any formula phi containing an individual constant alpha (phi(alpha))'.

Against the sixth manoevure whose demerit he does not care to explicate, Grice rejects '[any] additional premise' to be required. '[T]he steps licensed by U. I. and E. G. will NOT be subject to a marginal [i.e. 'extrasystematic' extra system G, that is] assumption or pretence', notably to the effect that a name occurring in such steps has a bearer.

----

SECOND PROPERTY:

On interpretations Z of the system (G), which assign no designatum to alpha,

for some

phi(alpha)

phi will be correlated with 1, i.e. it will be true.

----

Even: "some such phi(alpha) will be theorems" of [G]." -- i.e. conceptual truths which are true in any interpretation and valid simpliciter.

---

THIRD formal property:

ALGORITH. "[O]n [strictly] FORMAL grounds" it will be possible to deide, with respect to

any

phi(alpha)

'whether or not its truth requires that alpha should have a designatum'.

----

FOURTH formal property:

"Pegasus exists"

should be represented in sentences in the system.

---- (Grice proposes to do this via his weak notion of identity -- with a proviso in terms of a second-order system).

--

FIFTH property

a=a

alpha=alpha

should also be represented in at least an extension of the system. This is done, with a vengeance, by Myro (in Grandy/Warner, "Time and existence") where not only a=a gets a full representation, but one which is chronologically variable, too.