Why Ross Should Have Read Grice (But Didn't)

From Fox's online study on imperative logic citing, mis-citing, rather (:)) Grice's scalar implicatures.

In the section on the paradox identified by A. Ross back in the day (1941), Fox writes:

"It is often observed that adopting a rule corresponding to the disjunction
introduction rule may give rise to difficulties when it comes to justifying
actions by appeal to an imperative."

i.e.

----- My wife is in the kitchen
---------- Therefore, my wife is in the kitchen or in the garden.
(example by Grice, 1961, The causal theory of perception).

---

Fox:

"Ross (1941) was perhaps the first to raise
this issue, and it has subsequently become known as Ross’ Paradox."

"We
will discuss this point briefly and indicate why we don’t think it should be
considered a major concern."

---------- exactly. It ain't. But for different reasons. We think Grice saves the day.


Fox:

"The “paradox” is founded on a couple of assumptions."


(a)

"Imperatives are involved in entailment relationships that mirror those for
propositions."

------ and they are.

(b)

"An individual is obliged to carry out the entailments of an imperative."

---- indeed. With Jones we even believe that if you say that p, and p entails q, then you are committed to q.

----


Fox:

"Or even, an individual is justified in carrying out actions that satisfy the
entailments of an imperative."


e.g.

Grin and bear it!
----- therefore, grin!

Fox:

"The argument then proceeds as follows."

"As p implies (p _ q), so p! implies
(p! _ q!) [from (a)]."

"But then, performing an action that satisfies q! may then
be justified by appeal to p! — for any q!, and any p! [from (b)]."

"From this, the
argument goes, we should conclude that not all aspects of imperatives are
amenable to logical rules."

------------- if KANT COULD BE REDUCED EVER SO EASILY AND SIMPLY!

--


Fox:

"The force of the argument is perhaps made more
strongly if we interpret the imperative p! as meaning

“Make it the case that p”.35

"An alternative conclusion is that “justification” does not work this way,
in the same way that we are not justified in saying that q is true just because
p _ q follows from an assertion that p is true."

36

"Furthermore, it could be
argued that the paraphrase of an imperative as being a statement of the form
“make it the case that . . . ” might be a misleading and inappropriate reduction
from imperatives to propositions."

"It is also worth noting that this particular aspect of Ross’s argument does
not by itself rule out the possibility of a logic of satisfaction, but rather is
against the possibility of a logic of satisfaction that is simultaneously a logic
of justification (or validity, in the terminology of Ross)."

--- who perhaps should have read Tarski!

---


Fox:

"We will discuss a
specific objection he raises against such logics below."

"In the present theory, we would replace “p is true” — or rather p(a) True
—by the satisifaction statement p!a True."

"On a free choice reading, it would
follow that (p [ q)!a, but it would not follow that q!a."

"It is perhaps worth looking into the background behind the claim that a
logic of imperatives is problematic."

33

"This is what Ross describes as inferences of “validity” (Ross, 1941).
34

"Arguably it is these different possibilities for relating imperatives to propositional that have
been the source of some confusion in the literature. The problems described by Ross (1941) are essentially the difficulties of reconciling the inferences of satisfaction and validity. (Section 3.2).
35
See Kenny (1966) for example.

36

Indeed it may be possible to devise a form of Ross’ paradox for propositions, if suitable
assumptions are made.

"Ross considers several solutions to the problem of practical inference, focusing
on the issue of satisfaction (when do we judge that an imperative has
been fulfilled) and validity (what is it that is understood to have been ordered)."

"It is perhaps in this sense that he deems

disjunction introduction

to

be

“invalid” in that it does not then conform to what is understood to have
been ordered."

---- the cheek!

---

Fox:

"Ross advances the view that
(92)

“. . . the characteristic feature of the existing practical inferences
is that they purport to bring about a combination of the results to
which the logic of satisfaction and the logic of validity may lead
respectively, namely so that the transformation rules of the
logic of satisfaction are complied with, but that relevance
with regard to the validity of the imperative is ascribed to
the transformation.”"

"Ross (1945), author’s emphasis.

"He then goes on to show that in most cases, except for negation and subsumption,
it does not appear possible to reconcile the logic of satisfaction
with the logic of validity."

"Furthermore, he argues that even in the case of
subsumption and negation, the inferences make appeal to tacit assumptions
concerning the notion of validity (for example that it is is not possible for
someone to demand that A and its negation both hold)."


"Notice however that if we do not accept Ross’s hypothesis (92) — that is
that notions of validity and satisfaction must be applied together when determining
legitimate practical inference entailment patterns — then his argument
no longer applies."

"The alternative hypothesis that we offer in its place
is that, while agreeing that there are at least two distinct notions (which we
term satisfaction and refinement (Section 3.4) that are relevant when reasoning
with imperatives and when deciding how to comply with imperatives, these
notions can be formulated without making an a priori assumption that every
individual entailment or inference rule must simultaneously encapsulate both
notions in order to characterise practical entailments."

"We can demonstrate that there is a problem with the claim advanced in
(92) that practical inferences are required to combine the results of a logic
of satisfaction with those of a logic of validity."

"Consider a formal system"

--- such as System G_HP -- a system devised by J. L. Speranza, with the help of R. B. Jones, after Myro's System G (-- a 'highly powerful' one, Speranza says -- punning on Herbert Paul Grice. A 'hopefully plausible' one, the more cautious Jones comments.

---

".... that allows us to reason with the ways in which a computer program might
implement a specification."

"It is reasonable to suppose we would wish to be
able to make the following inferences."

(93)

(a)

Given the need for a program to meet the requirement (a _ b),
this can be achieved by making it meet the requirement a.
(b)

Given a program satisfies the requirement a then it also satisfies
the requirement (a _ b).

"In the case of (93a), (a _ b) can be “refined” to a, so we have rule of the form
. . . (a _ b) . . . 􀀀! . . . a . . .

In the case of (93b) from the satisfaction of a we can infer the satisfaction of
(a _ b), so we have rule of the form
. . . a . . . 􀀀! . . . (a _ b) . . .

"There is a connection between these two notions, and it is only right for the
disjunction itself to have a consistent interpretation in both cases."

"There is
not, however, any clear case for requiring that the rules we are following
in (93a) and (93b) be identical in form and interpretation.37"

"Indeed, it is not
immediately clear how this could be achieved.38

"The fact that we can consider
inferences of the form (93a) without having to combine it directy into a single
rule with inferences of the form (93b) suggests that the assumptions of (92)
are incorrect, especially if we view programs-implementing-specifications as
involving a form of practical inference."

"If there is any doubt about this last assumption, then an alternative approach
would be to present a formal account of practical inference which
does not require individual rules to encapsulate both satisfaction and “validity”."

"Such an account is sketched in Section 3.4."

"So the arguments can be summed up as follows: if (92) is the correct
characterisation of practical inference, then, according to Ross (1945) there
can be no formalisation of such a logic. The alternative is that if there are such
formalisations, then the characterisation of (92) is incorrect. The contention
made here is that there such formal systems are already used to reason about
computer programs and their specifications, among other things, and so the
latter characterisation appears to be the most appropriate."

-----


"Let us turn to Ross’s specific arguments against logics of satisfaction,
rather than practical inferences as such."

"He argues that logics of satisfaction
such as the one proposed by Hofstadter & McKinsey (1939) gain nothing, in
that imperatives just appear to be a syntactic decoration of propositions."

"This
view is independent of the hypothesis (92)."

"This may indeed be the case with
Hofstadter & McKinsey (1939), but that is arguably merely contingent on the
precise formulation."

"If, for example, such logics have behaviours that capture
insights into the meaning of imperatives and fiats then surely something
has been gained, especially if such behaviours diverge from that of a naïve
propositional re-interpretation."

"In the case of vernacular proofs, if the fiat
“Let f be a continuous function.” is to be interpreted as giving rise to some notion
of hypothetical conditionality, along the lines of Section 2.9, then surely
something has indeed been gained."

"Perhaps the real issue behind Ross’ Paradox is a desire to see inferences
involving imperatives as being a way of determining an answer to the question
what should be done? in the face of discourse involving imperatives."

"That
is to develop a theory of practical inference or satisfactoriness relationships, in
the terminology of Kenny (1966)."