H. P. Grice follows Hofstadter and McKinsey in choosing 'satisfactory' as a generic semantic evaluation for both buletic and doxastic utterances

Grice likes Hofstadter and McKinsey.

The sorry story, as he calls it, of Deontic logic, as it faces Jørgensen's  "Jørgensen's dilemma is one that fascinates Grice.

"Jørgensen's dilemma is best seen as a trilemma," Grice says. 

The following three claims are incompatible: An inference requires that each element (the premise and the conclusion) has what Boole, Peirce, and Frege call a truth "value."But an "imperative" does not have a truth-value. 

It is alleged that there may be an inference between this or that imperative. Responses to this problem involve rejecting one of the three premises. The input-output logics reject the first premise. They provide inference mechanism on elements without presupposing that these elements have a truth value. Alternatively, one can deny the second premise. 

One way to do this is to distinguish between the imperative itself and an indicative about the it. According to this response, only the indicative about the imperative has a truth value. 

Finally, one can deny the third premise. But this is to deny that there is a logic of imperatives worth investigating. 

Grice preferred to define 'value' =df 'satisfactoriness.' 

Thus, '.p' can be 0 or 1, '!p' can be 0 or 1. 

"The form of the utterance will guide you as to how to read 'satisfactoriness,' which is my jargon for 'value' applicable both to an indicative and an imperative." 

With 'satisfactoriness,' Grice offers a variant to Hofstadter and McKinsey's 'satisfaction.' In their "On the Logic of Imperatives," a syntax is elaborated for the imperative mode, using 'satisfaction.'' 

"We understand an imperative to be *satisfied* (as 'The door is closed' may also be said to be satisfied iff the door is closed) iff what is commanded is the case. Thus the fiat “Let the door be closed!” is satisfied if the door is closed. We shall thus refer to the satisfaction of an imperative." (p. 447). 

According to Hofstadter and McKinsey, the function is a satisfaction-function. 

This or that unary operator and this or that dyadic operator become this or that satisfaction-function. As Grice puts it, "an inferential rule, which flat rationality is the capacity to apply, is not an arbitrary rule. An inferential rule picks out this or that transitions of acceptance in which transmission of the predicate "satisfactory" (buletic or doxastic) is guaranteed or (in this or that non-deductive case) to be expected." 

As Grice notes, since the sentential form will indicate what species of value is involved, he uses the generic  'satisfactory'. 

He imports into the object-language this or that phrase:

'It is buletically satisfactory that...' 

and 

'It is doxastically satisfactory that...' 

Alla Tarski disquotational routine:

'!p is buletically satisfactory' just in case '!p' is buletically satisfactory.  

'⊢p is doxastically satisfactory just in case '⊢ p' is doxastically satisfactory.'  

Grice then introduces 

'It is ACCEPTABLE that...' (with the syntactical provisions which he is using).

On the buletic side:

'It is acceptable that !p' is DOXASTICALLY satisfactory just in case:

'!p' is buletically satisfactory' 

is DOXASTICALLY satisfactory.

This should not mean that the buletic reduces to the doxastic. In fact, for Grice the buletic has dominance (in the logical way) over the doxastic (vide below his treatement of this or that mixed imperative-cum-indicative utterance)

Grice goes on to provide this or that generic or  generalized version ('semantic evaluation') of this or that  'satisfactoriness-functor.'


Using 'φ' and 'ψ' to represent sentences (in either mode

"AND" 

SATISFACTORINESS TABLE FOR BOOLEAN CONJUNCTION:

As in 

"He went to bed and took off his boots."

Or

"Go to bed and take off your boots!"


"φ and ψ" is satisfactory just in case "φ" is satisfactory and "ψ" is satisfactory.


"OR" 



as in 

"Smith's wife is in the kitchen or in the bedroom!"

"Stay in the kitchen or the bedroom!"



"φ or ψ" is satisfactory just in case one of the pair "φ and ψ" is satisfactory.

"IF" (Grice stopped using 'negation,' 'conjunction' and 'disjunction' as "pretentious," preferring "not," "and" and "or," when he realised that there is no ordinary-language counterpart for whatever "if" is doing -- "not one that ends with "-tion," anyways [sic]." ("I refuse to use "material implication"!" -- ("And "condisionalisation" sounds absurd, since my whole point is that "if" has nothing to do with 'conditions' at the level of the explicatum!") 

"φ  ⊃  ψ" is satisfactory just in case "φ" is unsatisfactory or "ψ" is satisfactory. 

There are, however, a number of points to be made. It is not fully clear to me just how strong the motivation would be for introducing such connectives, nor whether, if they are introduced, restrictions should not be imposed. The problematic examples will be, of course, the mixed ones (those in which one clause is buletic and the other doxastic). It seems natural to look for guidance from 'ordinary' language. "The beast is filthy and don't (I shan't) touch it" seems all right, but "Don't touch the beast and it is filthy" seems dubious, and "Touch the beast and it will bite you", while idiomatic, is not a conjunction, nor a genuine invitation to touch the beast. And "Either he is taking a bath or leave the bathroom door open" is perhaps intelligible, but "Leave the bathroom door open or he is taking a bath" seems considerably less so. It is perhaps worth noting that, in unmixed cases, satisfactoriness would be specifiable either as buletic satisfactoriness or as doxastic satisfactoriness. But for mixed cases no such specification would be available unless we make a special case, as Grice does in "Method"  for the buletic mode to be dominant. The real crunch comes, however, with negation (which Grice has been carefully ignoring). 'Not ⊢p' might perhaps be treated as equivalent to '⊢ not-p', but what about 'Not ! p'? What do we say in cases like, perhaps, "Let it be that I now put my hand on my head" or "Let it be that my bicycle faces north", in which (at least on occasion) it seems to be that neither '! A' nor '! ~A' is either satisfactory or unsatisfactory? What value do we assign to '~ ! A' and to '~ ! ~A'? Do we proscribe the forms altogether (for all cases)? But that would seem to be a pity, since '~ ! ~A' seems to be quite promising as a representation for 'you may (permissive) do A'; that is, I signify my refusal to prohibit your doing A. Do we disallow embedding of these forms? But that (again if we use them to represent 'may') seems too restrictive. Again, if '! A' is neither satisfactory nor unsatisfactory, do we assign a third 'value' to '! A' ('buletically neuter'), or do we say that we have a 'practical value gap'? These and other such problems would require careful consideration; but Grice cannot see that they would prove insoluble, any more than analogous problems connected with Strawson's presupposition are insoluble; in the latter case the difficulty is not so much to find a solution as to select the best solution from those which present themselves.