One could fear that a new modality will have to be introduced every time we come upon a new excerpt. As it turns out, this is not the case. Three modalities, F, IMPR, UND, will be sufficient in our model to represent the logical meaning of replies. For convenience we will also use the modality DES (highly desirable fact) as a synonym of not UND, and also PROB instead of not IMPR. F stands for an ever false proposition, and is used to rewrite first order logic in a symmetrical way. So [p1 => p2] will be rewritten as [(p1 & not p2) => F]. [(a & b) => F] will thus mean that a and b are logically incompatible. We can sum up the semantics of this conversational logic:
p => F
|
p
is false
|
| p
=> IMPR
|
p
is highly improbable
|
| p
=> UND
|
p
is highly undesirable : the occurrence of p is sufficient to make the
speaker unhappy
|
| p
=> DES
|
p
is desirable: the occurrence of p is sufficient to make the speaker happy
|
Notice that with this semantics, formulas like p => PROB or p => T (with T = not F) are superfluous, since they impose no constraint on p.
In this representation, if p is very improbable, then (not p) is very probable (but nothing can be said when p is neither improbable nor probable). Again, if p is highly undesirable, (not p) is highly desirable. But nothing can be said if p is neutral. Notice that this is compatible with usual rules of formal logic. For instance if p is highly undesirable (p => UND), then (not p) is highly desirable in the sense that I need (not p) to be happy. In logical terms (DES => not p), which is the contrapositive of the previous implication.
We saw how this formalism can be successfully used to represent the logical meaning of logical contexts. Now we will see how some of the constraints that speakers have to place on the first intervention can be easily modeled using conversational logic.