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In propositional modal logic you can't express the distinction between "de dicto" modal statements and "de re" modal statements. Does this mean that all modal statements in propositional modal logic should be considered as being "de dicto"?

The difference between a "de re" and a "de dicto" statement concerns the place of the modal operator in a sentence and can be illustrated by means of the following example (which I borrow from a post by Fabio Somenzi):

◻∀x¬Cxx: It is necessarily true that nothing is the cause of itself (de dicto, the claim is necessarily true; i.e., the operator takes the whole sentence as its scope)

∀x◻¬Cxx: All things have the essential property of not being the cause of themselves (de re, all things have a necessary property; i.e., the modal property is ascribed to an object, not a sentence)

So my question is: given that in propositional modal logic you cannot express the difference between the two statements above (both "collapse" to ◻p) does this mean that every statement in modal propositional logic has to be considered as a "de dicto" statement? Or this question just doesn't make any sense?

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    $\begingroup$ What's your definition of "de dicto"? $\endgroup$ – grndl Mar 18 '17 at 22:37
  • $\begingroup$ Maybe useful The De Re/De Dicto Distinction $\endgroup$ – Mauro ALLEGRANZA Mar 19 '17 at 9:13
  • $\begingroup$ What about $\Box(p \vee q)$ vs. $(\Box p) \vee (\Box q)$? They are not equivalent, and, syntactically, one is asserting the necessity of $p \vee q$, while the other is asserting the disjunction of necessary "things." $\endgroup$ – Fabio Somenzi Mar 19 '17 at 15:07
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The de dicto/de re dichotomy is due to the Medieval Theory of Modalities in the context of categorical proposition analysis.

A categorical proposition, like "Some men are philosophers" has a quantity (universal, particular, singular) and a quality (affirmative, negative).

With the modalities, we have that the modal adverb qualifies the copula, and the structure of the sentence can be described as follows:

quantity/subject/modalized copula/predicate (for example: Some men are-necessarily philosophers).

In this case, the negation can be located in different places, either

quantity/subject/copula modalized by a negated mode/predicate (for example: Some men are-not-necessarily philosophers)

or

quantity/subject/modalized negative copula/predicate (for example: Some men are-necessarily-not philosophers).

This treatment of modalities has been called in the 13th Century: de re (in sensu diviso).

But it is possible a different treatment, called de dicto (in sensu composito):

In a de dicto modal sentence that which is asserted in a non-modal sentence is considered as the subject about which the mode is predicated. When modal sentences are understood in this way, they are always singular, their form being:

subject/copula/mode (for example: That some men are philosophers is necessary.)

From this point of view, the opposition de dicto/de re makes little sense in propositional language; at most, we can say that the treatment of modalities in propositional language is de dicto (i.e. in sensu composito) because the mode applies to the "undevided" sentence: there is no copula to apply the de re.

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