De dicto and de re

Context of desire

Consider the sentence "Jana wants to marry the tallest man in Fulsom County". It could be read either de dicto or de re; the meanings would be different. One interpretation is that Jana wants to marry the tallest man in Fulsom County, whomever he might be. On this interpretation, what the statement tells us is that Jana has a certain unspecific desire: what she desires is for Jana is marrying the tallest man in Fulsom County to be true. The desire is directed at that situation, regardless of how it is to be achieved. The other interpretation is that Jana wants to marry a certain man, who in fact happens to be the tallest man in Fulsom County. Her desire is for that man, and she desires herself to marry him. The first interpretation is the de dicto interpretation, because Jana's desire relates to the words "the tallest man in Fulsom County", and the second interpretation is the de re interpretation, because Jana's desire relates to the man to which those words refer.

Another way to understand the distinction is to ask what Jana would want if a nine-foot-tall immigrant moved to Fulsom county. If she continued to want to marry the same man that she had previously wished to marry (and perceived this as representing no change in her desires), then she could be taken to have meant the original statement in a de re sense. If she no longer wanted to marry that man, but instead now wished to marry the immigrant – i.e., the new tallest man in Fulsom County – and saw this as a continuation of her earlier desire, then she meant the original statement in a de dicto sense.

Context of modality

The number of discovered chemical elements is 118. Take the sentence "The number of chemical elements is necessarily greater than 100". Again, there are two interpretations as per the de dicto/de re distinction.

• The de dicto interpretation ${\displaystyle ◻\mathrm{\exists }{N}_{chem}(N_{chem}>100)}$ says that, in other accessible possible worlds, even if the inner workings of the atom could differ (so that ${\displaystyle N_{chem}}$ have different values), the number of elements still could not be 100 or fewer.
• The de re interpretation ${\displaystyle \mathrm{\exists }{N}_{chem}◻(N_{chem}>100)}$ says that, the inner workings of the atom are what they are (${\displaystyle N_{chem}=118}$ in all accessible possible worlds), and the number 118 is greater than 100.

Another example: "The President of the US in 2001 could not have been Al Gore".

• The de dicto reading ${\displaystyle ◻\mathrm{\exists }Pre{s}_{2001}(Pre{s}_{2001}\ne Al)}$ says that, in other accessible possible worlds, even if the result of the 2000 election could differ, the President of the US in 2001 still could not have been Al Gore. This claim seems false; presumably, in some other accessible possible worlds where the Supreme Court did not rule that Bush had won the election, Al Gore could have been the President of the US in 2001 in that possible world.
• The de re reading ${\displaystyle \mathrm{\exists }Pre{s}_{2001}◻(Pre{s}_{2001}\ne Al)}$ says that, the President of the US in 2001 is who he is, and that is George Bush in all accessible possible worlds, and George Bush could not have been Al Gore.

Representing de dicto and de re in modal logic

In modal logic the distinction between de dicto and de re is one of scope. In de dicto claims, any existential quantifiers are within the scope of the modal operator, whereas in de re claims the modal operator falls within the scope of the existential quantifier. For example:

Generally speaking, ${\displaystyle ◻\mathrm{\forall }xAx}$ is logically equivalent to ${\displaystyle \mathrm{\forall }x◻Ax}$, both meaning that all x in all the possible worlds are A (assuming that the range of quantification/domain of discourse is the same in all the accessible possible worlds); However, ${\displaystyle ◻\mathrm{\exists }xAx}$ means that each accessible possible world has its own x that is A, but they are not necessarily the same, whereas ${\displaystyle \mathrm{\exists }x◻Ax}$ means that there is a special x that is A in all accessible possible worlds.

Similarly, ${\displaystyle ◊\mathrm{\exists }xAx}$ is logically equivalent to ${\displaystyle \mathrm{\exists }x◊Ax}$, both meaning that in some accessible possible world, there is some x that is A; However, ${\displaystyle ◊\mathrm{\forall }xAx}$ means that in some accessible possible world, all x are A, whereas ${\displaystyle \mathrm{\forall }x◊Ax}$ means that for each x in the range of quantification/domain of discourse, there is some accessible possible world where x is A, but it can be true that no world has two x that are both A.