92 The Debate over the Consequence Argument
By appending the “□” to the original proposition p, we generate a distinct proposi-
tion, □p, asserting, not just that p obtains, but that it is necessary that p obtains.
There are various modalities, though the most commonly recognized and man-
ageable ones are logical necessity, represented by the box as indicated above, and
logical possibility, represented by a diamond (◊). Other modalities include such
notions as belief, knowledge, justification, obligation, and permissibility.
Modal propositions and the logical relations they can enter into have logical
properties that distinguish them from their non- modal counterparts. For instance,
their truth is not simply a function of the truth of the constituent propositions
embedded in them, not in the way that complex propositions in what is known as
first- order truth- functional logic are simply a function of their constituents. Con-
sider, for instance, conjunction in a non- modal context. If p is true and q is true,
then the conjunction p&q is true, given the semantic rules for assigning values to
conjunctions (a conjunction is true just in case all of its ingredient conjuncts are
true). But now consider a modal context. It is false that M: Mitt Romney is cur-
rently President of the United States (in October 2013). But it is true that ◊M: It
is possible that Mitt Romney is currently President of the United States (in
October 2013). Had he won the last presidential election, he would be. Note, fur-
thermore, that had he in fact won the election so that the non- modal claim M
was in fact true, it would remain true that ◊M, since everything that is actually
true is possibly true. Thus, the truth of this modal proposition, ◊M, is not simply
a direct function of the truth of the non- modal ingredient, M, embedded in it.
To offer just one more example of how modal contexts differ from non- modal
ones, note that the relations between modal propositions give rise to questions
that have no place in non- modal contexts. Consider, for example, the modal
positions ◊M: It is possible that Mitt Romney is President of the United States
(in October 2013), which is true, and ◊B: It is possible that Barack Obama is
President of the United States (in October 2013), which is also true. So we have
a true conjunction: ◊M&◊B. But can we validly “combine” the two distinct
claims of possibility into a single possibility claim by way of “agglomeration”
and thereby say ◊(M&B)? We cannot. Why? We risk transition from a true to a
false claim when, from ◊M&◊B we infer ◊(M&B). The latter tells us that it is
possible that the following conjunctive claim is true: that Mitt Romney is Pres-
ident of the United States (in October 2013) and that Barack Obama is President
of the United States (in October 2013). But, it is not possible that both of these
propositions obtain at the same time (at least not under the assumption that the
US Constitution has not been altered).
A further helpful point is this: Some modalities need indexing in some manner,
for example, to times or persons, or both. For instance, treat the capital letter “J” as
representing the modality “is justified in believing that.” Let us apply this to some
simple proposition. For instance, suppose that Jimmy Olson discovered at the
stroke of midnight on New Year’s Eve at the turn of the last millennium for incon-
trovertible reasons that Superman is Clark Kent. So consider the proposition:
r = Superman is Clark Kent.