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7-schema axiomatization of Free logic with identity which is sound and complete
with respect to this semantics. It incorporates the UI
Free
rule previously discussed, as
well as the axiom
∀
xE!x
.
1.5
Actual versus Possible
Another well developed framework for dealing with objects that can be referred to but
do not actually exist is supplied by modal logic, and the discussion of Western logic
will finish with a brief examination of possible world semantics. Although in the ac-
tual world Plato did not possess a beard, it's nonetheless
possible
that he could have
grown one, say like Aristotle's, and so there's a plausible sense in which Plato's beard
'exists' in alternative possible worlds. Similarly, there have never been flying horses in
this world, but if biological evolution had taken a somewhat different course then
there
might
have been. Indeed, the possibility of a winged horse seems no more out-
landish than the palaeontological fact of flying dinosaurs, and thus Pegasus is a possi-
ble though non-actual creature.
There are a number of options and technical choices that must be made when pro-
viding a semantics for quantified modal logic, and Kripke's [10] groundbreaking work
adopts some key choices that embody principles of Free rather than classical logic.
The most distinctive of these concerns the extensions of predicates. Each world
w
in a
modal structure has a domain
D
w
of objects that exist at that index. Let
UD
be the
union of all domains
D
w
for worlds in the structure. Then the binary interpretation
function
I
(
w
,
P
n
) can assign an object
e
to the extension of the predicate
P
n
at some
world
w
, even though
e
∉
D
w
and hence
e
does not exist at that world. The only re-
striction is that
I
(
w
,
P
n
)
⊆
UD
n
. Conversely, a predication can turn out to be
false
in a
world
w
, when evaluated with respect to an object
e
∉
D
w
, but where
e
does exist
at
another world
w
' which has access to
w
. In addition, Kripke upholds the principle that
the quantifiers have existential import and are thereby restricted at each world to the
set
D
w
. This combination of features is in harmony with the positive dual-domain
semantics for Free logic described above, where
UD
corresponds to the outer domain
D
o
, while
D
w
constitutes the inner domain
D
i
of locally existent objects over which the
quantifiers range. One of the prime advantages of this combination of choices is that it
allows both the Barcan formula,
∀
x
□
Ψ
x
→
□
∀
x
Ψ
x,
and its equally implausible con-
verse to be refuted, thereby yielding the maximum degree of articulation with respect
to scope interactions between the quantifiers and the modal operators. Neither the
Barcan formula nor its converse are derivable in Free logic, whereas both are valid in
straightforward modal extensions of classical logic (see Schweizer [11] for further
discussion).
2
The Analysis of Non-existence in Classical Indian Philosophy
In classical Indian philosophy, the riddle of non-being was a historical focal point of
controversy, particularly between rival Buddhist and Hindu schools. The remainder of
the paper will explore the polemical exchange between the Yogācāra-Sautrāntika
school of Buddhism and the orthodox Nyāya
darśana
of Hinduism. The exposition
relies primarily on Matilal [12,13], Siderits [14,15] and Tillemans [16] as sources.
