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the set of terms is itself partial. Second, it can map to something, but this 'something'
is not in the realm of actual or proper existents, and hence is outside the range of the
(classical) quantifiers. In this case the semantical interpretation function can be total,
but with a range that exceeds the scope of the quantifiers. This is in broad accord with
Meinong's [8] famous and influential distinction between existentent and subsistent
objects. Subsistence is a wide ontological category that includes both concrete and
abstract objects, where concrete objects both exist and subsist, while abstract entities
merely subsist. Meinong's idea serves as an inspiration behind a standard version of
Free logic in which the semantic structures have both an 'inner' and 'outer' domain,
and where the inner domain D i specifies the universe of existent objects over which
the quantifiers range. There are technical choices to be made concerning the relation
between D i and the outer domain D o , and it's possible to make them disjoint, or to
adopt the Meinongian picture and let D i D o . In the current exposition the latter op-
tion will be selected, and we will allow D i (although not D o ) to be empty, thereby
evading yet another philosophically dubious presupposition of classical logic, namely
that at least one object must exist, which presupposition is embodied in the logical
truth y ( y = y ). A straightforward semantics for this type of dual-domain Free logic
can be specified as a direct extension of the classical approach, where the objects not
belonging to the inner domain cannot be accessed by the quantifiers, but where such
objects can be accessed by the interpretation function, both to serve as the referents of
singular terms, and to appear in the extensions of predicate expressions.
A Free logic interpretation for the respective first-order language with identity L , is
a triple < D i , D o , f >, where D i is a (possibly empty) set of existent objects, D o is a
(non-empty) set of subsistent objects, and D i D o . f is an interpretation function
such that for every individual constant c of L , f ( c )
D o , and for every n -place predi-
cate P n of L , f ( P n ) D o n . Given an interpretation < D i , D o , f >, the valuation function
V assigns truth values to formulas Ө of L in the following manner (truth functional
combinations are evaluated as normal):
(i) if Ө is of the form P n c 1 ,…, c n , then V ( Ө ) = True iff < f ( c 1 ),…, f ( c n ) >
f ( P n ).
V ( Ө ) = False otherwise;
(ii) if Ө is of the form c 1 = c 2 , then V (Ө) = True iff f ( c 1 )= f ( c 2 ). V (Ө) = False
oth-
erwise;
(iii) if Ө is of the form E! ( c ), then V (Ө) = True iff f ( c 1 )
D i . V (Ө) = False other-
wise;
(iv) if Ө is of the form v ϕ v , then V (Ө) = True iff for every e
D i , V e a v / a ) = 1,
where a is a new individual constant, Φ v / a is the result of substituting a for
every free occurrence of v in Φ, and V e a is the valuation function on the interpre-
tation < D i , D o , f * > which is exactly like < D i , D o , f > except that f *( a ) = e .
V (Ө) = False otherwise.
In this 'positive' Free logic, predications involving nonexistent objects can be eva-
luated as true on the basis of set membership, in the typical Tarskian fashion. For
example, suppose the merely subsistent Pegasus is an element of D o but not D i , the
1-place predicate Wx stands for the property of 'being winged', f ( c 1 ) = Pegasus, and f
( W ) = {Pegasus, ... }. Then 'Pegasus is winged' is formalized as W c 1 and is evaluated
as True, while E! ( c 1 ) comes out False. Leblanc and Thomason [9] provide a
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