By a ‘singular term without existential
import’ I mean a ‘singular term that refers to no existent object’. Examples
are
the greatest natural number
and
Vulcan (the putative planet, not the god).
Some
people treat at least a subclass of such terms as referring but to non-existent
objects. For example, Alexius Meinong, Terence Parsons, and Dana Scott, are
three prominent members of this group. Others, however, eschew reference to
nonexistent objects entirely. For example, Bertrand Russell, W. V. Quine, and
Bas van Fraassen are three well-known exponents in this group. By an
‘elementary first order language’ I mean ‘a first order predicate logic with
singular terms with or without identity’.
Consider what Pierce would have called three leading principles in the formal
semantics of the elementary first order languages of many if not most
philosophical logicians.
(1) The value of a complex
expression is a function of the values of its parts;
(2) Co-valent
sentences, co-extensive predicates, and co-referential singular terms
substitute for each other in sentences salva
veritate;
and
(3) Every sentence is true or false.
(1)
is the principle of comprehensionality,
(2) is the principle of extensionality
(in the sense of salva veritate
substitution), and (3) is the principle of bivalence.
Frege
believed singular terms without existential import violate the principle of comprehensionality. What is not so often
recognized--apparently not even by Frege himself--is that, in the presence of bivalence, such singular terms may also
compromise the principle of extensionality.
Perhaps (1) can be formulated so as to imply (2), and thus support the Fregeian
intuition that singular terms without existential import violate comprehensionality in the presence of bivalence. But I leave this possibility
open here.
If the elementary first order language includes
atomic sentences containing singular terms without existential import, and if
the models of that language contain only a single domain of discourse (possibly
empty), then there is a forceful argument that (2) and (3) are, indeed,
incompatible, and that a confirmed extensionalist (like Frege, for example)
must, therefore, reject bivalence. Here it is.
I introduce the following notation and conventions
with their respective informal readings.
Let E1…En
be expressions, Ok (k ³ 0)
an operator (for example; negation, definite description, existential
quantification, and so on) that takes expressions into expressions.
Let VAL(EI,xI) mean the value of EI is xI.
I assume that the language contains irreferential
singular terms (hence singular terms without existential import). This is the
case exactly when
~$xVAL(E,x), where E is a singular term.
Also, it is assumed that E1…En are expressions in any of the three
categories of logical grammar--sentences, predicates or singular terms--and
that the values of the variables x1…xn
vary with the category of logical grammar to which Ei belongs—respectively, truth-values, sets, and
individuals. Finally, I make the conventional assumption that VAL is single
valued, that is, I assume
SV "x,y[(VAL(E,x)
& VAL(E,y)) É x = y].
SV rules out, for example,
truth-value gluts, sentences with more than one truth-value.
Suppose
E,F are expressions in the same
category of logical grammar, and Q(E)
and Q(F) be expressions differing at
most for occurrences of E,F.
Extensionality (as it applies to linguistic
entities) may then be expressed as follows:
EXT "x[VAL(E,x) º VAL(F,x)] É "y[VAL(Q(E),y) º VAL(Q(F),y)].
This
formulation of extensionality is much more general than Quine’s well known
rendition (which is essentially (2) above). It yields his salva veritate substititution standard when E,F are taken to be sentences, that is, as the sort of expression
in which VAL(E,x) (and VAL(F,x)) are such that x = T(ruth) or x = F(alsehood).
Now let t
be a singular term without existential import (= irreferential, in virtue of
the assumption about the models for the current first order language), say,
‘Vulcan’, and let Q(Pred) be Pred(t). Assume E is the predicate (open sentence) ‘x =x’, F is ‘x =
x & E!(x)’ and F* is ‘E!(x)
É x = x’. These three predicates (open sentences) are coextensive.[2]
So
(4) Q(E)
is t
= t,
(5) Q(F)
is t = t & E!(t),
and
(6) Q(F*)
is E!(t) É t = t.
Thus
if Q(E) is true, Q(F) is false. If, on the other hand, Q(E) is false (as in negative free
logics) then Q(F’) is true. Since E,F and F* are co-extensive, it follows from EXT that "y[VAL(Q(E),y) º VAL(Q(F),y) º VAL(Q(F*),y]. But, in view of the
immediately preceding, this is impossible if VAL(Q(E),y) is single valued. So, on the face of it, some singular
terms without existential import compromise extensionality given bivalence at
least for atomic sentences.
The argument thus seems to
substantiate Frege’s nonscientific inclination to abandon bivalence. However, rejection of bivalence is no unconditional
panacea.
Suppose Q(E)
is truth-valueless. Then Q(F) will
only be truth-valueless given the weak Kleene reading of & (roughly, if
part of a conjunction is truth-valueless, so is the conjunction). If & gets the strong reading (if part
of the conjunction is false, so is the conjunction), then the argument shows
that even in languages with truth-value gaps, that is, where bivalence is rejected, extensionality
will still fail.
Those who cleave to the Fregeian rejection of bivalence yet favor the strong Kleene
readings of the connectives as more intuitive might suggest that strictly
speaking there is no violation of EXT
in the case where t = t is assumed
to be truth-valueless because there is no alteration of truth-value. What is revealed rather is the necessity of
distinguishing between stronger and weaker readings of violations of EXT. If, for simplicity, one restricts
oneself just to sentences, something constitutes a violation of EXT strongly speaking only if there is
an actual change in truth-value and not merely a change in truth-value status. But this attempt to justify the
rejection of bivalence while maintaining extensionality on definitional grounds
is simply a shell game in the absence of any independent motivation.
An
extensionalist, not as sensitive as Frege to the idea of nonexistent objects,
is sure to point out that the master argument in the previous section rests on
the arguable assumption that a singular term’s having no existential import
amounts to its being irreferential. But this is an artifact of the single
domain models of the first order language under consideration. When this
assumption is dropped, the co-extensiveness of the pivotal predicates evaporates.
For instance, in the inner domain-outer domains semantics favored by many free
logicians, the expression ‘Vulcan’ refers, but to something in the outer domain
of non-existents, and the predicates (open sentences) ‘x = x’ and ‘x = x & E!x’
are no longer coextensive, since the former is true of everything (including
all the entities in the outer domain), but the latter isn’t.
The inner domain-outer domain strategy,
nevertheless, is subject to the following serious objection: it is always
possible to find a predicate P!--perhaps
simply the predicate ‘___ is an object’--
whose extension is the union of the inner and outer domains no matter how
widely the outer domain is conceived and is such that the sentence
P!ix(Qx &
~Qx)
(or
some other candidate definite description) is false. (A favorite candidate of
the later Meinoing was “the thought about a thought not about itself”.) But
then the incompatibility argument above can be resurrected putting P! in place of E!. One could be Frege-like about this, at least in his scientific
mood, and assign some value in the outer domain to
ix(Qx &
~Qx),
thus
making
P!ix(Qx &
~Qx)
true.
But then one could have adopted this strategy at the very beginning even in
single domain models (as Frege, in fact, did for the scientific language), a
strategy that amounts to abandoning one of the key conditions of the argument,
namely, that there are languages containing singular terms without existential
import.[3]
Another assumption involved in the
original argument is that predicates are indistinguishable from open sentences.
But this assumption has been attacked, on various grounds, both by some
philosophical logicians (Leonard, Quine, Scales, for example) and more recently
by some mechanistic mathematicians (Farmer, for example).[4]
For one thing, predicates can have complements, but open sentences do not. For
another open sentences get truth-values under an assignment to the variables,
but predicates do not. The upshot is that it could conceivably be the case that
a predication—say
(7)
So and so is an x such that it is not
the case that …x… ,
containing
the complement of the predicate ‘an x
such that …x…’, will not have the
same truth value as the negation of a predication—say
(8) It is not
the case that so an so is an x such that …x… .
Nevertheless,
in classical first order languages when variables in an open sentence are
replaced by constant singular terms, they get the same truth-values as certain
regimented complexes of general terms (predicates) concatenated with singular
terms (that is, predications). So the distinction just noted becomes
semantically idle.
To illustrate, consider first order languages with a
rule for substitution into predicate placeholders (as in the fourth edition of
Quine’s Methods of Logic[5]).
Predicates qua general terms are formed from open sentences by prefixing them
with predicate forming operators such as
{x:…} (read: is an x such that A).
A
law of predicate abstraction, namely,
PA {x:A},s º A(s/x), where s is a
singular term or a variable other than x
governs
predications. But then any significant semantical difference between
complementation and negation disappears because, for example, the predication
(9)
Chirac is such that he is a
non-unilateralist
and
the predication
(10) It is not the
case that Chirac is such that he is a
unilateralist
are
logically equivalent. Indeed, given the addition of predicate abstracts to the
language, a more perspicuous rendition of the counterexample would restrict Pred to expressions of the form {x:A}, and Pred(t) to expressions of the form {x:A},t. Then Q(E), Q(F) and Q(F*) would be respectively
(4*) {x:E!},t,
(5*) {x:E!(x)
& x =x},t,
and
(6*) {x:E!(x)
É x = x},t.
Application
of PA to these formulas would yield
the sentences sans abstracts (4)-(6), and hence, in the presence of bivalence,
the counterexample to EXT.
There are philosophical logicians whose
view of predication requires that a true predication imply that the singular
term purporting to specify the subject of the predication actually have
existential import. An example is Ronald Scales.[6]
Others are Tyler Burge, William Farmer and Solomon Feferman.
In Scales’ negative free logic, the sentences
(11) Vulcan is an x such that x does not exist (is a nonexistent)
and
(12) It is not the case that Vulcan is an x such that x exists (is an existent),
do
not have the same truth-value, the former being false and the latter true. This
requires abandonment of PA in favor
of
PA* {x:A},s º [E!s (or $y(y = s)) & A(s/x)], where t is a constant and not a variable.
(In
Scales treatment, variables are treated semantically as having existential
import; so PA, as well as its
universal closure, still hold where s
is a variable.) PA* implies that
only when the free variable in an open sentence is replaced by a singular term
with existential import does the result have the same truth-value as the
corresponding predication.
The effect on the above argument is dramatic. E becomes {x:x = x}, F is {x:E!x & x = x}, and F* is {x:E!x É x = x}. Moreover they are co-extensive
(because "y({x:A),y º A(x/y)) is logically true). Now Q(E), Q(F) and Q(F) are,
respectively, {x:x = x},t, {x:E!x & x = x},t and {x:E! É x = x},t. But these are all (and
always) false if t is irreferential
in virtue of PA*. The upshot is that
if predicates and open sentences are distinguished, and PA gives way to PA*,
then a first order language with irreferential singular terms that is
extensional (qua preservation of truth-value, extension or reference) need not
be non-bivalent.
One can expect this particular resolution of the
problem to be appealing to those extensionalistically inclined mechanistic
mathematicians whose logical foundation for partial functions is negative free
logic, a species of free logic in which all atomic sentences containing at
least one singular term without existential import are counted false.[7]
Another more modest proposal replaces PA* by
PA** $y(y = s) É [{x:A},s º A(s/x)] where s is a
singular term or a variable.
This
proposal leaves open the question how to regard the truth-value of many
predications containing singular terms without existential import. But it is
clear that EXT is compatible with
bivalence in the presence of PA**.
Of course, this proposal for restoring the compatibility of EXT and bivalence in languages
containing singular terms without existential import succeeds, if you can call
it success, by not letting the problem arise. Except, perhaps, for Quine, most
will regard this way of resolving the original problem as no solution at all…as
your friendly psychiatrist would be the first to point out.
I shall end with a quasi-historical note. The
recommendation for restoring the compatibility of EXT and bivalence inherent in the Scales strategy is latent in
Russell’s theory of definite descriptions. It was first exploited by Arthur
Smullyan to resolve certain problems in quantified modal logic and then later
by Thomason and Stalnaker in the same area of interest. Scales method is,
indeed, reminiscent of the de-dicto de-re distinction in modal contexts. It is Scales’
achievement to see that such a de-dicto de-re-like distinction applies to
predications in languages eschewing any operators beyond those of elementary
logic and the sole predicate of identity.
[1] The essential details of the argument are already present in Karel Lambert, ‘Predication and extensionality’, Journal of Philosophical Logic, 3 (1974) 255-264. Here the argument is given a more precise formulation, the use to which it is put is different, and an analysis of its cogency from the point of view of the extensionalistically inclined logician is provided.
[2] Conventionally when "x(VAL(E,x) º VAL(F,x)), then E and F are co-extensive if E,F are predicates, co-referential if E,F are singular terms, and co-valent if E,F are sentences.
[3] This assumes that a singular term has ‘t’ has existential import if and only if it is true that ‘$x(x = t)’, a condition that holds for all singular terms in classical first order languages.
[4] Henry S. Leonard, ‘Essences, attributes and predicates’, Proceedings and Addresses of the American Philosophical Association, University of Delaware Press, Newark, Delaware (1964), pp.33ff; Ronald Scales, Attribution and Existence, University of Michigan Microfilms, (1969); and Willard V. Quine, Methods of Logic (Fourth Edition), and Harvard University Press, Cambridge, Massachusetts (1982), pp. 132-35. See also William Farmer, ‘Reasoning about partial functions with the aid of a computer’, Erkenntnis, 43 (1995), pp. 279-294.
[5] Op. Cit., Methods of Logic, (1982)p.132-35.
[6] Op. Cit., Attribution and Existence, (1969).
[7] Cf. The reference to William Farmer in footnote 4. But see also Solomon Feferman, ‘Definedness’, Erkenntnis, 43, (1995), pp. 295-320.