1
Supervaluational consequence and modus ponens1
Javier Castro Albano
jcalbano@fibertel.com.ar
There are several ways to define the notion of logical consequence within a supervaluationary
framework. Timothy Williamson argued that a certain account, global validity, requires the
rejection of some rules from classical logic2. In a recent paper3Achille Varzi strengthened that
result by proving three theses: (i) global validity admits four different definitions of logical
consequence, not only one; (ii) Williamson’s argument applies to only two of them; (iii) there
are also counterexamples to the classicality of the other two definitions. In this paper my
intention is to approach the issue of the non classicality of global validity from a different angle
by showing that Varzi’s four definitions present some degree of conflict with modus ponens.
Two of these definitions simply declare modus ponens invalid; with the other two the situation
is much more puzzling: it seems that while every instance of the semantical law , ╞ 
should hold, the syntactical rule , ├ cannot be accepted4.
1. Williamson5 distinguished two notions of validity that may be defined within
supervaluationist semantics: global validity (A) and local validity ():
(A)
An argument is valid iff, necessarily, if every premise is supertrue, then its conclusion is
supertrue
()
An argument is valid iff, necessarily, on all precisifications, if every premise is true,
then its conclusion is true
Supervaluationist semantics may be presented by analogy with modal semantics. Standardly, a
model for a vague language L is a structure containing a number of points, which may be
thought of as admissible precisifications for L (that is, admissible ways of making precise the
vague words in L) 6. Sentences of L are true or false at points of the structure;  is true at a
point iff  is not true at that point,  is true at a point iff  and  are true at that point, etc. A
sentence is supertrue iff it is true at all points and superfalse iff it is false at all points; it is clear
1
I must thank Achille Varzi, Eduardo Alejandro Barrio, Eleonora Cresto, Federico Pailos, Ezequiel
Zerbudis and all the members of the Grupo de Acción Filosofica (GAF) of the University of Buenos Aires
for their comments on an earlier draft of this paper.
2
Williamson [1994]
3
Varzi, Achille: [forthcoming]. This is an article soon to be published in Mind which will be mentioned a
lot in the rest of this paper.
4
To simplify notation I will use symbols as names of themselves (therefore using concatenation itself to
express the concatenation of symbols of the object language). I will also drop square brackets and other
similar devices.
5
Williamson [1994, p.147-148]
6
See Williamson [1994, p.146-153] for a presentation of the standard account.
2
that some sentences will be neither supertrue nor superfalse. Usually (but not always),
supervaluationists identify truth in a vague language with supertruth and falseness with
superfalseness. This framework allows us to introduce a new operator D for ‘it is definitely the
case that’; Williamson defines D as ‘ is supertrue’ so D behaves analogously as the necessity
operator of modal semantics7.
But the analogy between modal logic and supervaluationism breaks down when we get to
logical consequence because within this framework Williamson finds two different ways of
understanding the idea that logical consequence is (necessary) truth preservation. The idea
behind (A) is that, given the supervaluationist identification of truth and supertruth,
supervaluationists should identify logical consequence with the (necessary) preservation of
supertruth. On the other hand, () is the result of the following line of thought: if what makes a
sentence true is that it is true at all precisifications, then what makes an argument truth
preserving is that (necessarily) every precisification that makes its premises true makes its
conclusion true. The ‘necessarily’ in both definitions may be understood as ‘in every model’ as
it is usually done in model theoretical approaches to logical consequence8.
2. As Williamson points out definition () is the analogous condition for modal validity so it is
not a surprise to find that every argument valid in classical logic is valid according to (). The
situation is different with (A) because Williamson showed that there are at least four classical
rules that are not valid according to (A)9:
Conditional proof:
,├ 
├ 
Proof: let  be D
Contraposition
,├ 
,├ )
Proof: let  be D
Indirect proof
,├ ( )
├ 
Proof: let  be (D) and 
be D
Proof by cases
,├  and ,├ 
, ()├ 
Proof: let  be  and  be
(D  D)
See Varzi [forthcoming] for an alternative account of D.
Nothing substantial in the present paper depends on this particular elucidation of the necessity condition,
so those who reject the model-theoretical approach may switch to their preferred account.
9
Williamson [1994, p.151-152]. See Fine [1975] for a previous mention of the invalidity of
Contraposition and Machina [1976] for a previous mention of the invalidity of Indirect Proof.
7
8
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3. From this result Williamson concludes that global validity delivers a non classical logic. But
the situation is not so simple. Achille Varzi observed that beside (A), there are three other
definitions of global validity available10:
(B)
An argument is valid iff, necessarily, if its conclusion is superfalse, then some premise
is superfalse.
(C)
An argument is valid iff, necessarily, if every premise is supertrue, then its conclusion is
not superfalse
(D)
An argument is valid iff, necessarily, if its conclusion is not supertrue, then some
premise is superfalse.
Varzi arrives at these new definitions after observing that in classical logic the idea that logical
consequence should preserve truth may be expressed in four different, but equivalent, ways: (i)
an argument is valid iff, necessarily, if every premise is true, then its conclusion is true (ii) an
argument is valid iff, necessarily if its conclusion is false then some premise is false; (iii) an
argument is valid iff, necessarily, if every premise is true, then its conclusion is not false; and
(iv) an argument is valid iff, necessarily, if its conclusion is not true, then some premise is false.
(i)-(iv) are equivalent in classical logic because classical logic is bivalent (every statement is
either true or false). But since supervaluationist frameworks are not bivalent (there are
statements that are neither supertrue nor superfalse), the global versions of (i)-(iv) (that is,
definitions (A)-(D)) present four non equivalent ways of understanding the truth preservation
condition11. On the other side, the local versions of (i)-(iv) are equivalent, and so we may regard
() as the definition of local validity12.
The important point about this distinction is that, as Varzi showed, definitions (A)-(D) behave
differently with respect to Williamson’s four rules of inference. (A), as we saw, rejects all of
them; (C) verifies conditional proof, contraposition and indirect proof but not proof by cases;
on the other side, (B) and (D) verify all of them. So, Williamson’s argument is not enough to
prove that global validity is, in itself, non classical; it only proves that definitions (A) and (C)
generate a non classical logic. There are, however, counterexamples to the classicality of (B)
and (D): Varzi provides the following: , ├ ( ) is classically valid but it is invalid
according to (B) and (C) because ( ) is superfalse, even when  and  are neither
supertrue nor superfalse.
10
Varzi [forthcoming]. His formulation is slightly different from the one I am presenting because Varzi is
working with multiple-conclusion arguments.
11
The distinction between definitions (A) and (C) was mentioned in Keefe [2000].
12
Varzi [forthcoming] also examines two versions of a different kind of definition called collective
validity.
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4. It is not difficult to prove that definitions (B) and (D) must also declare modus ponens
invalid. Let M be a model in which  is neither supertrue nor superfalse and  is super false. In
M the conditional statement  is neither supertrue nor superfalse, since it is true at every
point which makes  false and false at every point which makes  true. Therefore, argument ,
 /  is an instance of modus ponens which has a superfalse conclusion but no superfalse
premises. According to (B) it is an invalid argument. And, since its conclusion is not supertrue,
it is also invalid according to (D).
5. The situation is far more puzzling regarding definition (A). It seems clear that any system that
is complete according to (A) must reject the syntactical rule of modus ponens13. As Varzi
observed, all global and local notions “…coincide with the classical notion when it comes to
identifying logical truths and logical falsities…”14. So the following classical logical truths must
be valid according to (A):
(1)
()
(2)
[()] [()()]
In a complete system, (1) and (2) should be derivable; and it is well known that (1) and (2)
together with the syntactical rule of modus ponens suffice for proving conditional proof. This is
Herbrand’s proof of the Deduction Theorem that we may find in almost any logic text-book15.
So, if conditional proof is not valid in a system complete according to (A) 16, the syntactical rule
of modus ponens must be invalid.
But this is a puzzling result since it seems that if  and  are both supertrue,  must be
supertrue also. This means that, semantically, modus ponens (or each instance of modus
ponens) should be valid (supertruth preserving). Should we defend the syntactical modus ponens
by claiming that any logical system based on definition (A) must be syntactically incomplete?
Completeness, it may be argued, is not a universal requirement for logical systems. Maybe, but
we must take notice that this would be a very weird kind of incompleteness, a kind of
incompleteness that surfaces at the level of the propositional calculus. We should therefore
abandon the intuitive idea that propositional logical truths like (1) and (2) can be freely
introduced at any point of a derivation. To preserve the syntactical rule of modus ponens, then,
Up to this point I have been using ‘modus ponens’ to refer ambiguously both to the semantical law ,
╞  and to the syntactical rule of inference , ├. Since this distinction may be important
from now on I will explicitly indicate which notion is being considered.
14
Varzi [forthcoming]
15
See Mendelson [1997, p.37]
16
In fact, completeness is not needed for this proof; it requires only the derivability of a few specific
tautologies. The same remark applies to the results discussed in sections 6 and 7.
13
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we must reject at least one of the following two rules: (i) ├ (); (ii) ├ [()]
[()()]. It is far from clear for me on what grounds should we reject (i) or (ii). Their
semantical versions seem to be as clearly valid (according to (A)) as the semantical modus
ponens.
6. An analogous result holds for definition (C). If Varzi is right and (C) validates conditional
proof and contraposition and it rejects proof by cases, any logical system complete with respect
to (C) must also declare the syntactical modus ponens invalid since we can derive proof by
cases from conditional proof, contraposition and a few logical truths using modus ponens.
Proof: we must prove that  is derivable from  and (  ) if , ├  and ,├ . Suppose
. From contraposition and ,├  we obtain . Since (  )() is a classical
logical truth (and therefore derivable in a system complete according to (C)), we may apply
modus ponens two times to obtain . Now, from  and ,├  we get . So, from the
assumption of  we proved ; therefore, applying conditional proof we get (). Finally,
since () and ()[()] are logical truths (both classically and
according to (C)), we apply again modus ponens two times to obtain .
7. The general point may be expressed as follows. It is well known that (1), (2) and
()[()], together with the syntactical rule of modus ponens constitute a
complete system for classical propositional logic17. That means that all classical inference rules
that depend only of the behavior of the standard propositional connectives are provable in that
system. Therefore, if all tautologies are derivable in a complete system according to (A) and
(C), and some classical propositional rules of inference are not valid according to them, the
responsible for that failure can only be the syntactical rule of modus ponens. To avoid the
failure of the syntactical rule of modus ponens (and therefore keep the correspondence with the
semantical modus ponens which is valid) we may appeal to a weird sense of propositional
incompleteness; but the problem seems to reappear at a different point: we must reject the idea
that propositional logical truths like (1) and (2) are derivable (even if there is a decidable
procedure to determine its logical truth).
The standard approach to logical consequence is the semantical approach. But if a logical
system is to be used as a tool for actual reasoning, it seems that it must also provide an adequate
set of inference rules that mirrors (to some extent) the semantical patterns of validity. This is
where proof theory enters the picture. If my argument is correct, we are far from understanding
what the proof theory for a global account of supervaluational consequence might look like.
17
See Mendelson [1997]
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References
Keefe, R. [2000]: “Supervaluationism and Validity”, Philosophical Topics 28, p. 93–105.
Machina,, K. F. [1976]: ‘Truth, Belief, and Vagueness’, Journal of PhilosophicalLogic 5, p.
47–78.
Mendelson, E. [1997]: Introduction to Mathematical Logic, London, Chapman & Hall
Varzi, A. [forthcoming]: “Supervaluationism and its logics” Mind
Williamson, T. [1994]: Vagueness, London, Routledge.
Fine, K. [1975]: ‘Vagueness, Truth and Logic’, Synthese 30, p. 265–300.