Making Too Much of Possible Worlds
John Woods
1. Plurality and hegemony
A possible worlds treatment of the normal alethic modalities was, after classical
model theory, logic’s most significant semantic achievement in the century just past.1
Kripke’s ground-breaking paper appeared in 1959 and, in the scant few succeeding years,
its principal analytical tool, possible worlds, was adapted to serve a range of quite
different-seeming purposes – from non-normal logics,2 to epistemic and doxastic logics3,
deontic4 and temporal logics5 and, not much later, the logic of counterfactual
conditionals.6 In short order, possible worlds acquired a twofold reputation which has
steadily enlarged to the present day. They were celebrated for both their mathematical
power and their sheer versatility. This sets the stage for what I want to do here. I wish to
explore the extent to which the supposed versatility of a possible worlds semantics is
justified. In so doing, I shall confine my attention to its role in (1) logics of counterfactual
conditionals, and (2) logics of belief. The question I pose is, why and on what grounds
should we think that the device of possible worlds turns the semantic trick for these
logics?
My answer is that they do not turn the trick for them. Whereupon a further
question presses for attention. If possible worlds semantics don’t work there, why does
virtually everyone think that they do? Answering this second question is risky. Who am I
to say why virtually everyone thinks that the possible worlds approach is more successful
than I do? Who has vouchsafed me these powers? I shall try to mitigate the riskiness of
my answer by contextualizing the evaluation of this approach in the following ways.
First, the triumph of possible worlds occurred in the midst of a powerful general trend in
logical theory, especially, in the past 60 years. In that period, logical theory became
aggressively and widely pluralistic. Second, the versatility – the sheer ubiquity – of
possible worlds as a tool of semantic and philosophical analysis, gives to possible worlds
a kind of hegemonic standing. It is a paradigm of semantic analysis. These two factors
Saul J. Kripke, “A completeness theorem in modal logic”. Journal of Symbolic Logic 24 (1959) 1-14.
Notably, Kripke’s semantics for S2 in “Semantical analysis of modal logic II: Non-normal modal
propositional calculi”. In J.W. Addison, L. Henkin and A. Tarski, editors, The Theory of Models,
Amsterdam: North Holland 1965.
3
The first notable development was Jaakko Hintikka’s Knowledge and Belief, Ithaca: Cornell University
Press 1962. Hintikka constructs a semantics of model systems of model sets; not possible worlds strictly
speaking, but a near thing. For this same approach is alethic settings, see Hintikka, “The modes of
modality”, Acta Philosophica Fennica, 16 (1963), 65-81.
4
See here Georg von Wright “Deontic logic”, Mind, 60 (1951) 1-15. Considered the breakthrough essay,
von Wright’s system precedes Kripke’s possible worlds approach, but was soon adjusted to admit of it.
5
Here, too, Arthur Prior sees tenses and temporal indices as modalities, but in advance of Kripke’s possible
worlds. Still, such logics were also quickly adapted to a Kripke-style semantics. However, we might note in
passing that Prior’s 1954 notion of instant-propositions actually anticipates the idea of possible worlds. See
A.N. Prior, Time and Modality, Oxford: Oxford University Press 1957.
6
Robert Stalnaker “A theory of conditionals”, American Philosophical Quarterly Monograph Series, no. 2
(1968) Oxford: Blackwell. David Lewis “Completeness and decidability of three logics of
counterfactuals”, Theoria, 37 (1971) pp. 74-85.
1
2
1
form the context against which to judge possible worlds. As we proceed, I shall be
examining a pair of related claims:
(a) Logic’s pluralism is a natural, if unintended, disguise of the limitations
of possible worlds.
(b) The hegemony of the possible worlds approach is a natural, if unintended,
spur to their over-use, to applications that exceed their legitimate reach.
The remaining part of the paper is organized as follows. Pluralism occupies us in section
2, and hegemony in section 3. Section 4 is given over to counterfactuals, and section 5 to
belief. Section 6 brings the paper to a close with an appraisal of hypotheses (a) and (b).
2. Pluralism in logic
Pluralism is all the rage in logic, one of the subject’s most distinctive features in
the last half-century or so. Entirely apart from the influence I claim for it on the over-use
of possible worlds semantics, it is well worth the attention of philosophers of logic as a
quite general problem. So we will not go far wrong to tarry with it awhile.
Some of logic’s pluralism is benign. Logicians have displayed an impressive
versatility in finding a wide range of quite different things to apply their methods to. But
much of this multiplicity is rivalrous, at least on its face. It is far from uncommon for a
given theoretical target – entailment, say, or necessity – to attract numbers of systematic
workings-up that give every appearance of contradicting one another. Entailment theory
is a notable cause célèbre, dividing logicians into two main camps, each of which
subdivides into its own subcamps, with further divisions within these. Perhaps the main
principium divisionem is ex falso quodlibet, the theorem that asserts the equivalence of
negation and absolute-inconsistency.7 Logicians who favour it are classicists (that is,
classicists about inconsistency). Everyone else is a paraconsistentist; and there is, in turn,
a hefty plenitude of paraconsistent logics ranging from relevant logic to dialetheism.
All this conflicted abundance raises another question. Given logic’s historical
pretensions to objectivity, are there principled ways of adjudicating these rivalries – of
picking the winners and the losers – while retaining logic’s realist presumptions? Various
remedies have been proposed, some more satisfying than others. On some approaches,
the incompatibility of apparently rival accounts is denied, and with it the need for
adjudication. One of these is the ambiguation strategy, according to which seemingly
incompatible theories of some same target concept C aren’t in fact directed at the same C
but rather at different concepts C1, , Cn, each corresponding to a different sense of the
ambiguous term “C”. The pluralism of the (C.I.) Lewis-systems is a case in point.
According to the ambiguation thesis, S5 and S4 aren’t rival logics of necessity. They are
non-competing logics of different concepts of necessity, one “logical” and the other
“metaphysical”, or some such thing.
A formal system S is negation-inconsistent if and only if for some sentence , both  and ⌐¬ are Sderivable. S is absolutely inconsistent if and only if every sentence of S is S-derivable. Ex falso says that S
is negation-inconsistent if and only if it is absolutely inconsistent. “Falsum” here denotes a logical
falsehood.
7
2
A moment’s reflection reveals the weakness of the ambiguation strategy. If we
consider only the propositional logics of the alethic modalities extant at the end of the
1960s, we see that their numbers run to well over fifty, many of which conflict with one
another.8 It is perfectly true that “necessary” is ambiguous in English. It is perfectly false
that it is fifty-wise ambiguous. Then, too, there is the kind of case exemplified by S2. S2
is a non-normal logic in which every sentence is possibly possible, including all
contradictions. It strains credulity to suppose that there is any sense of “possible”
according to which contradictions are possibly possible.
A second way of denying the incompatibility of rival logics offers a way of
escaping these difficulties. It provides that what a system of logic says about a target
property is true in the system. If a different provision is made for that concept in a
different system, then the apparently conflicting result is true in that system, thus erasing
the incompatibility. Just as the ambiguation thesis preserves the objectivity of logic,
purporting that its rivalrous appearances are but reflections of objective truths about
different concepts, so the relativity thesis holds that the propositions of logic are
objectively true-in this system or that. The relativity thesis is itself a form of the
ambiguation theses, but with a difference. Whereas the ambiguation thesis postulates an
antecedently existing plurality of senses of target concepts, the relativity thesis holds that
different senses of a target concept are created by the different things made true of it by
the respective systems in which these truths are embedded. But here, too, there are
difficulties. One is that, on this view and contrary to what we would have supposed, there
is no fact of the matter about entailment or any other target concept. There are only facts
of the matter-in.
The second difficulty, relatedly, is that there appears to be no antecedent upper
limit on what a system of logic can make true-in-it, provided that logic in question is
“well-made”. A well-made logic has a certain kind of formal virtue. Let us say that a
logic is formally adequate to the extent that it has an effectively generable grammar, a
rigorous syntax of proof, a robust formal semantics, as well as some of the prized
metatheoretical properties such as soundness and completeness, together with reliable
procedures for demonstrating their presence or absence. With this description at hand, we
can define Cole Porterism in logic. Cole Porterism asserts that for formally adequate
systems there is no à priori limit on what can be made true-in them; hence Anything Goes
in logic.9
In the modern era, a good many logicians approach their respective targets with
two objectives in mind. One is to produce a system that is formally adequate. The other is
to provide some objectively-rooted elucidation of target concepts. Contrary to what is
implied by Cole-Porterism, logicians of the present stripe assume that there are
unrelativized objective facts about target concepts, and they take it for granted that a part
of their job is to bare those facts in suitably rigorous ways. When a system of logic
achieves this objective, we could say that it aspires to conceptual adequacy. Many
8
Hughes and Cresswell, A New Introduction to Modal Logic, London: Routledge, 1996. First published in
1968.
9
With apologies to Anything Goes, a Broadway musical with songs and lyrics by Cole Porter. Book by
Guy Bolton and P.G. Woodhouse . Revised by Howard Lindsay and Russell Crouse, 1934. And thanks to
David Stove from whom I’ve borrowed the notion of Cole Porterism. See David Stove, “The jazz age in
the philosophy of science”. In Roger Kimball, editor, Against the Idols of the Age: David Stove, pages 3-32,
New Brunswick, NJ: Transaction 1999. First published in Encounter in 1985.
3
logicians take it that their mission is to produce systems of logic that are both formally
and conceptually adequate. Let us say that an acceptable balance between formal and
conceptual adequacy is the default programme for logic. As Kripke observes of the set
theoretic mechanics of his own modal semantics, they are “also useful in making certain
concepts clear.”10
The mere fact of its sprawling pluralism places logic’s default programme under a
cloud. The recent history of logic suggests that while the two objectives of the
programme may remain in play, there are rather clear indications of a rank ordering in
which formal adequacy dominates over conceptual adequacy, often to the point of the
latter’s extinction. In this connection, it is revealing to consider the present situation in
dialetheic logic. A dialetheic logic is one in which some (few) contradictions are true.
Aside from a few pensioned-off Soviet hacks and a sprinkling of old-fashioned
Hegelians, no one believes that there is any such thing as a true contradiction.11 Certainly,
for the mainstream of logic the very idea is preposterous, and anyone seriously proposing
it has lost his intellectual purchase. By mainstream lights, even a formally adequate logic
of true contradictions would be a bust on the score of conceptual adequacy. For it
traduces the concepts of truth and contradiction alike to allow for their co-instantiation.
Still, dialetheic logicians publish their conceptual heresies in the best of the mainstream
journals and with the leading university presses. If this isn’t evidence of Cole Porterism
run amok, it is hard to imagine what would be, or could.
What explains the latitude shown such conceptual depravity? The answer is that
dialetheic logicians are clever technicians. In their formal work, they display a
commendable mathematical versatility. System LP, for example, has a recursively
specifiable grammar, a formal theory of proof, a well-worked out semantics, with respect
to which it is sound and complete. It has a straightforward three-valued semantics and
most, if not all, of its inference rules are nicely intuitive.12 It is in the judgment of one
commentator “a laboratory for the paraconsistent logicians.”13 If, as we just now
supposed, formal adequacy dominates over conceptual adequacy in modern logic, can the
case of dialetheic logic be anything but an especially aggressive form of this
domination?14
10
Naming and Necessity, Cambridge, MA: Harvard University Press 1981, n. 18, p. 19.
It is quite true that there is also a (distinguished) smattering of more recently minted Hegelians in the
intellectual descendent class of Sellars, mainly at Pittsburgh. But the Pittsburgh Hegelians aren’t
dialetheists.
12
LP is set out in Graham Priest, “The logic of paradox”, Journal of Philosophical Logic, 8 (1979), 219241.
13
Edward Mares, “Who’s afraid of impossible worlds?” Notre Dame Journal of Formal Logic, 38 (1997),
516-526; p. 517.
14
Of course, this is not the dialetheist’s own view of the significance of their work. They agree that their
logic is formally impressive, but they also insist on its conceptual or philosophical value. In In
Contradiction, 2nd edition. Oxford: Oxford University Press, 2006, Priest tells us that initially he had a
devil of a time in getting himself to believe that the Liar sentence is true, but that now he is free of his
former consistentist scruples. His The Limits of Thought, Cambridge: Cambridge University Press 1995, is
likewise an extended philosophical apologia for true contradictions.
11
4
All this lattitude is a bit strange. “Nice technical properties should not be taken as
arguments for [the] philosophical virtues of a theory.”15 The question of how to judge a
system of logic is now oddly unhinged. All will agree on the necessity that it pass muster
on the score of formal adequacy. But beyond that, what? By what appears to be the
prevailing methodological favoritism, no formally adequate system of logic can be
dismissed merely on grounds of conceptual inadequacy. Does this leave the possibility –
however slight – that a formally adequate logic might be rejected not merely but partly
on grounds of conceptual inadequacy? If so, what would the other parts of the grounds
for dismissal be?
One possibility is that there is a certain respect in which the system in question
failed to meet the objectives of its creator and its proponents. Suppose that the system’s
founder sought for a formally adequate means of providing an objectively-based
elucidation of some concept C  entailment say.16 This would be a revalation of what
entailment really is. Suppose that the ensuing formally adequate logic is formally
adequate but fails to fulfill the founder’s ambition. Suppose that its provisions for C-hood
are not conceptually adequate. Then, might we not judge the logic a failure in relation to
those intentions? Similarly, might we not also say that the success of a system of logic is
a function of the kind of thing it is wanted for? Consider, for example, Frege’s decision to
provide for the (intuitively) empty singular terms of his logic the null set as their common
denotation. In a logic designed to be the reductive home of arithmetic, perhaps this is no
bad thing. But in a logic for natural languages, it leaves a lot to be desired.
In noting its diminished role as a motivator of logical systems and as a criterion of
their success, it would be imprudent to overlook that in a quite general way conceptual
adequacy is a harder sell than the more mathematically tractable issue of formalization.
Conceptual adequacy is the stock-in-trade of philosophers, especially those of analytic
bent, for whom their subject’s main job is the provision of conceptually sound
explications of difficult and often puzzling issues. This is more easily said than done,
needless to say; and the history of philosophy is replete with pluralisms of its own
triggered by its practitioners’ inability to agree on what counts, case by case, issue by
issue, as conceptually adequate. There is in this a necessary methodological lesson. It is
that one should not impose on the technical logician harsher demands for conceptual
adequacy than the philosopher himself is able to meet. Neither should we resent logic’s
pluralism about a given theoretical target any more than we resent a like, and antecedent,
In the apt words of Reinhard Kahle, in “Against possible worlds”. In Cédric Dégrmont, Laurent Keiff
and Helge Rűckert, editors, Dialogues, Logics and Other Strange Things: Essays in Honour of Shahid
Rahman, pp. 235-247. London: College Publications, 2008; p. 241.
16
Recall the battles over ex falso between the early proponents of relevant logic and boosters of the strict
implication approach. See, for example, Woods “Relevance”, Logique et Analyse, XXV(1964) 49-53,
Donald Hockney and Kent Wilson, “In defence of a relevance condition”, Logique et Analyse,, 31 (1965),
211-220, Woods “Relevance revisited: A reply to Hockney and Wilson”, Logique et Analyse, 9 (1966)
364-371, and Woods, “How not to invalidate disjunctive syllogism”, Logique et Analyse, XXXII (1965)
312-320, see also A.R. Anderson and Nuel D. Belnap, Jr., Entailment: The Logic of Relevance and
Necessity, volume 1, Princeton: Princeton University Press 1975; p. 165, and Woods, Paradox and
Paraconsistency: Conflict Resolution in the Abstract Sciences, Cambridge: Cambridge University Press
2003; pp. 63-64. The tone of the earlier disputes is interesting, what with Anderson and Belnap claiming
the proof of ex falso to be “self-evidently preposterous”, and Lewis (and his co-author C.H. Langford),
insisting that the rules of the proof wholly conform to the common meanings of proof and inference.
(Symbolic Logic, New York: Dover, 1932).
15
5
philosophical pluralism about it. A case in point is the KK-hypothesis in epistemic logic
(According to the KK-hypothesis someone knows that Φ if and only if he knows that he
knows that Φ.) If, like Hintikka’s logic of Knowledge and Belief, one’s system is an
epistemicized version of S4, then the KK-hypothesis falls out as a matter of course; it is
the epistemic counterpart of S4’s distinguishing axiom. There is, even so, much
disagreement as to the conceptual adequacy of a logic that sanctions this principle. But
the principle is no less a matter of long-standing philosophical contention, well before the
technical innovations of modern logic.
No one doubts the attraction of yoking formal system-building to the demands of
conceptual adequacy. But, again, as the history of the formal sciences amply attest, there
is also something to be said for removing that girdle entirely. The paradigm case of the
formalization of a conceptual non-starter was Riemann’s reconstruction of plane
geometry in the absence of the parallel postulate. The last thing that Riemann or any of
his contemporaries thought was that hyperbolic geometry provided an elucidation of the
concept of space. Its virtues were entirely formal ones, notably its relative consistency
proof. But with advent of Einstein’s special theory, it was clearly advantageous to have
had Riemann’s geometry waiting on the shelf.17 Had it not been there, it would have been
necessary to invent it. The moral of this is that sometimes a conceptually-bereft formal
theory will take on conceptual significance after the fact. This is further reason not to be
over-hard on them before the fact, so to speak.
I shall not attempt a comprehensive answer to the questions that flow from logic’s
pluralism.18 It suits my present purpose to give them some due recognition here, as both
motivation of and prelude to the further business of this essay. Before turning to this
question, we must say a little something about the hegemonic or paradigmatic status of
possible worlds. Here too is another issue of more general importance than is caught by
our pair of problem logics  the logic of counterfactuals and the logic of belief..
3. Hegemony
It is a truism that successful theories are marked by their dominance. Dominance
is expressed in two ways: First, by the size of its proponent-class; with regard to a given
application; and second, by the range of its applications. Probability theory is a case in
point. Virtually everyone agrees that the standard probability calculus – fashioned in the
spirit of the Kolmogorev axioms – gives a deeply right account of the mathematics of
games of chance. Probability theory is therefore a dominant theory in the first of our two
senses. It is also the case that the calculus of probability is commonly held to be
canonical for all uses of probability admitting of theoretical articulation – probability in
the law, probability in decision theory, probability in statistical reasoning, probability in
inductive reasoning, and so on. Accordingly, the probability calculus is a dominant
theory in our second sense. Hardly anyone disputes the factual accuracy of these
dominance claims. It is a matter of descriptive fact that probability theory looms in these
17
Of course, a reverse inducement could also be considered. The existence of non-Euclidean geometry on
the shelf presented physics with a possibility to exploit that might not otherwise have obtained.
18
For efforts in this regard see my Paradox and Paraconsistency.
6
two ways. True, there is a body of opinion regretting the second kind of dominance on
normative/conceptual grounds,19 but it is very much a minority view.
Dominant theories have paradigmatic status; they have hegemonic reach.
Dominant theories are natural colonizers. They seek to draw into their analytical and
explanatory ambits phenomena for which they were not initially intended; they engineer
the take-over of rival theoretical approaches; and they provide both motive and
wherewithal for reductionism and the project of scientific unification. They have
impressive records in Mergers and Acquisitions.20 Dominant theories are underwritten by
a methodological principle which I have dubbed Can Do.21 It is a conservative principle,
bidding the theorist not to re-invent the wheel, and to make his theoretical
accommodations with resources that are ready to hand, well-understood and attended by
good performance-records. Can Do is methodological embodiment of a well-known pair
of maxims: Nothing succeeds like success, and imitation is the sincerest form of flattery.
It is almost impossible to overstate the power of the grip of Can Do and of the
benefits that flow from it. Even so, like all virtues, Can Do possesses a degenerate case,
which we might call Make Do. Make Do is Can Do gone wrong. It is Can Do put to
inappropriate uses – to engagements that exceed its reach – simply on grounds that the
resources it calls into play are well understood, available and, in some contexts,
successful. As strong a virtue as Can Do is, Make Do is equally a vice. It drives the
engines of dogmatism and it brokers deep misconceptions in the research programmes of
human enquiry. It is an especially odious kind of “procrusteanism”.22 There is a
particularly entrenched version of it that should be guarded against. It is expressed by the
silly idea that it is better to have a defective theory of something rather than no theory at
all. 23
19
See, for example, L. Jonathan Cohen, The Probable and the Provable, Oxford: Clarendon Press 1977;
“Some historical remarks on the Baconian conception of probability”, Journal of the History of Ideas, 41
(1980), 219-231, and “Whose is the fallacy? A rejoinder to Daniel Kahneman and Amos Tversky,
Cognition, 8 (1980), 89-92; Gilbert Harman Change in View, Cambridge, MA: MIT Press 1986; chapter 1;
Gerd Gigerenzer, Adaptive Thinking: Rationality in the Real World, Oxford: Oxford University Press 2000;
Dov Gabbay and John Woods, “Probability in the law: A plea for not making do”. In Cédric Dégremont,
Laurent Keiff and Helge Rűckert, editors, Dialogues, Logics and Other Strange Things: Essays in Honour
of Shahid Rahman, pp. 149-188. London: College Publications, 2008. See also James Franklin, The Science
of Conjecture: Evidence and Probability Before Pascal, Baltimore: The Johns Hopkins University Press
2001.
20
In some cases, two dominant theories will merge to create a new one – biochemistry, for example. More
recently, and still very much in progress, is the quantum integration movement, whose purpose is to
establish the bona fides of quantum theory in subject areas other than physics. See, for example, Peter
Bruza et al., Proceedings of the Second Quantum Interaction Symposium, University of Oxford, March 2628, 2008. London: College Publications 2008.
21
Paradox and Paraconsistency, pp. 277 and 325.
22
As Stephen Toulmin remarks in The Philosophy of Science: An Introduction, London: Hutchinson 1953,
p. 126. See also Bovens and Hartmann on Bayesian modelling: “Indeed it is easy to distort the problem
under consideration by making implausible assumptions that make the modelling easier. And it is often all
too easy to get carried away by the formulae. Not every partial derivative is meaningful – no matter how
pretty the result is.” (Luc Bovens and Stephan Hartmann, Bayesian Epistemology, Oxford: Oxford
University Press 2003; p. 130).
23
There are attempts at semanticizing the modalities that have nothing to do with possible worlds. One
such is the proof-theoretic of Reinhard Kahle. See his “A proof-theoretic view of necessity”, Synthese, 148
(2006), pp. 659-673. Interesting in its own right, my view here is that even if there were no alternatives to
consider, the possible worlds approach to counterfactuals and belief would still have to be rejected.
7
This is not the place to dwell at length on what might explain the appeal of this
sentiment. But perhaps it might quickly be observed that for anyone drawn to a certain
kind of epistemological holism, there is on occasion some case to be made for saving
strong theories at the cost of discounting or reconceptualizing dissident data. This alone
makes it extremely difficult to mark with reliable precision the point at which the virtue
of Can Do collapses into the vice of Make Do. Still, as best we can we should be on
guard against their confusion.
This, too, is more easily said than done. A common symptom of the onset of
Make Do is the degree of re-interpretation of target phenomena required to give the
appearance of appropriate fit. A rough rule of thumb is that the greater the necessity to
massage the data and the greater the extent of it, the likelier it is that the theory seeking to
account for them exceeds its reach. It hardly needs saying that data-massage doesn’t
dance to precise determinations. It is a commonplace among philosophers of science that
even when theories have legitimate application to their data, the data will have to have
been “primed” in the appropriate way. This is a lesson as old as Bacon24 and is one of the
founding precepts of modern science. If data-massaging is a theoretical necessity, how
much can it be a bad thing?
A further complication is the progressivity of theory. The history of science
discloses lots of cases in which an imperfect theory is retained not because its virtues
outweigh its defects, but because efforts to correct it are inducements to the eventual
construction of superior successor-theories. Here, too, we meet with the necessary but
vexed distinction between theories whose imperfections do and theories whose
imperfections don’t necessitate their abandonment or at least the curtailment of their
would-be reach. For all its lack of transparency and sharpness, it is a distinction for
whose application we have some, if not perfect, aptitude.
Make Do gives rise to a phenomenon that we could call “paradigm-creep”.
Paradigm creep is the dark side of hegemony. It is dominance to no good end. It lies in
the virtuous nature of Can Do to push the envelope in ways that are at risk of paradigmcreep. In an ironic twist, one could say that paradigm-creep is more likely to occur in
theories of greatest success. Paradigm-creep, or the risk of it, is the price to be paid for
the brilliance of our best theories. But paradigm-creep is not self-announcing. Again, one
needs to be on guard against it. Our mention here of the hegemony attaching to the
probability calculus is but one example of a number of dominant theories that appeal to
logicians and technically-minded philosophers. Here are some others. Bayesianism is a
hegemon of statistical inference; deductive logic is a hegemon of logical enquiry; rational
decision theory is a hegemon of practical reasoning; and  possible worlds are a
hegemon of modal semantics.
What a wild ride the standard semantics for intensional logics has had. The
possible worlds framework has cropped up all over the place as the favoured instrument
for handling an astonishing range of philosophical problems.25 For all its swash-buckling
More recently, see Patrick Suppes “Models of data”. In E. Nagel, P. Suppes and A. Tarski (eds.). Logic
Methodology and Philosophy of Science: Proceedings of the 1960 International Congress, pp. 252-261.
Stanford: Stanford University Press, 1962.
25
For sober reflections on this fad, see Roberta Ballarin, “The interpretation of necessity and the necessity
of interpretation”, Journal of Philosophy,101 (2004) 609-638.
24
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excitement, the logics of counterfactuals and belief-statements are victims of paradigmcreep. Let us turn now to counterfactuals.
4. Counterfactuals
A counterfactual conditional is a sentence in the form ⌐φ ⇝ ¬, in which ⇝ is
the sign for subjunctive conditionality. ⌐φ ⇝ ¬ is read: ⌐If it were the case that φ, it
would be the case that ¬. Counterfactuals have been a subject of interest to logicians
since the 1940s,26 but they had to wait twenty-five years or so for a well-accepted formal
semantics, with the appearance of a short burst of papers in the seventies, of which the
leading idea was this:
Were-would: A sentence ⌐If it were the case that φ, then it would be
the case that ¬ is true domestically, that is, in our world w, if and only if
every world w maximally similar to w, except that in w the
antecedent φ is true, is a world in which  is also true.
Were-would is a strong biconditional. It asserts a co-entailment between its relata. For
ease of reference “a Φ-world” abbreviates “a maximally similar world except that Φ is
true there”, and “max-sim” abbreviates “maximally similar”. The Were-would principle
animates Robert Stalnaker’s approach to counterfactuals and plays a similar role in a
number of other approaches, notably those of Lewis, Thomason and Sobel.27Systems
fashioned in the spirit of Were-would are what I mean by mainstream accounts. Everyone
concedes that the mainstream accounts are good value on the score of formal adequacy.
It is intuitively clear that some counterfactuals are true. A further presumption is
that some of them are false, albeit not the ones that are true (dialetheists aside). It may
also be supposed that the class of true counterfactuals cuts across the distinction between
the necessary and the contingent. For example, “If Harry were married, he would not be a
bachelor” is a necessary truth. On the other hand, “If Harry were married, he would be
radiantly happy” is a contingency whose truth relies in part on the neuropsychological
make-up of Harry. A fourth assumption is that for a certain range of true counterfactuals
there exists a kind of consistency requirement. It provides that should a given
counterfactual antecedent have  as its consequent, then it will not also have ⌐¬ as
consequent. For example, if “If Harry were married, he would be radiantly happy” is
true, then “If Harry were married, he would not be radiantly happy” is not.
Roderick Chisholm, “The contrary-to-fact conditional”, Mind, 55 (1946), 289-300, and Nelson Goodman
“The problem of counterfactual conditionals”, Journal of Philosophy, 44 (1947), 113-120.
27
Stalnaker, “A theory of conditionals”, and Stalnaker and Richmond H. Thomason, “A semantical
analysis of conditional logic”, Theoria, 36 (1970) 23-42. Lewis, “Completeness and decidability of three
logics of counterfactuals”, and J. Howard Sobel, “Utilitarianisms: Simple and general”, Inquiry, 13 (1970),
394-449. See also Mares, “Who’s afraid?”, p. 522, Jeffrey Goodman, “An extended Lewis-Stalnaker
semantics and the new problem of counterpossibles”, Philosophical Papers, 33, (2004) p. 38, and Woods,
“Subjunctive conditionals and middling modalities”, Manitoba Modern Languages Bulletin, 8 (1972) 2531. The views of Stalnaker and Lewis are further refined in Stalnaker’s Inquiry, Cambridge, MA: MIT
Press 1984, and Lewis’ Counterfactuals, Oxford: Blackwell 1973.
26
9
Not all counterfactuals honour this condition. Consider a particular sort of
counterfactual the counterlogical conditional, in which the antecedent φ is a formal
inconsistency or is otherwise necessarily false. Consider in particular the case in which φ
is a sentence in the form ⌐χ  χ¬. Then since χ and ⌐χ¬ each follow from ⌐χ  χ¬, we
have it that ⌐χ  χ¬ can serve as the antecedent of true contralogicals made different by
the fact that their respective consequents are indeed one another’s negations. Although
there are counterlogicals that don’t respect the constraint in question,28 there are other
contralogicals that do. For example, “If intuitive set theory were consistent, modern
mathematics would be less complicated than it is” is true, and “If intuitive set theory were
consistent, then modern mathematics would not be less complicated than it is” is false.29
It might be proposed as a first approximation that a condition on the conceptual
adequacy of a semantic theory of counterfactual conditionals is that it satisfies these four
assumptions. Perhaps the qualification is over-circumspect. Whether it is or not will be
influenced by the theorist’s intentions. If the objective is to produce a semantics for a
natural language, then it should account for native speakers’ “judgments of synonymy,
entailment, contradiction, and so on”.30 When “so on” includes counterfactual
conditionality, the pull of our four intuitions has a good deal more than firstapproximation force.
The Stalnaker-Lewis approach to counterfactuals is an adaptation of the
pivotal element of the semantics of the alethic modal logics of necessity and
possibility. In the alethic systems, it is formally sufficient that possible worlds be
members of a non-empty set W on which is defined a binary relation A susceptible to
various combinations of the relational properties of reflexivity, symmetry, transitivity and
extendability.31 From the point of view of the formal adequacy of a modal semantics,
nothing else need be known or assumed about W and A Here is Kripke on this point:
The main and the original motivation for the ‘possible worlds’ analysis – and the
way it clarified modal logic – was that it enabled modal logic to be treated by the
same set-theoretic techniques of model theory that proved so successful when
applied to extensional logic.32
While the building blocks admit of genuinely important philosophical questions –
such as the reality of mathematical abstraction – the metaphysical concerns typically
excited by possible worlds aren’t about the metaphysical legitimacy of these formal
elements.33 Such is also the understanding of Stalnaker:
We note in passing a further distinction implicit in the present examples. ⌐ (φ  ψ) ⇝φ¬ is formally true.
“If Harry’s shirt were crimson, it would be red” is materially true. Although there is some overlap here, the
three distinctions are extensionally inequivalent.
29
“If the axiom of choice were false, the cardinals wouldn’t be linearly ordered, The Banach-Tarski
theorem would fail and so on.” (Hartry Field, Realism, Mathematics and Modality, Oxford: Blackwell
1989; p. 238).
30
D.R. Dowty, R.E. Wall and S. Peters, Introduction to Montague Semantics, Dordrecht: Reidel 1981; p. 2.
31
There is a minority view contra the dyadicity of accessibility. See, for example Nathan Salmon, “Logic
of what might have been”, Philosophical Review 98 (1989) p. 19.
32
Naming and Necessity, n. 8, p. 19.
33
Concerning even impossible worlds, “[f]ar from violating our ontological intuitions, [their] existence 
is supported by our metaphysical theories. They are made from common-or-garden variety entities found in
28
10
Possible worlds are primitive notions of the theory, not because of their
ontological status, but because it is useful to theorize it at a certain level of
abstraction . The concept is a formal or functional notion, like the notion of an
individual presupposed by the semantics for extensional quantification theory.34
However, informally speaking, W is said to contain possible worlds and A is read as an
accessibility relation on worlds. It is not uncommon, in turn, for possible worlds to be
given informal construal as the totality of facts expressed by maximal consistent sets of
sentences from some common language.35 This is said to give us a degree of conceptual
insight into the internal structure of a possible world, formulable as an answer to the
question, “What’s in it?” What’s in it is what corresponds to every sentence from the
language in question consistent with the sentences in it. It is a recursive answer, of
course; and it suggests that possible worlds are made up of the references of a “base set”
of sentences followed by whatever can consistently be added to them. Call this the
“induction set”. But it should be emphasized that it is in no wise a condition on the
formal adequacy of modal semantics that these informal understandings be true.36
Counterfactuals impose a significant constraint on the accessibility relation, a
constraint that is not describable with the machinery of Kripke-frames. In approaches of
the Stalnaker-Lewis sort, the logic of counterfactuals provides that accessible worlds are
possible worlds semantics – relations, individuals and sets. Impossible worlds are ‘made from consistent
stuff’ available in possible worlds.” (Mares, “Who’s afraid?”, p. 518)
34
Stalnaker, Ways a World Might Be, Oxford: Oxford University Press 2003; p. 38.See also Stalnaker,
“Modalities and possible worlds”. In Jaegwon Kim and Ernest Sosa, editors, A Companion to Metaphysics,
pp. 333-337. Oxford: Blackwell, 1995: “The possible worlds representation of content and modality should
be regarded, not as a proposed solution to the metaphysical problem of the nature of modal truth, but as a
framework for articulating and sharpening the problem.” (p. 333). Cf. Dagfinn Follesdal, “Semantik”. In J.
Spech, editor, Handbuch wissenschaftstheoretischer Begriffe, volume 3, pp. 568-579. Berlin: Vandenhock
and Rumprecht, 1980: “All these semantics for modal logic employ  the concept of possible world as a
basic, undefined notion. For this reason it is permissible to consider them as merely algebraic structures
which clarify the logical interrelations between different modal concepts, but which do not provide any
criteria for what is possible and what is necessary.” (p. 572-573).
35
On some accounts worlds are identified outright with maximal consistent sentence-sets themselves, but
this is not Lewis’s way: “I emphatically do not identify possible worlds in any way with respectable
linguistic entities; I take them to be respectable entities in their own right.” (Lewis, “Possible worlds”. In
Michael J. Loux, editor The Possible and the Actual, pp. 182-189. Ithaca: Cornell University Press 1979.)
Lewisian worlds are “very inclusive” ways that actual things might be, ways that are accessible to
quantification.
36
Dale Jacquette is spot-on on this same point. “Hintikka and Kripke  replace all reference to logically
possible worlds in their original formulations of model set-theoretical semantics for modal logics by
syntactic structures consisting of ordered sets of sentences and operations on sets of sentences. A model in
these formulations is not a world .” (“The logic of fiction and reform of modal logic”. In, Kent A.
Peacock and Andrew D. Irvine, editors, Mistakes of Reason, pages 48-63. Toronto: University of Toronto
Press 2005; n. 2, p. 61.) Jacquette goes on to remark that the distinction between possible worlds (whatever
they might be) and these set-theoretic structures are “sometimes blurred in expositions of modal logic, as in
Alvin Plantinga’s The Nature of Necessity (Oxford: Clarendon Press, 1974), esp. 44-8, and G.E. Hughes
and M.J. Cresswell, An Introduction to Modal Logic (London: Methuen, 1972): 75-80.” I might add that
much the same confusion is to be found in Kripke’s own (rather Carnapian) attempts to see possible worlds
in the manner of probability spaces, at pp. 117-119 of Naming and Necessity.
11
answerable to the further requirement of maximal similarity.37 Max-sim worlds are a
significant departure from Kripke’s algebraic structures. In Kripke’s hands, accessibility
between worlds is formally describable in the elementary logic of relations. Maximally
similar worlds require substantial constraints on accessibility that far out-reach the
descriptive wherewithal of the logic of relations. It wouldn’t be far wrong to say that
getting the max-sim property sorted out is a problem of the same order, and of no less
difficulty, than the problem of counterfactuals itself.
Contralogical conditionals are a special problem for these logics. On Stalnaker’s
approach, all contralogicals turn out to be vacuously true. Likewise, on Lewis’ approach,
it is recognized that if the theory did recognize them, they would all be vacuously true; so
the theory refuses to recognize them.38 In “A classically-based theory of impossible
worlds”39, Zalta writes as follows:
[W]e seem to require the resources of worlds other than possible worlds to find a
proper subject matter and semantics for these [= counterlogical] counterfactuals (p.
24).
Other approaches are more accommodating, Mares for example.40
If the semantic intuitions of native speakers are anything to go on, it behooves a
logic of counterfactuals to take note of the fact that in natural speech the contralogicals
are not in fact vacuously true. Some are actually true, and some are actually false. Failing
to do so is a failure of the S-L approach. But it is not a failure that I want to emphasize
here. Rather I want to draw attention to the quite general difficulty posed by Were-would
for counterfactuals and contralogicals alike.
4.1. The overdetermination problem for counterfactuals
It is now time to develop the case against a possible worlds approach to
counterfactuals carrying the maximal similarity qualification on the accessibility relation.
The objection will take the following form: On any reasonable understanding of max-sim
worlds, any such semantics makes too many false counterfactuals true, and in so doing
violates the adequacy condition of paragraphs ago. A logic having this flaw is one whose
theorems overdetermine its conceptual target.
Proof.
“Lewis thinks that closeness can be best explicated  in terms of comparative overall similarity of
possible worlds; a concept that he claims is both remarkably vague and quite well understood. Stalnaker
thinks closeness is best explicated in terms of minimal change” (Goodman, “An extended Lewis-Stalnaker
semantics” p. 42).
38
Lewis: “There is at least some intuitive justification for the decision to make a “would” counterfactual
with an impossible antecedent come out vacuously true. Confronted by an antecedent that is not really an
entertainable supposition, one might react by saying, with a shrug: If that were so anything you like would
be true!” (Counterfactuals, p. 24). Does Lewis not see that his own counterlogical is, by his own classicist
lights, not vacuously true? Perhaps he was trifling with us.
39
Edward N. Zalta “A classically-based theory of impossible worlds”, Notre Dame Journal of Formal
Logic, 38 (1997) 644.
40
“Who’s afraid?”, p. 522.
37
12
Let W be a set of max-sim φ-worlds w1, , wn in which χ is any sentence
occurring in each. Then every max-sim world containing φ is also a world
containing χ. Φ-worlds are reckoned on the model of belief revision. If B is a
belief set and B(Φ) is a revision of that belief-set in face of a new input Φ, then
B(Φ) preserves as much of B as is consistent with the presence there of Φ.
Similarly, if w is the real world, then a Φ-world is a world that preserves as much
of the membership of w as is consistent with the truth there of Φ. In those cases in
which Φis itself inconsistent, and Φ is admitted to B, then B(Φ) will contain as
much of B as to provide for its minimal inconsistency in the face of Φ. Similarly,
if Φ is inconsistent, then a Φ-world contains as much of w as comports with the
same sort of paraconsistent accommodation of Φ. Presumably, then both the Φworld and the real world will contain all the truths of neuroscience, such as they
may be. Consider a particular case. Let “If Harry were a bachelor, he would be
unmarried” be true in w. On our present supposition, every “Harry is a bachelor”world, for example, contain the sentence “The primary role of the orbitalfrontal
cortex is the discrimination of rewarding and punishing stimuli”. Then by the RHclause of Were-would, “If Harry were a bachelor” then the primary role of the
orbitafrontal cortex would be the discrimination of rewarding and punishing
stimuli” is true. But it isn’t true in fact. “If Harry were a bachelor, the primary
role of the orbitalfrontal cortex would be the discrimination of rewarding and
punishing stimuli” is a false positive in w, and “The primary role of the
orbitalfrontal cortex is the discrimination of rewarding and punishing stimuli” is a
false-positive trigger. It is immediate upon reflection that, on the belief-revision
model Φ-worlds are replete with false-positive triggers. A logic of counterfactuals
is hyper-overdetermining if the number of its false positives is large. Accordingly,
the problem with Were-would is that any logic honouring it is hyperoverdetermining. It provides that there are masses of values of χ in the worlds of
W for which the corresponding conditional ⌐Φ ⇝ χ¬ is domestically false. ∎
4.2 A spurious difficulty?
The over-determination objection turns on the claim that S-L logic purports to
verify a large class of conditionals that are actually false, not true. Hence our example:
“If Harry were a bachelor, the primary role of the orbitalfrontal cortex would be the
discrimination of rewarding and punishing stimuli” comes out true in the theory but is
false in fact. But is it false? Consider two further examples:
1. If Harry were a bachelor, the sum of 2 and itself would be 4.
2. If Harry were a bachelor, Paris would be the capital city of France.
I daresay that there are native speakers for which sentences have interpretations which
make them true, not false. Certainly this would appear to be the case when they are
interpreted as “even-still” conditionals, as follows:
1. Even if Harry were a bachelor, the sum of 2 and itself would still be 4.
13
2. Even if Harry were a bachelor, Paris would still be the capital city of France.
The same reading likewise awaits our orbitafrontal cortex conditional, converting an
intuitively false sentence into an intuitively true on
William Lycan is right to say that “even-if” is a “very troublesome locution,
raising special and difficult questions in the semantics of conditionals.”41 Still, certain
features of it and its “still” complement are fairly easy to discern.
“Even-still” conditionals have an interestingly peculiar structure: “even”
implicates that the truth value of the antecedent is not presently known to the utterer, and
“still” implies the truth of the consequent. The connection asserted by ⌐Even if Φ were
the case, it would still be the case that ψ¬. This is an interesting twist. It looks for all the
world like a defining condition of S-L Φ-worlds, as follows:
Let ⌐Φ ⇝ ψ¬ be true domestically, that is, in our world w, and let  be any true
sentence of w except possibly for Φ and ψ, then the Φ-world w with respect to w
is constituted by Φ and ψ and all values of  for which ⌐Even if Φ were the case,
 would still be the case¬ is a domestic truth.
Thus the role of the “even-still” construction is to specify what remains invariant under
possibly counterfactual specific variation, whereas the role of the unadorned “werewould” is to indicate what varies under possibly counterfactual specific variation. The
latter admits of the former as an (innocently) limiting case. S-L domestic truths don’t
require either their antecedents or their antecedents to fail domestically. But if it were
known that they didn’t, the “were-would” idiom would lose its point. In contrast, an
“even-still” conditional fails domestically if its consequent fails domestically. It cannot
be true that even if Harry were a bachelor, it would still be the case that the primary role
of the orbitafrontal cortex is the discrimination of rewarding and punishing stimuli unless
it is also true that that is indeed its primary role. I conclude that “were-would” and “evenstill” sentences exhibit different logical forms. So “even-still” truth does not overturn
“were-would” falsity. The overdetermination difficulty for a S-L semantics is left
standing.
As presently construed, the maximality requirement is an open invitation to
hyper-overdetermination. If the true counterfactuals in w include contralogicals, and if
the verifying worlds w for these contralogicals are treated classically  which is the way
in which they are treated by Stalnaker  then the world that verifies the contralogicals C
is the trivial world T, the world in which everything is the case. Accordingly, every
contralogical whatever is true in w. Of course, as we saw, we might be drawn to the
paraconsistent option which promises the prospect of inconsistent worlds without
41
William G. Lycan, Real Conditionals, Oxford: Oxford University Press, 2001; p. 93. See also A. Hazen
and M. Slote, “’Even if’”, Analysis, 39 (1979), 35-41, and Jonathan Bennett, “’Even if’”, Linguistics and
Philosophy, 5 (1982), 403-418.
14
triviality.42 But paraconsistentizing φ-worlds is of no avail, as the following examples
show.

Let φ be any sentence. Then ⌐φ ⇝ φ¬ is true in w. We are examining
the requirement that any φ-world w with respect to a domestically true
counterfactual will contain its consequent. Since φ is arbitrary, w is T.

Let φ and χ be any pair of sentences. Then ⌐φ  χ ⇝ χ¬ is true in w. By
the requirement under review the associated φ-world w will contain χ. But χ is
arbitrary. So w is T.
At the first bullet the reflexivity of ⇝, is presupposed and at the second the validity of
the -elimination rule. It is hard to see how – short of a monomanaical ad hocery – we
could disabuse ourselves of these features of ⇝. But our discussion up to now has also
embodied a further assumption which we turn to now.
4.3. The same worlds and unique worlds
Again, let C be the set of domestically true counterfactual conditionals. Then the
assumption at hand is this:
SW: The antecedents of C are true in the same worlds to this one.
Call this “the same worlds assumption”.
It is clear upon reflection that the same worlds assumption is not very plausible
for counterfactuals. If it held true, then every world max-sim to this one would be an
impossible world. Consider for concreteness the following pair:
5. If Harry were married he wouldn’t be a bachelor.
6. If Harry weren’t married he would be a bachelor.
Both (5) and (6) are true domestically. So all worlds max-sim to this one containing the
antecedents of (5) and (6) are impossible worlds. The example generalizes alarmingly.
How might we rehabilitate the idea of maximally similar counterfactual worlds? Possibly
we could retain Were-would, but the same worlds assumption will have to go. The
question is, with what shall we replace it? Here is something to consider. It is the unique
world assumption:
5
See, for example, Graham Priest et al., editors, On Paraconsistency. Munich: Philosophia Verlag 1989;
Jean-Yves Béziau et al., editors, Handbook of Paraconsistency, London: College Publications, 2007, and
Bryson Brown, “Preservationism: A short history” and Graham Priest, “Paraconsistency and dialetheism”,
both in Dov Gabbay and John Woods, editors, The Many Valued and Nonmonotonic Turn in Logic, volume
8 of the Handbook of the History of Logic, Amsterdam: North-Holland, 2007.
42
Chris Mortensen, Inconsistent Mathematics. Dordrecht, Boston and London: Kluwer 1995.
15
UW: Let C = A1 , An be the set of domestically true counterfactual
conditionals, and let B1, , Bn,  be their respective antecedents. W = w1, ,
wn,  is the set of worlds in which respectively these antecedents are true.W is
subject to two further constraints. One is that the wi are pairwise distinct. The other
is that the relation of being-true-in induces a one-to-one correspondence between
antecedents Bi and worlds wi.
What UW tries to convey is that each domestically true counterfactual has its own unique
nearest verifying world.
4.4. Relevance
We see what’s wanted here. Any interpretation of maximal similarity that allows
for a large commonality of members between a given world and its Φ-worlds is one in
which the sentences of a large subset of that overlapping membership make
counterfactuals false in the given world true. They contain large numbers of falsepositive triggers. So if Were-would is to succeed in delivering the semantic goods for
counterfactuals, the max-sim relation must be tightened in such a way as to eliminate
from Φ-worlds engendering the false positives of the domestic world.
We must impose stricter constraints. We must find a filter that leeches out falsepositive triggers. One possibility is that we might require that any sentence in a φ-world
be relevantly connected to φ. There are many ways of construing relevance.43 If, for
example, we understood it as topical relevance, or sameness of subject matter, then
except for φ and ψ themselves, we would exclude from the corresponding φ-world any χ
not topically relevant to φ. Topical relevance is in its turn understood in a number of
different ways, whether as Anderson and Belnap’s sharing of propositional atoms44 or as
provided by the further refinements of Demolombe and Jones.45 But here too the details
are unimportant. No matter how the notion is precisified, topical relevance is not a strong
enough filter. Consider again “If Harry were married, he wouldn’t be a bachelor.” It is
also true that Hilary Clinton is married. This is a truth compatible with and topically
relevant to the antecedent of our conditional. It follows that if Harry were married, then
Hilary Clinton would be married. But this is not true.
4.5. Deductive closure
43
See Dov Gabbay and John Woods, Agenda Relevance: A Study in Formal Pragmatics, volume 1 of their
A Practical Logic of Cognitive Systems, Amsterdam: North-Holland 2005, and Deirdre Wilson and Dan
Sperber, Relevance: Communication and Cognition, 2nd edition. Oxford: Blackwell 1995. See also Greg
Restall, “Relevant and Substructural Logics”. In Dov Gabbay and John Woods, editors, Logic and the
Modalities in the Twentieth Century, pages 289-398, volume 7 of the Handbook of the History of Logic,
Amsterdam: North-Holland 2006.
44
See, for example, Nuel D. Belnap, Jr., “Entailment and relevance”, Journal of Symbolic Logic, 25 (1960)
144-146, and Anderson and Belnap, Entailment: The Logic of Relevance and Necessity.
45
R. Demolombe and A.J.I. Jones, “Sentences of the kind ‘sentence p is about topic t’”. In H.J. Ohlbach
and U. Eeyle, editors, Logic, Language and Reasoning, pages 115-133. Dordrecht and Boston: Kluwer,
1999.
16
Again, we know what’s wanted. We cannot allow Φ-worlds to contain any
sentence  for which ⌐Φ ⇝ ¬ is domestically false. No doubt that is just the right filter.
It is a filter that blocks the overdetermination problem. It tells us that a Φ-world w
contains every sentence contained in our world w, with the exception of Φ itself and any
further sentence which is the consequent of a false counterfactual in w whose antecedent
is Φ. What it gains by way of overdetermination-avoidance it loses by way of vicious
circularity: Φ-worlds are needed for the definition of ⇝, and ⇝ is required for the
present conception of Φ-worlds. So we must consider other candidates for Φ-worldhood.
One such is that we restrict the membership of φ-words to φ’s deductive closure
clos(φ) (assuming for now that where φ is inconsistent, a reasonable consensus over the
details of paraconsistent closure.) This is yet another case in which the devil is in the
general idea, not the details.
Proof:
While it has the attraction of evading the overdetermination problem, there are
two bad problems with it. For one thing, it excludes contingent counterfactuals
such as “If Harry were married, he would be radiantly happy.” And it wouldn’t be
a maximally similar worlds approach to counterfactuals. One could define ⇝
without the merest murmur of max-sim worlds. To wit:
⌐
φ ⇝ ψ¬ is true iff ψ  clos(φ). ∎
4.6. Forlornness
On thinking it over, perhaps there is nothing whatever wrong with the notion of
maximally similar worlds. The trouble is that the notion is of no use to the logic of
conditionals. Like the lovely girl of legend, maximal similarity is all dressed-up with
nowhere to go, waiting for the phone to ring. If this isn’t forlornness, what is?
There is reason to think that in founding the max-sim approach to counterfactuals,
its creators sought to tell us what sentences in the form ⌐If Φ were the case, ψ would be
the case¬ mean. That is to say, they wanted to tell us what such sentences mean in
English. The system which delivers the answer to this question is sound and complete. So
it succeeds in telling us what such sentences mean. But it does not succeed in telling us
what these sentences mean in English. It tells what they mean in L-S. Relative to the
intention to say what counterfactuals mean in English, the max-sim approach fails.
Relative to that intention, it is both conceptually inadequate and a failure. (Notice the
lack of a general equivalence here.)
We are now met with the quite general question about systems that fail the
conceptual ambitions of their creators. What, apart from their mathematical versatility are
they good for? Maximal similarity logics don’t give the meaning of were-would
sentences in English, but they do give the meaning of ⇝. What is the use of a theory that
gives the meaning of ⇝?
17
Consider again the case of Riemannian geometry. The lesson of Riemann’s
geometry is that a well-made mathematical object might sit on the shelf awaiting events
that may lend it conceptual legitimacy. If I am not mistaken, there is some evidence that
the formal success of dialetheic logic has set, or is setting the stage for a philosophical
circumspect reconsideration of the rejection out-of-hand of true contradictions.46 If this is
right, then a possible reaction to a formally adequate theory that fails the presumed
objective of getting the concept C objectively right, is a reconsideration of K itself. In the
dialetheic context, there are philosophers who are more open than was previously the
case to the idea that the concept of truth and the concept of contradiction do indeed admit
of co-instantiation under quite particular circumstances. There are two ways of seeing
this reconsideration. We might take the formal success of dialetheic logic as something
that fosters conceptual change. Or we might see it as something that brings about
conceptual clarification.47
Our question is what, apart from the failed intention of saying what were-would
sentences mean in English, is a max-sim logic of ⇝ good for? This is a question which
when put to Riemann in a suitably adjusted form, he could not possibly have answered. It
is a question which when put in a suitably adjusted form to the loathers of true
contradictions in 1979 could only have brought forth derision and scorn. There is no
particular harm in having on the shelf formally adequate theories that fail the conceptual
ambitions of their founders and for which no alternative validation seems ready to hand
or forseeably likely. Perhaps the day will come when having a formally adequate maxsim theory of the meaning of ⇝ will fall into the conceptual graces of some future
configuration of events. But it lies in the nature of these rehabilitations that they fall well
beyond the reach of credible prediction and prior individuation.
5. Belief
The expression “it is believed that” is a sentence-operator. When prefixed to a
sentence it yields a sentence. Sentences prefixed in this way are intensional contexts.
They are resistant to the intersubstitutivity of extensional equivalents. These same
remarks apply equally to the alethic expressions “necessarily” and “possibly”. They are
sentence-operators; they generate intensional contexts; and  they are modalities. Do we
not have it, then, that the belief-operator is likewise a modality? And, if so, would it not
stand to reason that, just as the necessity and possibility-operators admit of – some would
say demand – a possible worlds semantics, possible worlds are the right semantics for
belief as well? There is more haste in these suggestions than is seemly. Certainly not
every sentence operator generates an intensional context, and not every intensional
context embeds a modality. Classical negation makes sentences out of sentences without
intensionalizing them. And “Haphazardly, John balanced the knife on his nose” is an
Of course, Priest’s own The Limits of Thought is designed with just this purpose in mind. See also
Woods, “Dialectical considerations on the logic of contradiction I”, Logic Journal of the IGPL, 13 (2005)
231-260.
47
This, too is a philosophically vexed distinction. Recall Quine’s observation that one person’s explication
is another’s stipulation.
46
18
intensional context, with respect to which it is a stretch to take the “haphazard” as a
modality.48
I have already indicated that it exceeds the bounds of the present paper to discuss
in any detail the metaphysics of modality. Suffice it here to say that in the form in which
historically they have attracted the attention of logicians, modalities are qualifications of
truth and falsity. In the case of necessity, it is a way of being true. In the case of
possibility, it is a susceptibility to being true. This connection to truth is writ large in
Kripke’s semantics. As we saw, Kripke’s own view is that formally speaking the
machinery of the semantics is just some bits and pieces of set theory.W is a non-empty
set and A is a binary relation on W varying with regard to the logical properties of
relations – reflexivity, symmetry, transitivity and extendability. But the theory is more
than its underlying machinery. The theory is powered by an ascription of truth to
quantifications of that machinery, to quantifications of members of W relative to A. For
ease of exposition let us abbreviate the expression “quantifications of members of W
relative to A” as “Q(Set)”. Φ is necessarily true in w iff for all w alternative to w Φ is
true. Φ is possibly true in w iff for some w alternative to w Φ is true. Thus, where M is a
modality and w is a world, the modal structure of these operators is:
MS: MΦ (w) iff in Q (Set) Φ is true,
where it is understood that “iff” expresses a co-entailment relation. Accordingly MS
provides that it is the truth of the modal sentence that makes for the truth of its scope,
whether in all worlds or some as the case may be. Thus MS discloses an intrinsic link
between necessity here and truth everywhere, and between possibility here and truth
somewhere. An important feature of MS is its provision for the contextual eliminability of
modalities. Necessary truth is truth in the quantification of a set theoretic structure.
Possibility is truth in a different quantification of a set theoretic structure. The modality
that the rule governs disappears. It shows up in the LH of the rule’s biconditional sign;
and it has no showing in its RH.49
With these observations in mind, it is now possible to advance a claim of
fundamental importance for an understanding of Kripke’s contribution to modal logic:
It is not the business of a possible worlds semantics to provide a semantics of
possible worlds.
In some approaches, modal expressions are predicates. See here W.V. Quine, “Three grades of modal
involvement. ” In W.V. Quine, Proceedings of the XIth International Congress of Philosophy (1953) 14,
pp. 65-81. See also Volker Halbach, Hannes Leitgeb and Philip Welsh, “Possible worlds semantics for
modal notions conceived as predicates”, Journal of Philosophical Logic, 32 (2003), pp. 179-223.
48
We note in passing that counterfactuals fits MS. In Lewis’ formulation ⌐ ⇝ ¬ is true here iff in the
max-sim alternative to here ⌐  ¬ is true. As applied to necessity and possibility, MS provides that the
modal sentence is true here if and only if the result of scrubbing away its modal character is true in the
49
appropriate alternative world(s). What, then, is the result of scrubbing away the modal character of ⌐ ⇝
¬? Lewis’ implied answer is plausible. Scrub away the modalness of ⌐ ⇝ ¬, and you get the
extensional conditional ⌐  ¬.
19
Proof:
All the variations of the normal logics are got by altering the mathematical makeup of Set. This is done by varying the logical properties of the accessibility or
alternativeness relation – reflexivity, symmetry, transitivity and extendability.
Thus S5 is characterized by an accessibility relation that is reflexive, symmetrical
and transitive. S4 is got by an accessibility relation that is reflexive and transitive
but not symmetrical. T’s accessibility is reflexive only; and so on.
Notwithstanding these system-by-system variations on Set, the underlying model
structure MS of these modalities is invariant. Modalities are truths in
quantifications of set theoretic structures.
A semantics for necessity is a theory that gives the meaning of
“necessary” in English (or of some or other sense of “necessary” in English). This
is done by providing truth conditions for sentences in the form ⌐Necessarily Φ¬.
Integral to the process is the definition of an interpretation of the language, or
language-fragment in question. A Kripke interpretation for L a triple W, A, v
in which W and A are as before, and v is a function that maps sentences to truth
values in virtue of relations borne by elements of W to elements of L relative to
A. A semantics of necessity succeeds to the degree to which the logical truths
furnished in its interpretation are true to the meaning of “necessary” or of the
requisite sense of it in English. It is not by any means a condition on this success
that W be a set of possible worlds (whatever they might be) or that A be a
possible worlds accessibility relation (whatever it might be).
Accordingly, it is not the business of a possible worlds semantics to provide a
semantics of possible worlds. It is not its function to specify the meaning of
“possible world” in English. Beyond these formal specifications, a possible
worlds semantics has nothing to say about possible worlds. The concept that a
possible worlds semantics makes clear is not the concept of possible worlds, but
rather the concept of necessity. The great lesson of Kripke’s breakthrough is that
in achieving conceptual clarity about necessity50 it is unnecessary to attain
conceptual clarity about possible worlds.∎
We can now set up the problem with belief. The problem with belief is that a
possible worlds semantics requires for it that it instantiate the modal structure MS,
expressed is the biconditional Bel:
Bel: ⌐a believes that Φ¬ is true domestically if and only if there is a doxastically
alternative world w in which Φ is true.
Why in the world would we write Bel this way? Where is its intuitive appeal? The answer
is to never mind its intuitive appeal. The reason it is written this way is that MS demands
it. A sentence is believed in this world if and only if  is true in a quantification of a set
theoretic structure. MS, again, is a contextual eliminator of the modalities it governs.
50
See again, Kripke: one of the functions of the set theoretic apparatus of the semantics of necessity is to
make the concept of necessity clear. (Naming and Necessity, n. 18, p. 19).
20
Like Were-would and MS, Bel is also a strong biconditional. Its “if and only if”
expresses the co-entailment of its relata. Being true in a doxastic alternative of it
necessitates its belief in this world. There is no formal specification of worlds that
captures what is wanted here. On any formal specification Bel, like Were-would,
generates its own
hyper-overdetermination problem. It makes masses of this world’s belief-sentences false.
That is, given the meaning of “believes” in English, it makes masses of belief-sentences
false relative to that meaning. Formally adequate logics built around Bel get the meaning
of “believes” right. But they get the meaning of “believes” in English wrong. The
meaning they get right is the meaning they confer on “believes”. The meaning they get
wrong is the meaning that “believes” antecedently has in English.
Proof:
Here, too, doxastically alternative worlds of a given world are conceived of on the
model of belief-revisions. If w is the real world and D(w) is a doxastic alternative
of it, then D(w) contains as much of w as is compatible with the following pair of
facts: (1) For any Φ believed in w, Φ needn’t be true in w; and yet (2) Φ is true in
D(w). Consider a case. Suppose that “Paris is the capital city of France” is
believed in w. Then there is a D(w) in which that sentence is true. D(w) in turn
will contain all that is in w except for the conditions imposed by (1) and (2). Let
one such sentence of D(w) be “Julius Caesar cried a lot at his fourth birthday
party.” Bel provides that this sentence is believed in w. But it is not. No one
believes this, never mind that it might well have been true. The example
generalizes. Let  be any such sentence and in D(w) which, intuitively is not
believed in w. But Bel rules that it is believed in w. Thus Bel violates our
intuitions about what “believes” means in English. In the category of belief, Bel
creates for too many false positives in w; D(w) contains too many false-positive
triggers. So there is a hyper-overdetermination problem for Bel.∎
As with counterfactuals, so with belief-contexts. Blockage of the hyperoverdetermination problem requires that alternative worlds be suitably constrained. As
with the logic of conditionals, a good deal will have to be said about these worlds beyond
that they are members of W relative to A. Of course, informally speaking, here too we
know what is wanted. We want alternative worlds to be constrained in such a way that
being true in them necessitates the truth in our world of all and only those beliefstatements that are actually true here. This is just the right specification of worlds for
successful evasion of the hyper-overdetermination problem. But, as before, the cost of
our success is vicious circularity. Such worlds are needed to give the meaning of
“believes”, and the meaning of “believes” is needed for the specification of the worlds.
The problem with which we are now met is this: Are there independent, nonquestion-begging grounds on which to assert a connection between the belief that Φ and
Φ’s truth? The answer is:
Such grounds do exist, but the link they establish between the belief that Φ and
the truth of Φ is not one that answers to the co-entailment of Bel.
21
Proof:
There are three (alethic) modal status’ open to a proposition. It is necessary, it is
impossible or it is contingent. A contingent proposition is one that is not
impossible and not necessary. A necessary proposition is true everywhere. Hence,
it is trivial that if any such proposition is believed there is a world in which it is
true. Similarly, if Φ is a contingent proposition, then if Φ is believed there is some
world in which it is true. But in neither of these cases is the truth in a world of the
believed proposition a consequence of its being believed in this one. If Φ is
contingent, then its truth in a w flows from its contingency, not from its being
believed. If Φ is necessary, then its truth in all w flows from its necessity, not
from its being believed, even it is believed. Impossibilities are a case apart, as we
might expect. Since contradictions are true nowhere, either the definition of belief
is incorrect or no one believes a contradiction. In standard approaches to belief
contexts, it is the latter thesis that prevails.51 Accordingly, that contradictions
require non-belief is guaranteed by Bel. But here, too, the link between the
absence of truth and the absence of belief is that it takes us from the
contradictoriness of sentence to its truth in no world whatever. That the objects of
contradictory beliefs are true nowhere is a matter of what flows from their
necessary falsity, not from the (alleged) fact that they are unbelievable.∎
The point of largest importance is that a possible worlds semantics breaks the link
between a modal truth and the truth of its scope. There is no intrinsic connection between
believing that Φ and Φ’s being true. This presents the possible worlds diehard with an
interesting challenge. He must figure out ways of reconciling the belief-operator to the
modal structure MS. In the spirit of the deviations occasioned by non-normal logics,
doxastic logicians attempt to meet their challenge by imposing on the accessibility such
changes as effect the required reconceptualization of possible worlds.
As we saw with max-sim worlds, since no formal adjustment of the accessibility
relation will give us doxastic worlds, doxastic worlds have to be wrought extra-formally,
that is to say, conceptually or philosophically. For the standard definition of ⌐a believes
that Φ¬ to hold up, there must be some conceptual basis on which to rest the idea of
doxastic worlds. One way of doing this is, as we saw, to mandate the impossibility of
believing the impossible. Perhaps there are aggressively idealized epistemologies in
which it is “analytic” of the “rational” agent that he does not believe impossibilities. In
that case, it might be bruited that a philosophy of belief is a normative account of the
ideally rational believer. But this is not semantics of belief. Given the linguistic
judgments of native speakers, it is no part of the meaning of “believes” that rational
believers don’t believe inconsistencies. It might be conceded – but not by me52 – that if
there is any semantic lesson to be learned here, it is about the meaning of “rational”. For
myself, I doubt that there is a semantics of “rational” in English in which only people
who had lost their reason believed in 1901 that the axioms of intuitive set theory were
In Knowledge and Belief, if Φ is a necessary truth everyone knows it, hence everyone believes it.
Similarly, since everyone knows that ⌐ Φ¬ is the negation of , and since knowledge and belief are bound
by a consistency requirement in K & B, then no one believes a contradiction in K & B..
52
Dov Gabbay and John Woods, “Normative models of rational agency”, Logic Journal of the IGPL, 11
(2003), 597-613.
51
22
true. Any judgment to the contrary is the result of a normatively motivated semantic
prescription.
Since it is a distinctive feature of Kripke’s possible worlds semantics for the
alethic modalities that their truth conditions be delivered by the modal structure MS in
ways wholly independent of any conceptualization of possible worlds, belief logics are a
significant departure from this. To get a basis of belief to conform to the modal structure
MS it is essential that doxastic worlds be conceptualized in ways that are descriptively
inadequate or viciously circular, never mind the presumption that they might be
normatively ideal. The job of a semantics of “believes” is to produce truth conditions that
reflect the linguistic practices of competent speakers of English. Doing this gives us the
meaning of “believes” in English. The modal semantics of “believes” doesn’t do this. It
preserves the modal character of belief-statements by getting the meaning of “believes”
in English wrong. The culprit here is the structure MS. Belief-sentences don’t conform to
it. If we allow that the modal character of a sentence operator always involves its
instantiating the MS-structure, then there is reason to think that belief is not a modality
after all. Which would give it, in turn, that the possible worlds approach is wrong in
principle for the semantics of belief.
Still, we can’t change the fact – nor should we underappreciate it – that the going
logics of belief bristle with mathematical versatility. With commendable mathematical
verve, such systems assign a meaning to “believes” which is not the meaning it has in
English. It is the meaning it has in doxastic modal logic. Perhaps there is value in having
this sense of “believes” waiting on the shelf, all aglow with formal adequacy. Perhaps
some future turning of events will throw up occasion to deploy this sense of “believes”.
Perhaps there is some waiting Riemannian vindication of that meaning. But, as with the
meaning of ⇝, such vindications exceed the reach of credible prediction and prior
individuation.
6. Conclusion
I conjectured at the beginning that the inadequacies of the standard approach to
counterfactuals and belief-contexts had something to do with logical pluralism and the
hegemonic status of possible worlds. I said that pluralism tended to disguise those
inadequacies and that hegemony prompted semanticists to engender them. In bringing
this paper to a close, let us give these conjectures the final say.
The overdetermination problem is an obvious feature of the L-S biconditional that
sets out the truth conditions for the “were-would” connective53 in English. Its
obviousness makes it problematic both as to why it is not common knowledge and as to
why the L-S approach hasn’t long since been given up on. How might the rich pluralism
of logic have a bearing here? As we saw, there are two prominent responses to
pluralism’s putative damage to the objectivity of logic. According to the ambiguation
thesis, provided that they are formally adequate, apparently conflicting systems of logic
are true of different senses of the terms expressing the logics’ target concepts. In the case
of “were-would” sentences, this carries the suggestion that there is some sense of “werewould” according to which the intuitively false counterfactuals of our native semantics
53
If indeed, syntactically, that’s what it is. See here Lycan, True Conditionals, chapter 1.
23
are actually true. According to the relativity thesis, provided again that they are formally
well-made, the theorems of conflicting systems are true in those respective places, and
the question of their extra-systemic truth doesn’t arise. By these lights, there is no fact of
the matter as to whether the falsehoods claimed by the overdetermination thesis are
indeed false. If they are provable in an L-S logic of conditionals, then they are true there,
and that’s all there is to it. Much the same can be said for the kindred inadequacy of Bel.
There is little point in dwelling on the unsatisfactoriness of these apologiae.
Perhaps they exercise some causal influence on the persistence of our affection for a
modal semantics that goes too far. But their causal significance is more distal than
proximate. It is better, I think, to cut to the chase. The reason that the overdetermination
problem is not much bothered with is that, having taken the mathematical turn well over a
hundred years ago, modern mainstream logic so favours mathematical versatility as to
make of conceptual adequacy an engendered species.
What now of hegemony? The criticisms of sections 4 and 5 are to the effect that
the inadequacies of max-sim treatment of counterfactuals and of the doxastic
alternativeness treatment of belief-contexts proceed from the paradigm-creep of the
dominant semantic theory. Unlike the possible worlds approach to necessity, a condition
of its success is that it provide the meaning(s) of “necessary”; it is not required that it
provide the meaning of “possible world”. But counterfactuals and belief-contexts are
different. There it is required that the logics furnish a semantics for two things, not one.
In the case of counterfactuals, the two things are “were-would” and max-similar
worldhood. In the case of belief-contexts the two things requiring semantic analysis are
“believes” and doxastic alternativeness. Therein lies the way of chaos. The implied
semantics of maximal similarity is either incompatible with the express semantics of
“were-would”, or it savages the theory of counterfactuals with vicious circularity.
Similarly, the implied semantics of doxastically alternative worldhood clashes with the
express semantics of “believes”, or it visits on the theory of belief a vicious circularity.
Both the ambiguation and relativist strategies offer a way out of these binds, but at the
cost of giving up on the meaning of “were-would” and of “believes” in English. Here,
too, there may be some distal causal influence, but the proximate cause is that formal
adequacy out-ranks conceptual fidelity.
Belief-contexts are also an example of paradigm creep working in the opposite
direction. Paradigm-creep induces the over-massaging of data. If P is a dominant
theoretical paradigm for data of type d, then if different data d could be represented as
data of type d, then P would be an authoritatively accessible theory for d-data as well. It
all turns on the warrant we have for representing the d-data as d-data. Counting in favour
is accessibility to a powerfully successful theory. Counting against it is that the d-data
aren’t data of the d-kind, so much so that they won’t brook construal as such. But, again,
with ambiguation and relativity hovering nearby, the idea that there is a fact of the matter
as to whether or not the d-data are indeed d-data is open to question. Ambiguationists
will be drawn to the idea that there is a sense of “d” according to which the d-data are
indeed construable as d-data. And relativists will look with favour on the suggestion that
whether the d-data are construable as d-data is a matter of whether there is a formally
adequate theory that makes them so. It is easy to locate belief-contexts in this d-data/ddata landscape. The d-data are the uses of the belief-operator in competent English, and
the d-data are data of a type whose structure is exemplified by modal expressions such as
24
“necessarily” and “possibly”. Concretely, the claim that the d-data can be construed as
data of the d-type is the claim that the belief-operator is a modality. And the inducement
to construe it thus is that there awaits a formally strong and conceptually not unsuccessful
theory of the (normal) modalities.
But, here too, once the smoke has cleared, the central explanatory fact continues
to be the domination of formally adequate theories over conceptually satisfying ones. In
the dynamic pull between technique and substance, substance isn’t doing all that well
these days.54
The Abductive Systems Group
Department of Philosophy
University of British Columbia
jhwoods@interchange.ubc.ca
Group on Logic and Computational Science
King’s College London
For helpful criticisms and suggestions I thank this volume’s editors and Roberta Ballarin, Patrick
Blackburn, Reinhard Kahle, Alex Liu Cheng, Kian Mintz-Woo and Alirio Rosales. I am especially
indebted to Kahle for allowing me to read a pre-publication version of his “’Modalities’ without worlds”, to
appear.
54
25