Synthese (2012) 188:487–498
DOI 10.1007/s11229-011-9940-6
Prior on an insolubilium of Jean Buridan
Sara L. Uckelman
Received: 13 April 2011 / Accepted: 13 April 2011 / Published online: 17 May 2011
© The Author(s) 2011. This article is published with open access at Springerlink.com
Abstract
We present Prior’s discussion of a puzzle about valditity found in the
writings of the fourteenth-century French logician Jean Buridan and show how Prior’s
study of this puzzle may have provided the conceptual inspiration for his development
of hybrid logic.
Keywords
Arthur Prior
· Hybrid logic · Insolubilia · Jean Buridan
1 Introduction
Elsewhere in this volume, we give a historical overview of Arthur Prior’s work on
medieval logic, focusing on the unpublished material in his archives. In this paper we
take a more conceptual approach towards showing his debt to the medieval logicians.
In
Uckelman
(
2011
) we saw how the works of the Stoic logician Diodoros Chronos
and medieval logicians such as Jean Buridan and Peter de Rivo provided inspiration for
Prior’s development of temporal logic. In this paper we show that an insolubilium (puz-
zle) of Jean Buridan may also have provided the original inspiration for hybrid logic.
2 A insolubilium about validity
2.1 Medieval conceptions of validity
Jean Buridan was a French logician working at the University of Paris in the early
fourteenth century.
1
Buridan, like many medieval logicians, defined ‘proposition’ in a
1
For further discussion of Buridan’s works and Prior’s research on Buridan, see
Uckelman 2011
.
S. L. Uckelman (
B
)
Institute for Logic, Language, and Computation, Amsterdam, The Netherlands
e-mail: S.L.Uckelman@uva.nl
123
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Synthese (2012) 188:487–498
fashion antithetical to the standard modern definition. By the term propositio, Buridan
meant not an abstract entity, always existing and eternally true or eternally false, but
rather a specific mental, spoken, or written token declarative sentence. On such a view,
propositions are transient objects, which come into and go out of existence and are
not necessarily sharable.
2
Such a view naturally has consequences for how validity is to be defined. According
to standard modern definitions of validity, an argument is valid if whenever proposi-
tions expressed by the premises are true, the propositions expressed by the conclusion
must be true, or that the truth of the premises forces the truth of the conclusion. Since
truth or falsity, and hence possible truth and possible falsity, can only be ascribed
to things which exist, this definition builds in the assumption that the propositions
expressed by the premises and the conclusion always exist. On the medieval concep-
tion of propositions that we’re considering now, validity can no longer be defined
as a relationship between necessarily existing and unchanging entities, because truth
values do not attach to necessarily existing abstract objects but instead attach to con-
tingently existing transient concrete tokens. In order for a proposition to be true, it
must first exist. However, we can easily imagine circumstances in which true premises
are written down, but where the conclusion is not written down (or otherwise spoken
or thought), making the premises true and the conclusion not true. So, the modern
definition of validity is obviously not suitable.
Medieval definitions of validity are also couched in terms of necessity and truth
and a certain relation between the truth of the premises and the truth of the conclu-
sion holding in some necessary fashion.
3
However, these definitions when combined
with the contingent nature of propositions on the medieval view give rise to some
paradoxical results. Two of these paradoxes, or insolubilia, Prior investigated in detail
in (
1969
). In this paper, we present the results of his research, and show a natural
extension of these results to basic hybrid logic. We conclude that, though there is no
explicit textual evidence for this, Prior’s attempt to provide a semantics for Buridan’s
distinction between ‘possible’ and ‘possibly-true’ may have provided a conceptual
grounding and inspiration for his development of hybrid logic.
We make one note on terminology before turning to the details of Prior’s research
on Buridan. Because the modern conception of propositions is so strongly entrenched,
and because the medieval view of propositions presented above is so foreign and anti-
thetical to the modern view, we attempt to lessen any confusion which might arise
because of the differences of these views by calling token propositions by their Latin
name, propositio (pl. propositiones). Throughout the following we use propositio to
refer to a specific spoken, mental, or written token, and we use the English word
‘proposition’ in its modern sense.
4
2
For more details on ancient and medieval theories of the proposition, see
Nuchelmans
(
1973
).
3
Broadie
(
1993
, pp. 88–90) presents three definitions of validity which are discussed in medieval literature,
each of which have different problems connected with the fact that the validity relation must hold between
contingent entities.
4
Note, however, that this convention is not followed in the texts that we are quoting. We trust that context
will make it clear which sense is being used.
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489
2.2 The insolubilium
Prior (
1969
) considers an insolubilium or sophism of Buridan’s. An insoluble is a
“proposition arrived at by apparently valid forms of reasoning, which nonetheless
implies its own contradictory” (
Buridan 1966
, p. 5). The standard presentation of an
insoluble in a medieval logical text is to first state the insoluble, then give an argument
for its truth, and another argument for its falsity, and then an argument for the correct-
ness of one of these arguments and the incorrectness of the other. The specific insoluble
discussed by Prior is the first one of the eighth chapter of Buridan’s Sophismata:
SOPHISM. (1) Every proposition is affirmative, so none is negative.
It is proved, first, by the argument from contraries, for just as it follows that if
every man is ill, then no man is healthy, because it is impossible for the same
person to be both healthy and ill, so it follows in the proposed [case] that it is
impossible for the same proposition to be both affirmative and negative at once.
…
The opposite is argued, because from a possible proposition there does not fol-
low an impossible. And yet the first proposition is possible, namely, “Every
proposition is affirmative.” For God could destroy all negatives, leaving affirma-
tives. Thus, every proposition would be affirmative. But the other is impossible,
namely, “None is negative”, for in no case could it be true. For whenever it is
not, it is neither true nor false, and whenever it is, then some [proposition] is
negative, namely, it. Hence, it is false to say that none is negative (
Buridan 1966
,
pp. 180–181).
Both arguments are intuitively plausible. The first one turns on an equivalence repre-
sentable in modern notation as
∀x(Px → Ax) ≡ ¬∃(Px ∧ ¬Ax)
This is just the interdefinability of the quantifiers.
The second argument is more interesting because it turns on properties specific to
Buridan’s propositiones. The propositio “No propositio is negative” can never be true,
because in order to be true, it must first exist (i.e., be thought, written, or spoken),
and as it is itself negative, as soon as it exists, it contradicts itself. (A bit later on in
Ch. VIII, Buridan follows this line of effect in considering the argument from “No
propositio is negative” to its contradictory “Some propositio is negative”.)
Buridan accepts the argument in favor of the inference, and rejects the argument
against it. He concludes that if one allows there to be premises which are possible
even though they are never true, then we are not dealing with a case of an impossible
conclusion following from a possible one. He says that
it is manifest that a proposition is not called possible because it can be true, nor
impossible because it cannot be true (
Buridan 1966
, p. 182).
5
5
The diametric opposition of this view to Diodoros’s, discussed in
Uckelman
(
2011
), will hopefully have
struck the reader forcefully.
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This conclusion leads him to make a distinction between sentences which are possi-
ble and those which are possibly-true. It is this distinction which Prior formalizes in
(
1969
).
When Prior discusses the inference from “No propositio is negative” to “Some
propositio is negative”, he notes that such examples may appear to be evincing “some
sort of confusion between use and mention, or between object-language and metalan-
guage” (
Prior 1969
, p. 481). The thought is that there is merely some fundamental
confusion going on and that somewhere Buridan is doing something illicit. Prior goes
on to note that this is not the case, and that “there is nothing against a language
containing its own syntax, though there may be plenty against its containing its own
semantics” (
Prior 1969
, p. 481). Indeed, we have examples of languages containing
their own syntax, namely, Peano arithmetic, which is expressive enough to express
syntactical notions such as ‘proof’, ‘provable’, etc. To show that there is no problem
with a language containing its own syntax, and that the move which Buridan is making
by having truth attach to individual tokens of sentences is not illicit, Prior constructs
a language containing some of its own syntax, where the object-language and the
meta-language are sharply distinguished. He uses this to give a semi-formalization of
the paradox, and we now present this semi-formalization.
3 Prior’s semi-formalization
In this section we present a slightly modified version of the semi-formalization which
Prior used to analyze Buridan’s insoluble. We have two languages, an object language
L and a metalanguage M. Our metalanguage M is standard English. We define L
syntactically.
L is composed of three types of strings of marks: terms, signs of quantity,
and copulae.
Definition 3.1 The terms of
L are the strings
propositio
affirmativa
negativa
We shall, as needed, use capital Roman letters, A
, B, C,…, as variables for these
terms.
Definition 3.2 The signs of quantity of
L are the strings
omnis
quaedam
nulla
Definition 3.3 The copulae of
L are the strings
est
non est
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491
Definition 3.4 A string of characters is a sentence of
L or an L-sentence
6
if and only
if it is a sign of quantity followed by a term followed by a copulae (which can be
non est only if the sign of quantity is quaedam) followed by another (not necessarily
distinct) term.
Thus, all
L-sentences will be of one of the following four forms:
• omnis A est B
• nulla A est B
• quaedam A est B
• quaedam A non est B
Strictly speaking, these strings are syntactic entities which, as yet, have no mean-
ing. However, the choice for
L-strings of these particular standard Latin terms was
meant to be transparent. When we give the truth conditions for these
L-sentences,
their meanings will correspond to the meanings of the Latin sentences.
Before we can present the truth definitions for
L-sentences, we need to introduce a
notion which will allow the distinctive nature of Buridan’s propositiones as concrete,
existing objects to be reflected. To do so, we assume we have at our disposal an unlim-
ited number of sheets of paper. On these sheets, certain
L-sentences may be written.
These inscriptions on sheets of paper are our tokens. We will define two types of truth
with respect to tokens on a sheet of paper: a sentence may be true on a sheet of paper
and a sentence may be true of a sheet of a paper. In order to define these two types
of truth, we must first define what each of the
L-terms refers to, or, in Prior’s words,
connotes. Prior says:
Each term is associated with a particular group of shapes, which it may be said
to connote, though this means no more than that the presence on a sheet of marks
of certain shapes will determine…whether or not sentences containing certain
terms are to be counted as “true on their sheets” (
Prior 1969
, p. 483).
Definition 3.5 The connotation of a term is defined as follows:
• The term propositio connotes all L-sentences.
• The term negativa connotes all L-sentences whose sign of quantity is nulla or
whose copula is non est.
• The term affirmativa connotes all L-sentences which are not connoted by the term
negativa.
We often say that a sentence is of type A rather than that it is connoted by term A.
We now give the truth conditions for true on a sheet for each type of sentence.
Definition 3.6 (Truth on a sheet)
• A sentence of the form Omnis A est B is true on a sheet iff
1. It is written on the sheet.
2. There is at least one sentence on the sheet which is of type A.
3. There is no sentence on the sheet which is of type A which is not of type B.
6
When clear, we will drop ‘
L’ and refer to these strings simply as ‘sentences’.
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• A sentence of the form Nulla A est B is true on a sheet iff
1. It is written on the sheet.
2. There is no sentence on the sheet which is of both type A and type B.
• A sentence of the form Quaedam A est B is true on a sheet iff
1. It is written on the sheet.
2. There is at least one sentence on the sheet which is of both type A and type B.
• A sentence of the form Quaedem A non est B is true on a sheet iff
1. It is written on the sheet, and either
2. There is at least one sentence on the sheet which is of type A and is not of type
B, or
3. There is no sentence on the sheet of type A.
7
Before we turn to examples of this definition, we make a few notes on the relationships
between the four types of sentences. These four types of sentences are the four found
in the traditional Aristotelian Square of Opposition. The truth definition given above
satisfies the standard relationships in this square, particularly that sentences of the
form Omnis A est B and Quaedam A non est B are contradictories of each other and
sentences of the form Nulla A est B and Quaedam A est B are also contradictories of
each other. Hence, if both of a pair of contradictories occur on a sheet, then one will
be false on the sheet and the other will be true on the sheet.
8
We now give examples of this truth definition.
-
On Sheet 1, both sentences are false: They are both false because there are no negative
propositiones on the sheet.
7
Note that in this definition, one of the conditions of a sentence of the form Omnis A est B being true is
that at least one sentence of form both A and B occurs on the sheet. Since Quaedam A non est B is the
contradictory of Omnis A est B, when it is present on a sheet, its truth can be triggered merely by there
being no sentences of type A. This phenomenon, ‘existential import’, is another widely-discussed difference
between medieval and modern logical theories, but one which will not occupy us here.
8
The proofs of these claims are straightforward and are not proved here.
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493
On Sheet 2, the first sentence is false and the second is true. The second is true
because the first propositio is affirmative and the first is false because the second
violates the truth conditions of the first.
And on Sheet 3, both sentences are true. The first is true by the connotations of
the terms; the affirmative sentences are defined as those which are not negative. The
second is true because neither sentence is affirmative.
It is now clear how we can use
L-sentences to make claims about the syntax of other
L-sentences, in a completely unproblematic manner which involves no confusion of
metalanguage and object language. Because we can do so, we can make a distinction
between sentences which are possible and those which are possibly true.
First, note that this language contains some sentences such, that any time they
are present on a sheet, they are true on that sheet, and some sentences such that any
time they are present on a sheet, they are false on that sheet. Consider the following
sentences:
Nulla affirmativa est negativa
Quaedam affirmativa est negativa
The first says that no affirmative sentence is negative. Given the connotations of affir-
mativa and negativa, the first sentence is clearly tautological, and the second sentence,
being its contradictory, will equally clearly be false on any sheet on which it appears.
This is nothing surprising, as we expect any standard logical system to contain tautol-
ogies and contradictions. But contrast these two sentences with the following:
Quaedam propositio est affirmativa
Quaedam propositio non est affirmativa
Nulla propositio est negativa
The first two will be true on any sheet on which they occur, and the last will be false on
any sheet on which it occurs. But these cases differ relevantly from the first two. Con-
sider the first sentence and its contradictory, Nulla propositio est affirmativa. While
the first is true on any sheet it is written on, its contradictory is not false on every sheet
it is written on. For example, if it is the only sentence written on a sheet, then it is true
on that sheet.
If we move into the metalanguage, we can see another difference between the two
groups of statements. We cannot imagine any sheet of paper which has a sentence
which is both negative and affirmative. The sentence Quaedam affirmativa est nega-
tiva is a logical contradiction, because as a sentence is affirmative if and only if it is
not negative, this is simply the assertion
∃x(N x ∧ ¬N x)
likewise, its contradictory is the logical necessity
∀x(N x → N x)
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But we can easily imagine a sheet of paper which is correctly described by “No prop-
osition is negative” (to adapt a reason of Buridan’s given later in the same text, God
could easily have annihilated all negative propositions, so that the meta-claim “No
proposition is negative” would be true), and ones where “Some proposition is affir-
mative” is an incorrect description. As Buridan points out, there is nothing logically
contradictory about the claim
¬∃x N x, nor anything logically necessary about the
claims
∃ x Ax and ∃ x¬ Ax.
Sentences of these types are the ones which underpin Buridan’s conclusion that
there are some sentences which are possible, but not possibly-true, and some which
are impossible but not necessarily-false.
4 A hybrid logic approach
The presentation in the previous section corresponds, with a few changes of notation,
to Prior’s original consideration of the sophisma. However, his discussion of the dif-
ference between ‘possible’ and ‘possibly-true’ is done in an informal, metalanguage
setting. It is possible to give a natural and straightforward extension of the previous
semantics to a hybrid semantics in order to give a formal presentation of the distinction
in the object language, which we do now.
We begin by extending
L to a new syntax L . L contains all the terms, signs of quan-
tity, and copulae of
L, plus an infinite set of numerals N and the modalities @
n
, ♦, ♦·, ,
and
. We call elements of
N nominals, and let the variables n, m, , k…range over
the nominals.
Definition 4.1 A string of characters is a propositio of
L (or an L -propositio) iff it
is a sentence of
L.
9
We correspondingly modify def.
3.5
so that propositio connotes
all
L - propositiones.
Definition 4.2 A string of characters is a sentence of
L (or an L -sentence) iff it is
one of the following:
• It is an L -propositio.
• It is @
n
followed by an
L -propositio.
• It is ♦ followed by an L -propositio.
• It is ♦· followed by an L -propositio.
• It is followed by an L -propositio.
• It is followed by an L -propositio.
A model
S is formally defined as a tuple S, V, N , where S is a (possibly infinite)
set of sheets s
1
, s
2
, s
3
…, V is a function assigning
L - propositiones to subsets of S,
and N is a function from
N to S such that N(n) = s
n
. We abuse notation and say
that V
(s
n
) is the set of propositiones which are written on s
n
. We define two notions
of truth. The first applies to propositiones only, and is a simplification of the truth
definition presented in the previous section.
9
As before, we will omit the
L when ambiguity will not occur.
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495
Definition 4.3 (Truth of a sheet)
• A propositio of the form Omnis A est B is true of a sheet s
n
iff
1. There is at least one propositio on s
n
of type A.
2. There is no propositio on s
n
which is of type A which is not of type B.
• A propositio of the form Nulla A est B is true of a sheet s
n
iff
1. There is no propositio on s
n
which is of both type A and type B.
• A propositio of the form Quaedam A est B is true of a sheet s
n
iff
1. There is at least one propositio on s
n
which is of both type A and type B.
• A propositio of the form Quaedem A non est B is true of a sheet s
n
iff
1. Either there is at least one propositio on s
n
which is of type A and is not of
type B,
2. Or there is no propositio on s
n
of type A.
Let
ϕ be an arbitrary propositio. We then say that s
n
ϕ iff ϕ is true of s
n
.
With this definition of truth of a sheet, we can give a complete truth definition for
arbitrary
L formulas. Let ϕ be an arbitrary L -propositio.
Definition 4.4 (Truth)
• s
n
@
m
ϕ iff ϕ ∈ V (s
n
) and s
m
ϕ
• s
n
ϕ iff s
n
@
n
ϕ
The addition of the @
n
operator allows us to distinguish between possibility and possi-
ble truth. Formally, possibility and possible truth, like truth, are evaluated with respect
to a sheet of paper:
Definition 4.5 (Possibility & Possible Truth)
• s
n
♦ϕ iff ϕ ∈ V (s
n
) and there exists an m s.t. s
n
@
m
ϕ
• s
n
♦·ϕ iff ϕ ∈ V (s
n
) and there exists an m s.t. s
m
@
m
ϕ
• s
n
ϕ iff ϕ ∈ V (s
n
) and for all m, s
n
@
m
ϕ
• s
n
ϕ iff ϕ ∈ V (s
n
) and for all m, ϕ ∈ V (s
m
) implies s
m
ϕ
Like
L-sentences above, strings like ♦ϕ are strictly speaking syntactic entities which
have no meaning. However, it should be intuitively clear that the truth conditions for
these four operators will allow us to treat them as object-language renderings of stan-
dard meta-language notions, and read
♦ϕ as ‘ϕ is possible’; ♦·ϕ as ‘ϕ is possibly-true’;
ϕ as ‘ϕ is necessary’; and ϕ as ‘ϕ is necessarily-true’. We can then conclude that
ϕ is possible but not possibly-true if ∀m, n s.t. s
m
@
n
ϕ, m = n, and there are m, n
s.t. s
m
@
n
ϕ, as well as the following:
Lemma 4.6
(a) s
n
ϕ implies s
n
♦·ϕ implies s
n
♦ϕ, and not vice versa.
(b) s
n
ϕ implies s
n
ϕ implies s
n
ϕ, and not vice versa.
Proof
(a) Assume s
n
ϕ. This means that s
n
@
n
ϕ, that is, ϕ ∈ V (s
n
) and s
n
ϕ.
Hence there exists m s.t. s
m
@
m
ϕ, namely n. So s
n
♦·ϕ. Further there exists
m s.t. s
n
@
m
ϕ, namely n again. So s
n
♦ϕ.
For the reverse direction, consider the following model: Let
ϕ = Nulla propositio
est negativa.
ϕ ∈ V (s
4
), and s
5
ϕ. Hence, there exists s
4
@
5
ϕ, so s
4
♦ϕ.
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Synthese (2012) 188:487–498
However, s
4
♦·ϕ because neither s
4
@
4
ϕ nor s
5
@
5
ϕ. Let ψ = Omnis
propositio est affirmativa. s
5
@
5
ψ, and ψ ∈ V (s
4
), so s
4
♦·ψ. However,
s
4
ψ, since s
4
ψ.
(b) Let
ϕ be arbitrary and assume s
n
ϕ. Then ϕ ∈ V (s
n
) and for every m, s
n
@
m
ϕ. Suppose s
n
ϕ. Then either ϕ ∈ V (s
n
) or there is an m, ϕ ∈ V (s
m
)
but s
m
ϕ. The first disjunct results in an immediate contradiction. So, fix m.
Then s
n
@
m
ϕ. But then s
m
ϕ, which is also a contradiction. That s
n
ϕ is
obvious from the preceding.
For the reverse direction, let
ψ be as above. Since ψ ∈ V (s
5
) and s
5
ψ, s
5
ψ.
However, s
5
@
4
ψ, so s
5
ψ. To show that s
n
ϕ doesn’t imply s
n
,
consider the following model:
Let
ϕ =Omnis propositio est affirmativa. Then s
7
ϕ, and ϕ ∈ V (s
7
). Since
ϕ ∈ V ( s
6
), s
7
ϕ. However, s
6
ϕ, so s
7
@
6
ϕ. Hence, s
7
ϕ.
Throughout his paper, Prior is scrupulous about keeping meta-language notions such
as truth and possibility strictly within the meta-theory and outside of his syntax, so that
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497
one could not make semantic claims such as “Every propositio is possible” or “Some
negative is not true” within the object language. With the introduction of the possibil-
ity operators into the syntax, there is a natural worry that we have somehow violated
this strict division, and introduced something illicit and potentially problematic into
our logic. This is not the case, because in
L we distinguish between propositiones
and sentences. Propositiones can refer to themselves; they cannot refer to sentences,
and sentences do not refer to propositiones. In this fashion, we are able to provide a
higher layer of formalization extending the formalization originally presented by Prior
without introducing any of the problematic issues he wished to avoid.
This hybridization of the system presented in
Prior
(
1969
) as a way to model
Buridan’s distinction is eminently natural, which makes it noteworthy that there is no
evidence in his paper that Prior ever considered such a development himself. Prior
was translating and reading Buridan, and writing this paper and another on similar
sophisms, at the same time that he was exploring the four grades of tense logical
involvement which led to his hybridization of tense logic.
10
Given how natural it is
extend the notion of sentences on sheets to propositions in possible worlds, and to
formalize how propositiones can make claims about propositiones on other sheets by
means of sheet-indices and operators on these indices, it is plausible that Prior’s work
with medieval discussions of medieval problems provided some inspiration for his
later development of hybrid logic.
Regardless of what the actual relationship between Prior’s work with Buridan and
his development of hybrid logic, the hybridization of Prior’s framework is interesting
not only as a model of a perhaps antiquated medieval theory, but also in its own right.
At the end of his paper, Prior notes that
Part of the interest of these results is that relations between the truth, falsehood,
possible-truth, necessary-truth, possibility and necessity of sentences on sheets
may be thought of as “mirroring” certain relations between features of what is
or is not the case in the world…If there can be a somewhat more sophisticated
semantics than some of the stock ones, there can also be a more sophisticated
modal logic (
Prior 1969
, pp. 491–492).
Hybridization of the sheets-of-paper models is one natural next step towards this
more sophisticated semantics.
11
10
See
Prior
(
1968
) and
Blackburn
(
2006
).
11
Hybrid logic is not the only semantics that can make this distinction. An alternative semantics, called
double-index semantics, which could be used to model this phenomenon was developed and published a few
years after Prior’s introduction of hybridization; Creswell says that “[double-indexing] seems to have been
first investigated by Frank Vach in his UCLA Ph.D., but the first published use is by
Kamp
(
1971
). It was later
used in
Åqvist
(
1973
) and
Segerberg
(
1973
), who called it two-dimensional modal logic” (
Cresswell 1985
,
p. 154). [The references are
Kamp
(
1971
),
Åqvist
(
1973
), and
Segerberg
(
1973
).] However, given Prior’s
close connection with hybrid logic and lack Of evidence that he was familiar with the double-indexing or
two-dimensional approach, we feel that the hybrid semantics is more natural for extending his account of
Buridan’s sophisma. My thanks to the anonymous referee for pointing me towards these references.
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Synthese (2012) 188:487–498
Acknowledgments
The author wishes to thank Peter Øhrstrøm for the invitation to write this paper, and
Benedikt Löwe for encouraging investigation of this topic. The author was partially funded by the project
“Dialogical Foundations of Semantics” (DiFoS) in the ESF EuroCoRes programme LogICCC (LogICCC-
FP004; DN 231-80-002; CN 2008/08314/GW).
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncom-
mercial License which permits any noncommercial use, distribution, and reproduction in any medium,
provided the original author(s) and source are credited.
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