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Abstract

Diodorean modalities are logical notions that specify, in a precise way, how sentences may be true with respect to time: a sentence is diodoreanly necessary at a given instant iff it is true since that instant on. Arthur Prior has treated them as sentential operators and built up a logic for such modalities (DIOD) conjecturing that the frame for such a logic (the "diodorean frame") was the frame for S4. The Conjecture was soon proved false, through a number of counterexamples that played a role in the research on modal logics between S4 and S5. The present paper aims at showing that (i) the search for the diodorean frame benefited from such a research, and that (ii) there has been a mutual interaction between the search of the diodorean frame and some characterisation results. The paper is divided into five parts. In section 1, I will introduce diodorean modalities, while in Section 2 I will be focusing on Prior's reconstruction of the Master Argument and his characterisation of DIOD. In section 3, I present a conjecture Prior advanced about the characterisation of DIOD and some counterexamples to it. The notions of "frame" and "frame for" will be also introduced. In section 4 I summarise the connections between the search of the diodorean frame and some researches in modal logic. Section 5 presents a short conclusion.
47
The Search for the Diodorean Frame
Roberto Ciuni*
ciuniroberto@yahoo.it
ABSTRACT
Diodorean modalities are logical notions that specify, in a precise way, how sentences may be true with
respect to time: a sentence is diodoreanly necessary at a given instant iff it is true since that instant on.
Arthur Prior has treated them as sentential operators and built up a logic for such modalities (DIOD)
conjecturing that the frame for such a logic (the "diodorean frame") was the frame for S4. The
Conjecture was soon proved false, through a number of counterexamples that played a role in the
research on modal logics between S4 and S5. The present paper aims at showing that (i) the search for
the diodorean frame benefited from such a research, and that (ii) there has been a mutual interaction
between the search of the diodorean frame and some characterisation results. The paper is divided into
five parts. In section 1, I will introduce diodorean modalities, while in Section 2 I will be focusing on
Prior's reconstruction of the Master Argument and his characterisation of DIOD. In section 3, I present a
conjecture Prior advanced about the characterisation of DIOD and some counterexamples to it. The
notions of "frame" and "frame for" will be also introduced. In section 4 I summarise the connections
between the search of the diodorean frame and some researches in modal logic. Section 5 presents a
short conclusion.
1. INTRODUCTION
Diodorean possibility and necessity constitute the diodorean modalities, and are defined as
follows: p is diodoreanly possible (since now on, d-possible) at a given instant t iff p is true at t
or at some later instant, and p is diodoreanly necessary (since now on, d-necessary) at a given
instant t iff p is true at t and at every later instant.
d-necessity and d-possibility are comprised in the family of modalities, i.e. those notions
that specify the truth-value of sentences in a non-extensional way. A list of such notions
usually include epistemic and doxastic predicates ("... is believed", "... is known"), notions as
"possible" and "necessary". Remarkably, also tenses are included in the list, since they specify
the way a sentence is true with respect to time.
Today all these notions receive an essentially uniform treatment as sentential operators,
that is operators that transform sentences in other sentences (e.g. "I eat" in "It is possible that
I eat"). Such an approach is due to the work of Saul Kripke
1
, and is considered one of the major
results of contemporary logic. The operators that aim at expressing tenses are called temporal
* Delft University of Technology
1
(Kripke, 1959) is a milestone of modal logic. There, Kripke focuses on the alethic notion of necessity,
but in doing this he provides a semantics that has soon been used for any kind of modality (temporal,
epistemic, doxastic, deontic). The formal tools used to Kripke are now the standard ones in modal logic,
and I will employ them in the present paper.
Humana.Mente Issue 8 January 2009
48
operators, and the logics and languages that comprise them are called temporal logics
2
and
temporal languages. When referring to the field in its entirety, the label "temporal logic" is
often used.
In what follows, we will deal with temporal logic, since diodorean modalities have been
defined on the basis of temporal notions. The temporal language will be given in section 2.1.
Throughout the paper we will also meet logics where the notions of logical or metaphysical
"necessity" and "possibility"
3
are expressed. They are the so-called "alethic (modal) logics". The
main difference between these logics and the temporal ones is that, while
p p
is taken as valid when "" is given an "alethic" reading (as "necessarily"), it is invalid in any
plausible temporal reading
4
. Today, "alethic logics" are simply called "modal logics". This may
create some ambiguities, because temporal logics are often referred to as "modal logics".
In this paper, I will mainly use the label "modal logics" for those logics whose operators can
be read in an alethic way, and "temporal logics" for those that fit with a temporal reading. In
section 3.1 my use of "modal" will be more ambiguous, but what I say there applies to both
temporal and modal logics.
A neat difference in label may not mirror a dramatic divide in the labelled subjects. This is
exactly the case with temporal and modal logics: some alethic logics may be built up as
fragments of temporal logics. In other words, we may take a temporal logic L and define there
a "modal operator" on the basis of temporal operators. Then we may extract a modal logic L’
by taking that fragment of L that contains all and only the sentences where just the modal
operator appear
5
. Thus, there may be no real divide: some alethic logics may have a temporal
reading.
It is easy to see that a logic for diodorean modalities will have this reading, since d-necessity
and d-possibility are defined by temporal notions. However, I consider the diodorean logic as a
fragment where only operators for d-necessity and d-possibility. In the paper I will also
mention two modal logics: S4 and S5. As we will see, S4 has been related with diodorean
2
Some insisted on the opportunity of calling them tense operators and tense logics, in order to
distinguish them from other logics that express relations between instants, or between sentences and
instants) and drop away tenses. These logic would be "temporal logics" (since dealing with time), but
not tense logics. However, today the label "temporal logics" is used for both of them. It is clear by what
follows, that here take into account only logics containing sentential operators for tenses.
3
The notion of a "logical necessity (possibility)" can be characterised as follows: p is a logically necessary
(possible) truth if it follows from (is compatible with) the laws of logic. The notion of a "metaphysical
necessity" strikes many as unclear. Probably, one of the most perspicuous characterisation is: p is a
metaphysically necessary (possible) truth iff p is true in virtue of the objects it is about (if it is compatible
with the nature of the object it is about).
4
"If always (at least once) in the future (past) [it is the case that] p, then p" is clearly false.
5
To be more precise, we take the fragment that contains all and only the sentences where there are just
those combinations of tenses that define the modal operator.
Roberto Ciuni The Search for Diodorean Frame
49
modalities, and both logics are very important in other fields of logics: they have been used to
study the relations between Intuitionistic logic, Classical logic and logics between the two.
As many modal notions, diodorean modalities have philosophical roots. They represent a
view on possibility that was supported in Antiquity, and they was involved in the debates
about determinism and free will. Arthur Prior rediscovered such modalities in the half of the
past century, and propose a formal approach to them
6
.
In what follows, my aim is to highlight the (usually) neglected connections between some
researches in modal logic -in the Fifties and the Sixties- and the search for the diodorean
frame. More precisely, I will show the benefits that Prior's investigation received by researches
in characterisation results (see Section 3) and in the logics between S4 and S5 (that since now
on I will call "intermediate logics"). Before doing this, in Section 2 I present Prior's
reconstruction of the Master Argument, an argument that the greek philosopher and logician
Diodorus Cronus used to support the view that gave rise to the modalities that bring his name.
Some assumptions made by Prior decisively influenced the search for the diodorean frame.
Section 3 presents a conjecture Prior advanced about the characterisation of the diodorean
logic DIOD, and introduces as well counterexamples to this conjecture. This counterexamples
shed light on the existence of previously unknown logics. Section 4 summarises the content of
the paper, and is followed by a short conclusion (Section 5). A last remark before starting. In all
his works, Prior uses the so-called polish notation, a symbolism where logical connectives are
prefixed to the sentences they connect. Such a notation is very hard to read and somewhat
unfamiliar today. For these reasons, in this paper I will use the contemporary notation (with
connectives appearing in the sentences, and not prefixed to them).
2. FROM THE MASTER ARGUMENT TO THE DIODOREAN LOGIC
Diodorean modalities have been named thus after the ancient greek philosopher and logician
Diodorus Cronus, who defended his conception of modalities in an argument that became
famous as "the Master Argument" (see (Denyer, 2009), this volume). By the latter, Diodorus
aimed at showing that the only plausible meaning of "possible" is "true either now or at least
once in the future". The Master Argument is well-known to us for its quite puzzling character:
we have just indirect sources of it, and all of them mention two premises and the supposed
conclusion, without reporting the inference from the former to the latter. This is quite
problematic, since any reasonable derivation of the conclusion from the given premises seems
to require further assumptions.
The premises are:
a) If a sentence p held, then it is a matter of necessity that it held (the past is somehow
necessary).
6
In (Prior, 1955).
Humana.Mente Issue 8 January 2009
50
b) If necessarily q and p q, then necessarily p.
and the conclusion is:
z) if p is and will always be false, then p is impossible.
In other words, by assuming a) and b), it should derive that the notion of "necessity"
appearing in the premises collapses on the notion of "d-necessity". Here, it is worth noting that
the notion of necessity employed in the premises and steps of the Argument is not d-necessity.
Indeed, the aim of the argument is reducing an otherwise characterised notion of modality to
the diodorean one, and employing the latter as "the" notion of modality through the
Argument would jeopardise it with circularity. This fits well with the fact that the Argument
supports the Diodorean Modalities as the right way of conceiving "modalities" (intended on a
more general way), and it does not aim at proposing the definition of the d-modalities.
The success of the Master Argument is ascribable to the fact that a) and b) were widely
accepted by ancient philosophers, and that the lost inference from them to the conclusion was
considered correct. To us, the problematic aspect of the Argument is due to the fact that we
have no direct or decisive evidence for guessing how it should be suitably restored. The
combination of the two things has promoted a variety of attempts to reconstruct the
Argument in order to make its inferences explicit. Even if we restrict ourselves to the past
century, a wide number of such attempts have been proposed, and the debate about them is
lively. In addition, since a decisive evidence about the Argument lacks, it is difficult to foresee a
conclusion.
Reconstructions of the Master Argument have today the form of precise formalisations
7
.
This situation is due to the fact that for about ten years (1955-1965) the Master Argument has
held a main position in the crossover field of philosophy and modal logics. This field was fed by
traditional philosophical problems but was concerned at the same time with the properties of
the formal logics and languages that were employed at that time to shed light on the notions
of necessity, eternity, knowledge and the like. Diodorean modalities and the Master Argument
have been a very specific example of how philosophical problems have been readdressed by
formal tools.
Since we are here interested in diodorean modalities and their connections with the search
on frames for modal logics, I will focus just on Prior's reconstruction
8
. Indeed, it has been the
starting point of a work on a diodorean logic and on the frame for it (see Section 3.1 and 3.2).
7
For this aspect, see (Denyer, 2009), this volume.
8
For comparing Prior's reconstruction with other formal attempts, see (Denyer, 2009), this volume.
Roberto Ciuni The Search for Diodorean Frame
51
2.1 PRIORS RECONSTRUCTION OF THE MASTER ARGUMENT
Arthur Prior is the first to put together the exegetical problem and the tools of contemporary
modal logic. First he deals with diodorean modalities in (Prior, 1955), where he provides a
characterisation of these modalities and his reconstruction of the Master Argument by giving
the guidelines he will always follow thereafter. Then, he comes back on the issue in various
papers or book's chapters (see (Prior, 1958), (Prior, 1962), (Prior, 1957) and (Prior, 1967)),
often correcting or making more precise what he had previously stated.
In order to give a perspicuous and straightforward description of Prior's reconstruction, we
need to introduce a formal tool: a language that is able to express tenses and give the
definition of the diodorean modalities. The temporal language (lT) we need is an expansion of
the language of propositional classical logic by the operators P and F. P and F mean "at least
once in the past" and "at least once in the future", respectively. H and G are their duals
9
, to be
read as "always in the past" and "always in the future". Thus, PGp means "at least once in the
past [it is the case that] always in the future p"
10
. Let me use L for the operator of d-necessity,
that is defined as follows:
Lp := p Gp
L will be the operator of d-necessity (do be defined as M's dual)
11
. (lT) must include as well
a way for expressing the notion of necessity involved in the premises of the Argument. As we
have seen, it cannot be expressed by L, on the pain of circularity. Ancient sources give us no
precise hint on how to interpret such a notion
12
, but it is clear that it should have an intuitive
or theory-laden reading (i.e. it should correspond to some common or philosophical view on
necessity). Indeed, the Argument is interesting as far as it reduces to d-necessity an otherwise
conceived notion of necessity, as I have already suggested above. If this was denied, the
Master Argument would loose any intuitive or philosophical appeal.
Since facing the Master Argument and its problems is beyond the tasks of this work, I will use
here the symbol NEC for expressing the necessity to be reduced, while keeping myself neutral
on the viable interpretations of it. In conformity with the contemporary modal machinery, NEC
will be treated as an operator. In addition it is implicit in Prior's reconstruction (as in any other
one), that the symbol obeys the rules of inference:
9
In symbols: Hp := Pp and Gp := Fp
10
Another example: HFp is "always in the past [it is the case that] at least once in the future p" is true.
Notice that this sentence is nothing but PFp, and its negation is consequently PGp.
11
In symbols: Mp := Lp.
12
In any case, the reduction looks implausible if logical necessity involved in the premises: a) would not
sound feasible, and in any case the conclusion would sound hardly acceptable if it was "p is true by
virtue of the laws of logic iff p is and will always be true". In addition, it is clear that the notion to be
reduced is not the notion of "possibly" as "at least once", because if "necessary" is read as "always",
then the premise a) is patently false: we may have that p was true and that, in some earlier instant, p
had been false up to that instant.
Humana.Mente Issue 8 January 2009
52
RNEC p NECp
(in other words, if p is a theorem, also NECp is ) and:
(Modus Ponens and Uniform Substitution). In symbols, a) and b) become:
a') Pq NECPq
b') NEC(p q) NECq) NECp
a’)-b’) contribute to settle the general framework on which the Argument has to run, and
yet tell us nothing of the "inferential gap" that stands between the Argument's premises and
its conclusion. To restore the Argument, Prior added two premises to those mentioned by the
ancient sources:
c) PG
p p
d) (p G
p) PGp
It is worthy to include premise c) in the set of sentences that should hold under the
Diodorean conception of truth in time. Indeed, it is part of our most basic intuitions about time
that, if once in the past [it is the case that] p is going to be true at any subsequent instant,
then p is true now (otherwise
Gp should hold at any instant previous than now). Things are
not that easy for d), as we shall see below. Once this is settled, Prior's reconstruction reshapes
the Argument as follows:
1
(p q) ((q r) (p r)
by propositional logic
2
(p (q r)) (q (p r))
by propositional logic
3
PGp NECPGp
by a’), with G p substituting q
4
(p G
p) NECPGp
by 1, 3 and d) via MP, with p Gp
substituting P, PGp substituting q and
NECPGp substituting r
5
(p Gp) (NEC(PG
p p))
by c) and 1 via RNEC and MP
6
NEC(PGp p)
by c) via RNEC
7
(p Gp) NECp
by 3, 5, 6, b’) via MP and RNEC
8
Lp NECp
by 7 and the definition of L
MP
(p q) p q
US
p q
where q results from uni form substitution of propositional variables r1, ..., rn in p
with formulae whichever b1, ..., bn
Roberto Ciuni The Search for Diodorean Frame
53
Thus, if a’)-d), RNEC and MP are embarked together, then Diodorus' reduction follows. No
doubt can be cast on the validity of the argument.
Nevertheless, some perplexities may arise if we consider premise d). Indeed, d) is valid only
if time is discrete. Suppose that time is dense or continuous, and that p is false from t on. Now
take any instant t' earlier than t. Since there are infinite instants between t' and t, we cannot
exclude that p is true in one of such instants, say t''. The same for t'' and t, and so on. Thus,
PGp and hence d) are falsified. On the contrary, if time is discrete t must have an immediate
predecessor. The above situation standing, the predecessor of t verifies PGp, since p is false
from t on. Thus d) is verified. Useless to say, the imposition of a discrete time cannot but rise
doubts. However, it seems plausible in a reconstruction of the Argument. Indeed, there is
some evidence that Diodorus proposed a form of temporal atomism that included the
discreteness of time
13
.
It is not the aim of this paper to determine how this should influence our evaluation of
Prior's attempt
14
. The main point here is that discreteness had a major historical role in
dismissing a conjecture that Prior advanced about the frame for a diodorean logic, and that I
will introduce in the next section. Consequently, the acceptance of it had an influence in the
search of the diodorean frame. In other words: the inclusion of discreteness in Prior's
reconstruction of the Argument has been a reason for conceiving diodorean modalities as
satisfying them, and consequently for looking at a frame where the condition is fulfilled.
2.2 THE DIODOREAN LOGIC
On the basis of his reconstruction, Prior outlined a logic for the diodorean modalities, i.e. a
logic where all and only the diodorean tenets (as emerging by Prior's reconstruction) and their
consequences where theorems. This is the main task of (Prior, 1955) and (Prior, 1958), and one
of the main topics in (Prior, 1957) and (Prior, 1967). The logic was meant to be a modal logic
based on a temporal one, and this is one of the reasons for some confusion we find in the
above texts. Indeed, Prior insists on the temporal character of diodorean modalities, but at the
same time the frame he proposes for them (see section 3.1) is not suitable for temporal logics
(for the notion of frame, see again section 3.1). Thus the reader may have the impression that
Prior stresses the "temporal meaning" of diodorean modalities just when he deals with them
in a non-formal way. When formal topics are considered, Prior seems to treat them with no
regard to such a "meaning". This is due to the fact that, when explaining what diodorean
modalities are, he presents them through the notions of presentity and futurity. Otherwise, it
would be difficult to understand the rationale of introducing them among the modal notions.
On the contrary, Prior considered diodorean modalities "in isolation" (as they were joined by
no tense operator or defined by no temporal notion) when he aimed at investigating their
13
See (Denyer, 1981).
14
In any case, Prior's reconstruction is still one of the most convincing. For this, see (Denyer, 2009), this
volume.
Humana.Mente Issue 8 January 2009
54
formal properties. This is clear by the fact that, when Prior writes about the diodorean logic, he
describes it as a logic where just L and the dual M are the operators.
This twofold approach to them should not induce us to believe that a real division holds
here. Indeed, for Prior the diodorean logic should be in accordance with the properties of time
that make the Master Argument valid. This is clear by the fact such an accordance is used by
Prior to admit or dismiss hypothesis on the diodorean frame. In proposing the principles of
such a logic, Prior relied on a very basic intuition about time: the earlier/later relation between
instant is transitive. Obviously, discreteness must be imposed, for the reasons I have suggested
in the above section. Given this, Prior settled the following principles settled for the tense
operators and the diodorean modalities:
AG1 G((p q) Gp) Gq and mirror image
AG2 Gp GGp and mirror image
AG3 (p Gp) PGp and mirror image
AG4 PGp p and mirror image
AG5 Gp Fp and mirror image
AL1 L((p q) Lp) Lq
AL2 Lp p
AL3 Lp LLp
together with the following rules of inference:
MP as above.
RG p Gp
RL p Lp
where the mirror image of a sentence p is the result of substituting any occurrence of P (or
F) with F (or P). AG1 together with RG corresponds to the condition that is usually called
normality
15
, and its presence in DIOD is justified by the fact that it was allegedly accepted by
the greek logicians. AG2 is due to the transitivity of the earlier/later relation on time, while
AG3 expresses in the language the discreteness of time. AG4 is premise c) under the
substitution of p by p. AG5 expresses the infinity of time: if every instant later than t verifies
p, then there is an instant later than t that verifies p. If time had an end, this would not be true:
in this case, t could be the last instant, Gp would be vacuously true at it. Indeed, no instant
later than t would falsify p, since there is no such instant. But for the same reason, Fp would
not be true. The infinity of time may be found questionable. However, Prior explicitly
embarked it
16
, and I will follow him on this point. In dealing with the Argument, Prior does
15
Normal temporal logics are those logics where ((p q) p) q is valid (where is G or H).
16
The matrix Prior uses in (Prior, 1957) to represent diodorean modalities is infinite, and since each
number of the matrix should be read as if it is associated to an instant, we must conclude that the
matrix suggests a reading of time where infinity is comprised.
Roberto Ciuni The Search for Diodorean Frame
55
never hint for some form of "non-homogeneity" between the past an the future, this meaning
that validity might not be preserved by the mirror image of a sentence. Thus, I the mirror
images of AG1-AG5 to be valid. The need of including AL1 among the principles is clear by the
Master Argument: since (NEC(p q) NECq) NECp and NECp Lp are valid in the
diodorean perspective, one can easily infer that (L(p q) Lq) Lp is valid too. But it is
easy to see that the latter is equivalent to AL1. This proves as well the validity of AG1
17
. AL2 is
made valid by the definition of L, since Lq is nothing but p Gp and (p Gp) p is valid. AL3's
validity is due to the definition of L and AG2. The validity of the rules may be maintained on
the ground of what we know about logic in (greek) antiquity (Modus Ponens was universally
taken as a correct rule, a sentence that is proved to be true was taken ipso facto as always
true).
The axiomatic and inferential apparatus settled above is enough to build a diodorean logic.
However, before doing this, something must be said on how Prior read AG4. Suppose that
each instant may be followed by different, incompatible courses of events. Each "course of
events" (or branch) is made by linearly ordered instant and is maximal w.r.t. such instants
18
.
Well, how should we read PGp p in this case? If we conceive time as linear, reading the
sentence is straightforward, but if time branches in the future, the sentence may look
ambiguous. What does its antecedent mean? It means that there is a past such that Gp is true
with respect to a given instant and a given branch (or all branches)? Or does it mean that there
is a past such that Gp is true with respect to a given instant and some branch
19
? According to
Prior, AG4 should be read on a linear time. However, the linearity of time is usually taken as
the main way of representing determinism, that is (in temporal contexts), the view that
DET There is no alternative to what happens, happened or will happen.
In other words, not only the past and the present are beyond any possible attempt to modify
them: also what will happen is completely determined
20
. The link with linear time is
17
Indeed, by AL1 and the definition of L, it derives that (((p q) p) q) (G((p q) Gp) Gq), by
which AG1 follows.
18
This means that if b is a branch, then for every t and t', if they belong to b, they are comparable (i.e.
the one is either earlier, or later than the latter, or they are the same instant).
19
In the first case, AG4 is true, while in the second it is false: if things could have gone as verifying p
forever after a certain instant, this does not mean that they have gone in such a way. Hence we could
have PGp p. Today we have a number of different semantics that allow us to express all this options.
Ockhamist semantics are able to express all options: at a given instant t and w.r.t. the branch b, "Gp" is
read "in every instant later than t and belonging to b, p is true", while "in every instant later than t and
belonging to all (some) b, p is true" is expressed by ¬◊¬Gp (◊Gp), respectively. For these semantics and
their developments, see (Zanardo, 2009) and (Øhrstrøm, 2009), this volume. The first, important work
on semantics for non-linear time has been carried out by Prior. A good overview of this work is present
in (Prior, 1967).
20
When embarking time-reduced modalities as we are doing here, determinism should not be confused
with the idea that Mp Lp. The latter is stronger than determinism, since stating that what happens
now or later, always happens in the future (or that what sometimes happens, always happens, if
"possible" is read as "at least once in time".
Humana.Mente Issue 8 January 2009
56
straightforward: given t and t' , either they are identical, or the one is earlier or later than the
latter, to the effect that any instant is followed only by one "possible development" of the
events
21
.
Determinism and linearity seem far from being conceptually needed in the Argument. Yet
for Prior the aim of the Master Argument "was to refute the Aristotelian view that while it is
now beyond the power of men or gods to affect the past, there are alternative futures
between which choice is possible. Against this, Diodorus held that the possible is simply what
either is or will be true" ((Prior, 1962), p.138). In other words, the Master Argument was as
well an argument for determinism)
22
. As with discreteness, the very important issue here is
that linearity is important to falsify a conjecture by Prior on the diodorean frame. It is for
these reasons that I will assume that the diodorean logic requires linear time.
As a consequence of the above, I call DIOD* the logic resulting by AG1-AL3, RG-RL, the
theorems of propositional classical logic and by accepting DET. Analogously, I call DIOD the
modal logic obtaining by the fragment of DIOD* where temporal operators per se are excluded
(i.e. the fragment where the only operators are M and L
23
). The latter is what Prior calls "the
Diodorean Logic"
24
.
3. FROM THE FRAME FOR S4 TO THE DIODOREAN FRAME
When one builds a logic L, a very natural question is: "which kind of structure does verify all
and only the theorems of L"?. In modal logics, finding an answer to such a question means
finding a characterisation result. After building up DIOD
25
, he proposed a conjecture in (Prior,
1957). A wrong one, as we shall see.
21
It should be noticed that the linearity of time does not imply determinism: if we build a many-valued
logic where a sentence about contingent future events is given an "undefined" truth-value, then we can
endorse linearity while escaping the commitment to DET. Yet, as Prior points out in (Prior, 1955) (p.
211), the task is not straightforward as it seems. In addition, Diodorus and the majority of philosophers
of his time seemed to adopted a two-valued logics. Even in the case of Aristotle (the main philosopher
that could have been open to may values with respect to statements about the future), his endorsement
of a many-value position is far from clear (for this point, see (Mariani 2009), this volume). Consequently,
to the purposes of this work I will accept the idea that linearity gives a good temporal representation of
determinism.
22
In other writings, Prior confronted the Argument with non-linear (and thus indeterministic) time. He
did it by reading the antecedent of AG4 as "in every instant later than t and belonging to some branch b,
p is true", probably because the Argument should have tried to reduce this reading to "in every instant
later than t and belonging to all b, p is true". With such a reading, the Argument turns out to be false.
Obviously, we know (as Prior, actually) that other readings of "at least once in the past, it is always in the
future [the case that] p" make AG4 true in non-linear time. See (Braüner & Øhrstrøm & Hasle, 2000) for
this and others issue concerning Prior's reading of the Argument and non-linearity.
23
Such a choice may look strange, since the two operators conceals temporal ones. However, in this
fragment G and F may not appear alone, but just in sentences p Gp or p Fp. Since Gp and Fp cannot
be disentangled by such sentences, G and F are not here acting properly as operators.
24
Prior called such a logic D, but I prefer not to use that name, since it may cause confusion with the
basic deontic logic, usually called D.
25
A task that he accomplished in (Prior, 1955), even with some difference with my presentation.
Roberto Ciuni The Search for Diodorean Frame
57
3.1 PRIORS CONJECTURE
A structure characterises a (modal) logic iff the former validates all and only the theorems of
the latter. To find a characterisation result may be difficult, but it cannot even be pursued
without setting a precise formal machinery. In investigating the characterisation of a modal
logic, Prior mainly used the device of matrices. Each sentence p is endowed with a sequence of
truth-values 0 or 1. In temporal logics, we may say that this sequence represents the truth-
value of p at the different instants in time. Lp (Mp) is given value 1 in a certain position of the
sequence iff p's value is 1 from that position on (at that position or some subsequent one). If a
sentence is given value 1 in each position of any possible sequence of a matrix, then it is valid
w.r.t. that matrix. We may say that a given matrix characterises a logic L if it validates all and
only the theorems of L.
Matrices have been proven themselves in many formal results about modal logics.
However, they are quite complex to handle, at least if compared with another tool that has
been elaborated for the semantic of modal logic: kripkean semantics
26
. In these semantics,
sentences are interpreted on the basis of a Kripke frame (or simply a "frame"), i.e. a structure
made by a set of points and an accessibility relation imposed on the set. The latter determines
if a given point has, so to speak, access to the information of another point.
To the sake of simplicity, here I will use frames, while neglecting matrices, since this will
make the assessment of the results easier, and will achieve it by a formal tool many readers
are more familiar with.
In the temporal case, sentences are interpreted on frames made by sets t of instants and
the earlier/later relation < ( := T, <). In order to establish the truth-value of the sentences,
we use a function that assigns each sentence p a set of instants (intuitively, the set of the
instants where p is true). We then introduce the function that assigns each pair (sentence,
instant) to a truth-value, according to the condition that a sentence p is true at the instant t iff
t (p):
TC1 (p, t) = 1 iff t (p)
TC2 (Fp, t) = 1 iff t' (t < t' and (p, t') = 1
TC3 (Pp, t) = 1 iff t' (t' < t and (p, t') = 1
The truth-clauses for p or p q (with a dyadic connective) are straightforward, and the
ones for G and H easily derive from TC2 and TC3. A model based on is a pair m := , . A
26
Such formal tools have been introduced by Saul Kripke (in (Kripke, 1959) and (Kripke, 1963)), usually
considered as the founder of contemporary modal logic. Actually, before (Kripke, 1959) was published,
Prior had elaborated a set of truth-clauses for tensed sentences that are similar to Kripke's semantics.
This kind of semantics is also known as possible world semantics. Here, I prefer not to use it, since the
structures employed by this semantics may be made by sets of instants, or event points of space,
depending on the context where the logic has to be applied. The notion of "possible world" is then
unessential to correctly refer to that semantics.
Humana.Mente Issue 8 January 2009
58
sentence p is true in (or verified by) a model m iff it is true at any instant comprised in m, and
false in it (falsified by it) otherwise.
Validity A sentence p is valid w.r.t. a frame iff it is true in any m based on .
I will also say that a frame validates or verifies (falsifies) a sentence p if p is valid w.r.t.
(if some models based on falsifies p). If the relation < comprised in has the property A, we
will say that is an A-frame. Since < is transitive, the frames for the temporal logics are
transitive-frames
27
. Concerning a logic L, I will say that
In A sentence p is in L iff p is a theorem of L (L p, that is, either an axiom of L, or the
transformation of an axiom via the admitted rules of inference).
Here, it is important to notice that we need to adjust the above presentation, if we wish to
deal with DIOD in isolation. Indeed, if we have to consider just an accessibility relation that is
suitable for L, we cannot use <, since a frame including the earlier/later relation would not
verify AL2. Instead, we have to use , the "earlier/later (or identical)" relation. We may think of
as imposed on the set t of instants I have mentioned above. Thus we have that DIOD is T, ,
and the truth-clause for Lp is :
TCL (Lp, t) = 1 iff t' (t t' then (p, t')) = 1
the clause can be easily shown to be equivalent to the one for p Gp if the relation of the
frame is <. The truth-clause for M is straightforward (since M is L). The problem we will
address on this section is: which frame is a frame for DIOD? This meaning nothing but "which
frame characterises DIOD?" Some technical notions are helpful here:
For 1 A frame is the frame for a logic L (L) iff characterises L (relatively to a given
language l).
For 2 The frame for a logic L is the frame for a logic L’ iff it is the frame for L and it is the frame
L’.
28
It is clear that the frames for DIOD are reflexive and transitive (since is). In (Prior, 1955)
(p. 209), Prior had already -correctly- guessed that the diodorean frame verifies all the
27
A remarkable exeption is the frame for the minimal temporal logic, whose theorems do not include a
sentence expressing transitivity. However, the temporal reading of such a logic is somehow
questionable.
28
Please notice that the last definition does not imply that L and L’ coincide: indeed, they may be based
on two different languages, and thus the former has as its frame relatively to the language L while the
latter has as its frame relatively to the language L’. However, if L and L’ are based on the same
languages and is the frame for both, then L and L’ coincide (since they validate the same sentences).
Roberto Ciuni The Search for Diodorean Frame
59
theorems of S4, since the relation in S4 is reflexive and transitive. These conditions correspond
in the logic to AL2 and AL3, that is to Lp p and Lp LLp respectively. In addition, AL1 ((L(p
q) Lp) Lq) is valid w.r.t. to S4. Since the rules of inference are shared by the two logics
and preserve validity, all theorems of DIOD are verified by S4.
In (Prior, 1957) Prior tries to go look beyond this simple result. There, he conjectures that that
frame verified all and only the theorems in DIOD. Rephrasing Prior's investigation in the
terminology and by the tools employed in this paper, we have the following conjecture:
Prior's Conjecture: The frame for S4 is the frame for the Diodorean modalities: DIOD = S4.
The original point of Prior's Conjecture is stating that only the theorems of DIOD are
verified by S4.
With our current knowledge of the frames for modal logic, it is not difficult to foresee that the
conjecture is incorrect. However, it was a reasonable option at those times. Indeed, when Prior
was studying the diodorean modalities, the only known logic between S4 and S5 was S4.5.
Prior knew that such a logic includes a sentence that has no plausible diodorean reading
29
.
Thus, the frame for S4.5 had been immediately excluded. In addition, in those very years S4.5
was later found equivalent to S5 (thus there exists no "frame for" S4.5). The frame for S5 does
not go, since the latter includes Mp LMp, and such a sentence is clearly false in a diodorean
reading
30
. The only candidate left was S4
31
.
3.2 COUNTEREXAMPLES: FROM THE FRAMES FOR S4 TO THE FRAMES FOR DIOD
As we have seen, DIOD was designed by Prior to be a deterministic logic, on the basis of the
idea that DET was essential in the diodorean conception of modalities. It turns out that the
principle, though very vague, has been enough to expose Prior's Conjecture to relevant
counterexamples. Let us consider the following sentence:
lin Mp Mq (p q) M(p Mq) M(q Mp)
It is easy to prove that lin is valid in a frame where the accessibility relation is transitive and
linear. Take a linearly ordered set of instants: if Mp Mq is true at t, then either p q is true at
29
For this, see (Prior, 1967), p.23-24.
30
The fact that now or in the future p is true, does not imply that the same holds for every future
instant. If P is true now and false thereafter, Mp is true, while MLp is false.
31
It should also be considered that modal logic and its formal results were then at their beginnings, and
many issues, though looking obvious today, were still hypothesis waiting for a proof or a
counterexample. In addition, the device of matrices makes it harder to find counterexamples as the one
we have presented. While it is easy for a single researcher to find all them using frames and models, a
much more articulated work is needed if using matrices, and just the contribution of many researchers
may help to find counterexamples in a short time.
Humana.Mente Issue 8 January 2009
60
t itself, or the instant that verifies p (or q) is earlier than the one that verifies q (or p), or
identical to it. This possible combinations give us the consequent of lin.
To see that a non linear frame falsify lin, suppose there is an instant t where M(p q) is
true and (p q) is not. Now take two later instants t' and t'' that are incomparable (they are
not earlier, later or identical one with another), satisfying the following:
(i) In all the instants between t and t' (both excluded), p q is true. The same at all the
instants between t and t'' (both excluded).
(ii) At t' we have that p is true but q is false thenceforth (thus having that Lq is true at t' ).
(iii) At t'' we have that q is true but p is false thenceforth (thus having that Lp is true at t'').
Since t' and t'' are incomparable, (ii)-(iii) are compatible one with another. But as a
consequence of (i)-(iii), our sentence is false. Indeed, Mp Mq is true at t (because p and q are
true at t' and t'', respectively), but (p q) M(p Mq) M(q Mp) is false at t (since p q is
and no instant from t on verifies (p Mq) or (q Mp)). The counterexample shows as well that
there are transitive but not linear. This has tow main consequences.
(l.1) lin is not in S4. Otherwise, the implication from AG2 to lin should be in S4. But this does
not hold, since some transitive frame falsities lin.
(l.2) lin is not valid w.r.t. S4, since there is a model that is transitive and yet falsifies lin (and
since S4 validate all and only the sentences in S4).
Prior's attention on lin was first driven by (Hintikka, 1958) (a review of (Prior, 1957)), where
it is suggested that a temporal interpretation of S4 cannot by given without adding lin to it
32
. In
any case, (l.1) leads to the conclusion that DIOD S4: the frame for S4 is not the diodorean
frame. This
(l.3) led Prior to dismiss his own conjecture in (Prior, 1958), where he explicitly admit that lin
must be in DIOD (in accordance with the links between linearity and DET, see section
2.1)
33
.
32
lin is not the only sentence that readdresses the search for the diodorean frame toward linear frames:
L(Lp Lq) L(Lq Lp) (lin*) requires linearity as well to be valid. The sentence had been pointed out
to Prior by Lemmon (see (Prior, 1958), p.226). Prior later proved that lin and lin* are equivalent ((Prior,
1964)) and that lin* is valid in DIOD. The last proof seems to assume that linearity as a condition that is
plausible for time in se, even out of the diodorean conception of modality.
33
Actually, Prior's position about lin is somehow unclear: in (Prior, 1958) and (Prior, 1967), he defends
its endorsement in DIOD because of its intrinsic "tense-logical plausibility". A consequence is that a
linear (and hence deterministic) representation of time is imposed not by the diodorean logic, but by
what our intuitions about time take to be plausible. If one argues this way, linearity should be suitable
for any temporal logic (DIOD included). However, in this way the Master Argument and the diodorean
Roberto Ciuni The Search for Diodorean Frame
61
(l.4) helped to understand that there is a logic that is stronger than S4 and yet weaker than
S5. Indeed, lin cannot be derived by any axiom of S4 (see (l.2) above). At the same
time, no axiom of S5 can be derived by it.
A new modal logic was de facto discovered through the falsification of Prior's Conjecture.
The new logic was called S4.3 (today the most widespread name for it). Establishing the
fatherhood of the logic is beyond the purpose of this paper. In any case, it should be case that
at least two works reached to lin (or equivalent sentences). One is Hintikka, that simply
mention it as a sentence that is not in S4 (see above), the other is actually a duo: Michael
Dummett and Edward Lemmon, that in (Dummett & Lemmon, 1959) found the sentence
independently from Hintikka and gave the name to S4.3. The interesting thing to notice is that
the work by Dummett and Lemmon focus on intermediate modal logics, and that its rationale
is completely independent from Prior's research. Indeed, the two authors focused on
intermediate modal logics because they can be used for establishing properties of logics that
are stronger than the Intuitionistic one but weaker than the Classical one
34
. Finding out that lin
is not in S4 has been useful for finding one of such logics and extending the class of modal
logics.
Thus, the same discovery had led to a progress both in the search of the diodorean frame
and in our knowledge of intermediate modal logics. The philosophical topic of the diodorean
logic has benefited from research that was undertook for more specific and technical reasons.
(Dummett & Lemmon, 1959) crosses with the search of the diodorean frame also in
another way: it is the first study where it is noticed that S4.3 is not discrete. This is important
for us, since the diodorean logic should go together with the second condition Prior added to
the Argument (that is discreteness).
Now let us take the sentence:
disc (MLp (L(p M(p Mp)) p
It is easy to see that if is non-discrete, the sentence is false, while the discreteness of
makes it true
35
. Indeed, take the situation:
(i') There is an instant t that verifies both MLp and p.
(ii') There is an instant t' such that t t' and that verifies Lp.
(iii') At any instant from t on, p M(p Mp).
conception of modality would cease to be a relevant argument and conception for determinism, in
contrast with (Prior, 1962), p.138.
34
This field of study has its roots in the Gödel-Tarski-McKinsey theorem, that states that a sentence p is
a theorem of Intuitionistic Logic iff its modal translation is a theorem of S4. In those years one of the
main works on the topic was (Dummett, 1959).
35
In (Dummett & Lemmon, 1959) the relevant sentence is: (disc*) (L(L(p Lp) p) MLp) p. The
equivalence of disc and disc* has been proven by Prior in (Prior, 1967).
Humana.Mente Issue 8 January 2009
62
Now, M(p Mp) is true at t, by (i') and (iii'). As a consequence, there must be an instant t''
that verifies p Mp. Such an instant is later than t, since the latter falsifies p. But it is also
earlier than t', since Mp is always false from t' on. For the same reason, there is an instant t'''
between t' and t'' where p is true: it cannot be t'', t' or any instant later than t'', since they all
verify P. At the same time, it must be later than t', in order p Mp to be true there. But in t''',
M(p Mp) is true, by (i') and the fact that t''' verifies p. As a consequence, a further instant
(strictly) between t''' and t' is needed, and so ad infinitum. This is perfectly consistence with
density and continuity, since between any two instant there are infinite instant. Hence the
situation may hold in frames that are based on a dense or continuous . Thus proves that disc
is not valid w.r.t. non-discrete frames. On the contrary, if is discrete there will be a last
instant between t''' and t' . In this last instant, even if having p, M(p Mp) cannot be but
false, since the instant is followed by t' , where Lp is true. As a consequence, if we have MLp
and L(p M(p Mp) at t, we must also have p at t. This shows the validity of disc w.r.t.
discrete frames. This means that:
(d.1) disc is not in S4.3 (for reasons analogous to the ones in l.1).
(d.2) disc is not valid w.r.t. S4.3, since there is a model that is transitive, linear and yet
falsifies lin (and since S4.3 validate all and only the sentences in S4.3).
As a consequence, S4.3 is not the diodorean frame. Here, we have a situation that
resembles the one we had with lin: a new logic was discovered. Or better, it had been clarified
what axioms DIOD needs. And once again, the investigation on diodorean modalities had
benefited from some other researches, namely those on intermediate logics.
However, at this point discreteness is the only condition to be unfulfilled. Thus, it is enough to
added discreteness to a reflexive, transitive and linear frame to have DIOD. This is what Prior
implicitly suggests in (Prior, 1967), p.29. It is clear that such a new frame validate all the
theorems of DIOD. But does it validate only them? In other words: is it a frame for DIOD. Prior
does not prove it in (Prior, 1967), but anyway that was not a conjecture at that time. Robert
Bull had already proved in (Bull, 1965) that the frame for DIOD is discrete, reflexive, transitive
and linear
36
. As a consequence, we may say that (Prior, 1967) (p.29) concludes the search for
the diodorean frame.
Few time later, DIOD resurfaced in the research on intermediate logics. In (Zeman, 1968)
the logic is introduced (together with a cognate logic) with the name most often used today:
S4.3.1. It was already clear that S4.3.1 was discrete. In any case, the success of the name is well
deserved, since it helps in immediately grasping the place DIOD has in the logics between S4
and S5.
36
Two further different proofs of that are given in an unpublished work by Kripke and in (Segerberg,
1970).
Roberto Ciuni The Search for Diodorean Frame
63
4. THE SEARCH FOR THE DIODOREAN FRAME AND MODAL LOGIC
We have seen that the search for the diodorean frame has benefited from two different
researches in modal logics: (i) the research on intermediate logics, and (ii) the research for
characterisation results. Thus shows how the work in progress on technical issues of logics
helped Prior's investigation.
It is now time to see how Prior investigation stimulated some technical result. We may
distinguish two different contributions: (1) indirect ones (mainly to characterisation results),
(2) stimulus to works that explicitly mention diodorean modalities. Let look at them separately.
(1) The direction of the benefits has not been just from characterisation results to the search of
diodorean frame. Prior's conjecture has promoted some researches in that field.
Kripke, in private correspondence, presented to Prior a matrix for S4, and that resembles some
frames for branching time. The issue is mentioned in (Prior, 1967), p.27, and discussed in detail
in (Øhrstrøm & Hasle, 1993). Kripke was also able to find a characterisation result for S4.3.1,
and contributed as well to the falsification of Prior's Conjecture with finding that LMp LMp
that is not valid in the frames for S4 (the proof is straightforward and so I omit it).
As clear from the same correspondence, Kripke's interest in the characterisation for this kind
of logics is rooted in his reading of (Prior, 1957), and on the philosophical relevance of a
temporal interpretation of some modal logics. In particular, Kripke thought that temporal
specifications are not relevant in scientific theories
37
. This shows that his interest to such logics
was linked to the philosophical issues Prior has addressed by using formal methods about
modalities and temporal specifications.
Another result came from Lemmon. In (Dummett & Lemmon, 1959) he presented a
modification of Kripke matrix to verify all and only the theorems of S4.2, that is S4 plus
MLp LMp. In (Prior, 1967), Prior presents the sentence as a result of Lemmon's own work,
and as preceding the work with Dummett. We may hypothesise a connection between
Lemmon's matrix and Prior's work. Lemmon interest in modal logic was triggered by (Prior,
1957)
38
, and in addition MLp LMp had a role in the search of the diodorean frame, since it is
valid in all linear frames and is falsified by the frame for S4
39
.
(2) In addition, some works in modal logic take DIOD explicitly into account. Examples of this
are (Bull, 1965) (already mentioned) and (Makinson, 1966). Beside proving Bull's paper
undertakes an algebraic treatment of all the logics that had been involved in Prior's search (S4,
37
see (Øhrstrøm & Hasle, 1993).
38
To be more precise, it was triggered by the John Locke Lectures that Prior delivered in Oxford (1956).
39
Indeed, MLp LMp expresses the condition of convergence, that is implied by linearity (while the
converse does not hold) and it is not implied by transitivity.
Humana.Mente Issue 8 January 2009
64
S4.3, and obviously DIOD). (Makinson, 1966) shows that infinite non-equivalent formulae are
contained in the sentences in S4.2, S4.3 and DIOD
40
, as it is for S4. Many years later, Robert
Goldblatt applied the notion of diodorean modalities to Minkowski spacetime (see (Goldblatt,
1980)), finding some interesting characterisation results, with the collaboration of Johan Van
Benthem.
We may now sum up what has emerged through the paper. The search of the diodorean frame
has entwined with research of other fields of modal logics through:
(1)
(2)
(3)
5. CONCLUSION
In this paper, I argued that the search for the diodorean frame entwined with the researches
on intermediate logics and on characterisation results, that it has benefited from this, and that
in some cases stimulated them. Thus, the history of the diodorean modalities can be taken as a
fruitful case of interaction between philosophy and logic, and as an example of how
philosophical topics have interacted with technical investigations in modal logic.
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Symbolic Logic, 30/1: 58-64.
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Article
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In the field of logic, our century has seen a most striking rediscovery of the importance of time and tense. This is first and foremost due to the works of Arthur Norman Prior, who took a primary inspiration from his studies in ancient and medieval logic. In the 1950's and 1960's Prior laid out the foundation of tense logic, and showed that this important discipline was intimately connected with modal logic. He revived the medieval attempt at formulating a temporal logic for natural language. Prior also argued that temporal logic is fundamental for understanding and describing the world in which we live. He regarded tense and modal logic as particularly relevant to a number of important metaphysical problems; in particular, Prior analyzed the fundamental question of determinism versus freedom of choice by using the insights gained from the development of formal temporal logic. Tense logic should not be seen merely as a new branch of logic like for instance deontic logic. In Prior's opinion, logic in general should be understood as tense logic. He almost single-handedly introduced and formally developed this conception of logic, which revived ancient and medieval ideas regarding time and logic. For that reason Prior should be regarded as one of the greatest logicians of our century. The relevance of tense logic has now become clear not only to logicians but also to computer scientists and even to some physicists. In this paper we intend to discuss: (1) The history of Prior's first formulation of ideas regarding tense logic in the early 1950's, (2) His idea of a new (or revived) approach to logic, (3) His use of tense logic in metaphysics, (4) His view on the relation between tense logic and natural science in general, (5) The conflict between the tensed view of time and the special theory of relativity. The discussion introduces hitherto unknown or unnoticed aspects on Prior's work, stemming from an interview with Prior's widow, Dr. Mary Prior, and studies of unpublished papers kept at the Bodleian Library, Oxford. But our approach is not only historical. We also intend to
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The Diodorean interpretation of modality reads the operator □ as “it is now and always will be the case that”. In this paper time is modelled by the four-dimensional Minkowskian geometry that forms the basis of Einstein's special theory of relativity, with “event” y coming after event x just in case a signal can be sent from x to y at a speed at most that of the speed of light (so that y is in the causal future of x). It is shown that the modal sentences valid in this structure are precisely the theorems of the well-known logic S4.2, and that this system axiomatises the logics of two and three dimensional spacetimes as well. Requiring signals to travel slower than light makes no difference to what is valid under the Diodorean interpretation. However if the “is now” part is deleted, so that the temporal ordering becomes irreflexive, then there are sentences that distinguish two and three dimensions, and sentences that can be falsified by approaching the future at the speed of light, but not otherwise.
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It is well known that the modal calculus S4 has infinitely many non-equivalent formulae in a single proposition letter (in standard terminology, infinitely many modal functions ), whilst S5 has only finitely many. However, the situation regarding the intermediate modal calculi S4.2, S4.3, and Prior's Diodorean tense-logic D does not seem to have been settled. In this note we show that each of these systems, together with a certain proper supersystem D* of D, has infinitely many modal functions. This is in contrast with the fact that in the intermediate propositional logics KC and LC, which correspond under the McKinsey-Tarski translations to S4.2 and S4.3, there are only finitely many non-equivalent formulae in a single proposition letter.
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Attention was directed to modal systems in which ‘necessarily α ’ is interpreted as ‘ α . is and always will be the case’ by Prior in his John Locke Lectures of 1956. The present paper shows that S4.3, the extension of S4 with ALCLpLqLCLqLp , is complete with respect to this interpretation when time is taken to be continuous, and that D, the extension of S4.3 with ALNLpLCLCLCpLpLpLp , is complete with respect to this interpretation when time is taken to be discrete. The method employed depends upon the application of an algebraic result of Garrett Birkhoff's to the models for these systems, in the sense of Tarski. A considerable amount of work on S4.3 and D precedes this paper. The original model with discrete time is given in Prior's [7] (p. 23, but note the correction in [8]); that taking time to be continuous yields a weaker system is pointed out by him in [9]. S4.3 and D are studied in [3] of Dummett and Lemmon, where it is shown that D includes S4.3 and CLCLCpLpLpCMLpLp . While in Oxford in 1963, Kripke proved that these were in fact sufficient for D, using semantic tableaux. A decision procedure for S4.3, using Birkhoff's result, is given in my [2]. Dummett conjectured, in a conversation, that taking time to be continuous yielded S4.3. Thus the originality of this paper lies in giving a suitable completeness proof for S4.3, and in the unified algebraic treatment of the systems. It should be emphasised that the credit for first axiomatising D belongs to Kripke.
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§1. In [1] Gödel proves the non-existence of a finite matrix characteristic for the intuitionist propositional calculus IC by the use of the finite matrices , where n is a natural number and
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The present paper attempts to state and prove a completeness theorem for the system S5 of [1], supplemented by first-order quantifiers and the sign of equality. We assume that we possess a denumerably infinite list of individual variables a, b, c, …, x, y, z, …, x m , y m , z m , … as well as a denumerably infinite list of n -adic predicate variables P ⁿ , Q ⁿ , R ⁿ , …, P m ⁿ , Q m ⁿ , R m ⁿ ,…; if n =0, an n -adic predicate variable is often called a “propositional variable.” A formula P ⁿ ( x 1 , …, x n ) is an n -adic prime formula; often the superscript will be omitted if such an omission does not sacrifice clarity.