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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 PRIOR’S 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 PRIOR’S 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.

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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).

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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)

benefits from the research on intermediate logics, as witnessed by the fact that works in

that field contributed to falsify Prior's Conjecture ((Dummett & Lemmon, 1959)).

(2)

interaction with characterisation results, as witnessed by the fact that (a) the result in

(Bull, 1965) ensures that the frame for DIOD is reflexive, transitive, linear and discrete, a

result that Prior acknowledged in (Prior, 1967), p.31, (b) the research on characterisation

results for S4 and S4.2 by Kripke and Lemmon (respectively) was probably motivated by

Prior's Conjecture or by other issues addressed by Prior.

(3)

explicit consideration in technical works on modal logics, as shown by a variety of studies

that focuses on logics that extend S4. In these studies, the modalities under account are

called "diodorean modalities".

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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