Existential Graphs and Tense Logic

Peter Ohrstr0m Department of Communication

Aalborg University Langagervej 8

9220 Aalborg Ost Denmark

e-maih poe@hum.auc.dk

Abstract. In this paper it is argued that the founder of modern tense logic, A.N. Prior was very much inspired by the logic and philosophy of CS. Peirce. It is also demonstrated how the most important tense logical systems of Prior, K t, K o and K b, can be reformulated in terms of the existential graphs invented by Peirce.

In the 1950s and 1960s A. N. Prior succeeded in re-establishing a logic of time and tenses. It is obvious that the study of Peirce's philosophy meant a great deal to Prior and in his brief presentation of the history of the modern tense logic, to be found in the appendix to Prior's first great tense logical work Time and Modal i ty [1957], elaborations of the importance of Peirce for the logic of time makes up about one fourth of the work. On the basis of his studies of Peirce's philosophy Prior even proposed a specific Peircean system of logic [Prior 1967, p. 132]. This system he evidently found very attractive.

There are many indications that Peirce, as one of the earliest modern philosophers, realised that tenses could, and even should, be reflected in our logic. He formulated his position in the following way: "Time has usually been considered by logicians to be what is called 'extra-logical' matter. I have never shared this opinion." [CP 4.523] As it appears from this statement, Peirce made himself spokesman for an open and undogmatic understanding of logic. This openness, which was obviously due to his extensive knowledge of classic and scholastic logic, also meant that he would not accept a logic which conceived of truth as timeless. He could easily imagine that some later development of logic would take time seriously. Such a logic would just follow the lead of Aristotelian and Scholastic logic by taking seriously such expressions as 'E will happen', 'E happened', 'Y happened when X started'. Peirce, however, held that around the turn of the century logicians were not ready to (re)introduce time in logic: "But I have thought that logic had not yet reached the state of development at which the introduction of temporal modifications of its forms would not result in great confusion; and I am much of that way of thinking yet." [CP 4.523]

The human mind is tense-oriented, i.e. thinking in tenses - past, present and future - is essential to the human mind. According to Peirce the future "appears in mental forms, intentions and expectations", whereas memory supplies us with a knowledge of the past [CP 2.86]. Peirce's thinking makes it natural to link the tenses with the concepts of modality. He stated explicitly that the fact that "time is a particular variety of objective Modality is too obvious for argumentation" [CP 5.459]. Modal notions seem to be essential for the understanding of the Peircean ideas which came to be a great inspiration for A. N. Prior in his development of

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temporal logic. In particular, the definition of the possible was very essential to Peirce. Early in his authorship he defined it using semantically negative expressions, but later he emphasised the positive character. On the 18th of March 1897 Peirce wrote: ".. my old definition of the possible as that which we do not know not to be true (in some state of information real or feigned) is an anacoluthon. The possible is a positive universe, and the two negations happen to fit in, but that is all ... I found myself arrested until I could form a whole logic of possibility, - a very difficult and laborious task." [CP 8.308]

Later he formulated something similar even clearer: "Potentiality is the absence of Determination (in the usual broad sense) not of a mere negative kind, but a positive capacity to be Yea and to be Nay; not ignorance but a state of being ... Actuality is the Act which determines the merely possible ... Necessitation is the support of Actuality by reason ..?' [MS 277, 1908]

1. Prior's Tense Logic in a Peircean Context Prior's tense logic Kt includes two operators P and F, corresponding to past and future respectively (see [~hrstrCm & Hasle 1995, p.203 ff.]). From these the dual operators H and G, may be defined by means of H A - ~P-A and GA--- -F~A. The axioms of Kt are

(A1) G(A DB) ~ (GA D GB) (A2) H(A DB) ~ (HA D HB) (A3) A D HFA (A4) A D GPA

The rules of inference in K t are modus ponens and necessitation for H and G. It is well known that Kt corresponds to a class of temporal models (TIME,<) in

which no restrictions on the before-after relation < is made. When dealing with the question regarding the topological structure of time Peirce was very open- minded. He even accepted the possibility of a "sudden stoppage of everything" [CP 4.547]. In this context he did not accept the supposition of a law of nature which would forbid the closure of time. It is now well known that the claim that there wilt be no last time is equivalent to the further assumption that, GA ~ FA, is a thesis.

Peirce's views on the relation between time and cognition as well as his idea of time in general are very complex, and it must be admitted that his statements do not unambiguously point in any single direction. A few quotations will illustrate that: "For time is itself an organized something, having its law or regularity; so that time itself is a part of that universe whose origin is to be considered. We have therefore to suppose a state of things before time was organised. Accordingly, when we speak of the universe as 'arising', we do not mean that literally." [CP 6.214] And: "The idea of time must be employed in arriving at the conception of logical consecution; but the idea once obtained, the time element may be omitted, this leaving the logical sequence free from time. That done, time appears as an existential analogue of the logical flow. '~ [CP 1.491]

Statements like these are typical for Peirce's philosophyl They form a good inspiration for further speculation regarding the concept of time. It must be admitted, however, that his ideas of time become very complicated when it is

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added that Peirce sometimes argued in favour of the possibility of what Milic Capek [1991, p.265] has termed a 'self-returning nature of time'. Peirce stated: "The other question is whether time is infinite in duration or not. I f it has no flaw in its continuity, it must, as we shall see in Chapter 4 return to itself. This may happen after a finite time, as Pythagoras is said to have supposed, or in infinite time, which would be a doctrine of consistent pessimism." [CP 1.498] Peirce formulated similar views in [CP 1.498] and in [CP 6.210]. It is very hard to see how the idea of eternal recurrence could ever fit with the rest of the Peircean thinking. Peirce may in fact have had problems with this question himself, since the 'Chapter 4' to which he refers in the above quotation was apparently never written. However, in 1902 he wrote the following in his application to the Carnegie Institution for support for his research in logic: "For example, many persons would say that a man's being the father of his own father was inconceivable. But there are various ways in which such an event may be conceived, as, for example, by simply supposing that all time forms a closed cycle which the two lives completely exhaust. Certainly, unless there is some abstruse reason to the contrary which does not at once strike me, it is quite possible that, as a matter of fact, time does form a closed cycle. At the same time, until some positive reason shall appear for believing that it is so, it will be shown in another memoir that we are justified in disbelieving it..." [Manuscript L75, Draft D (315-324)] So, it appears that although Peirce maintained that there is no a priori argument against time as a closed cycle, he also argued that the burden of proof falls on anyone who will claim such a view.

However, even if there should be an ending time or even if time should in fact be viewed as a circle, there can be little doubt that the temporal ordering should be conceived as transitive. It is easily verified that if we add the axiom, FFA D FA, to K t we get the tense logic Kc, which corresponds semantically to the transitive models. But Peirce clearly wanted to go even further than that. He maintained that the past can be characterised as factual. According to Peirce a fact should be understood as a "fait accompli; its esse is in praeterito" [CP 2.84]. Hence facts of the past should be viewed as now-unpreventable. The future is in Peirce's opinion quite different, since "all our knowledge of the future is obtained through the medium of something else" [CP 2.86]. The medium mentioned here might perhaps be some natural law. That is, in some cases the future can be present in its causes, and in these cases we can have knowledge of the future if we know the causes. In other cases we must confine ourselves to other kinds of law-like statements. It should, however, be mentioned that Peirce was not ready to conceive of natural laws as being at the same level as logical laws.

We see that Peirce was clearly in favour of an asymmetrical conception of time. He further distinguished between three modes of being: "My view is that there are three modes of being. I hold that we can directly observe them in elements of whatever is at any time before the mind in any way. They are the being of positive qualitative possibility, the being of actual fact, and the being of law that will govern facts in the future." [CP 1.21-1.23]

Thus the three modes of being in Peirce's philosophy are: actuality, possibility and necessity. With respect to tenses, 'actuality' covers both the past and the present. We cannot (any longer) influence the forming of the past, But Peirce thinks of the future as a possibility sphere with certain predetermined incidents (necessary, or determined by law). At the same time he rejects the idea that the

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truth about the contingent and (undecided) future could be known by now. In his own words: "The Past consists of the sum of faits accomplis, and this Accomplishment is the Existential Mode of Time .. . . the Mode of the Past is that of Actuality. Nothing of the sort is true of the Future .... (The future) is not Actual, since it does not act except through the idea of it, that is as a law acts; but is either Necessary or Possible..." [CP 5.459]

Peirce did not define the past as 'necessary', but reserved this definition for 'the preordained'. However, he maintained that psychologically the relationship of the present to the past and the future, respectively, are decisively different. Whilst in principle any past event belongs to the domain of the memory we have no possibilities of obtaining a similar insight in future events. On the contrary, the future lies openly before use enabling us to influence the forming of the future events within certain limits. A similar possibility of influencing the forming of the past events is not available. Peirce formulated this in the following way: "I remember the past, but I have absolutely no slightest approach to such knowledge of the future. On the other hand I have considerable power over the future, but nobody except the Parisian mob imagines that he can change the past by much or by little." [CP 6.70] And Peirce wrote: "A certain event either will happen or will not. There is nothing now in existence to constitute the truth of its being about to happen, or of its being not about to happen, unless it be certain circumstances to which only a law or uniformity can lend efficacy. But that law or uniformity, the nominalists say, has no real being, it is only a mental representation. If so, neither the being about to happen nor the being about not to happen has any reality at present ..." [CP 6.368] It may be concluded that according to the Peircean view of time the future is open in the sense of offering a branching of possibilities, whereas the past as now-given can be modelled as a single line. If we add the axiom

FPA D (FA v P A v A)

we get the tense logic Kb, in which the before-after-relation is backwards linear. For this reason it seems reasonable to assume that a Peirce tense-logic will have to include at least the axioms of K b, where F is interpreted as 'possible in the future' (Further details on Peircean tense logic can be found in [OhrstrCm & Hasle, 1995, p. 211 ff]).

Semantically, the system Kb corresponds a model of branching time formally invented by Saul Kripke in 1958, although the idea can in fact be said to be much earlier (see [OhrstrOm & Hasle, 1995, p. 189 ff]). It is very interesting that Peirce comes close to the idea of branching time when discussing the possibility of 'thinking machines': "If such a machine were constructed, it would have to supply not only the logic, but also would have to determine arbitrarily what one out of an infinity of logical conclusions should be drawn first. In the logic of relatives, choice has to be made between different lines of reasoning equally logical and equally direct. Therefore, until you can make time branch, or think two different things together, it is absurd to talk of a development being determined by logic alone." [Quoted from Peirce Project Newsletter, Vol. 1, No. 2, June 1994, Indiana University]

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It should be noted that this thought is rather similar to ideas involved in the modern treatment of parallel systems with temporal logic based on branching time semantics (see [0hrstrCm & Hasle, 1995, p. 344 ff]).

I f tense logical systems are to be represented in a proper Peircean manner, there can little doubt that one ought to make the attempt in terms of the existential graphs which Peirce invented in 1896 and which he continued developing in the following years.

Peirce was obviously aware of the fact that tempo-modal problems cannot be handled in a satisfactory way in terms of Alpha or Beta graphs (see [Ohrstr0m et al. 1994]). In order to obtain a graphical representation of logics with operators we have to introduce the so-called Gamma graphs which involves at least one new kind of cut. Peirce himself admitted that the Gamma graphs are unfinished: "The Gamma part has been a good deal considered, but has not been settled . . . . What the Beta part of existential graphs fail to express is Possibility and Necessity. Subjective possibility is simply the character of that which we do not know to be false." [MS 507]

It should be noted that the Gamma graphs were invented at a time when Peirce still preferred a 'negative definition' of possibility. In the following we are going to suggest graphical systems corresponding to K t, Kc, and K b.

2. The minimal Tense Logic Kt The Gamma graphs may be seen as a generalisation of the Alpha graphs. It is well known from studies of Peirce's Alpha graphs that propositional logic is equivalent to a system based on the empty sheet of assertion (SA) as its only axiom supplemented by the following five rules:

RI ' . The rule of erasure. Any evenly enclosed graph may be erased. R2'. The rule of insertion. Any graph may be scribed on any oddly enclosed area. R3'. The rule of iteration. If a graph Gr occurs in the SA or in a nest of cuts, it may be scribed on any area not part of Gr, which is contained by the place of Gr. R4'. The rule of deiteration. Any graph whose occurrence could be the result of iteration may be erased. R5'. The rule of the double cut. The double cut may be inserted around, or be deleted from, any graph on any area.

For a traditional modal logic a single extra kind of cut will do (see [van den Berg 1993]), but if we want a system of graphs corresponding to a Priorean tense logic we shall need two extra kinds of cut. It would be in agreement with Peirce's approach to the Gamma graphs to use colours in order to distinguish between the three kinds of cut. For practical reasons, however, we shall represent the three kinds formally using a linear notation as follows:

(f-cut): future negation (corresponding to the tense logical F-) (p-cut): past negation (corresponding to the tense logical P-)

The traditional negation may be likewise presented as

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(n-cut): present negation

Graphically, the three kinds may be represented as follows:

n-cut f-cut

/ \ \ t

p-cut

' )

It is also possible to introduce a number of complex cuts like

nfp-cut:

The n-cut is clearly the cut involved in (RI'-R5'). It turns out, however, that in the context of K t the rules of erasure and insertion can be applied to any graph. We shall call the generalised rules (R1) and (R2). The generalisation of the rule of iteration (and deiteration) is more complicated since inferences like [A,{B}] -> [A,{A,B}] are invalid in K t. For this reason we restrict the rule to:

R3. The rule of iteration. If a graph Gr occurs in the SA or in a nest of cuts, it may be scribed on any area not part of Gr, which is contained by the place of Gr and only separated from it by n-cuts.

We shall use the name R4 for the rule of deiteration corresponding to R3. The rule of double negation is only valid for n-cut. In consequence, we shall use the restricted version of the rule:

R5. The rule of the double n-cut. The n-double cut may be inserted around, or deleted from, any graph on any area.

Using the rules (R1-5) it is possible to derive a number of other rules like implicative transitivity i.e.

Transitivity. If [A,[BI] and [B,C] are both provable graphs, then [A,C] will also be a provable graph.

In addition to (R1-5), the following rules are easily seen to be valid in the system:

R6. The rule of modal conversion. An evenly enclosed n-cut may be replaced by a nfp-cut or a npf-cut. An oddly enclosed nfp-cut (or npf-cut) may be replaced by an n-cut.

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RT. The necessitation rule. If the graph q is provable then the graphs { [q] } and ([q]) are also provable. R8. The distribution rule. In any context, the graph [{p,q}] may be replaced by the conjunction of the graphs [{p}] and [{q}], and this conjunction may be replaced by the graph [{p,q}]. Likewise, the graph [(p,q)] may be replaced by the conjunction of the graphs [(p)] and [(q)], and this conjunction may be replaced by the graph [(p,q)].

It is easy to verify that (R6) implies that any evenly enclosed fp-cut or pf-cut may be removed. It also follows that (R8) means that if [A,[B]] is a provable graph, then the graphs [[(A)],[[(B)]]I, [[{A}],[[{B}]]], [([A]),[([B])]], and [{ [A] },[{ [B] }]] are also provable. One may speak about distribution of rip-, nf-, pn- and fn-cuts over the implication [A,[B]].

The validity of (R6) follows from the axioms (A3) and (A4), whereas the validity of the distribution rule is a consequence of the equivalences

H(A A B) = (HA A HB) G(A A B) = (GA ^ GB)

which may be proved in Kt. Conversely, in a system of existential graphs with the rules (R1-8) and the empty SA as an axiom, all the axioms and rules in K t can be derived. Modus ponens, necessitation and the axioms (A3-4) follow immediately from the (R1-8) and the empty SA. In this system it is possible to derive the graphs corresponding to (A1) and (A2). This can be done in the following way:

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By (R4) & (R5)

By (R1 & (R5)~........~

It is easily verified that this graph corresponds exactly to the axiom (AI). Following a similar line of reasoning the graph corresponding to (A2) can also be deduced.

In consequence it may be concluded that the system of existential graphs with SA as an axiom is equivalent with Kt.

3. The Causal Tense Logic, Kc Now let us turn to the tense logic Kc. The crucial axiom FFA D FA, corresponds to the graphical equivalent:

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It is easy to see that this axiom corresponds to the rule:

R9. The duplication rule. An evenly enclosed nf-double cut around a graph may be duplicated. An evenly enclosed duplication of a fn-double cut around a graph may be replaced a single fn-double cut.

It is well known that the axiom FFA ~ FA, semantically corresponds to the transitivity of the earlier-later-relation. It seems that this property could just as well be expressed in terms of the axiom PPA D PA, or the corresponding graph. This graph can, however, be proved using (R1-9). The Kc-axiom corresponding to (R9), can be transformed into the following graph using the rule of np-distribution and substituting [(A)] for A:

By ( R 7 ) ~ [ ~ , ~ ~ \ ~

By (R6) & ( R 5 ) . ' ~ ~ /

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This graphs corresponds to the axiom, PPA D PA. The proof seems to be somewhat shorter and also more straight forward than the corresponding demonstration in ordinary tense logic.

Using the rules (R1-9) a number of interesting theorems may be proved. The proofs of the following two lemmas are left to the reader.

m m

4. The Backwards Linear Tense Logic K b In Kb the crucial axiom is FPA D ( F A v P A v A). It is easy to see that this axiom can be expressed in terms of the following rule for existential graphs:

R10. In an evenly enclosed area the conjunction of the graphs A, [{A}], and [(A)] can be replaced by [{[([A])]}].

Sometimes the axiom

(PA/x PB) ~ (P(B A A) v P(B A PA) v P(PB A A))

J

is presented as the crucial axiom for K b. It must be admitted that this axiom appeals to backwards linearity in a very obvious way. In the following we shall prove the axiom in the graphical system with the rules (RI-10) . - From the axioms we immediately deduce:

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1

214

By the rule deiteration:

/

i

f f

A

r

B

By (R10):

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By the rule of distribution:

.<.. it@)) ) ",) V.-:~c2/ \x,j//

By the rules of modal conversion and double cut:

/ !

\

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By lemma 1 and the rule of transitivity:

/ /

By lemma 2:

This graph corresponds exactly to

(PA ~', PB) D (P(B ~\ A) v P(B A PA) v P(PB ,x A))

The above proof may be compared with the one given by Prior, cf. [OhrstrCm & Hasle p. 207 ff.]. The number of steps needed in the present graphical proof is a bit smaller than in Prior's proof, but it is a matter of taste whether it is easier to follow.

5. Conclusion Peirce described the situation concerning the study of Gamma graphs as follows: "... the gamma part is still in its infancy. It will be many years before my successors will be able to bring it to the perfection to which the alpha and beta parts have been brought. For logical investigation is very slow, involving as it does the taking up of a confused mass of ordinary ideas, embracing we know not what and going through with a great quantity of analyses and generalizations and experiments before one can so much as get a new branch fairly inaugurated." [CP 4.5111

As he suspected the task was to be completed by his successors. In the present paper it has been studied how some of the new tempo-modal logics can be represented in terms of his Gamma graphs, and rules corresponding to the systems Kt, Kc, and Kb have been suggested.

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Even if it turns out to that such graphical systems are neither easier to use nor more intuitive than the corresponding formalisms in terms of traditional tense- logical systems, it may still be an interesting transformation, since it makes it possible to incorporate tense-logical reasoning in modern systems based on existential graphs. The present version of tense logic may also give rise to an interesting temporal reasoning component in systems of conceptual graphs such as that ofB. Moulin and D. Cot6 [1992].

The tense-logical approach may be seen as an alternative to the use of temporal frameworks or time coordinate systems, cf.. [Moulin 1994]. In such systems, temporal objects (instants or durations) have a crucial rrle to play. In ordinary tense logic there is no need for temporal objects and for quantifications over such objects. In an extended system, however, temporal objects may be introduced as some rather complicated kind of propositions, cf. [~hrstr~m & Hasle p. 216 ft.]. There is still a lot to be done before such a system can be constructed in terms of existential graphs. But when such a system has been established it will be possible to compare the tense-logical approach in this paper with the other graphical representations of temporal reasoning.

References: Berg, Harmen van den: 1993b, 'Modal Logic for Conceptual Graphs', in Mineau, Guy W.; Moulin, Bernard; Sowa, John (editors), Conceptual Graphs for Knowledge Representation, Springer-Verlag, pp. 411-429. Capek, Milic: 1991, The New Aspects of Time, Boston Studies in the Philosophy of Science Vot. 125, Kluwer Academic Publishers. Keeler, M. The Philosophical Context of Peirce's Existential, University of Washington, Seattle. Moulin, B..; Cotr, D.: 1992,'Extending the Conceptual Graph Model for Differentiating Temporal and Non-Temporal Knowledge', in [Nagle et al.], pp. 381-389. Moulin, B..: 1994, 'The Temporal Structure of a Discourse and Verb Tense Determination', in W.M. Tepfenhart et al. (eds.), Conceptual Structures: Current Practices, Springer-Verlag 1994, pp. 45-68. Nagle, T.; Nagle, J.; Gerholz, L.; Eklund, P. (editors): 1992, Conceptual Structures, Current Research and Practice, Ellis Horwood Workshops. Peirce, C.S.: Collected Papers, 8 volumes (eds. P. Weiss, A. Burks, C. Hartshorne), Cambridge: Harward University Press, I931-1958, (CP). Peirce, C.S., Application to the Carnegie Institution , July 15, 1902, Manuscript L75, Analytical reconstruction and editorial work by Joseph Ransdell, Indiana University, 1994. Prior, A. N.: 1957, Time and Modality, Oxford. Prior, A. N.: 1967, Past, Present and Future, Oxford. Roberts, Don D.: 1973,The Existential Graphs of Charles S. Peirce, Mouton. ~hrstrCm, P., van den Berg, H., Schmidt, J.: Some Peircean Problems Regarding Graphs for Time and Modality, Second International Conference on Conceptual Structures, University of Maryland, 1994, p.78-92. OhrstrCm, P. & Hasle,P.F.V.: 1995, Temporal Logic. From Ancient Ideas to Artificial Intelligence, Studies in Linguistics and Philosophy 57, Kluwer Academic Publishers.