Between the Time of Physics and the Time of Metaphysics, the Time of Tense Logic? 2 The Pythagorean Table of Physics

According to Russell, Being is divided into atemporal universals and temporal particulars. But, to the extent that the Antisthene ’ s caballeity pops up in the topos noeton , the ancestors of the horse must precede the horse in the sublunar. Thus, Being must be divided into Sosein , Dasein and Zeitsein. Only the Whitehea-dian “ eternal objects ” , such as geometrical forms and colours, are atemporal universals, while caballeity is in the Zeitsein together with Conquérant (one among Napoleon ’ s horses). In this ontology, according to the Lautmanian shift, Spatiotemporality pertains to the Sosein and has a two-fold structure: logical and mathematical. The tense logic of A.N. Prior describes it, while its link with the mathematics of spacetime is fixed by the Boolean kernel in the von Neu-mann algebra for quantum logic. This speculation will be preceded by a narra-tive exposition of elementary tense logic. time? for invi-tation and generous hospitality. Great thanks due for his marked concerns, to Elie During for the courtesy of his discussion and to Alessandra Campo for the truth of her philosophical enthusiasm. A special thanks is for


The Leibnizian Foundation
In ad raft² written in 1671 or 1672, Leibniz discloses, in Russelliant erms,t he structural similarityb etween three versions of the Apulean square AEIO.T his is the Leibnizian This table of similarities has two main consequences.The first is the well-known thesis thatan ecessary truth is true in all possiblew orlds.³ In symbols: p ⇔ w |w p ("Necessarily p iff for each world w it is true at w that p") This Leibnizian analysis of necessity (generalizable to the other modes of the square) is of such significance thati td eserves an ame. Iw ill call it Leibniz's Bridge. It is ab ridge because it connects al eft-side modality with ar ight-side quantifier.I nC arnapian terms, modality is the explanandum and quantification is the explanans. Of course, the quantifier operates in the possible worlds of Molina, dubious entities surrounded by am etaphysical flavour.H owever,i nt he view of such logicians as Carnap, quantification theory enjoys al ess controversial status than modal logic.

The KripkeanM athematization
With respect to modal logic, we must distinguish between its axiomatization and its mathematization. The well-known axiomatization of modal logic was laid down in its main lines by C.I. Lewis in 1932 (Lewis and Langford1 932).B ut the mathematization of modal logic is another story.This concept was introduced by Robert Goldblatt (2005) and the understanding of its true meaning requires a philosophical elucidation.
The mathematization of modal logic has at least twob irths. The first,a swe shall see, is probablyd ue to ArthurP rior.T he second, instead, is due to Saul Kripke,⁴ the philosopher who provided its canonical formulation. The cornerstone of the latter is a Kripke-model or K-model. AKripke-model Mmust be written in two main versions. The original (Kripke 1963)⁵ one is: M=〈〈W, P〉,V 〉  Here we cannot discuss the question of priority between authors.I na ddition to the already mentioned paper by Goldblatt on the evolution of mathematical modal logic, see Copeland (2002).A nd it is enough to sayt hat the mathematizations of Prior and Kripkea re independent from each other.The publication of the results of the young Kripke'sw ork has been postponed because of bad advicesf romh is academic masters.  Original version but not in literalterms. The essential point is the differencebetween this original version, with its philosophical significance, and the new one which has vanished in the rarefied air of abstract formalism.

Between the Time of Physicsa nd the Time of Metaphysics
In this version of the model M, W is aset of possibleworlds w, P is the relation of relativep ossibility between possiblew orlds,a nd Vi st he so-called "function of valuation" which takest he proposition p of al anguageLas argument and the subsets of W where p is true as values.
From atheoretical point of view,inKripke'smodel the two members have a very different status (Blackburn, De Rijkeand Venema 2001, Df 1.19,17). The couple 〈W, P〉,i.e., the frame of the model, givesthe mathematical picture of its ontology,while the valuation function Vgives the additional information endowing this ontology with alogic. In Husserlian terms, which are derivedfrom Aristotle, one can sayt hatt he frame is the Ontology of the model and the valuation is its Apophantics. In the frame, usually,t herei sn or elation of "relative possibility", but ar elation of accessibility. This, at least,i sw hat Peter Geach suggested to Prior in aletter of 1960.Yet,relative possibility and accessibility are the converse of one another: w has access to w' if and onlyi fw 'is possible relatively to w.
Therefore, since modal logic has evolved, we maywrite Kripke'smodel in the following terms: In this new version E is any non-emptyset,whose members are not onlythe old "worlds" of Molina or Leibniz, but as well "points",instants, words, or whatnot; and R is any binary relation. This acceptationd efines an abstract or "formal" Kripke model. Now,when modallogic is endowed with aKripkemodel, what is the effect of this model on those concepts of modal logic illustrated by the definition of necessity thatw ef ind in Leibniz? The definition of necessity becomes M|v p⇔ ww Pv → M|w p ( " In Mitistrue at v that necessarily p iff for all w,ifwis possible relatively to v, then in M it is true at w that p") In other words, a footbridge is added on the right side of Leibniz'sbridge. It conditionsorcontrols the relevanceofpossibleworlds in the definition of amodality as ac onstraint imposed on relative possibility,s ot hat the relation R of an abstract model can be said to be ar elation of "parametrization". Now,why is this footbridge or parametrization a mathematization of modal logic?Acrucial example willhelp us to answer this question. Yet, in order to find it,w eh avet or eturn to the second modald iscovery made by Leibniz, i. e., the structural similarity between the metaphysical modalities, such as possibility and necessity,onthe one hand, and the deontic modalities, such as permission and obligation, on the other.N evertheless,this structural similarity does not go without restrictions.
Indeed, when the Apuleans quare is decoratedw ith metaphysical modalities, we find in its A-Is ide, that is between the necessity p and the possibility ⋄p,t he halfwayh ouse of the naked p: This writing presupposes the law p → p (an ecesse ad esse valet consequentia) accordingtowhich, if CiceroisnecessarilyT ullius Marcus,then CiceroisTullius Marcus.But if we resort to deontic modalities, thingschange. From the fact that Babbitt has the duty to payt axes, it doesn'tf ollow thatB abbitt pays taxes, because metaphysical and normative modalities have ad ifferent meaning.B ut this fact has also a mathematical accompaniment.Ifthe relation of parametrization P or R is compelled to be reflexive,then we have wPw and so, by applying our previous definition of necessity,wen ow obtain Nonetheless,t here is no corresponding principle for the deontic modalities. Now,reflexivity,aswell as transitivity,symmetry (etc.) are mathematical features of arelation. So that, if one controls the modal laws using these mathematical attributes of the parametrizingr elation R,m odal logic becomes mathematical modal logic in the sense of Goldblatt.

The Creation of TenseL ogic by ArthurP rior
Arthur Norman Prior was the main founderoft emporal logic and the creatorof tense logic. Overall, its work can be broken down into three major momentsmovements: 1. an anachronistic encounter between Diodorus Chronos and St. Thomas Aquinas;2 .am athematization of temporal logic and 3. the encoding of tense logic.

St. ThomasA quinasM eets Diodorus Chronos
The foundation of temporal logic coincides with the erection of an Apulean column containing a pedestal (I) and a capital (A).P rior founds the pedestal Io f temporallogic in Aquinas definition of past and future: the Thomas Aquinas Theorem of temporall ogic: Past is what has been present and futurei sw hat willb ep resent (Est enim praeteritum quod fuit praesens, futurum autem quod erit praesens) More profoundlythis definitionmeans that the present appears in the three "parts" of time: in the past as past present,i nt he present as present and in the future as future present. As aresultofthis,onlytwo symbols will be required on the pedestalp osition I: Pp for "It is past that p" and Fp for "It is future that p"; for the present time, the simple p will be enough, since its illustrations will be, for example, "It rains" in order to reportapresent fact equallyreported in the p of Pp and Fp accordingt ot he Thomas Aquinas theorem.
Herearemark is imperative.Ifacensus of the symbols is taken as the criterion for ac ensus of the modalities, then each post on the Apulean columnw ill require, in temporall ogic, two modalities, such as Pp and Fp. The capital Ao f temporallogic has been designed by taking adefinition of necessitygiven by Diodorus Chronos and remedying its hemiplegia.
Thanks to Boethius we know the four definitions of the four modalities given by Diodorus Chronos.Each one fits with aposition of the Apuleansquare AEIO. But before Prior'swork, faced with these Diodorean definitions, the historians of logic were commeune poulequi atrouvé un couteau. Among these definitions, in fact,wef ind for example this definition of necessity: Necessary is what,b eing true, will not be false.
If we put this Diodorean definition of necessitya tt he top Ao fo ur AI column, what we obtain as capital of the column is onlyahalf-Janus which, camped in the present,can onlysee the future. And all Diodorean modalities are affected by this semi-blindness. Yet, from Prior'spoint of view,this means that Diodorus has provided onlyo ne half of the couple required in position A. Thus, the Diodorean capital must be completed with the Aquinate pedestalinorder to obtain on its topawell-balanced Janus-logic. The required symbols are Hp and Gp,with the mnemonics "it always Has been the case" and "it is always Going to be the case";i nF rench: Auparavant et Dorénavant.
When all this is done, we gett he completeC olumn of Temporal Logic, with its Aquinate pedestal and its Diodorean capital: One of the laws of this logic is Pp → GPp ("Whatever has happened will always have happened").

The Mathematization of TemporalL ogic
On 27 th August 1954,inhis Presidential Address at the second New Zealand Congress of Philosophy (held in Wellington) Prior unveiled ac alculus which, in its publishedv ersion of 1958, contains the following formula: then it is true at u that p") Since in this formula the truth of p at u is conditioned by the relation of succession (<) between t and u,thanks to it Prior accomplished the mathematization of both temporal and modal logic. In otherwords, buildingaPrior'sbridge for temporal logic as acase of the Leibnizian bridge built for modal logic, Prior adds on the right sideofthe latter the footbridge of the "before-after relation" in its function of parametrization.
As an illustration of this function, let'stake the formula Fp → FFp. Is it alaw of temporall ogic?Y es, if, generally, Now this condition of density can be presented either as an attribute of the parametrizing relation (<), or as an attribute of the set on which this relation plays (here it is as et of temporal moments). This shows that,i naKripke model 〈〈E, R〉,V 〉 ,t he frame 〈E, R〉 which is denoting the ontology of the model, enjoys astrongunity because the mathematical attribute (such as density), that is the reason of its inclusion in the alleged model, can be attributed either to E or to R.

The Emergenceo fT ense Logic
Among Prior'sb ridges, let'sn ow consider the following one: ("It is true at t that p is past iff there is a u which precedes t and it is true at u that p") Herewemustremember that,accordingtoMcTaggart (1908), anyevent E can be grasped in twoseries: it can be said to be future, present and past-the three attributes of the A-series-and it can be temporallyr elated with other events with respect to which it can be said to come before,b econtemporary with or come after -the three relations of the B-series. Now,a sw ec an see if we consider, for example, the succession of births of Homer,M ilton and Borges, there is no changeinthe B-seriesbut onlyinthe A-series. Onlyinthe latter,infact,isafuture event present and then becomes past. Prior thematized this metaphysical watershed in his temporal logic. In the last equivalence, in fact,wefind that,onthe left side, the A attribute P of McTaggart has become a modal one, and that,o nt he right side, the B relation <o f McTaggart has become a parametrizing relation of mathematical modal logic. So, we are no more in the situation of the Kripkean models 〈〈E, R〉,V 〉 where any binary relation R was doing the job of the parametrizingone. In the temporal logic of Prior,i no ther words, not onlym ust "<" have its orderingm eaning,but this relational meaning has the same philosophical and metaphysical relevance givent oi tb yM cTaggart.
Arthur Prior is the logician who has introduced the metaphysics of time into temporall ogic. But this contamination is onlyt he first step of his revolution.
In fact,f rom Leibniz to Carnap (and his followers), Leibniz'sb ridgeh as been conceivedasapath connectingleft-shore modalities with right-shore quantifications. Yet, if we follow Prior,L eibniz'sbridge becomes a palindrome that can be crossed in both directions, with different results.Ifwegofrom the Aattributes on the left shore to the Brelations on the right shore, we are in temporallogic. But if we go from the Brelations to the Aattributes as modalities, we enter into (pure) tense logic.⁶

2T he PythagoreanT able of Physics
Physics has its Pythagorean table⁷ (Table T) century'srevolutionary physics is fully spatio-temporal or,moreexactly, doubly spatiotemporal: therei sarelativistic spatio-temporality and a quantum spatio-temporality.The first is notorious;the second needs to be made explicit.Whatisquantized in Quantum Mechanics is, in fact,the mechanical action and the mechanical action can be expressed in both a spatial and a temporal form. Thus, a quantized actionisalink between time and space: afact which is quite ironical with respect to the history of the development of physics. Leibniz, as it is well-known, has made of continuity one of the most important principles of nature (Naturanon facit saltus)and something like aquantized action,preciselytothe extent that it is adiscrete action, seems to refute this principle. Yet, Leibniz is also the philosopher-physicist who sawm echanicala ction (the so-called "moving-action")a st he heart of physics.T his fact is duly registered by Martial Gueroult in his Leibniz. Dynamique et Métaphysique (1934). Here, Gueroulth as also argued thatL eibniz was aware of the spatio-temporal character of mechanicala ction. What he called "action mortice",i nf act,i s " the product of the amount of movement multiplied by the space travelled or of the living forcem ultipliedb yt ime" (Gueroult, 1967, 50,o ur translation). So that,for the momentum and the energy,the best Begriffsschrift is the Leibnizian couple 〈mv, mv 2 〉. Now,l et us consider Heisenberg'sr elations⁸ of indeterminacyt ransliterated in this Leibnizian Begriffsschrift:  Ihavepresentedafirst version of this thesis when Iwas invited by Elie Duringtogivealecture in his seminar series on physicsa nd philosophya tD iderot University.  Where d shortens "distance" as ac oordinateo fs pace.

Between the Time of Physics and the Time of Metaphysics
These relations can be viewed as asimple corollary of our Pythagorean table: H1 stating the spatial Heisenbergr elation, H2 its temporal mate. But this fact can also work as the occasion for ar eflection: which account does Heisenbergg ive for the two members in each relation?W hat is common to d and t on the one hand and to mv and mv 2 on the other?I nb rief, what is the ratio essendi of the spatial or temporal factor on the one side, and of the energy or momentum factor on the other?
With regard to this issue, Leibniz provides,i fn ot the concepts, at least the vocabulary.AsGueroult explains: "If extensionisconsidered in time,intensity is force; if it is considered in space, intensity is speed"⁹ (Gueroult, 1967, 130,o ur translation). Here we have the Leibnizian watershed of Nature: in Nature, space and time are the extensive components;momentum and energy are the intensive ones.
This Leibnizian vocabulary will receive its full conceptual expression when it will be housedi nL autman'sp hilosophyo fs cience (Dumoncel2 008) and in its systematic and speculative network. Lautman, in fact,i sadirect heir of Plato and it is well known that Plato, trying to give ab lueprint of his allegory of the Cave,t raced an analogical Line between the visible and the intelligible worlds which undergoes as econd analogical division.T hus, the scale of beingso r realms we obtainedh as the following four steps: From aphilosophicalpoint of view,wefind that,onthe LautmanLine, the mathematical and the physical floors are alreadyg iven and defined by their wellknown scientific status. What is problematic are the floor of the "Ideas" and the floor of Natural philosophy. The former overcomes what concerns our pre-sent¹⁰ but the latter,a ccordingt oL autman, plays ac rucial role in our theory of time. In the Lautmans cale, Natural philosophyw orks in fact as a lock between Mathematics and its application in Physics. And it is here that the Leibnizian watershed of Nature plays akey-role: we leave mv and mv 2 ,aswell as energy and momentum,tothe physicist,keepingfor the philosopher onlythe Leibnizian couple of extensive and intensive. Thisisthe Lautmanian sampling of Natural Philosophy.

3T he Zeitsein in itsD iagram
From Plato to Russell, passingt hrough Leibniz, metaphysics has been crudely dualist.I nt he exsanguinous languageo fR ussell, metaphysics boils down to the φx and the a to be substituted to the x in φx. So, in the vocabulary of Nicolaï Hartmann, Being is split in the Sosein of the "universals" φx and the Dasein of the "particulars" a. Yet, symbolic logic has been constructed by mathematicians, and in mathematics the typical a is an umber,a gain in the Sosein. Conversely, the Sosein is supposed to include the caballeity. Darwin has changed all this. To the extent that caballeity sits in the Sosein, the ancestors of the horseand the horses must previouslypop up in the Dasein. Accordingt oW hitehead'sv ision, onlyg eometrical formsa nd colours are "eternal objects".Then, the Zeitsein mayb ed efined as the union of the Dasein and of the quiddities which, in the Sosein,a re derivedf rom the evolution at work  Deleuze has understood this point well. Barothas givenagood accountofitinhis book on Lautman (Barot 2009), discussed in the symposiumonLautman (Marquis 2010). Formoreonthe topic we must refert oo ur Cavaillès et Lautman,a gaini no ur scriptorium. in the Dasein. The Zeitsein requires its Übersistlich Darstellung in the form of Figure 1, p. 341. This representation is ad oubled cone with its upper sheet and its lower sheet, separated by ap lane. The lower sheet represents time;t he upper sheet with the plane represents temporality. At the top of temporality,o nt he empyrean of metaphysics,wef ind the Prototime: in order to represent it we must imagine a compass rotating in the drawingo facircle.
The center of the circle, which is marked by the puncture of the compass, represents the protopresent. The radius of the circle, measured by the opening of the compass,r epresents the protopast. The compass' rotation represents the protofuture. The threep arts of proto-timea re divided into two subsets: the protopresent and the protopast are united in the happened (advenu), while the future is àv enir.
In this construction, we find af irst contingency:t he circle can be traced either dextrogyre,inthe sense of the handsonawatch, or levogyre,inthe sense of the trigonometric circle. This is onlyabinary contingency:t he two possibilities correspond to the twotickingsofthe prototime-clock: if the prototime turnsdextrogyre, the ticking is tick-tack; if it turns levogyre, the ticking is tack-tick. Now,the prototime is taken in a "bushing" operation b. We can have an intuitive idea of its effect onlybyanalogy. We can imagine prototime by comparing it with an elementaryp article that has to go through aY oung split: before it crosses it,t he path of the particle is straight; when it passes through the Young split, its localisation is determined.
Yet, we shall suppose that, because of this localisation, the velocity-vector of the particle (with its three dimensions, and so with its direction too) is underdetermined. Each state of the particle,the one before and the one after the Young split,corresponds to an effect of the bushing b. The first one is a linearisation of what has happened: the protopast is mapped onto an actualised part of the protofuture, so that the protopresent gets its chronological standard position: being after the protopast.The second effect is a ramification of the future in af an of "future contingents".
The resultofthese two bushing operations on prototime is what Ipropose to call Protoduration: b(prototime) =p rotoduration From this hypostasis forth, one can conceive the genesis of the Zeitsein exploiting the affinities between the works of three philosophers: Elisabeth Anscombe, Peter Geach and Arthur Prior,¹¹ rememberingt hat physics alsoh as its demons: Laplace'sd emon, with his fatal determinism and Maxwell'sd emon, with his charge of neguentropy.B ut,s ince physics givesr ights to its demons, we can give the same rights for metaphysics' demons.
With respect to protoduration, the tutelar demon is¹² Miss Anscombe'so ne. Her 1964'sp aper Before and After provides her contribution to tensel ogic. The title of the paper allows us to define Miss Anscombe'sd emon. Itse ssence is dipolar,that is made of two different possibilities: being an angel and then a devil; being a devil and then an angel. In symbols, A stands for the Anscombe tense operator "and after",s ot hatt he two main possibilitiesf or her demon are: angel A devil ("he is an angel, and then he is ad evil") devil A angel ("he is ad evil, and then he is an angel") Of course, these two possibilities are onlytwo branches of abinary contingency amid the full fan of the contingents-futures openedb yt he protoduration. Thus, comingfrom Temporality,that is from Prototime and Protoduration,toTime,one has to distinguish between what is written in capitals and what is written in lower case.
In capital letters,wefind the Lautman'snon-commutativity (Lautman2006, 269). Andso, when Miss Anscombe'sdemon applies his metaphysical functionto natural philosophy, the Leibnizian couple of the extensive and intensive becomes useful and relevant.The binary contingency will in fact take preciselythe form of the two abovem entioned natural possibilities: intensive A extensive or extensive A intensive  Fort he relationship between Anscombe and Prior,s ee Gardies 1975,1 20.I ti si naletter to Prior that P.T. Geach defined the Transworlds Airlines TWA of his relation of accessibility between possible worlds in the same waya sP rior did in 1962.  Anscombe was alreadymarried to Geach when my friend GeorgesK alinowski met her at the Manchester conclave (closed colloqium) where Arthur Prior gathered together the few pioneers of deontic logic; but,a sh et oldm el ater,h el earned on this occasion that,i na cademic circles, G.E.M. Anscombe must onlyb en amed "Miss Anscombe".A nd it is onlyu nder this name that she has ad emon.
Lautman asks to himself: is it possible to describe, in the womb of mathematics, astructure which would be likeafirst drawing of the temporalform able to qualify the sensiblep henomena?( Lautman2 006,2 77). Such as tructure mayb e named mathematical prototemporality. Yet, our thesis is that there is alsoalogical prototemporality,a nd that logical prototemporality is the coreo ft ense logic.
In lower case we register the fact that Elisabeth Anscombe married Peter Geach. And P.T. Geach tells us that: "Smith committed seven burglaries,t hen a murder and then he wash anged" (Geach 1957,7 1). But in the Zeitsein Geach's Smith becomes ac oncrete hypostasis of the twot ickingsi nt he prototimeclock. In miss Anscombe's Begriffsschrift: school A prison ("he was at school, and then in prison") prison A school ("he was in prison, and then at school") Now,our Zeitsein is displayedw ith all its hypostasises. Therefore, we can insert on its scale, each one in its proper place, the main laws of tense logic, also underliningt he paradigmatic value thatP rior attributedtothem.¹³ At the top, that is on the level of Prototime, we find the definition of the modality Always given by Prior.I ti snot defined by an instants-quantification ("Always p" =f or all instants t it is true at t that p)b ut as ac onjunction of temporal modalities: Always p = Df p ∧ Hp ∧ Gp ("Always p means by definition that p and auparavant p and dorénavant p") At the level of Protoduration Prior givest he following definition: t < t' = Df @ t Ft' "'ti sb efore t'' means by definitiont hat at ti ti sf uturet hat t'" Here, we see that,onthe Prior palindrome, the Carnapian order between explanandum and explanans is inverted: now,t he explanandum is "before",i .e., in McTaggart'st erms,aBrelation of the B-series; while the explanans is future, i. e., an Aa ttribute of the As eries. The latter has become at ensed modality.
At the level of Time, we find Findlay'sl aw: the emblematic lawo ft emporal logic which is alsothe first from an historicalpoint of view.Itwas enunciated by  Foradetailed expositiona nd explanation of this point,s ee Dumoncel (2018).
Between the Time of Physics and the Time of Metaphysics J.N. Findlayi nafootnote of his paper on time (Findlay1 941). And, when Prior read it,h ef ound the impetus to investigate botht emporala nd tense logic. Herei sP rior'sf ormulation of Findlay'sl aw: p ∨ Pp ∨ Fp → FPp ("If p or it is past that p or it is future that p,then it is future that it is past that p")

Comparative Conclusion
Faced with the question "What is time?",the Zeitsein thesis needstobespecified in the light of both Elie During'sa nd Etienne Klein'sp apers. Elie During has enunciatedanew Tertium non datur: "There is no third time" between natural time and psychological time. But,i ns pite of the title of the present paper,t he Zeitsein thesis is compatible with During's tertium non datur. Yet, the thesis of During seems to entail ap lain answer to Etienne Klein's question: "Who is entitled to talk about time?" If there is no third time in addition to physical and psychological time, then the answer to the question is that nobodyisentitled to talk about time, except for physicists such as Julian Barbour and psychologists such as Marc Wittmann.
However,w em ay sustain am ore balanced conception. This is because, in During'st erms, therei sa"'natural philosophy' stemmingf rom contemporary physics". "Natural philosophy" is asort of constant in the development of physics from Newton to Louis De Broglie and AlbertLautman. In the Lautman scale, as we have seen, natural philosophyfinds its place, as ahalfwayhouse between mathematics and physics. Moreover,when Lautmant ook into account vonN eumann'sa lgebra, he recognized that the mathematical stratum of his Platonic scale is a logico-mathematicale difice. From Klein'sv iewpoint, the logical layer included in the logico-mathematical edifice defines exactlyt he place of tense logic.
Within temporallogic, tense logic runs the risk of being reducedtothe logic of our phenomenal time.But this is amisunderstanding.The first mathematization of modal logic coincides,i nt he work of Arthur Prior,with the birth of temporal logic as ap aradigm for modal logic and, in this mathematization, the frame represents the ontology of the modal model. Besides, since temporal logic is akind of modal logic, the contribution of logic in answering the question "What is time" primarilyc oncerns the part of temporall ogic wheret he modal involvement (in Quine'ss ense) reaches its maximum point: tense logic.