Simple partial logic (=SPL) is, broadly speaking, an extensional logic which allows for the truth-value gap. First I give a system of propositional SPL by partializing classical logic, as well as extending it with several non-classical truth-functional operators. Second I show a way based on SPL to construct a system of tensed ontology, by representing tensed statements as two kinds of necessary statements in a linear model that consists of the present and future worlds. Finally I compare that way with other two ways based on Łukasiewicz’s three-valued logic and branching temporal logic

Kachi, Daisuke

English

PhilPapers

Tensed Ontology based on Simple Partial Logic∗Daisuke KachiAbstractSimple partial logic (=SPL) is, broadly speaking, an extensionallogic which allows for the truth-value gap. First I give a system ofpropositional SPL by partializing classical logic, as well as extendingit with several non-classical truth-functional operators. Second I showa way based on SPL to construct a system of tensed ontology, byrepresenting tensed statements as two kinds of necessary statements ina linear model that consists of the present and future worlds. Finally Icompare that way with other two ways based on ÃLukasiewicz’s three-valued logic and branching temporal logic.1 Simple Partial LogicSPL is a system of truth-functional logic that generalizes (propositional)classical logic (=CL) by allowing for the truth-value gap. At the sametime the SPL given in this paper is an extension of CL by adding severalnon-classical truth-functional operators which are reducible to one. Here Irestrict my arguments to its syntax and semantics, omitting its proof theory.There are two kinds of logical operators in SPL: the basic operators,which are also included in the syntax of CL, and the modal operators, whichare not included. I show the operators and their semantics in the followingtables:Table 1: Basic operatorsA ∼At f- -f tA&B t - ft t - f- - - ff f f fA∨B t - ft t t t- t - -f t - fA→B t - ft t - f- t - -f t t tA↔B t - ft t - f- - - -f f - t∗Reprinted from Proceedings of Ninth International Symposium on Temporal Repre-sentaion and Reasoning: TIME-02, pp. 141-145, 2002.1Table 2: Modal operatorsA TA FA LA MAt t f t f- f f f tf f t t fA 7→B t - ft t f f- t t tf t f fThey are reducible, for example, to ‘&’, ‘∼’, ‘T’ in the following way:A∨B =df ∼(∼A&∼B)A→B =df ∼(A&∼B)A↔B =df (A→B)&(B→A)FA =df T∼ALA =df TA∨FA (‘A’ has a truth-value.)MA =df ∼LAA 7→B =df LA→TB (‘A’ presupposes ‘B’.)I call the intermediate value ‘gap-value’, distinguishing it from ‘truth-values’, which refer only to truth and falsity. The formula that does notinclude modal operators are called ‘basic formula’.The definitions of the basic operators coincide with those of Kleene’sstrong three-valued logic and have the following semantic properties:1(1)Reliable Determinability (=Reliability)The truth-tables that define the meaning of the base-operators in SPLcoincide with those of CL under total valuation.2 In other words, a basicformula has the same truth-value with that of CL, if all the atomic formulaeincluded have truth-values.(2)MonotonicityIf the basic formula that includes gap-valued (=gapped) atomic formulaehas a truth-value, it does not make its truth-value change by altering thegap-value to a truth-value.[Definition]One operator is stronger than another under partial valuation. =df Theformer gives a truth-value to the formula which it bounds, whenever thelatter gives one to the same formula.(3)Strength1In this paper I adopted most of Langholm’s terminology for partial logic, except for‘monotonicity’, which he calls ‘persistence’. [Langhholm,T. 1988][Langhholm, T. 1996]2I call the truth-valuation where every atomic formula is assigned a truth-value ‘totalvaluation’. The valuation which is not total is ‘partial valuation’.2The above basic operators are the strongest ones that are reliable andmonotonic among those based on CL.Modal operators have the following properties:3(1) Strong DeterminabilityModal operators always bring about a truth-value under partial valua-tion.(2)Weak MonotonictyIf a basic formula has a truth-value, it does not make its truth-valuelost (namely, be changed to the gap-value) by altering the gap-value to atruth-value (this is a corollary of strong determinability).We can understand the strong determinability as resulting from a kind ofself-referring function of the modal operators, which describe the partialityof truth-valuation itself. Accordingly it destroys the monotonicty of SPLbut it keeps the weakened monotonicity given above.Now I define two conceptions of validity in SPL:4(1)Weak Validity (=being w-valid)Σ |=w C iff there are no partial valuations which make all the formulaein the set Σ∪{∼C} true.(2) Strong Validity (=being s-valid)Σ |=s C iff there are no partial valuations which make all the formulaein the set Σ∪{∼C} non-false (=true or gapped).Adopting these definitions of validity, the following semantic theoremshold:[Th1]Σ |=w C if Σ |=s C[Th2w] Weak Deduction Theorem (WDT)P1, P2,—, Pn |=w C iff |=w (P1&P2&—&Pn)→C[Th2s] Strong Deduction Theorem (SDT)P1, P2,—, Pn |=s C iff |=s (P1&P2&—&Pn)→C[Th3w] Weak Contraposition Theorem (WCT)P1, P2,—, Pn |=w C iff |=w ∼C→∼(P1&P2&—&Pn)[Th3s] Strong Contraposition Theorem (SCT)P1, P2,—, Pn |=s C iff |=s ∼C→∼(P1&P2&—&Pn)3Most of the modal operators coincide with the operators which Woodruff adopted forhis three-valued logic ‘System Q’. [Woodruff, P. 1970]4I have given the justification of these definitions and the proofs of the following se-mantic theorems in my paper: [Kachi, D. 2002]3[Definition]Σ |=c C =df The argument which derives C from Σ is valid in CL.(This implies that the argument includes only the basic formulae.)[Th4]Σ |=w C iff Σ |=c C[Th5]There are no s-valid arguments that include only the basic formulae.The holding of both the deduction theorem and the contraposition the-orem is one of the reasons why I adopt these definitions of validity. Otherpossible definitions lack this property.5 [Th4] establishes that SPL is analternative way to the supervaluational logic to keep classical validity al-lowing for the truth-value gap. It is preferable at least in keeping truth-functionality. In contrast [Th5] shows that strong validity is proper to SPL.The following are among the s-valid arguments:TA |=s A (cf. A 6|=s A, A 6|=s TA)LA |=s A∨∼A (cf. 6|=s A∨∼A)|=s TA∨∼TA (cf. 6|=s TA∨T∼A)(A 7→B) & LA |=s TB2 Application of SPL to the Ontology of TimeOne of the fields of the application of SPL is the theory of presupposition(indeed one of the meta-operators ‘A 7→B’ is to be read as ‘A presupposesB’). We can interpret ‘TA’ as a statement with the presupposition of havinga truth-value; it insists that (it is presupposed that) ‘A’ has a truth-valueand it is the case that A. In fact the following theorem holds:|=s TA↔(LA&A)In contrast, we should take ‘A’ as a statement which lacks such a pre-supposition, and so it is valued as ‘truth-valueless’ or ‘gapped’ when thepresupposition fails. Or we can also understand ‘A’ as a conditional state-ment, ‘If ‘A’ has a truth-value it is the case that A.’ However, it should benoted here that we must not take this conditional as a two-valued materialconditional. If we did, ‘A’ would become true whenever it lacks a truth-value, which appears contradictory. Rather it means that the statement ‘A’has effects only if it has a truth-value and that it will be cancelled when itdoes not.5As for the examples of other definitions, see the following: [Blamey, S. 1986] [Busch,D. 1996] [Fenstad, J. E. 1997]4Let’s call ‘TA’ and ‘A’ respectively ‘a strong statement’ and ‘a weakstatement’. For example, we can interpret a historical statement ‘Caesarcrossed the Rubicon’ as a strong statement, while a fictional statement ‘Alicewent into Wonderland’ as a weak statement. Usually we use the formerstatement presupposing that it actually has a truth-value, while it dependsupon the context or the convention whether the latter has a truth-value andso we take it to be sometimes truth-valueless.Moreover we can use the above two kinds of statements to distinguishstrong necessity and weak necessity in the theory of modality. A strongnecessary statement ‘[]TA’ insists on the holding of a fact in every possi-ble world unrestrictedly. On the other hand a weak necessarily statement‘[]A’ does it restrictedly to the possible worlds where the statement has atruth-value. Usually the distinction depends on whether the referents in thestatement exist in every possible world. For example, ‘It is necessary that2+2=4’ and ‘It is necessary that God is almighty’ usually insist on strongnecessity, while ‘It is necessary that water is H2O’ and ‘It is necessary thatSocrates is mortal’ assert weak necessity.Finally there is a way to apply SPL to the ontology of time, which wecan take as a special case of the above modal application. Firstly let’sassume that the possible worlds concerned here are the present and futureworlds. However, the worlds considered here are not the presentistic onesas in the semantics of tense logic, but rather four-dimensionalistic ones.Put concretely, this means that the atomic formulae supposed here are notthe present-tensed ones such as ‘It is cloudy in Saitama now’, but tenselessones with dates, such as ‘It [is] cloudy in Saitama at noon on the 14thFebruary 2002(‘[is]’ signifies that it is tenseless).’ At the same time tenselessstatements with earlier or later dates are also included among the atomicformulae which describe the same world.Next let’s adopt the model 〈W, R, V 〉 which I call ‘the future model’,where W is a set whose elements are the present and future worlds, and Ris a binary relation between the worlds, and V is a partial valuation in theworlds. The future model is prescribed according to the following rules:(1)All the worlds are lineally ordered according to their dates that areuniquely indexed to each world.(2)There is a world that has the earliest date among the worlds, namelythe present world.(3)If an atomic formula has a truth-value in a world , it has the sametruth-value in all the worlds that are later than the world.(4)Every atomic formula has a truth-value at least in one world.5Assuming this model, let’s introduce the distinction between strong ne-cessity and weak necessity. If we adopt the operator ‘G’ as meaning ‘in everypresent and future world’, we can define the truth-condition of ‘GTA’ and‘GA’ in the following way:[Definition]‘GTA’ is true in the future model. =df ‘A’ is true in every world in thefuture model.‘GA’ is true in the future model. =df ‘A’ is true in every world where‘A’ has a truth-value in the future model.The monotonicity of the atomic formulae given by (3) and the monotonic-ity of the basic operators, together with (4), make the following theoremshold for the future model:[M1]If ‘A’ is a basic formula, a strong necessary statement ‘GTA’ is true inthe future model iff ‘A’ is true in the present world.[M2]If ‘A’ is a basic formula, a weak necessary statement ‘GA’ is true in thefuture model iff there is a world in which ‘A’ is true and in all the worldsearlier than which ‘A’ is gapped (if there are).By [M2], if ‘A’ is a basic formula, we can take a weak necessary state-ment ‘GA’ in this model as a future-tensed statement within the frameworkof dynamic ontology according to which it lacks a truth-value until the mo-ment when the stated event is determined to occur or not. In contrast, astrong necessary statement ‘GTA’ whose ‘A’ is a basic formula correspondsto a past- or present-tensed statement. If we can characterize our ordinarytensed statements in such a way, it turns out that all of our ordinary tensedstatements are (weak or strong) necessary statements in the sense definedfor the future model.So if we confine ourselves within our ordinary tensed statements, namely‘GTA’ and ‘GA’ whose ‘A’s are only basic formulae, we can dispense withthe intensional operator ‘G’, since we can take it as always accompanyingthe tensed statements implicitly. Moreover, by the combination of [M1] and[M2], we can reduce the difference between the statements ‘GTA’ and ‘GA’to the difference as to whether they insist that the present world is theearliest world in which ‘A’ is true or it is not necessarily so, although bothinsist that ‘A’ is eternally true from some world on. That is to say, we canreduce the difference between the statements about many worlds into the6difference between their description about one world, namely the presentworld.3 Comparison with ÃLukasiewicz’s Three-valued Logicand Branching Temporal LogicAs is well known, ÃLukasiewicz’s three-valued logic, which I call the systemÃL3, was intended to deal with the future contingency. He tried to constructa truth-functional modal logic which does not result in determinism whenits necessity is construed as the determination of events. However, he aban-doned it in his later years for the reason that it does not make contradiction(‘A&∼A’) necessarily false, while it makes identity (‘A→A’) necessarily true,though he positively accepted the failure of excluded middle (‘A∨∼A’) asthe denial of determinism.The truth-table for the operator ‘T’ in SPL is the same as that for thenecessity operator ‘L’ in the system ÃL3. In fact under the interpretationgiven above, they play the same role, though I would rather interpret thelack of truth-value of a statement not as the result of causal indeterminationof the stated events, but as the result of the lack of its truth-maker.The truth-tables of the basic operators of SPL only differ from those ofthe system ÃL3 in terms of the value of the conditional (and biconditional)when both of its subformulae are gapped. The former makes it gapped, whilethe latter makes it true. Accordingly SPL does not make identity necessarilytrue either. In fact it is the corollary of [Th5] in the first section that noclassical tautology is a strong tautology, which means that it is always trueunder partial valuation. ÃLukasiewicz’s conditional is non-monotonic and wecan define it as ‘(MA&MB)∨(A→B)’ using Kleene’s monotonic conditional.And so we should rather understand it as a derivative operator.On the other hand, the laws of excluded middle, identity and contradic-tion are all preserved in SPL as weak tautologies, which is the corollary of[Th4]. We can understand a weak tautology as a formula which is alwaystrue whenever it has a truth-value (as I have stated above, we must not takethis conditional as a two-valued material conditional). In other words, SPLtakes the failure of the classical tautologies not as the destruction of thelaws of CL but only as the result of the lack of a truth-maker, which is nota logical but an ontological matter. In that sense SPL is not an alternativeof CL but just its generalization by allowing for the truth-value gap.Another way of dealing with the future contingency is to adopt a branch-ing model in which the future worlds are partially ordered by dates. There7are two views on the way to represent a future-tensed statement in thebranching model. One is ‘Peirce’s view’, which takes a future statement asa kind of necessary statement which insists on the occurrence of a futureevent in every branch. The other is ‘Ockham’s view’, which takes a future-tensed statement as a contingent statement which insists on the occurrenceof a future event in one of the branches that is called a ‘prima facie futurebranch’.However, both have defects. Firstly they do not reflect correctly our ordi-nary usage of the future-tensed statements. Peirce’s future-tensed statementseems too strong, because usually we do not insist on the present determi-nation of the occurrence of a future event, when we make a future-tensedstatement. If we did, it would be nothing but a present-tensed statement.On the other hand, Ockham’s future-tensed statement appears too weak,because there is no formal distinction between the prima facie branch andthe other branches, and so the statement is not different from the statementwhich only insists on the possibility of the occurrence of the future event.Secondly, I think that the branching model is ontologically improper asa model that represents the dynamic change of the world itself–the so called‘temporal flux’. In my view the future contingency should be interpretednot as the result of the present causal indeterminacy of future events but asthe result of the present lack of its truth-maker. Therefore SPL, which dealswith the truth-value gap, seems more relevant with the dynamic ontologywhich takes account of the passage of time.And finally, from a formal point of view, a linear model of the futureworlds is simpler than a branching model. So it is at least worth investigatinghow far we can represent the complexity of our tensed statements within theframework of a linear model. Indeed we saw at the end of the last sectionthat we can dispense with intensional operators and can truth-functionallyrepresent our ordinary tensed statements in a linear model of the futureworlds. In that sense I believe that ÃLukasiewicz was right in his motive forconstructing his system ÃL3 to deal with the problem of future contingencyextensionally.As for the first problem of the branching model, I believe that my in-terpretation of the future-tensed statements as a weak necessary statementis free from both the defects of Peirce’s view and Ockham’s view. My viewgoes midway between Peirce’s and Ockham’s. On the one hand it is closerto the former, because it takes a future-tensed statement as a kind of neces-sary statement. But it is different in that it takes a future-tensed statementnot as the description of the determination in the present world but as thedescription of all the present and future worlds. On the other hand, my8interpretation is similar to Ockham’s view, in that my view also seems toimply that insisting on the occurrence of a future event is compatible withadmitting its contingency. However, my view does not appear to threaten afuture-tensed statement to degenerate into just a possibility statement.REFERENCES1. Blamey, S. (1986), Partial Logic, in [Gabbay, D. and Guenthner, F.(Eds.) 1986] pp.1-70.2. Busch, D. (1996), Sequent Formalization of Three-Valued Logic, in[Doherty, P. (Ed.) 1996] pp.45-75.3. Fenstad, J. E. (1997), Partiality, in [Van Benthem, J. and Ter Mulen,A. (Ed.) 1997] pp.649-682.4. Gabbay, D. and Guenthner, F. (Eds.) (1986), Handbook of Philosoph-ical Logic 3, D.Reidel.5. Kachi, D. (2002), Validity in Simple Partial Logic, to appear in Annalsof the Japan Association for Philosophy of Science 10-5.6. Langhholm, T. (1988), Partiality, Truth and Persistence, CSLI.7. Langhholm, T. (1996), How Different is Partial Logic?, in [Doherty, P.(Ed.) 1996] pp.3-43.8. Lambert, K. (Ed.) (1970), Philosophical Problems in Logic, D.Reidel.9. Van Benthem, J. and Ter Mulen, A. (Eds.) (1997), Handbook of Logicand Language, Elsevier Science B. V.10. Woodruff, P. (1970), Logic and Truth-Value Gaps, in [Lambert, K.(Ed.) 1970] pp.121-142.9

Tensed Ontology Based on Simple Partial Logic

https://philpapers.org/archive/KACTOB.pdf

Tensed Ontology Based on Simple Partial Logic

Abstract

Similar works

Full text

Available Versions

PhilPapers