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.2.There will exist someone who will have everyone now existing as an an-cestor.Assuming, for simplicity, that we are quantifying only over persons, it isclear that (1) and (2) cannot be represented by:3.P("ex)("y)Ancestor-of(x, y)4.F("ex)("y)Ancestor-of(y, x).22Cf.Prior, 1967, Chapter 8. 2.8.PAST AND FUTURE OBJECTS 43What (3) and (4) represent are the different sentences:5.There did exist someone who was an ancestor of everyone then existing.6.There will exist someone who will have everyone then existing as an an-cestor.Of course, in temporal possibilism, referential concepts are available thatenable us to refer directly to past and future objects.Thus, for quantificationover past objects we have the quantifier "p and for quantification over futureobjects we have "f as the future-counterpart of "p.Using these quantifiers, theobvious representation of (1) and (2) is:7.("px)("y)Ancestor-of(x, y)8.("fx)("y)FAncestor-of(y,x).We should note here that the relational ancestor concept is such that:x is an ancestor of y only at those times when either y exists andx did exist, though x need not still exist at the time in question, orwhen x has continued to exist even though y has ceased to exist.When y no longer exists as well as x, we say that x was anancestor of y; and where y has yet to exist, we say that x will be anancestor of y.Now although these last analyses are not available in actualist tense logic,nevertheless semantical equivalences for them are available once we allow us theuse of the now-operator, N.N : It is nowthe case that.The now-operator is unlike the simple present tense in that it always bringsus back to the present even when it occurs within the scope of either the past-or future-tense operators.Thus, although the indirect references to past andfuture objects in (3) and (4) fail to provide adequate representations of (1)and (2), the same indirect references followed by the now-operator succeed incapturing the direct references given in (7) and (8):9.P("x)N ("y)Ancestor-of(x, y)10.F("x)N ("y)FAncestor-of(y,x).In other words, at least relative to any present-tense context, we can in generalaccount for direct reference to past and future objects, and hence to all of theobjects of temporal possibilism, as follows:("px)Õ ”!¬P¬("ex)NÕ("fx)Õ ”!¬F¬("ex)NÕ 44 CHAPTER 2.TIME, BEING, AND EXISTENCE("x)Õ ”! ("px)Õ '" Õ '" ("fx)Õ.These equivalences, it should be noted, cannot be used other than in apresent tense context; that is, the above use of the now-operator would beinappropriate when the equivalences are stated within the scope of a past- orfuture-tense operator, because in that case the direct reference to past or futureobjects would be from a point of time other than the present.Formally, whatis needed in such a case is the introduction of a so-called backwards-looking operator, such as the then-operator, which can be correlated with occurrences ofpast or future tense operators within whose scope they lie and that semanticallyevaluate the formulas to which they are themselves applied in terms of the pastor future times already referred to by the tense operators they are correlatedwith23.Backwards-looking operators, in other words, enable us to conceptuallyreturn to a past or future time already referred to in a given context in the sameway that the now-operator enables us to return to the present.In that regard,their role in the cognitive schemata characterizing our conceptual orientation intime and implicit in each of our assertions is essentially a projection of the roleof the now-operator.We will not formulate the semantics of these backwards-looking operatorshere.But we do want to note that by means of such operators we can accountfor the development of referential concepts by which we can refer directly topast or future objects.Such an account is already implicit in the fact thatsuch direct references to past or future objects can be made with respect tothe present alone.This shows that whereas the reference is direct at least ineffect, nevertheless the application of any identity criteria associated with suchreference will itself be indirect, and in particular, not such as to require a presentconfrontation, even if only in principle, with a past or future object.2.9 Modality Within Tense LogicIt is significant that the first modal concepts to be discussed and analyzed in thehistory of philosophy are concepts based on the distinction between the past,the present, and the future, that is, concepts that can be analyzed in terms ofthe temporal modalities that are represented by the standard tense operators.The Megaric logician Diodorus, for example, is reported as having argued thatthe possible is that which either is or will be the case, and that the necessary isthat which is and always will be the case.24 Formally, the Diodorean modalitiescan be defined as follows:f&fÕ =df (Õ ("FÕ)fÕ =df Õ '"¬F¬Õ4" fÕ ”!¬f&f ¬Õ23Cf.Vlach,1973 and Saarinen 1976.24See Prior 1967, chapter 2, for a discussion of Diodorus s argument. 2.9.MODALITY WITHIN TENSE LOGIC 45Aristotle, on the other hand, included the past as part of what is possible;that is, for Aristotle the possible is that which either was, is, or will be the casein what he assumed to be the infinity of time, and therefore the necessary iswhat is always the case25:f&tÕ =df PÕ (" Õ ("FÕtÕ =df ¬P¬Õ '" Õ '"¬F¬Õ4" tÕ ”!¬f&t¬ÕBoth Aristotle and Diodorus assumed that time is real and not ideal.Inother words, the Diodorean and Aristotelian temporal modalities are under-stood to be real modalities based on the nature of time.In fact they provide aparadigm by which we might understand what is meant by a real, as opposedto a merely formal, modality such as logical necessity.These temporally-basedmodalities contain an explanatory, concrete interpretation of what is called theaccessibility relation between possible worlds in modal logic, except that worldsare now construed as momentary states of the universe as described by the mod-els associated with the moments of a local time [ Pobierz caÅ‚ość w formacie PDF ]