# A token-based semantic analysis of McTaggart's paradox.

1. IntroductionAccording to an untold story of Humpty Dumpty, in addition to claiming that he can make a word mean whatever he wants it to mean, Humpty Dumpty once remarks that a contradiction is the cheapest thing that one can ever get. "See, I now sit on the wall, and I now have a great fall, oops, a contradiction right away!" As can be expected, no one pays much attention to his remark. This "cheap contradiction" conjoins two token-sentences that report about different stages of a person, thus does not amount to a "contradiction" in the first place. Being amused by Humpty Dumpty's innocent remark, we are confident that we know what a "genuine" contradiction is and can spot a fake one when we encounter it.

However, in 1908, when McTaggart came up with his argument for the unreality of time, we were caught stunned. Presumably, McTaggart's contradiction was on the same footing as that of Humpty Dumpty's, yet we found that it could not be disposed of as easily as we had expected. McTaggart's Paradox (MP) has taught us a lesson that when it comes to the question of whether some sentence is a contradiction, intuition is not something we can count on. Rather, we need to have a well-defined notion of contradiction to prevent us from accidentally buying a cheap contradiction home.

In Section 2, I sketch a semantic framework for tensed sentences upon which our notions of validity and contradiction will be built. As a methodological principle, I shall consider only concrete entities such as type-sentences or token-sentences that can be explicitly described or uttered in certain contexts, and leave abstract entities, such as propositions, out of the picture entirely. More specifically, given a conjunctive type-sentence, I will distinguish a static truth-condition from a dynamic truth-condition, and, based on this distinction, I will prescribe two notions of validity and two notion of contradiction. Finally, in Section 3, I introduce three possible solutions to MP.

As was remarked in Christensen (1974), "... so many philosophers are not sure that it [MP] has ever really been given a proper burial, and so from time to time someone digs it up all over again in order to pronounce it really dead". In a sense, what I shall be doing here is to give MP a belated semantic postmortem examination.

2. Token-Based Semantics for Tensed Sentences

Given a linguistic expression e, we can, borrowing Frege's terms, associate with it a sense s(e), and associate with each of its tokens a referent [r.sub.w]([e.sub.t]), where et stands for a token of e and the w stands for the world in which the token is uttered. Both the senses s(e)'s and the referents [r.sub.w]([e.sub.t)'s can be specified in terms of the truth conditions of token-sentences, and they are related by [r.sub.w]([e.sub.t) = s(e)(t, w). (1) In other words, the sense of an expression can be identified with something that, given a world, allows us to find out what the referent for each of its possible tokens is.

2.1 The Truth-condition of an Atomic Tensed Sentence

To describe the truth-condition for a type-sentence (2) is to specify the truth-values for all of its possible tokens in all possible worlds. More specifically, following the spirit of Frege and adopting Kaplan's terms, we can say that the character of a type-sentence is a function from its possible tokens to functions that map possible worlds to truth-values, and the content of one of its possible tokens is just the latter function that maps possible worlds to truth-values. The character of a type-sentence gives us a (pt, pw)-truth-condition, where the "pt" and "pw" here stand for "possible token" and "possible world" respectively; while the content of a token-sentence gives us a plain pw-truth-condition.

Let e be an event. The (pt, pw)-truth-conditions for type-sentences "e is past", "e is now" and "e is future" can be described as follows:

(pt, pw) -truth-conditions for atomic tensed sentences (3)

A token u of "e is past" uttered in a world w is true iff e is earlier than u in w.

A token u of "e is now" uttered in a world w is true iff e is simultaneous with u in w.

A token u of "e is future" uttered in a world w is true iff e is later than u in w.

One should bear in mind that the u and w here are, in a sense, independent of each other, that is, the token u tells us nothing about the world (4) just as the world w does not presuppose the existence and the whereabouts of u. Note also that what really concerns us here is the semantics of type-sentences, while the possible tokens of a type-sentence are only introduced as a means to help us grasp the sense of the type.

2.2 Two Versions of Truth-condition for a Conjunction

Having prescribed what the character of an atomic tensed type-sentence is, I now turn to the prescription of the character of a conjunctive type-sentence. A conjunctive type-sentence is the conjunction of two type-sentences by the type-connective "and", and a token of the conjunctive type-sentence is the conjunction of two token-sentences with a token "and".

Now I outline two versions of truth-condition for the type-sentence "A and B":

Static Truth-condition for a Conjunction:

A point-token u of the type-sentence "A and B" uttered in a world w is true iff both the point-token [u.sub.a] of A and the point-token [u.sub.b] of B are true in w.

Here, by a "point-token" I mean some idealized token that can be uttered at a single space-time point. The role such a token plays is similar to that of a point-like test-particle in physics. In particular, the spatial-temporal locations of [u.sub.a], [u.sub.b] and u are taken to be the same.

Dynamic Truth-condition for a Conjunction:

A physical token u of the type-sentence "A and B" uttered in a world w is true iff both the sub-token [u.sub.a] of A and the sub-token [u.sub.b] of B are true in w.

The tokens under consideration here are possible physical tokens that take space and time to utter. Typical examples of these tokens include Humpty Dumpty's utterance of "I now sit on a wall and I now have a great fall", and a token-sentence painted on a long wall stating "the exit is to the right and the exit is to the left", the truth of which depends on where those words are situated. Note that the spatial-temporal locations of [u.sub.a] and [u.sub.b] are usually different and the token u is taken to be composed of the pair ([u.sub.a], [u.sub.b]).

2.3 Static versus Dynamic Notions of Validity

A type-argument (5) consists of a collection of type-sentences, one of them being the conclusion, and a token-argument consists of a collection of token-sentences, one of them being the conclusion. Evidently, the "validity" for a type-argument demands more than the validity of a token-argument as, in addition to quantification over possible worlds, it involves quantification over possible tokens. Hereafter, unless stated otherwise, when I say "arguments", I mean type-arguments. We have two notions of validity for type arguments.

Static Notion of Validity for an Argument:

An argument [A.sub.1], ..., [A.sub.n] /[therefore] B is valid (6) iff for every world w and for every point-token u = ([u.sub.1], [u.sub.n], [u.sub.b]) of the argument in w, whenever all the point-tokens [u.sub.i] of [A.sub.i] are true in w, the point-token [u.sub.b] of B is true in w.

Here, again, by a point-token, I mean some token that is localized at a space-time point. In particular, the spatial-temporal locations of [u.sub.1], ..., [u.sub.n], [u.sub.b] and u itself are taken to be the same, and this is the notion of validity we are familiar with. Next, consider the following. (7)

Dynamic Notion of Validity for an Argument:

The argument [A.sub.1], ..., [A.sub.n] / [therefore] B is valid iff for every world w and for every possible token u = ([u.sub.1], ..., [u.sub.n], [u.sub.b]) of the argument in w, whenever all the sub-tokens [u.sub.i] of [A.sub.i] are true in w, the sub-token [u.sub.b] of B is true in w.

The tokens in question are possible physical tokens that take space and time to utter. A typical token-argument u is a sequence of token-sentences ([u.sub.1], ..., [u.sub.n], [u.sub.b]) uttered in this particular order in time.

To my knowledge this dynamic notion of validity has not been much discussed in the literature and I think it deserves more attention from the theorists of time. Consider the following arguments ([alpha]) and ([beta]):

([alpha]) The event e is now. /[therefore] The event e is now.

([beta]) The event e is now. /[therefore] The event e is past.

According to the static notion of validity, the argument ([alpha]) is valid and ([beta]) is invalid, which is expected when we assume that every pair of token-sentences of ([beta]) are uttered simultaneously. However, according to the dynamic notion of validity, the argument ([alpha]) is invalid and ([beta]) is valid. When it comes to the resolution of McTaggart's alleged paradox, this distinction helps us to have a better grasp of what a contradiction really is.

2.4 Static versus Dynamic Notions of Contradiction

Now, consider the notion of contradiction for tensed conjunctive sentences. First, we need to bear in mind that we are not interested in the truth of a particular token, but are concerned with the truth-values of all possible tokens of a type-sentence in all possible worlds. Second, we should remember to distinguish the two kinds of tokening of the type "A and B" that I mentioned before. I obtain the following two versions of contradiction.

Static Notion of Contradiction:

"A and B" is a (static-) contradiction if none of its point-tokens can ever be true in any possible world.

Dynamic Notion of Contradiction:

"A and B" is a (dynamic-) contradiction if none of its physical tokens ([u.sub.a], [u.sub.b]) can ever be true in any possible world.

Thus, we have that "The event e is now and the event e is past" is a contradiction in the static sense, while "The event e is now and the event e is now" is a contradiction in the dynamic sense. (8)

3. Static and Dynamic Solutions to McTaggart's Paradox

Based on the two notions of contradiction described in the last section, I now proceed to analyze MP and provide three solutions (two static and one dynamic) for it.

3.1 McTaggart's Paradox Recast

McTaggart famously distinguishes two characterizations of time, see McTaggart (1927) for instance: A-series refers to "that series of positions which runs from the far past through the near past to the present, and then from the present through the near future to the far future, or conversely"; and B-series refers to "the series of positions which runs from earlier to later or conversely." The properties being past, being present and being future are often called "A-properties", and the relations being earlier than, being later than, and being simultaneous with are called "B-relations". According to McTaggart, B-series cannot be a characterization of time, as time necessarily involves change, yet B-series does not account for change--"if N is ever earlier than O and later than M, it will always be, and has always been, earlier than O and later than M, since the relations of earlier and later are permanent". (McTaggart 1908) But A-series would lead to the following inconsistent results, hence time is unreal.

(*) Past, present, and future are incompatible determinations. Every event must be one or the other, but no event can be more than one. This is essential to the meaning of the terms ... The characteristics, therefore, are incompatible. (italic mine)

And

(#) But every event has them all. If M is past, it has been present and future. If it is future, it will be present and past. If it is present, it has been future and will be past. Thus all the three incompatible terms are predicable of each event which is obviously inconsistent with their being incompatible, and inconsistent with their producing change. (italic mine)

In the literature, to demonstrate the inconsistency, the above two theses are often analyzed in symbolic forms. Mellor's treatment in Real Time II is a typical example. (9)

Let P, N and F be respectively the A-times past, present (now) and future, and let e be any event. Then e's being past, present and future I write respectively as "Pe", "Ne" and "Fe"...."~", "&" and "[??]" mean respectively "not", "and" and "entails".

Then McTaggart's basic argument is that while the three times P, N and F are mutually incompatible, so that

(5) Pe [??] ~Ne; Ne [??] ~Fe; Fe [??] ~Pe; etc.,

the flow of time requires every event to have all three of them, i.e.

(6) Pe & Ne & Fe.

But (5) and (6) cannot both be true; since if (6) is true, two of the entailments in

(5) fail, making (5) false. But our A-concept of time commits us to both (5) and

(6) ; so it entails a contradiction and cannot apply to reality.

One natural reaction to Mellor's (6) would be to reject it, as apparently there is no way that we can utter a true token of "e is past, e is present and e is future". Thus, the alleged MP could not take off in the first instance. However, as McTaggart's seemingly acceptable (#) indicates, we normally would not mind accepting some thing like "e will be past, is present and has been future", which in Mellor's terminology can be written as "FPe & NNe & PFe". But, then the contradiction between (5) and (6) disappears. Anticipating this reaction, McTaggart claims that even if this iterated interpretation is granted, the event will still admit all of the nine second level determinations, namely, PP, PN, PF, NP, NN, NF, FP, FN and FF, and at least some of them are incompatible. (10) So the contradiction remains. Introducing a further level of tensed predicate only pushes the contradiction to a higher level.

I shall now adopt the machinery that we have developed in the last section to examine what is going on in these discussions. In particular, we shall ask: "What do we mean by 'incompatibility'?" "Are PN, FP etc. well-defined predicates of events?" and "How should we symbolize (*) and (#) properly?"

We are in a far better position than McTaggart in 1908 to tackle these problems, as we have a hundred more years of experience with analytic philosophy. We know better how to achieve clarity and precision in our use of terms, we know how to distinguish tokens from types and distinguish referents from senses, furthermore, we are more at home with using a formal logical language to formulate an argument than ever before. This is definitely to our advantage. Nevertheless, we need also to remember that this advantage can easily be misused by us to explain the MP away at the very beginning by simply "translating" the paradox into something non-paradoxical.

To avoid this, we should ask ourselves from time to time "Are the translation faithful to the original argument?" In other words, "Is a philosopher as clever as McTaggart likely to have asserted that translated version of his claim?" For example, the above two passages (*) and (#) quoted from McTaggart (1908) can be naively translated into

Pe [??] ~Ne [conjunction]~Fe;

Ne [??] ~Fe [conjunction]~Pe;

Fe [??] ~Ne [conjunction]~Pe;

and

(Pe [right arrow] (PNe [conjunction] PFe)) [conjunction] (Fe [right arrow] (FNe [conjunction] FPe)) [conjunction] (Ne [right arrow] (PFe [conjunction] FPe)),

respectively, where e is an event and "P", "N" and "F" stand for predicates being past, being present, and being future respectively, to avoid the inconsistency. But this seems to do no justice to McTaggart, as it is very likely that he himself would not admit that the latter stands for his (#). He could, as Lowe did in Lowe (1987), ask "What do these iterated tenses, such as PF mean?" and insist that he would only accept non-iterated determinations of an event, and hence Mellor's (6) "Pe & Ne & Fe" suits him better; or he could, as he did in McTaggart (1908), claim that even if the iterated determinations make sense, an event exhibits all nine secondary determinations and they, when taken together, still form an incompatible set of determinations. In other words, he can stress that what he has in mind is a conjunction of predications--be it first order or second order--rather than the three ad hoc conditionals in sight here.

I shall now present three interpretations of MP, two in terms of the static notion of contradiction and one in terms of the dynamic notion of contradiction, so that both the following two theses, namely

Incompatibility Thesis

I Past, present, and future are incompatible determinations,

and

All-inclusive Thesis

A Every event has them all,

are correctly translated, i.e. true to the spirit of McTaggart, yet the two theses are consistent with each other.

3.2 Two Static Solutions

The DNF (11) Formulation

What do we mean by "past, present, and future are incompatible determinations"? This statement needs to be spelled out in concrete terms. To begin with, I can describe it in terms of the (static-) validity of an argument, as follows: for any e, [alpha] [??] ~[beta] [conjunction] [sup.~][gamma], where [alpha], [beta], [gamma] are any permutation of "Fe", "Ne", "Pe"; and "P", "N" and "F" stand for "being past", "being present" and "being future" respectively. However, for simplicity in analysis, I shall demand the minimal incompatibility [alpha] [??] ~[beta][disjunction] ~[gamma] only. In other words, the incompatibility here applies to the collection of predicates {F, N, P} as a whole, rather than to any pair of them. Furthermore, for simplicity of notion, we would write this Incompatibility Thesis in terms of "contradictions" rather than in terms of "[??]" (i.e. validity), as follows:

Static-I For any event e, 'Pe [conjunction] Ne [conjunction] Fe' is a (static-) contradiction.

Note that this thesis is expressed as a sentence in the meta-language, (12) in the sense that the "Pe", "Ne" and "Fe" here are not "used" but only "mentioned." This thesis affirms something about the truth of all possible point-tokens of a particular conjunctive type-sentence, namely "Pe[conjunction]Ne [conjunction]Fe", claiming that no such token can ever be true. Note also that the truth of the statement Static-I itself is independent of the space-time location at which it is tokened--for instance, whether as a printed sentence on this paper or as a token-sentence spoken by someone.

Next, for the All-inclusive Thesis, we recall McTaggart's own remark first: "If M is past, it has been present and future. If it is future, it will be present and past. If it is present, it has been future and will be past. Thus all the three incompatible terms are predicable of each event". (McTaggart 1908)

Observe that: 1) the three "If"'s employed by McTaggart here are only to itemize three mutually exclusive cases rather than to convey three conditionals; 2) despite that here he has employed phrases such as "has been present" and "will be past" that have the flavor of a secondary iterated predicate, when it comes to the All-inclusive part, he seems to be still concerned only with the three primary first-order predicates "being past", "being present" and "being future"; 3) while "Pe" is an unproblematic tensed sentence, "PNe" may not make any sense. As has been criticized by Lowe, due to indexical fallacy, "PN" fails to be a proper predicate for an event e, just as "here over there" is not a proper predicate for an object. (13) Furthermore, "P" is not a predicate for "Ne", as the latter is a sentence rather than an event. We need to find a suitable symbolization for the seemingly unproblematic sentence "e has been present".

I shall understand it as the statement that there exists a time in the past so that a possible token of "Ne" uttered at that time would have the truth-value T, and then symbolize the statement as "P[Ne]". In other words, the "Ne" here is not "used" but only "mentioned", and "P[Ne]" says something about possible true tokens of Ne in the past. (14) Similarly, for a type-sentence [alpha], the type-sentences "N[[alpha]]" and "F[[alpha]]" assert the existence of possible true tokens of [alpha] in the present and in the future respectively. (15)

We can then, based on McTaggart's remark (#) quoted earlier, rephrase the All-inclusive Thesis as,

Static-A (DNF) For any event e, ((Pe [conjunction] P[Ne] [conjunction] P[Fe]) [disjunction] (Fe [conjunction] F[Ne] [conjunction] F[Pe]) [disjunction] (Ne [conjunction] P[Fe] [conjunction] F[Pe])). (16)

Note that the "P", "N" and "F" are used in two ways here--as predicates for events and as predicates for type-sentences.

Would McTaggart assert this version of the All-inclusive Thesis? As this formulation is based on his own remark, and the problem of iterated tense has been avoided, McTaggart seems to have no reason not to do so. Now, clearly Static-A (DNF) is consistent with Static-I. But if it is really so, then why did McTaggart claim that (A) was inconsistent with (I)? One possible answer is this: in Static-I, it is claimed that for any event e, whenever the type-sentence "Pe[conjunction]Ne [conjunction]Fe" is used (i.e. tokened at a specific space-time point), the token is false. Yet, being fooled by the appearance of all the three incompatible expressions, namely the "Pe", "Ne" and "Fe" in each of the disjuncts of Static-A (DNF), namely in "Pe [conjunction]P [Ne] [conjunction]P [Fe], Fe[conjunction]F[Ne] [conjunction]F[Pe] and Ve[conjunction]P[Fe] [conjunction]F[Pe]", McTaggart fails to notice that in each of these disjuncts, only one of "Pe", "Ne" and "Fe" is actually used, while the other two are merely mentioned. Thus no inconsistency arises here. In other words, McTaggart himself was a victim of the notorious "use-mention trap". This is our first static solution to MP.

The CNF (17) Formulation

Some might object to the previous solution by insisting that McTaggart makes the remark "if M is past, it has been present and future ..." only reluctantly--all he wants is to convince us that his original All-inclusive Thesis really is acceptable. According to them, we should aim at directly translating McTaggart's original claim that every event has all the three tensed properties rather than translating his remark. In particular, for any event e, we should have a "conjunction" instead of a disjunction. Now, a statement of the form that

(1) For any event e, Pe [conjunction] Ne [conjunction] Fe

is no doubt a good candidate for a "literal" translation of his original All-inclusive thesis. However, as I have mentioned earlier, our immediate reaction to this proposal would be to ask, so far as the All-inclusive thesis is concerned, "Did McTaggart really want to see the 'Pe [conjunction] Ne [conjunction] Fe' in (1) uttered as a point-token in space-time?" and the answer is surely a "No!"--every pointtoken of "Pe [conjunction] Ne [conjunction] Fe" is evidently false as has been asserted by the Incompatibility Thesis. We can safely say that what McTaggart had in mind was definitely something more believable than (1). As admitted by McTaggart himself,

It is never true, and the answer will run, that M is present, past and future. It is present, will be past, and has been future. Or it is past, and has been future and present, or again is future and will be present and past. The characteristics are only incompatible when they are simultaneous, and there is no contradiction to this in the fact that each term has all of them successively. (18)

What he says here is consistent with his earlier remark in (#) that "If M is past, it has been present and future. If it is future, ..." and can be precisely captured by our non-contradictory Static-A (DNF), and a conjunction is to be found within each of the three disjuncts. Furthermore, in each conjunction, the three conjuncts generally involve different tenses, that is, the characteristics present, past and future are to be applied at different times.

However, when we read on to his next sentence, "But this explanation involves a vicious circle...." we find that what McTaggart wants from (#) is presumably still a conjunction that, on the face of it, is acceptable, yet after some specification, turns out to be inconsistent with (*). In sum, to make his original conjunctive (#) acceptable for the reader, McTaggart introduces tenses (or second order temporal predicates) to explain how the three characteristics can indeed be applied to an event at different times, thus turning (#) into a disjunctive normal form. However, after the reader has accepted his DNF explanation, he generalizes (#) to be the conjunctive claim that, again, all the nine secondary predicates are applicable to an event. As (*) can be similarly generalized to be the claim that the nine characteristics are not compatible, the generalized (*) and the generalized (#) are still inconsistent.

To be more precise, concerning the interpretation of (#), McTaggart has shifted his position from to (i)~(iii) to (i)'~(iii)'.

(i) the existence of a true "Pe[conjunction]Ne[conjunction]Fe",

(ii) the existence of true "Pe", "Ne" and "Fe",

(iii) the existence of a true "N[Pe] [conjunction]P[Ne] [conjunction]P[Fe]", "F[Pe] [conjunction]N[Ne] [conjunction]P[Fe]", or "F[Pe] [conjunction]F[Ne] [conjunction]N[Fe]",

(i)' the existence of a true "P [Pe] [conjunction]P [Ne] [conjunction]P[Fe] [conjunction]N[Pe] [conjunction]N[Ne] [conjunction]N[Fe] [conjunction]F [Pe] [conjunction]F[Ne] [conjunction]F[Fe]",

(ii)' the existence of true "P[Pe]", "P[Ne]", "P[Fe]", "N[Pe]", "N[Ne]", "N[Fe]", "F[Pe]", "F[Ne]" and "F[Fe]",

(iii)' the existence of a true "(N[P[Pe]] [conjunction]N[P[Ne]] [conjunction]N[P[Fe]] [conjunction]N[N[Pe]] [conjunction]P [N[Ne]] [conjunction]P[N[Fe]] [conjunction]P[F[Pe]] [conjunction]P[F[Ne]] [conjunction]P[F[Fe]])", "(F[P[Pe]] [conjunction]F[P[Ne]] [conjunction]F[P[Fe]] [conjunction]F[N[Pe]] [conjunction]N[N[Ne]] [conjunction]P[N[Fe]] [conjunction]P[F[Pe]] [conjunction] P[F[Ne]] [conjunction]P[F[Fe]])", or "(F[P[Pe]] [conjunction]F[P[Ne]] [conjunction]F[P[Fe]] [conjunction]F[N[Pe]] [conjunction]F [N[Ne]] [conjunction]N[N[Fe]] [conjunction]N[F[Pe]] [conjunction]N[F[Ne]] [conjunction]N[F[Fe]])",

while his (*) shifts from (i)* to (i)'*

(i)* the non-existence of a true "Pe[conjunction]Ne[conjunction]Fe",

(i)'* the non-existence of a true "P[Pe] [conjunction]P [Ne] [conjunction]P[Fe] [conjunction]N[Pe] [conjunction]N[Ne] [conjunction]N [Fe] [conjunction]F[Pe] [conjunction]F[Ne] [conjunction]F[Fe]".

Now, (i) and (i)' are essentially Mellor's understanding of McTaggart's All-inclusive statement, and they are apparently inconsistent with (i)* and (i)'*, hence cannot be correct translations of the statement. (19) Next, (iii) and (iii)' resemble our Static-A (DNF) approach described earlier. Note that the All-inclusive statement that all the three primary (or the nine secondary) predicates are applicable to every event is supposed to be token-independently true. However, although the truth of the disjunction in Static-A (DNF) is token-independent, the truth of each of its disjuncts is token-dependent and, for any token of the disjunction, only one of the disjuncts is true. Therefore, if we want to stress that "Pe", "Ne" and "Fe" (or the nine secondary ones) can all be truly tokened, as claimed by the statements (ii) (or (ii)'), and express it in a conjunctive form, then we need to think more about the truth of these type-sentences.

The specification of when those tokens are to be located is as follows: if Pe then A [Pe] [conjunction]P[Ne] [conjunction]P [Fe]; if Ne then F[Pe] [conjunction]N[Ne] [conjunction]P [Fe] ; if Fe then F[Pe] [conjunction]F[Ne] [conjunction]A[Fe], where "A" is a derived predicate, and "A[[alpha]]" means that there exist possible past, present and future true tokens of [alpha]. What is important here is not the technique of finding a suitable space-time region for each of the tensed sentences to be truly uttered, but the fact that all such possible true tokens exist--as a consequence of the real line topology of time. And, on the one hand, this assertion of the existence of true tokens grants McTaggart the token-independence needed of his statement, and, on the other, it generates a prima facie inconsistency that "all these incompatible characterizations can be truly tokened".

Now, to say that there exists a true token of "Pe", proceed as follows. At any context, a token u = ([u.sub.p], [u.sub.n], [u.sub.f]) of "Pe [disjunction] Ne [disjunction] Fe" is true. No matter which of [u.sub.p], [u.sub.n] and [u.sub.f] is true, a token of "F[Pe]" is true. Secondly, for a true token of "Ne", if the sub-token [u.sub.n] is true, then we have "N[Ne]"; if [u.sub.n] is not true, then either [u.sub.p] or [u.sub.f] is true, and for the former, we can write "P[Ne]" and for the latter "F[Ne]". This case can then be summed up as "P[Ne] [disjunction] N[Ne] [disjunction] F[Ne]". Lastly, for the case of a true token of "Fe", a token of "P[Fe]" is always true. In this way, the Thesis (A) can be written as "for any event e, F[Pe] [conjunction] (P[Ne] [disjunction] N[Ne] [disjunction] F[Ne]) [conjunction] P[Fe]", which in the real life is often phrased as "every event will be past, and every event was, is, or will be present, and every event has been in the future".

Taking the real line topology of time into consideration, we observe that F[Pe] statically implies P[Pe] [disjunction] N[Pe] [disjunction] F[Pe] and vice versa, (20) while P[Fe] is statically equivalent to P[Fe] [disjunction] N[Fe] [disjunction] F[Fe]. As a consequence, we arrive at a simpler way of specifying the Thesis (A): (21)

Static-A (CNF) For any event e, (P[Pe] [disjunction] N[Pe] [disjunction] F[Pe]) [conjunction] (P[Ne] [disjunction] N[Ne] [disjunction] F[Ne]) [conjunction] (P[Fe] [disjunction] N[Fe] [disjunction] F[Fe]), (22)

and this is similar to Lowe's solution to MP outlined in Lowe (1992) and Lowe (1993). (23)

Let "E[[alpha]]" be a shorthand for "P[[alpha]] [disjunction] N[[alpha]] [disjunction] F[[alpha]]", which says that there exists a possible past, present or future true token of [alpha], then Static-A (CNF) can be simplified to be "for any event e, E[Pe] [conjunction] E[Ne] [conjunction] E[Fe]", and it is evidently compatible with Static-I, which says that for any event e, "Pe [conjunction] Ne [conjunction] Fe" is a contradiction. Thus, in terms of "E[[alpha]]", we have

Static-I For any event e, ~ E[Pe [conjunction] Ne [conjunction] Fe],

Static-A (CNF) For any event e, E[Pe] [conjunction] E[Ne] [conjunction] E[Fe].

It is easy to generalize this to higher order predicates. When McTaggart claims that while the nine secondary predications are incompatible, e has all nine of them, what he really asserts is

[I.sub.9] For any event e, ~ E[[[alpha].sub.1] [conjunction] [[alpha].sub.2] [conjunction] [[alpha].sub.3] [conjunction] [[alpha].sub.4] [conjunction] [[alpha].sub.5] [conjunction] [[alpha].sub.6] [conjunction] [[alpha].sub.7] [conjunction] [[alpha].sub.8] [conjunction] [[alpha].sub.9]]

[A.sub.9] For any event e, E[[[alpha].sub.1]] [conjunction] E[[[alpha].sub.2]] [conjunction] E[[[alpha].sub.3]] [conjunction] E[[[alpha].sub.4]] [conjunction] E[[[alpha].sub.5]] [conjunction] E[[[alpha].sub.6]] [conjunction] E[[[alpha].sub.7]] [conjunction] E[[[alpha].sub.8]] [conjunction] E[[[alpha].sub.9]]

here [[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3], [[alpha].sub.4], [[alpha].sub.5], [[alpha].sub.6], [[alpha].sub.7], [[alpha].sub.8] and [[alpha].sub.9] stand for "P[Pe]", "P[Ne]", "P[Fe]", "N[Pe]", "N[Ne]", "N[Fe]", "F[Pe]", "F[Ne]" and "F[Fe]" respectively. Apparently, there is no incompatibility between [I.sub.9] and [A.sub.9] .

3.3 A Dynamic Solution: The Token-in-Time Formulation

In this subsection we shall look at one further attack from Le Poidevin and Mellor (1987), which claims that Lowe's thesis fails to capture the passage of time. Lowe's position is that he acknowledges this accusation, but he has no intention to get into it. He stresses that "defending the reality of the A-series and developing an account of passage in A-series terms are two quite different projects", and admits that "it is the A-series that I seek to defend against McTaggart's attack on it, not the account of passage in A-series terms which he proposes, and whose demonstrable incoherence McTaggart illegitimately (in my view) projects on to the A-series itself". (24) However, I think we can do better than that.

The two solutions of MP that I describe in subsection 3.2 are both static in the sense described in Section 2. Now we turn to the dynamic aspect of the paradox. We have seen earlier that with a static interpretation, there is no chance for a point-token of the type-sentence "Pe [conjunction] Ne a Fe" to be true. This holds for the type-sentence "Fe [conjunction] Ne [conjunction] Pe" as well. However, so far as a dynamic interpretation of contradiction is concerned, there is no point in accepting this. It is indeed very easy to find a true token of "Fe [conjunction] Ne [conjunction] Pe". For instance, my token of "I am about to utter the sixteenth word of this sentence, I am now uttering it, and I have uttered it" is undoubtedly true. The sub-tokens [u.sub.f], [u.sub.n] and [u.sub.p] of a real world conjunction u= ([u.sub.f], [u.sub.n], [u.sub.p]) can be uttered at different times so that all of them are true at their respective times of utterance.

Under this interpretation, McTaggart's Incompatibility Thesis fails, because as we utter the conjuncts of a conjunction, or the premises and conclusion of an argument, time flows--we are not considering point-tokens that are all uttered at the same point-like moment of time, but rather are considering real world tokens that take space and time to utter. It is therefore not obvious at all why the predicates F, N and P should be incompatible. On the other hand, the All-inclusive Thesis now has a new way out: for any event e, the sentence type "Fe [conjunction] Ne [conjunction] Pe" is assertible, since there exists a possible physical token of it that lasts for a period of time so that the three conjuncts are all true at their respective times of utterance.

One might object that this "dynamic stuff" is surely not the mode of paradox that McTaggart has in mind, because if it is, then instead of embracing Ne [??] ~Pe, he would have embraced Ne [??] Pe, but there is simply no sign that McTaggart has ever taken this position. Indeed, even if we manage to find a convincing dynamic thesis such as

Given an event e, there exist no possible true tokens of "Ne [conjunction] Ne" or any conjunction of "Pe", "Ne" and "Fe" in this order,

it is clearly not the Incompatibility Thesis that McTaggart had in mind. These worries are all very true. However, we should note that these worries are all centered around the Incompatibility Thesis rather than the All-inclusive Thesis. In other words, we are still left with the following fallback position: McTaggart has mistakenly adopted two different notions of semantics in his presentation of the Incompatibility Thesis and the All-inclusive Thesis--a static one for the former and a dynamic one for the latter. As the static Incompatibility Thesis has been described in subsection 3.2, here we only need to be concerned with the formulation of a dynamic All-inclusive Thesis.

According to the dynamic notion of contradiction, the truth of a conjunction is dependent of the ordering of its conjuncts. For example, for any e, "Fe [conjunction] Ne [conjunction] Pe" is possibly true, but there won't be any true token of "Pe [conjunction] Ne [conjunction] Fe". Is this an unsatisfactory feature of the new framework? In a sense, yes, because it deviates from our standard conception of conjunction. But in order to make sense of tensed sentences, aren't we supposed to take the feature of this world, at least that of time itself, into account? Perhaps it is precisely the passage of time that we should incorporate into our semantics of tensed sentences so as to resolve MP successfully.

Based on the notions introduced in Section 2, we can formulate the dynamic All-inclusive Thesis as follows.

Dynamic-A Given an event e, there exists a possible true token of "Fe [conjunction] Ne [conjunction] Pe". (25)

This amounts to a demonstration of how time "moves", via the employment of token-sentences in time. I might as well put it:

The passage of time manifests itself through the functioning of a language. Therefore, doing "logical" analysis on type-sentences without referring to how they are to be actually tokened in the space-time is a fatal mistake.

Recall that the two solutions discussed in the preceding subsection both interpret McTaggart's All-inclusive statement as an objective statement about the world--asserting the existence of true tokens of "Pe", 'We" and "Fe" respectively in the space-time--and the truth of the statement itself is independent of when or where it is to be tokened. (26) However, one might suspect that this is not what McTaggart has in mind, because in stating his All-inclusive thesis he seemingly intends to use the type "e is future, e is present and e is past" as a whole, rather than merely to mention the three of its components and assert that they can each be truly tokened.

This objection in effect casts doubts on our first two solutions of MP and, on the face of it, makes MP a genuine threat to the A-theory. We can, of course, refuse to read McTaggart's All-inclusive thesis this way and decide to stick to the two original solutions that we have. However, the interesting thing here is that even if we take up the challenge of the objector, (27) things are still to our advantage. The objector's version of MP not only poses no threat to the A-theory, but also becomes a strong supporting evidence for the A-theory --MP is solvable only if we adopt a dynamic viewpoint of time. While McTaggart's Incompatibility Thesis is established through the employment of "point-tokens", the modified All-inclusive Thesis, i.e. the existence of possible true tokens of "Fe [conjunction] Ne [conjunction] Pe", can be established only if we take into account the passage of time when we utter a "real world token". The two prima facie inconsistent theses now pose us no threat, because the notions of "tokens" involved in them are of completely different nature, one being a point-token while the other being a possible-physical-token, furthermore they are governed by static and dynamic truth conditions respectively.

3.4 Summary

McTaggart's alleged paradox consists of two theses, the Incompatibility Thesis and the All-inclusive Thesis, and the two theses seem inconsistent. In order to analyze these theses, we have defined the notions of "validity" and "contradiction" in two ways--statically and dynamically. For the former, there are two options for us to (statically) resolve MP. Both of them assert the Incompatibility Thesis, while explaining away the "inconsistent" All-inclusive Thesis by suitably re-interpreting the iterated tensed sentences and stressing the use-mention distinction of sentence tokens and types. For the latter, we find that the All-inclusive Thesis can be dynamically interpreted, and as a result the alleged MP fails to generate the inconsistency that it claims to have generated.

The static interpretation guarantees that an A-theorist located in space-time has a way to describe the temporal feature of the world consistently. The dynamic interpretation, on the other hand, maintains that even if an A-theorist is concerned only with the actual uttering of tensed tokens, there is no contradiction involved in his or her conception of time. Moreover, the latter resolution of MP captures the essence of the passage of time, which in turn is closely related to change--the most important feature of time. However, a full-blown formal semantics of tensed sentences in the dynamic vein awaits further exploration.

REFERENCES

Christensen, Ferrel (1974), "McTaggart's Paradox and the Nature of Time," The Philosophical Quarterly 24(97): 289-299.

Kaplan, David (1989), "Demonstratives," in J. Almog, J. Perry, and H. Wettstein (eds.), Themes from Kaplan. Oxford: Oxford University Press, 481-563.

Le Poidevin, Robin, and Mellor, D. H. (1987), "Time, Change and the 'Indexical Fallacy,'" Mind 96: 534-8.

Le Poidevin, Robin (1993), "Lowe on McTaggart," Mind 102: 163-170.

Lowe, Jonathan E. (1987), "The Indexical Fallacy in McTaggart's Proof of the Unreality of Time," Mind 96: 62-70.

Lowe, Jonathan E. (1987a), "Reply to Le Poidevin and Mellor," Mind 96: 53942.

Lowe, Jonathan E. (1992), "McTaggart's Paradox Revisited," Mind 101: 323-6.

Lowe, Jonathan E. (1993), "Comment on Le Poidevin," Mind 102: 171-173.

Lowe, Jonathan E. (1998), "Tense and Persistence," in Robin Le Poidevin (ed.), Questions of Time and Tense. Oxford: Oxford University Press, 43-59.

McTaggart, John M. E. (1908), "The Unreality of Time," Mind 18: 457-84.

McTaggart, John M. E. (1927), The Nature of Existence. Cambridge: Cambridge University Press.

Mellor, D. H. (1998), Real Time II. New York: Routledge.

Savitt, Steven F. (2001), "A Limited Defense of Passage," American Philosophical Quarterly 38(3): 261-270.

Smith, Quentin (1993), Language and Time. New York: Oxford University Press.

Torre, Stephan (2009), "Truth Conditions, Truth-Bearers and the New B-Theory of Time," Philosophical Studies 142(3): 325-344.

NOTES

(1.) Here we assume that the sense of an expression is exhausted by its referential properties.

(2.) Many authors talk about truth-conditions for token-sentences. See, for example, Torre (2009), where Steve Torre distinguishes the truth-condition for a type-sentence from that for a token-sentence.

(3.) Note that on the right-hand side, we use B-relations exclusively. But this does not imply that we have adopted the B-theory. We did it this way only to make the truth-condition look simpler. In fact, one can try defining those B-relations in terms of A-properties alone. However, we shall not get into the details here.

(4.) If, on the other hand, one takes the realistic position that the uttering of a token fixes the world that is concerned then we can simply drop the phrases about the world w in the three truth-conditions above, and obtain the following three pt-truth-conditions,

pt -truth-conditions for atomic tensed sentences:

A token u of "e is past" is true iff e is earlier than u.

A token u of "e is now" is true iff e is simultaneous with u.

A token u of "e is future" is true iff e is later than u.

(5.) It is not to be confused with an "argument form". The word "type" here only indicates that the argument is composed of type-sentences.

(6.) It can also be written as [[A.sub.1], ..., [A.sub.n]} [??] B.

(7.) Note that we have to distinguish this notion of validity from the validity for a token-argument. An actual token of a type-argument is said to be valid iff for all possible worlds such that the premise tokens are true, the conclusion token is true. But, for simplicity, we shall not consider the latter notion of validity in this paper.

(8.) Note that we shall not consider possible worlds whose space-time structures are so different from ours that the "past", "present" and "future" have lost their usual meanings.

(9.) See 73-74.

(10.) For instance, PP, NN and FF.

(11.) Here DNF stands for Disjunctive Normal Form.

(12.) That is, English plus the language of first order logic.

(13.) See Lowe (1987, 1987a, 1992, 1993).

(14.) Therefore, predicate "P" can take up two kinds of arguments: events and type-sentences. For an event e, the sentence "Pe" is true if e is past; while for a type-sentence, "Ne" say, "P[Ne]" is true if there is a space-time location in the past where a token of "Ne" there would have been true.

(15.) Recall how Le Poidevin has elaborated for McTaggart on the iterated tense issue. According to Le Poidevin (1993), even if we grant Lowe that sentences involving iterated tenses, such as "FPe", do not make sense in the first place, MP still holds, as one can, instead of writing "FPe", just write "FT 'Pe'" meaning that it will be true that Pe, and then mimic McTaggart's original argument to arrive at a contradictory "NT 'Ne' [conjunction] PT 'Ne'[conjunction] FT 'Ne"', which is as contradictory as before.

Lowe maintains that "FT 'Pe"' is dubious, and I totally agree with him, as, after all, according to Le Poidevin's interpretation, "Pe" inherits a strong flavor of a "proposition", while we have not admitted propositions into our ontology. In contrast, in my "F[Pe]", the "Pe" is just a type-sentence, having nothing to do with a proposition at all

(16.) Note the three disjuncts happen to correspond to the three properties of being past, being future and being present of the Incompatibility Thesis. The reader is advised to compare it with Lowe's account sketched in Le Poidevin and Mellor (1987) and Le Poidevin (1993).

(17.) Here CNF stands for Conjunctive Normal Form.

(18.) McTaggart (1908). Note that the truth of the sentences "it is present, will be past, and has been future", "it is past, and has been future and present", and "it is future and will be present and past" are all token-dependent!

(19.) McTaggart's All-inclusive statement is supposed to be true--otherwise why would he expect other people to accept it in the first place.

(20.) Both P[Pe] and N[Pe] statically entail F[Pe].

(21.) See Lowe (1998) and Savatt (2001) also for similar but different ways of interpreting (#).

(22.) Note that CNF thesis is weaker than the DNF thesis.

(23.) The reader is, however, reminded that it differs from Lowe's solution in a significant way: the "Pe", "Ne" and "Fe" inside the brackets [...] here are "type-sentences" instead of "propositions".

(24.) See Lowe (1993), 171-173.

(25.) We have taken advantage of the order-insensitivity of McTaggart's thesis "But every event has them all" to choose an ordering of "Fe", "Ne" and "Pe" to our favor. Note also that we do not expect to see the actual uttering of such tokens. What's important is that we can conceive of the possibility of such an in-time long-lasting token.

(26.) To be more precise, the truth of "There exist true tokens of 'e is future,' 'e is present' and ' e is past"' is independent of its time of tokening.

(27.) That is, assuming that McTaggart indeed wants to use the type-sentence "e is future, e is present and e is past".

[C] Cheng-chih Tsai

CHENG-CHIH TSAI

cctsai@mmc.edu.tw

Mackay Medical College-New Taipei

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Author: | Tsai, Cheng-chih |
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Publication: | Linguistic and Philosophical Investigations |

Article Type: | Report |

Geographic Code: | 4EUUK |

Date: | Jan 1, 2011 |

Words: | 7898 |

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