Cross-Temporal Relations

The first kind of statement that is supposed to resist formulation in a presentist-friendly formal language are statements expressing cross-temporal relations. Cross-temporal relations come in two varieties. The first are relations between different things exiting at different times, such as the relation expressed by the statement “Susie admires Plato.” The second are relations between an object and itself at an earlier or later time, as in “John is better looking now than he was as a teenager.”

Cross-temporal relations of the first kind Ciuni and Torrengo (2013) call ontic cross-temporal relations, where an ontic cross-cross-temporal relation is “A relation between a presently existing entity and a non-presently existing entity.” (Ciuni and Torrengo 2013, 213) Cross-temporal relations of the second Ciuni and Torrego call factive cross-Cross-temporal relation, where a factive cross-temporal relation is “A relation that is cross-temporal exemplified by it terms.” (Ciuni and Torrengo 2013, 213) We can show that statements expressing ontic and factive cross-temporal relations can both be expressed using a presentist-friendly formal language.

Let us begin with the statement “Susie admires Plato.” Since Plato no longer exists, any quantifier binding a variable taking Plato as a value must be within the scope of a past-tense operator. Assuming Susie does now exist, the quantifier ranging over her will not be within the scope of such an operator. This may seem to lead to a problem (Sider 1999) (Sider 2006) (Szabo 2006). If we parse “Susie admires Plato” like so

∃x(x=Susie & P ∃y(y=Plato & x admires y),

the resulting sentence says that Susie admired Plato at a time long before she was born. The problem is that the predicate ‘x admires y’ appears within the scope of the operator ranging over Plato. If we remove the predicate from this position, we get

∃x(x=Susie & P ∃y(y=Plato) & x admires y),

which is ungrammatical, as the rightmost instance of ‘y’ is free.

Some philosophers have been led to try to analyze such sentences so as to eliminate the relation (M. Davidson 2003) (T. M. Crisp 2005) (De Clercq 2006). This is not necessary. In fact, there is a way to represent cross-temporal relations within a presentist formal language. We can parse “Susie admires Plato” like so:

∃t(at t ∃x(x=Susie & P ∃y(y=Plato & F ∃t`(t`=t & at t`(x admires y))))).

This says

There is some time t, at t there is some x such that x is Susie and it was the case that there is some y such that y is Plato and it will be the case that there is some time t` such that t` is identical to t, and at t` x admires y.

Since in a presentist formal language a quantifier not in the scope of either ‘P’ or ‘F’ ranges over things existing at the present time (or outside of time), t is the present time. Since t` is stipulated to be the same time as t, the predicate ‘x admires y’ is assigned to the present time. Note

importantly that this sentence assigns the 2-place predicate ‘x admires y’ to the time t`. It does not assign the variable y to t`. Hence, while this says that Plato is admired at t`, it does not say that Plato exists at t`.

Alternatively, another way to symbolize statements expressing cross-temporal relations in our presentist formal language would be to introduce a new tense operator, ‘N’, for “It is now the case that …” Any sentence within the scope of ‘N” is to be understood as a sentence about the present time, no matter what tense operators precede ‘N’. The ‘N’ operator effectively makes whatever is within its scope exempt from all previous tense operators. Using this operator, we would symbolize “Susie admires Plato” as

∃x(x=Susie & P ∃y(y=Plato & N (x admires y))), or, more simply, as

P ∃x(x=Plato & N(Susie admires x)).

Both of these say Susie exists at the present time, it was the case that Plato exists, and it is now the case that Susie admires Plato.⁴¹ Note importantly that the former sentence assigns the 2-place predicate ‘x admires y’ to the present time, and not the variable y, and the latter sentence assigns the 2-place predicate ‘Susie admires x’ to the present time, and not the variable x. Hence, both entail that Plato is admired at the present time, but not that Plato exists at the present time.

41 Using ‘N’ was the first solution that occurred to me for how to parse statements expressing cross-temporal relations. The other solution, involving reference to previously quantified times was inspired by Prior (1968). Prior notes that in a statement like “It will be the case that it is now the case that I am sitting down” (102) the word ‘now’

cannot be removed without altering the meaning of the sentence. However, Prior proposes that this sentence is equivalent in meaning to “It is now the case that for some proposition p which is true at one instant only, (i) it will be the case that [p and I am sitting], and (ii) it is now the case that p.” (106), and that ‘now’ can be eliminated from this statement without loss of meaning. While Prior’s use of ‘proposition’ is idiosyncratic, the basic proposal is straightforward, and we could easily replace “proposition which is true at one instant only” with “state of affairs which obtains at one instant only.” I have chosen to first indicate the time at which x occurs or obtains, and then refer back to that time, thus, “It is t and it will be the case that it was the case that t and I am sitting,” or “∃t F P

∃t`(t`=t & I am sitting).”

Those philosophers who have discussed this sort of problem will likely reject my solution on the grounds that it entails that objects which do not now exist can nonetheless now exhibit properties or stand in relations: Plato is admired by Susie now, despite not existing now. In fact, it is often assumed that presentism entails serious presentism, according to which only things existing at the present time (or outside of time) can exhibit properties or stand in relations. Thus, the serious presentist contends that only what exists at the present time (or, perhaps, outside of time) can be a subject of predication at the present time. If correct, this would imply that if ‘Plato’

in the foregoing statements in our presentist formal language is intended to refer to the historical philosopher, the presentist must denounce those statements as either false or meaningless.

If presentism is a substantive thesis, and if presentism entails serious presentism, then perhaps my construal of “Susie admires Plato” is not available to the presentist. At this juncture, however, our question is not whether the presentist can make sense of the claim that Susie admires Plato, but rather whether this statement can be expressed in a formal language in which quantifiers ranging over things existing at times other than the present are always within the scope of past-tense and future-past-tense operators. I have shown that they can be. If such statements are problematic, it is not a problem of logical form.

Consider next the statement “John is better looking now than as a teenager.” This statement differs from the last insofar as it relates not two things that exist at different times, but a single thing as it is and was at different times. Brogaard (2006) (2013) suggests that such relations should be taken as primitive. Using “My daughter is now taller than my son was,” as an example, Brogaard writes:

Where the property of having been nice can be represented as λx(x has been nice), the tensed binary relation ascribed by ‘My daughter is now taller than my son was’ can be represented as λxλy(x is now taller than y was). The former reads: the property of being an

x such that x has been nice; the latter reads: the relation between x and y such that x is now taller than y was. (Brogaard 2006, 197)

That is, there are persons, my daughter and my son, such that there obtains between them the relation of the first being taller than the second was.

In fact, such cross-temporal relations can be analyzed in a presentist formal language without being taken as primitive. “John is better looking than he was as a teenager” could be symbolized as:

∃x ∃t(x=John & P ∃t`(at t` x is a teenager & N ((x at t) is better looking than (x at t`)))).

That is, “There is (now) a person, John, and a time t, such that there was a time t` such that John was a teenager at t`, and now, John at t is better looking than John at t`.” It is worth commenting on the sentence “(x at t) is better looking than (x at t`).” This sentence is not to be read as meaning that that there are entities, x at t and x at t`, such that one is better looking than the other. Rather, it is to be read as saying x, given one set of circumstances (those that obtain when time t is present), is better looking than itself, under a different set of circumstances (those that obtain when time t`

is present).

If the foregoing construal of “John is better looking than he was as teenager” seems odd, we could also construe this statement as involving quantification over physical appearances, in the sense of the way someone or something appears, as follows:

∃x ∃A ∃A`(x=John & A=a physical appearance & A`=another physical appearance & Ax

& P (A`x) & ∀y ∀z)((Ay & A`z) y is better looking than z)

This says that there are two physical appearances, A and A`, such that John has A, it was the case that John had A`, and someone who has appearance A is better looking than someone who had appearance A`.

Finally, “My daughter is now taller than my son was” could be symbolized:

∃x ∃y(x=my daughter & y=my son & P ∃tN(x is taller than (y at t))).

As before, the sentence “x is taller than (y at t)” is not to be read as indicating that there is an entity y at t, such that x is taller than it. Rather, this says x is taller than y under a certain set of circumstances (those which obtained at time t). Alternatively, we could construe this using quantification over heights, as in

∃x ∃y ∃H ∃H`(x=my daughter & y=my son & H=a height & H`=another height & Hx &

P(H`y) & H is larger than H`).

That is, there are persons, my daughter and my son, and heights H and H`, such that my daughter has H, it was the case that my son had H`, and H is a larger height than H`.