Abstract
In this chapter I briefly give a philosophical introduction to truthmaker semantics and I present Fine’s logic and semantics for imperatives, I discuss two philosophical difficulties for Fine’s account (but really, for every truthmaker semantics similar to his) and propose some technical solutions.
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Notes
- 1.
The literature on truthmakers in the chiefly metaphysical sense is enormous. For this reason I do not note any references on this.
- 2.
For this line of thinking and a very useful introduction to the semantical project, see Fine (forthcoming[b]). For a recent unified (i.e. metaphysical and semantical) use of truthmakers see Jago (2018).
- 3.
- 4.
There would be an interesting question to be discussed at this point, which would however take us further away from imperative and deontic logic, namely the question of possible versus impossible states.
In previous work, Fine (Fine forthcoming[a]) came up with a mathematically elegant construction to represent possible and impossible states starting from a state space. In other terms, impossible states are those for which a natural fusion does not exist, and so we should force an arbitrary fusion of states. In this series of papers, a state space is called “topsy” if there is only one impossible state, namely, the full state (the fusion of everything); and “turvy” if there is only one necessary state, the null state(the fusion of nothing, which is part of everything). A space can also be “topsy-turvy”. A way to (or the only way?) to recover classic entailment is via a notion of compatibility, which is definable only with reference to possible states, i.e. to intensional constructions over the algebraic structure.
- 5.
The question easily becomes one about the relation between actions and times, and if actions happening at the same time can ever be kept separate. This question cannot be settled with Fine’s papers alone, as in fact it depends on further details which are not provided, such as “how injective” an hypothetical function from actions to times would be.
Suppose there is in fact a map f from actions to times (better, time intervals). This assumption is not only safe, but even (implicitly) required by Fine’s characterization of actions and imperatives. Suppose moreover f is order-preserving, a perfectly natural requirement. Now it is very easy to construct ker(f) such that we can induce equivalence classes of actions happening at the same time.
It is clear that if f is injective, there will be only one element for each equivalence class, and therefore no two distinct actions will happen in the same time interval.
But this is empirically false, f is not injective, and equivalence classes are populated by more than one action-equivalent.
Is this a problem for Fine’s account? It becomes one, I think, once we know that “we should always suppose that our imperatives relate ... to a fixed moment of time (p.6)”. On this topic, see the next section.
- 6.
- 7.
A different, disjunctive notion of consequence can be defined and studied.
- 8.
The reasons why T and S are not just sets of times and of space points are discussed in details by Link (1998), pp. 245–6. Roughly, if T were a set of time intervals, it will not be closed under unions, i.e. the union of any two intervals will not normally be an interval itself. But if we take T to be a union of time intervals, then any union of two unions is a union, so that T is closed under arbitrary union and has a (join)-semilattice structure. The same applies for intersections and for S.
- 9.
We may also offer as an additional reason for the partiality of f the considerations Fine (1994) makes about compounds existing at a time if all their parts exist at that time conjunctively.
- 10.
See for instance Alonso-Ovalle (2005).
- 11.
Thanks to Malte Willer who brought this class of examples to my attention.
- 12.
Should we wish to give clauses for iterated deontic operators, we can just adopt the clauses described earlier in this chapter.
- 13.
For a similar construction, see for instance J.B. Nation, Notes on Lattice Theory, http://www.math.hawaii.edu/~jb/.
- 14.
This endeavor should be distinguished from those of Leitgeb (2017), who starts from fundamentally inexact, monotonic structures, and of Johannes Korbmacher, ms.
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Faroldi, F.L.G. (2019). Truthmaker Semantics for Deontic Modals. In: Hyperintensionality and Normativity. Springer, Cham. https://doi.org/10.1007/978-3-030-03487-0_5
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