Thursday, May 21, 2015

Essence and Counterpossibles

In my last post I made the point that the predicate position in real definitions is hyperintensional. So, even if two predicates have the same intension, i.e. necessarily apply to all the same things, they might not be able to be substituted for each other in the real definition while preserving truth value. This means that, among the necessary properties of a thing, we have to distinguish those which are essential to the thing from those which are not.

Maybe one way to do this is by using counterfactuals with impossible antecedents, also known as counterpossibles. 

The general idea is this. Counterpossibles, according to a certain semantics, are also hyperintensional. You can insert intensionally equivalent antecedents into the same counterfactual, but only some of these counterfactuals will be true while others will be false. So maybe we can use a particular counterpossible schema (as we will see, that in (1*)) to discriminate between properties that are essential and those that are non-essential. 

More specifically, we can take two intensionally equivalent properties F and G, insert F into the antecedent, insert G into the antecedent, and the counterfactual may have a different truth value depending on which of F or G is substituted. Thus, the essential vs. non-essential distinction will be able to be defined in terms of the truth or falsity of instances of a certain counterfactual schema. To make the point more vivid, the counterfactual schema will be like a box we can insert intensionally equivalent properties such as F or G into. If we put in G for instance and the box outputs TRUE then G is essential; if it outputs FALSE then G is not essential. Since counterpossibles are hyperintensional, the 'box' won't always give the same output for properties with the same intension.

So: Consider two intensionally equivalent properties F and G. This means that, necessarily, if anything has F in a world then it also has G in the same world, and vice versa. If F and G are intensionally equivalent then the properties λz[□Fz] and λz[□Gz] are also intensionally equivalent, and thus the formulas λz[□Fz](x) and λz[□Gz](x) are intensionally equivalent. Now, consider a counterfactual of the form:

  • (C) If φ had been the case then A.
Since we are going with an interpretation allowing for non-trivially true counterpossibles, we can substitute in for φ either of two necessarily false propositions, P or Q, but it won't automatically follow that (C) will come out true under both substitutions.

So suppose it is true that a is necessarily F and necessarily G.

Let P be '¬λz[□Fz](a)', let Q be '¬λz[□Gz](a)' and let A be '¬λz[∃yy=z](a)'.

Again, it doesn't follow automatically from the semantics of counterpossibles that substituting P in for φ will give you the same truth value as substituting Q for φ, despite the fact that these two formulas are intensionally equivalent. What we can do then to distinguish whether F is essential or G is essential (or neither) is substitute in 
P for φ and Q for φ in (C). If (C) comes out true, then the property is essential; if it comes out false, then it is not.

With this hypothesis in mind, here is a very rough first stab. For any x:

  • (1) For any P, if P is a de re necessary property of x, then P is an essential property of x if and only if had x lacked P then x would not exist.
More formally, let |x| be λz[z=x]: 
  • (1*) For any P: If λz[□Pz](x) then: |x|Px iff (¬λz[Pz](x) □→¬λz[∃yy=z](x)])
Again, keep in mind that the counterfactuals here are to have a semantics where they can be read as non-trivially true counterpossibles, and thus not according to the standard Lewis-Stalnaker semantics. Also, the 'essentialist box', F, comes from Kit Fine's 'Logic of Essence'. Roughly, FA means that in virtue of the essence of the F's, A holds. So if |x| is the property of being identical to x, then |x|Px turns out to mean that in virtue of the essence of x P holds of x. Fine parses out the logic this way for its semantical and logical elegance.

Consider one example given by Kit Fine: Suppose we have the property λz[z∈{z}]. Intuitively, this is the property satisfied by something whenever it is in its singleton. Assuming that, necessarily, I am a member of my singleton, then this is a de re necessary property of me, i.e. λz[z∈{z}](a) However, it seems to not be an essential property of me.

So, is it true that the following holds?

  • (A) Had Alfredo lacked λz[z∈{z}], Alfredo would not exist.
  • (A*) ¬λz[z∈{z}](a) □→ ¬λz[∃yy=z](a)
It seems that (A) does not hold. For it's irrelevant to my nature whether there are any abstract objects, such as sets, at all. After all, it seems that if nominalism were true, I would still exist. This lends some weight toward thinking that, if I had lacked λz[z∈{z}]I would still exist. But then by the criterion in (1), since the counterpossible does not hold for the property λz[z∈{z}], it must follow that λz[z∈{z}] is not essential to me. 

Given a de re necessary property of x, P, it might also be that the truth of the appropriate counterpossible is just a necessary condition for a property's being essential. In other words:

  • (NEC) If P is essential to x, then were x to lack P x would not exist. 
  • (NEC*) If |x|Px, then (¬λz[Pz](x) □→¬λz[∃yy=z](x)]).
Or it might be a sufficient condition as well, i.e.:
  • (SUFF) Given that if x were to lack P x would not exist, then P is essential to x.
  • (SUFF*) If (¬λz[Pz](x) □→¬λz[∃yy=z](x)]), then |x|Px.
Keeping in mind of course the sense of the term 'essence' in mind, and the relevant semantics for counterpossibles, (NEC) seems definitely true, and probably uncontentious. I suppose the interesting question is whether (SUFF) is true. I think the examples lend some support to the idea, such as the case of the singleton set given above.

In the case of (SUFF) it is particularly important that we use the right semantics for counterpossibles. If (SUFF) is true then this is very useful when talking to those who don't recognize the sense of essence at stake here: If they already know how to evaluate the counterpossible in the antecedent according to a 'non-trivial' semantics, then we simply say, "Plug in the property for the antecedent; if the counterfactual holds non-trivially, the property is essential. Now you know what I mean." This might also be a nice way to interpret people who give multiple definitions of 'essential' and 'accidental' properties, such as Aristotle. Aristotle gives a modal definition of essential properties which can sound like it might contradict other definitions of his; but (1) is 'modal' too, and it might be a way to interpret Aristotle that makes him consistent.

Probably potential counter-examples to the hypothesis come to mind. I know that I already see some issues. But it might be useful to see how far this hypothesis can go. And maybe if the hypothesis doesn't hold in general (I bet it doesn't) it might at least help us pick out an important class of the essential properties. After all, it seems in part that the reason we recognize λz[z∈{z}] as non-essential to me is because in some (non-trivial) sense had I lacked it λz[z∈{z}] I would still exist. Had nominalism been true, I'd have still been real (save if you're a Platonist/Pythagorean of a certain sort, in which case maybe you'd have good, non-trivial reason to deny that the counterfactual holds).

It might seem overly complicated to do this quasi-formally as I have, but one thing I'd like to do is look more at the formal semantics of essence such as Fine's for instance (hence the essentialist 'box' operator from Fine's papers), and see how this relates to the formal semantics of counterpossibles (whatever that happens to be). 
Given the inter-dependence of the two notions, maybe the correct semantics for counterpossibles will help us find the correct semantics for essence, and vice versa. Maybe the notion of essence will help us give a more principled similarity metric for counterpossibles in certain contexts. Also, maybe the notions of essence and counterpossible will relate closely to other notions, such as grounding, dependence and explanation. It might be an interesting project to see how far formal methods can help us here in finding relations between these concepts.

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