Wednesday, March 18, 2015

"Whatever is Moved is Moved by Another"

In this post I am going to try to defend Aquinas's First Way, specifically against the attacks brought against it by my friend Alex. Alex has written a fine explanation and critique of Aquinas's first and most famous Way, the argument from motion. The paper can be found here. Unlike many attacks on Aquinas's argument, Alex's reading of Aquinas is sympathetic and charitable, and thus at the same time his criticisms are incisive and well-taken. Anyone who wants to fully understand my post should read Alex's paper first; nevertheless, I will summarize some of his most important results.

To be specific, I'm going to defend Aquinas's premise that whatever is moved is moved by another, which we shall call (MOV). I do not claim that Aquinas ever made the defense I am making. In fact, I think the argument I give is in some respects new. But when all is said and done what I am concerned with is whether Aquinas's premise is defensible and true.

First of all, Alex points out that the term 'motion' in scholastic philosophy really means change. And to say that an object is changing with respect to some feature P is to say that it is going from being potentially P to actually P (more on this terminology here). I will take this for granted in everything I say about change. Now, in summary, Alex's main objection to Aquinas's defense of (MOV) is that either it is (a) valid but palpably unsound or (b) all its premises are true yet it is invalid, i.e. does not prove the premise (cf. his paper for details). However, Alex does think that Aquinas can defend the following more modest premise, which David Oderberg attributes to Aquinas:

(ACT) If something changes from being potentially F to being actually F then there must be some actual being that initiates this change.

The problem is that the more modest and highly defensible premise (ACT) is not equivalent to (MOV), leaving (MOV) undefended and the First Way ultimately uncompelling.

Before I present my argument in favor of Aquinas's (MOV), we need some definitions. First, the definition of an external object, (EXT):

(EXT) x is an object external to y just in case x is not y and x is not a part of y. [def.]

Let's also define what I will call the notion of change per se. (This is my own terminology.) Intuitively, something changes something else per se if it is the most immediate and fundamental efficacious cause of the change, [or the only sufficient cause such that you can't get any 'closer' to the change]. For instance, my hand pushes a stick which pushes a rock; the idea is that the stick, specifically its tip, is what changes the location of the ball per se. Here is a somewhat more formal definition of changing per se, which we will call (CPS):

(CPS) x is a cause per se of a change in something y with respect to feature P by action E just in case (i) x changes y with respect to feature P by action E, (ii) if at the same time as action E there is an action F of some parts of x, and these parts also change y with respect to feature P by action F, then the action F taken alone is not sufficient for changing y with respect to P, and (iii) x's action E taken alone is causally sufficient for changing y with respect to P [def.]

This definition can be made more precise, but the concept should be somewhat clear. The idea behind what I've called change per se is that whatever changes something else per se is the thing that changes y in the most immediate sense and a sense more proper than other things. So, for instance, take the following objects: Me, my arm, my arm's atoms, and a stick. When my arm changes the location of the stick, I can be said to change the location of the stick; however, I cannot be said to change its location per se, since, arguably, if somehow my arm persisted in its motion without the rest of my body (maybe by a miracle it was detached and could float, pushing things around), it would still be sufficient for the stick's changing with respect to its location (contra (ii)). On the other hand, arguably, my arm, or at least some part of it, changes the stick's location per se by its motion, since clearly it can be said to be changing the stick's location, thus satisfying (i). It arguably satisfies (iii) for the reasons stated, and it arguably satisfies (ii) since intuitively if you removed most of the arm but left a chunk of it or a few of its atoms, and they did the same thing as they did when my whole arm's motion occurred, then they would not be able to bring about the stick's change of location.

Now maybe you will disagree with my example and say that given my definition the arm does not change the stick's location per se. But the example is simply to illustrate what I'm trying to get at. If you deny the example is an example of change per se then you should understand what I mean. Also, I would not be surprised if my definition requires chisholming; nevertheless, I think it is on the right track, and helps get my point across. What is most important is just that we have some intuitive understanding of what I mean by something's changing something else per se.

Now we need the following premises. I will translate them into predicate logic, and from my translations it should be clear which formulas correspond to which English phrases.

(1) For all x, if x is changed with respect to P by something y then there is some actual thing z which changes x with respect to P.

Translation 1: ∀x[∃yCxy→∃z(Az∧Cxz)]

(2) For all x and y, if y is actual and x is changed per se with respect to P by y, then y is either an object external to x or y is a proper part of x.

Translation 2: x∀y[(Ay∧Dxy)→(Eyx∨Pyx)]

(3) For all x, if x is changed with respect to P by something actual y, then there is a z which is actual and changing x per se with respect to P.

Translation 3: x[∃y(AyCxy)→∃z(Az∧Dxz)]

(4) For all x and y, if y is changing x per se with respect to P, then y is changing x with respect to P

Translation 4: x∀y(Dxy→Cxy)

(5) For all x and y, if x is external to y, then x is not identical to y.

Translation 5: x∀y(Exy→x≠y)

(6) For all x and y, if x is a proper part of y, then x is not identical to y.

Translation 6: x∀y(Pxy→x≠y)

Let's examine whether these premises are plausible or not. 4, 5 and 6 can easily be shown to follow from the definitions of 'change per se', 'external object', and 'proper part' respectively, so I will not talk about them any more. 1 is basically just a more precise statement of (ACT), so I won't say too much in its defense, but the premise is eminently plausible: Upon a small amount of reflection it is simply obvious that what is merely potential cannot have any power to bring about something actual. Merely potential chemical reactions do not bring about any actual chemical reactions. So the only thing which can bring about something is something which actually exists, and doesn't merely potentially exist. 

The crucial premises then are 2 and 3. 3 is quite plausible on the face of it. For surely if something is changed at all, then there is something which changes it in the most immediate sense i.e. changes it per se. There must be some most immediate explanation or cause of a change right? If there isn't, then the change can never come about. This seems intuitive enough.

(The intuition is this: There seems to be some sort of infinity problem here, though the problem isn't with an infinite regress but rather with what we can call an infinite "progress" of causes. If there is no immediate cause, there has to always be another cause that's "closer" to the effect, but never one that actually "gets" to the effect. If it isn't clear what I mean I can elaborate.)

What about 2? The idea behind 2 is that some things can truly be said to bring about changes in themselves in some sense, but they can't be said to bring about per se changes in themselves; properly speaking, it is the parts which are bringing about the change in the whole. For instance, dogs can move themselves only because their legs do. So, the only thing which can bring about a per se change in something is either something external to it or else one of its parts.

Suppose to the contrary that the cause x of the per se change in y with respect to P is not one of y's parts and is not something external to y. Then since clearly whatever is not a proper part of y and is not external to y is identical to y, it follows  x = y. So y brings about a per se change in y. Now either (a) some of the parts of y bring about the change in y or (b) none do (either way, definitely no parts bring it about per se, as per our assumption).

Assume (a). If none do, then y's parts remain completely the same, yet there is a change in y. But surely y taken alone is not sufficient for explaining the change in y, and thus y does not cause a per se change in itself! After all, how could y change itself with no external influence and no action of any of its parts at all? It would have to be a spontaneous causa sui! So on the supposition that the parts do not act in any way so as to bring about the change in y, it follows y cannot be a per se cause of a change in itself. Since we assumed however that y does cause a per se change in itself, it follows we must reject this supposition and conclude that some of the parts do in fact bring about a change in y. In other words, we must reject (a) and assume (b).

Assume (b). Suppose on the other hand that some of the parts do help cause the change in y. By the definition of per se change, the action of these proper parts of y is not sufficient for explaining the change in y; but nevertheless the action of y taken apart from any external cause is. This seems to make little sense; y still appears to be acting as a causa-sui, since it is still causing a change in itself at least in part independently of the action of its parts. Since this is impossible--nothing can be a self-cause except by the action of its parts--we must conclude that the parts do not help cause the change in y. Thus (b) is false.

Since both (a) and (b) are false, and either (a) or (b) must be true given our assumption that y causes a per se change in itself, we must reject our assumption that y caused a per se change in itself. But if that is the case, then given that there is no external cause of y then x (the cause of the change in y) must be a proper part of y, as we set out to prove.

So much for premises 1 and 2 then. Now, given that all the above premises 1-6 are true we can prove:

(7) For all x, if x is changed with respect to P by some y, then x is changed by some z non-identical to itself.

Translation 7: x[∃yCxy→∃z(Dxz∧x≠z)]

I won't explain the proof here; instead, for anyone who doubts me, I have attached a formal proof below. From 7 and 4 of course it can be shown quite easily that whatever is changed with respect to P is changed with respect to P by some non-identical z: That is to say, whatever is changed is changed by another. Hence, given my 1-6, Aquinas's premise is secure.

Proof of 7:

[Note: If you can't see the proof, right click and either open in a new tab or else save the image and zoom in with some image viewer. I did the proof rather quickly so it is not the most elegant and could be done in fewer steps, but it gets the job done.]

Sunday, November 16, 2014

Lowe, Mental Causation and Laws of Nature

E.J. Lowe has an interesting theory of dualistic mental causation. He proposes a model where mental causation doesn't happen by mental events directly causing particular physical events. Instead the mental exerts its influence by explaining the existence of the entire neural causal chain leading to physical movement.

Suppose you have a series S of neural events in the actual world which lead to a physical movement P. Given Lowe's view that the mind does not directly cause any neural event there must be an external physical event E which causes the first event in the neural chain. So it'll look something like this:

  • S: E > (N1 > N2 > ... > Nn) > P

Suppose Lowe is right and mental events do not directly cause any particular neural event, but instead explain why this whole series of neural events exists instead of another. For comparison, Lowe proposes that the mind acts in the same way that God could act as sustainer of the universe. Suppose the universe is an infinite series of physical events where God doesn't directly cause any of the events in the series:

  • U: ... > P0 > P1 > P2 > ...

Now, every event in the series has a physical cause within the series, but there is still the question about why it is U which exists in the first place as opposed to another, distinct, series (say U*):

  • U*: ... > P0* > P1* > P2* > ...

On this picture God does not directly cause any particular physical event in U, but rather explains why this whole series of physical events exists in the first place. Here God sustains the universe, as opposed to interacting with it.

If the mind's influence on the body is like this, then the mind would have to have counterfactual control over which series of events occurs given E. In other words, if the mind had decided differently, then had E happened a different chain of neural events would have existed than S.

Now presumably one chain of events is identical to another if all the events in the one chain are identical to the corresponding events in the other. Somewhat more precisely:

  • (E1 > E2 > ... > Em) = (E1* > E2* > ... > Em*) if and only if (a) E1 = E1* and E2 = E2* and ... and Em = Em*; and (b) Ei > Ei+1 if and only if Ei* > Ei+1* for 1 < i < m

Obviously an identity criterion for infinite causal chains would be pretty easy to give too, but we're dealing with causal chains in the brain which lead to physical movement, so presumably this is unnecessary for our purposes. If you like you can think of this criterion as being restricted to causal chains in the brain.

Now then, suppose the mind does cause a different chain than the actual one. Call this chain S*. In order for a different causal chain to happen than the actual one it'd have to look something like this (for convenience I put the original chain above the new one):

  • S: E > (N1 > N2 > ... > Nn) > P 
  • S*: E _ (N1* > N2* > ... > Nm*) _ P with Ni =/= Ni* for some i, 1≤ i ≤ n

The reason there must exist an i such that Ni =/= Ni* is because of our identity criterion above. If all the corresponding events in the chains are identical then the chains themselves are identical. But we supposed that some different chain was brought about, which again is possible given Lowe's view, since Lowe's view implies the mind has counterfactual control over which series of events occurs given E.

I leave blanks in between E and N1* and between Nm* and P since we don't want to assume too much; maybe the mind will cause a chain to exist which is uncaused or which does not lead to the same physical movement P. Now, there are a few possibilities here.

(i) If E does not cause N1* then a law of nature is violated, since (in the actual world) it is a law that E will cause N1. (Actually, if I were being more precise a bit more detail and argument would be necessary here, but this is right.)

(ii) If E does cause N1* and N1 =/= N1* then a law of nature is violated, since it is a law that E will cause N1. Moreover, the laws of nature are changed, since E causes something else than it normally would.

(iii) If E does cause N1* and N1 = N1* then no law of nature is violated yet, and we must look for the smallest i in the chain such that Ni =/= Ni*.

(iv) Again, such an i exists because of our identity criterion and the assumption of a different series S*. So Ni =/= Ni*. Then it will follow that Ni-1 = Ni-1*. So Ni-1 does not cause Ni. So a law of nature is violated, since it is a law that Ni-l will cause Ni.

Moreover, Ni-1 must cause Ni*, since this is a chain of neural events leading to a physical movement. By a metaphysically motivated syntactic rule for these representations of causal chains, there must be a '>' between every node in the chain. (If there were causal 'gaps' how could you really call it a causal chain; the only real causal chains would be the ones before and after the 'gap'.) So that means a law of nature is changed, since Ni-1 causes something different than it normally would.

So, no matter what, if the mind ever exercises the power it has to bring about another neural causal series, a law of nature must be violated. Moreover, if the first event in this series is caused then the laws of nature must be changed by the mind at some point. This is bad.

However, there's more to say than just that it's bad. Lowe could reply that, in the actual world, there are psychophysical laws which hold and which tell us the mind never actually uses this power of counterfactual control over laws. Thus, the normal patterns of events we observe still obey the laws of nature. The laws of nature continue to hold in this world.

But that leaves two options: (1) Does that mean minds never actually do anything? Are they causally effete and just let the world stay its course? Or else (2) Do minds still do something, and thereby contribute some causal influence? If they do, does that mean they cause the actual laws of nature to hold in the case of neural causal chains?

Both possibilities seem weird and unpalatable. If the first view is true that means mental causation never actually occurs in this world, though it could. On the second view there are two possibilities: Either (A) mental causation is superfluous, or else (B) laws of nature are really really weird.

On the one hand, it could be that the laws of nature by themselves are enough to explain why each event in the series causes the next, in which case the mental decision is superfluous and just 'backs up' the law of nature by its influence. The mind gives the laws of nature more 'oomph', though they are completely sufficient even without this 'oomph' at all. This would be case (A).

On the other hand, it could be that, once we move from E to N1, all of a sudden the mind is needed to hold up the laws of nature from then on. This is option (B). And that seems weird. That would mean when the brain comes into play, all of a sudden the laws of nature by themselves aren't sufficient for producing later events and the influence of the mind is needed to 'keep them going'. When it comes to the brain, the laws of nature need crutches; the mind is a metaphysical crutch. This is very metaphorical of course, but the point could certainly be made more precisely.

To be fair, this theory can't be disproven by the science. Neither option (1) nor (2) above is even possibly ruled out by the actual scientific evidence. So maybe this is one way to reject causal completeness while holding a theory that is empirically equivalent. I'm not sure it's a plausible one though.

Saturday, October 18, 2014

The 'How Does it Work?' Objection to Dualism: #1 Different Substances

"How can the mind move the body?" This is one of the most familiar rhetorical questions for dualists about the mind. It is rhetorical since it is meant to make the point that dualists can't explain how the mind moves the body, and that this is a problem for dualism.

One problem is that this question is highly ambiguous. This is reflected in the literature by the fact that the phrase 'interaction problem' is used by different philosophers to refer to quite different arguments. When someone poses this question there are multiple objections someone could have in mind. Someone could be saying it's in principle impossible for dualists to provide an explanation. Or they could simply be saying there is no plausible candidate for an explanation. Or they could be saying it's very likely that non-physical causation never happens in our world. Moreover, they could pose these arguments for different reasons. So it's important to get clear on which objection we're talking about. I'll try to distinguish a few ways we can formulate this objection and then reply to these objections. In the first post I'll identify and respond to what I call the 'Different Substances Objection'.

#1: The Different Substances Objection

First, one might object that, since mind and matter on a dualist view are completely separate there is no way for them to interact. Everything is mental or material, but not both. So there could not be interaction between the two. The problem is that it's not generally true that entities from two mutually exclusive categories cannot interact causally. Everything is either a proton or a non-proton for instance, but that doesn't mean protons and non-protons cannot interact.

Another very similar form of this objection is that mind and matter are different types of substances, so the two cannot interact. Or a more particular version of this objection would say that the mind is non-extended and the body is extended, so the two cannot interact. One problem is that, in general, it is false that in order for substance x to interact with substance y of kind F, x must itself be of kind F. You don't need to be a human to interact with a human; you don't need to be a proton to interact with a proton; etc.

A final and more sophisticated way to formulate this version of the objection is that since mind and matter share no intrinsic properties, the two cannot interact. First of all, the main premise of this argument is not true. Being a substance is common to both our minds and our brains. Having causal powers is intrinsic to both (note: one need not presuppose causation between mind and body occurs to hold this is true, since one should at least admit mental to mental causation). They both have the property of having metaphysical components. Of course, they don't share any intrinsic physical properties (since the mind does not have physical properties). But at least some of what I have listed are plausibly intrinsic properties.

More importantly though, why do two substances have to have common intrinsic properties to causally interact? This principle would have to be motivated by some more general theory where two substances' having common intrinsic properties P1, ... , Pn plays a relevant role in their ability to engage in causal interaction. In particular, to be relevant, if x causes change C in y in some way, then these P1, ... , Pn must contribute some causal influence to C. I'm not sure how else one would motivate the main premise here.

But we can imagine situations where the common intrinsic properties of agents plays no relevant role in the interaction of the two. For instance, even though a Jedi and a table share the intrinsic properties of having mass or volume or other physical properties, when the Jedi uses the Force he doesn't in any way seem to take advantage of his having mass or volume or his other physical properties. Of course, this is just an imaginary situation, but it seems perfectly coherent and thus there can be no a priori reason for thinking common intrinsic properties must play a relevant role in all causal interaction (of course, this isn't to deny that sometimes they do).

In fact, there might be some cases which are actual counterexamples. For instance, consider the EPR paradox cases from quantum mechanics. Suppose you have a source emitting an electron-positron pair in a state of quantum entanglement, where the spin of each is anti-correlated with the other. In other words, if electron e has upward spin then p has downward spin, and if e has downward spin then p has upward spin. Suppose moreover you have two observers A and B in different locations who can measure the spin of the particles along some axis Z, and e is sent to A while p is sent to B. If A measures e as having an upward spin then B will measure p as having a downward spin with 100% probability. On the other hand, if A measures e as having downward spin, B will measure p as having upward spin with 100% probability. Since experimentation and Bell's Theorem rule out local causal explanation here, and supposing there is causal interaction between e and p, there must be some causation here where local intrinsic properties such as mass, volume, velocity, etc. do not play a role in the causal interaction.

Of course, the particles do share the intrinsic properties of having spin. But it's not e's simply having spin which contributes causal influence to p's particular spin; rather, it is e's having an upward or downward spin which does so. For what the spin of p is depends on the actual spin of e. So it isn't clear that we can identify some common intrinsic property of e and p where's e's having this property causally influences p's having a particular spin.

This seems to me to be an adequate reply to the different substances objection. In the next post I'll talk about what I call the 'No Mechanism Objection', which poses the problem that there seems to be no familiar model which could make causal interaction between the mind and the body intelligible.

Monday, September 29, 2014

An Issue With Metaphysical Reduction

Take a fact F. In general, what does it mean to say that fact F metaphysically reduces to fact F'? Note I am speaking of metaphysical reduction as opposed to conceptual reduction. First of all, the latter has to do with concepts and propositions rather than facts. For example, when we say that being a bachelor just means being an unmarried male, or when we say the proposition that Alfredo is a grandfather just means that Alfredo is the father of a parent, these count as examples of conceptual reduction. These explications of meanings are just the result of fully specifying the nature of our concepts as they stand. These are very simple examples, but the more complex instances of conceptual reduction in philosophy follow the same general idea as these ones.

Metaphysical reduction on the other hand has to do with facts in the world and how they stand in relation to each other. I take it that the following necessary condition imposes a restriction on the relation of metaphysical reduction:
  • (R) If fact F metaphysically reduces to fact F' then (i) fact F holds in virtue of fact F' holding and (ii) the holding of fact F is nothing over and above the holding of fact F'.
As an example, physicalists often say that all mental facts are reducible to physical facts. I take it that this at least means that the mental facts hold in virtue of the physical facts and that they are nothing over and above the physical facts.

Now, (i) and (ii) seem to me to be in tension with each other. In fact, on the most straightforward reading of (ii) their simultaneously holding leads to a contradiction. Hence, we must find some other way to explain (ii), since it does not seem like a primitive relation. This is rather difficult. Let me explain.

By (i), reducibility must be an asymmetrical relation. This means that if F reduces to F' then F' does not reduce to F. For suppose F reduces to F'. Then F holds in virtue of F'. But the 'holding in virtue of' relation is asymmetrical, since otherwise there would be circular chains of ontological dependence. So if F holds in virtue of F', then F' does not hold in virtue of F, and thus by (R), F' is not reducible to F.

The problem is that the most straightforward reading of (ii) is that the holding of fact F is identical with the holding of fact F'. After all, suppose F and F' are not identical and we are dealing with a world of just F and F' (here I'm abbreviating, and I should really be saying the holding of F and the holding of F'). Then there is a perfectly clear sense in which F is something over and above F', viz. there are more things in the world than F! For if F =/= F', then for some x, x =/= F'. So there is something out there in the world which is extra-mentally distinct from F'. That seems to be a legitimate sense in which F is something over and above F'. So if F is not something over and above F' then F = F'.

But of course, if that were the case, then the 'in virtue of' relation here would not be asymmetrical, since if F = F' and F holds in virtue of the holding of F', then by substitution of equals F' holds in virtue of the holding of F. So reducibility would not, in fact, be asymmetrical. And that is a contradiction, since we earlier established it was.

One option is to say that the 'in virtue of' relation is not asymmetrical. But that seems deeply problematic insofar as it doesn't allow us to capture the reducibility we want to pick out. After all, every materialist will accept that all mental facts reduce to physical facts, but no materialist would ever dare say the physical facts reduce to the mental facts! (Personally I find the latter suggestion more plausible than the former, but regardless it is not something the materialist would ever claim.)

Instead, we have to find a sense in which one could say fact F is nothing over and above F' even though F is not identical to F'. And I'm not sure how to explain this. No idea if this works or not, or whether it is at all helpful, but here's a thought: Let us denote by 'a full truthmaker of P' a truthmaker of P which is not a constituent or part of some other truthmaker of P. Let Q be the proposition expressing the holding of F. Maybe we can say F is nothing over and above F' if the set of all full truthmakers of the proposition Q contains only F'. That would make (i) superfluous it seems. Or at least from pretty uncontentious premises (i) would follow as a consequence. This theory is a little weird though, since the question arises as to what, metaphysically speaking, explains why Q would be distinct from the proposition expressing the holding of F'.

With that said, I don't know if that's on the right track. And even if it gets the extension of the relation right it might not even produce a deeper understanding. The point being, I don't myself know how to explain (ii). Like I said though, it doesn't seem like this is a primitive or undefinable relation. I wonder then what we can say about it.

Monday, September 1, 2014

What God Knew and Abraham Didn't

The traditional story of Abraham and Isaac is one of the most perplexing parts of the Bible, at least for philosophers. There seems to be some sort of implicit contradiction in the idea of God commanding someone to sacrifice a human being, especially an innocent boy. Moreover, it seems like if one were commanded to do this one should not do it. Before thinking about this more we should review the story very quickly.

In the recounting of the story in Genesis 22, God wishes to test Abraham and see whether he "fears God." To this effect, God commands Abraham to go and sacrifice his only son, Isaac. On the third day of their journey Abraham takes Isaac up to a mountain to sacrifice him. Before going up, Abraham tells his servants with him, "We shall worship and come back to you." (Genesis 22:5) Abraham then binds Isaac and prepares to sacrifice him. When Abraham grabs his knife to kill Isaac an angel sent from God stops him by telling him not to kill the boy. God speaks through the angel and says that he now knows that Abraham fears him, and because of his actions God will shower blessings upon Abraham and his descendants.

Sometimes opponents of Christianity will say that this verse proves an inconsistency in the Christian conception of God. On the one hand, God is supposed to be a perfect being, and a perfect being, it seems, would never command something intrinsically evil such as sacrificing an innocent person to him. On the other hand, the Bible says he does. Let's give a precise argument which captures the force of this more vaguely formulated one.

Sunday, August 31, 2014

The Universe is Contingent (And Therefore Needs an Explanation)

One common fallacy is the fallacy of composition, where one argues from the fact that each part of a thing has a certain feature to the conclusion that the whole thing has that feature. For instance, one could argue that every brick of the house is cube-shaped, therefore the house is cube-shaped. Or one could argue that each part of one's brain is unconscious, therefore the whole brain is unconscious. These inferences are fallacious.

However, I think it is worth noting that not all inferences from properties of parts to properties of the whole are invalid. If each part of a wall is made entirely of stone, then the whole wall is made entirely of stone. Similarly, if each part of the ball is entirely red, then the whole ball is entirely red. And so on.

Contingency seems to be like this, at least in this case. So here's an argument that the universe must be contingent:

Friday, August 29, 2014

Quantifier Variance and the Semantics of Quantifiers

In my previous post I explained the basic idea behind quantifier variance. Now I want to criticize it. In particular, I said I want to point out some problems with the quantifier variantist's simultaneously affirming the following two statements:

(i) the different quantifiers behave the same logically; and

(ii) the different quantifiers have different meanings.

Let's do a little basic semantics. Let's define the truth function τ[ψ]U,g relative to models U and g for the cases of quantified formulas ψ as follows. The following definitions are true for all models M, all variable assignments s, all variables x, and all formulas φ. If a formula is not assigned to T it is assigned to F:

τ : {<ψ,U,g>|ψ is a formula, U a model, g a var. assign.} → {T,F}
  • (τ-)τ[∀xφ]M,s = T ⇔ for all variable assignments s′, if for all variables v, s(v) ≠ s′(v) ⇒ v = x, then τ[φ]M,s′ = 
  • (τ-)τ[∃xφ]M,s = T ⇔ for some variable assignment s′, for all variables v, s(v) ≠ s′(v) ⇒ v = x, and τ[φ]M,s = 

Monday, August 25, 2014

Basics of Quantifier Variance

When I say that there are tables is it unambiguous what I'm saying? Quantifier variantists say no. Or at least they would say that in certain contexts it is not. In particular, the sentence is ambiguous when we are engaging in metaphysical debate about the existence of the table, as in the following case.

Consider the debate between what I will call compositionalism and anti-compositionalism. Compositionalism is the thesis that there are composite material objects, while anti-compositionalism is the thesis that there are not. Take the case of a world with just a table and its parts, and suppose we are considering a form of compositionalism which says there are tables. Assume further that there are exactly n atoms which, according to this form of compositionalism, are proper parts of the table. Note that we are using a philosophical definition of 'atom', according to which an atom is a material object which has no proper parts. Anti-compositionalism says there is no table; there are just the n atoms. 

In essence, compositionalism says (A) there are n+1 distinct things (viz. the n atoms, plus the table), while anti-compositionalism says (B) there are n things and there are no more than n things. Note that (A) and (B) can be adequately translated into a quantified language which only contains variables, quantifiers, sentential connectives, and the identity sign with the usual interpretation. For example, (A) would be translated as follows:

∃x1∃x2...∃xn((x1≠x∧ ... ∧ x1≠xn+1) ∧ (x2≠x3 ∧ ...  x2≠xn+1) ∧ ... ∧ (xn≠xn+1))

Thursday, August 21, 2014

Pure Actuality

Many scholastic theologians, most notably Aquinas, make the claim that God is "pure actuality." This is supposed to do a lot of philosophical and theological "work"; it is by showing that there exists a being which is pure actuality that Aquinas is able to deduce many of the divine attributes. However, it is not immediately clear what this even means if one is not familiar with the metaphysical context of medieval philosophy.

A charitable interpreter who has read some medieval philosophy may be able to see how scholastics use this claim and identify certain inferences from this claim as being valid and others not. But it'd be nice if we had a more precise characterization of what it means to say God is 'pure actuality', so that we can see if all that Aquinas says follows actually does follow from this claim. Moreover, once we have a precise characterization of what Aquinas is even asserting, we can begin to more clearly assess the plausibility of the claim itself and whether Aquinas has established it. I propose the following definition:

  • x is pure actuality if and only if for all (intrinsic) P, if x is P then x is actually P.

For completeness and wider scope of application, I also propose the following definitions of a thing's being 'composed of' or 'having' actuality and potentiality:

  • x is composed of potentiality if and only if for some (intrinsic) P, x is P and x is potentially P
  • x is composed of actuality if and only if for some (intrinsic) P, x is P and x is actually P.