Philosophy of Logic

58th Philosophers’ Carnival

Richard Brown 3 Comments

Welcome to 58th edition of the Philosophers’ Carnival!

I am happy to be hosting the carnival again and glad to see that it seems to be doing well. I always liked the way that Avery did the 46th (international) Carnival and so I modeled this edition on his ‘psuedo-conference’ format. What follows is, indeed, a ‘narrow cross-section of philosophy from accross the web’.

Special Session on the Employability of Philosophers

  1. Presenter: Tom Brooks, The Brooks Blog
    The truth is out there: employers want philosophers
  2. Respondent: Rich Cochrane, Big Ideas
    The Value of a Philosophical Education

Symposium on Philosophy of Science

  1. Sharon Crasnow, Knowledge and Experience
    Is Science Based on Faith?
  2. Matt Brown, Weitermachen!
    Common Sense, Science, and “Evidence for Use”

Symposium on Race and Liberty 

  1. Richard Chapell, Philosophy, et cetera
    Implicit Interference
  2. Joseph Orosco, Engage: Conversations in Philosophy
    It’s Only Racism When I Say It Is

Invited Session

 Symposium on Philosophy of Consciousness

  1. Tanasije Gjorgoski, A brood comb
    The Myth of ‘Phenomenal/Conscious Experience’
  2. Richard Brown, Philosophy Sucks!
    Priming and Change Blindness
  3. Gabriel Gottlieb, Self and World
    Pre-reflective Consciousness: A Fichtean Intervention

Symposium on Metaphysics and Epistemology

  1. Marco, El Blog de Marcos
    Truthmaking and Explanation
  2. Kenny Pearce, blog.kennypearce.net
    What Does Bayesian Epistemology Have To Do With Probabilities?

Symposium on Philosophy of Religion

  1. Dave Maier, DuckRabbit
    D’Souza vs. Dawkins
  2. Enigman, Enigmania
    Is the Free-will Defence Defensible?
  3. Chris Hallquist, The Uncredible Hallq
    What’s the deal with philosophy of religion?

I hope you enjoyed! Be sure to check out future editions of the Philosophers’ Carnival.

    Submit your blog article to the next edition of philosophers’ carnival using our carnival submission form. Past posts and future hosts can be found on our blog carnival index page

A Counter-Example to the Cogito?

Richard Brown 6 Comments

Descartes famously argued that the one undoubtable truth is that when he is thinking he exists. This idea, I think therefore I am, is clear and distinct, which are the marks of self-evident necessary truths. Descartes’ idea still has a lot of pull, but isn’t there an obvious kind of counter-example to it?

Couldn’t it be the case that the Evil Demon has multiple personality disorder and that I (or you) am a figment of this fragmented consciousness? Couldn’t it be the case that the Evil Demon has made me up in the telling of some story to ease his boredom? Or maybe the Evil Demon is a Solipsist. In reality He is the only thing that exists and all of us are just a backdrop his all-powerful mind has concocted…It would then be the case that I wonder whether I exist and yet I do not exist…aren’t these kinds of things  counter-examples to the Cogito?

One response that might be made is that, while it is the case that I do not technically exist as I thought I did (as a mind-independent entity), I still exist (as a fictional mind-dependant entity). So, I still exist, just not in the way that I thought I did. This would allow us to keep the general truth that whenever there is some thinking there has to be a thinker (it would just be the Evil Demon himself who is actually doing the thinking), but it does seem to do violence to clearness and distinctness as a criterion of self-evident necessary truths.

Does anyone know if this kind of objection is ever dealt with by Descartes or any of his objectors/commentators?

Flamming LIPS!

Richard Brown 4 Comments

So I just got back from the Long Island Philosophical Society meeting, where I presented Language, Thought, Logic, and Existence (the virtual version is here if you missed it, which considering that there was 10 people there, you probably did) it was early but I had a good time…in the afternoon I commented on a paper by Glan Statile called ‘Mind, Matter, and Religious Experience’ which argued that materialism about the mind was empirically false as shown by the near death experience of Pam Reynolds.

I argued that there was no evidence that she had had any experience during the one hour time that she was actually brainsead and that the details of her experience suggest that she had experience before and after the time she was literally dead. During the discussion I was asked if she was brain dead for the whole seven hours and had had some experience would I be convinced that materialism was false. I said that I thought I would and he said that I had conceeded too much.

So suppose that Pam had no electrical activity in her brain at time T1 and that later when she is awake she is able to recount details from T1 that she would only be able to know if she had experienced the events she described at T1. Glen was arguing that this would be empirical evidence that materialism was false, and I had been agreeing with this premise. But the suggestion was, why wouldn’t this instead be evidence that there was some other (physical) property of the brain, which we weren’t monitoring and which was responsible for generating experience. So, maybe electricity is just an accidental feature of the brain, and something else is responsible for generating experience (maybe spin, or whatever). So, if materialism is an empirical hypothesis, how could it ever be falsified?

I also had a very interesting discussion with Jonathan Adler about my claim that most moral truths are analytic, but I plan a seperate post for that.

Re-Inventing the Wheel

Richard Brown 13 Comments

In discussing the ideas of the previous post with a colleague (Chris Steinsvold) I found out that I had basically re-discovered (a simple kind) of hybrid logic. Here is a nice introductory paper by Patrick Blackburn aptly subtitled ‘A Hybrid Logic Manifesto’….there is even a hybrid logic website! (where, incidently, I got the paper from…there are all kinds of papers there!)

The basic idea behind hybrid logic is that you first add new constants, called nominals and usually represented by i, j, k,… which are supposed to ‘denote’ or name particular worlds, instants, or whatever is required by the modality one is working in. One then introduces the ‘@’ operator, with a nominal subscript. So, @ip says that p is true at the world/instant/whatever that the nominal i denotes.

So, the ‘@’ operator that I introduced ( v(@a,w)=T iff (a, actual world)=T and v(~@a,w)=T iff (~a, actual world)=T) can be understood as the standard ‘@’ of hybrid logic where the nominal is the actual world and so assumed. To keep things simple I think I will change this @ with the assumed actual world nominal to ‘Θ’ this will give me a three operator modal logic [] (true at all possible worlds), ◊ (true at some possible world), and Θ (true at the actual world)…One reason to do this is that I am still not sure whether @ip means that p is true ONLY at i…that might be acceptable for tense logics but I wouldn’t want to take that to be true for Θp…this just says that p is true at the actual world, whether it is true at other possible worlds doesn’t seem to matter too much for the purposes that I want to use it for.

Then the ‘simpler’ proof that God’s omniscience is incompatible with my free will that I gave in the comments of the previous post can be put as follows

1. ΘB(k,a)
(assume that God actually knows beforehand what I will do)

2. Θa    (from 1)
(if He actually knows beforehand that a, then a is true at the actual world)

3. Θa –> []Θa
(if a actually happens, then it is true in all possible worlds that a happens in the actual world)

4. []Θa (2,3 MP)
5. ΘB(k,a) –> []Θa   (1-4 CP)
(if God actually knows what I will do before I do it then it is necessary that I do it in the actual world)

So, standard modal logic is not expressive enough to capture the intuition that God’s foreknowledge is incompatible with Human free will. We need to move to hybrid logic to do so…but this isn’t worrisome, since Blackburn shows (in the above linked paper) how we can translate any hybrid logical formula into standard first-order logic…

Third Time’s the Charm (or: This Time I Really Got It!!!)

Richard Brown 17 Comments

OK, so I am basically obsessed with this stuff about God’s omniscience and Free Will. I have been having some very interesting, and helpful, discussion about whether Plantinga’s defense, which I take it is the standard defense, of their compatibility is any good or not. I have a sneaking suspicion that the two are incompatible and I have been trying to construct a poof to that effect, with mixed results…but I think I got it this time…if it turns out that I don’t then I promise that I will give up!

It seems to me that the problem is that “If God knows what I will do before I do it then it is necessary that I do it” does not really capture what the person who says that God’s foreknowledge is incompatible with our free will is trying to say. This is because, as we have seen, it must be the case that all my actions are necessary, but this doesn’t sound right at all (however, I do think some people are committed to it).

So, to make it clearer what I am actually trying to say, let me introduce a new modal operator ‘@’  with the following truth condition, where ‘v(x,w)’ is the valuation of x at world w,

v(@a, w)=T iff v(a, actual world)=T

v(~@a,w)=T iff v(a, actual world)=F

this says that @a is true if a is in fact true at the actual world and ~@a is true if a is in fact false at the actual world (~@a<–>@~a) so there is no need to introduce a fourth operator). ‘[]’ and ‘<>’ are given their usual interpretations.

Then I can say that God actually knows before I do a certain action that I will in fact actually do it. To avoid getting involved in tense logic let us introduce a predicate ‘B’ for before (though I think we could define ‘B’ in terms of the standard tense operators F, P, H, and G). Let ‘k’ be ‘God knows that’ and ‘a’ be some action of mine, then I can symbolize ‘God actually knows before I do action a that I will in fact actually do action a’ as @B(k,a), then the proof goes as follows

1. @a & @B(k,a)   (this says that God actually knows what I did before I did it)

2. []@B(k,a) –> []@a  (necessary truth)

3. @a –> []@B(k,a)      (necessary truth)

4. @a       (from 1)

5. []@B(k,a)     (4,3 MP)

6. []@a     (5,2 MP)

7. (@a & @B(k,a)) –> []@a   (1-6 conditional proof)

Since 7 says that if it is the case that I actually do a and God knows beforehand that I actually do a then it is necessary that I actually do a, and God’s actually knowing that I do a entails that I actually do a (7) reduces to

7′ @B(k,a) –> []@a

which says that if God actually knows what I will do beforehand then it is necessary that I actually do it.

Now one may wonder what the difference between ‘a’ and ‘@a’ is. Ordinarily there will be no difference, but there will be a huge difference when we examine the modal properties of the two. []a will be true iff a is true in all possible worlds, whereas []@a will be true if @a is true in all possible worlds, or in other words if it is the case that at every possible world it is true that, in the actual world, I do a. This is why (3) above is a necessary truth but (3′) is not,

(3′) a –> []B(k,a)

(3′) says that if I do a then in every possible world God knows beforehand that I will do a. This can be false because there are possible worlds where the antecedant turns out false because in that world I do not do a and so God does not know it. But (3) can’t be false. For if it were then it would be the case both that I actually do a and that God did not actually know beforehand that I did a, which is just to deny that God is omnicient (so enigman will be happy).

Whew! So, if this is right then God’s foreknowledge is indeed incompatible with my having free will. If not then I will finally have to admit that there is at least one metaphysical interpretation on which it can both be true that God knows what I will do before I do it and that I am free…and I will then actually be very depressed!

(I Think) I Got It!

Richard Brown 17 Comments

If you have been following the discussion in Plantinga on Free Will and Omiscience you will have seen that I have been struggling to construct a proof of (1), which says that if God knows that I will do some action before I actually do it then it is necessary that I actually do it, from (2), which says that it is necessary that if God knows what I will do some action in advance then I will actually do it.

(1) K(G,R,a) –> []D(R,a)

(2) [](K(G,R,a) –> D(R,a)

So far the two attempts that I have made have both been invalid because of some bonehead mistakes. This has been driving me crazy for the past couple of days, but now I think I got it, in fact it almost seems too simple (which probably means I made another bonehead mistake!)…

I thought that it would be easier if I did not include quantifiers, but I think that is what actually confused me. So, what I really want to prove is (1′), which says that for any action x if God knows that I will do it in advance then it is necessary that I actually do it, from (2′), which you can figure out for yourself.

(1′) (x)(K(G,R,x) –> [](D(R,x))

(2′) (x)[](K(G,R,x) –> D(R,x))

this actually turns out to be quite easy (I *think* 🙂 ).

1. ~(x)(K(G,R,x) –> []D(R,x))          assume as a theorem

2. (Ex)~(K(G,R,x) –> []D(R,x))         1, by definition

3. (Ex)~~(K(G,R,x) & ~[]D(R,x))   2, by def

4. (Ex) (K(G,R,x) & ~[]D(R,x))       3, by def

5. K(G,R,a) & ~[]D(R,a)               4, EI

6. K(G,R,a)                          5, CE

7. []K(G,R,a)                       6, necessitation

8. ~[]D(R,a)                       5, CE

9. (x)[] (K(G,R,x) –> D(R,x))           assumption (2′)

10. [](K(G,R,a) –> D(R,a))              9, UI.

11. []K(G,R,a) –> []D(R,a)              10, distribution

12. ~[]D(R,a) –> ~[]K(G,R,a)         11, contraposition

13. ~[]K(G,R,a)                                 8,11 MP

14. []K(G,R,a) & ~[]K(G,R,a)          7,13 CI

15. (x)(K(G,R,x) –> [](D,R,x))      1-14 reductio

What I didn’t notice before was that since we are assuming 1 as a theorem and we can get K(G,R,a) from that then we can use the rule of necessation, which says that if phi follows from a theorem then phi is necessary, to get []K(G,R,a).

So, free will is incompatible with God’s omniscience…

Logical Skepticism

Richard Brown 7 Comments

I was reading this (older) post on the Law of Non-Contradiction over at Philosophy, et cetera. Is it really the case that it is not rational to question LNC? One might think that (P v -P) is an analytic truth. Indeed, I think that it is. But this is true only if one is operating within the confines of a two-valued logic. If one takes a many-valued logic then -(P & -P) is not analytic at all! So then the issue is whether or not classical two-valued logic is The One True Logic or not. How would we know this? I take it that someone like Richard would say that classical logic is what a maximally rational ideal agent would subscribe to. But how could we possibly know what kind of logic such an agent would subscribe to?

In an earlier post (Why Does 1+1=2?) I argued that there is nothing that could possibly decide between whether or not mathematics is an empirical science or an a priori one. The same seems true of logic. The basic laws of logic only seem rational to us because of the kind of experience that we have in this world filled with ‘medium sized dry-goods’ and this means that coherence as a criterion of truth is only valuable because of the world we live in and the kind of experience that we have. So the fact that the LNC seems to us to be denied at our own peril cannot possibly be evidence that it cannot be denied. We have no way of knowing whether it is actually ideally rational or not, since what we can concive of is limited to the kind of experience that we have.