Tuesday, February 20, 2018

Is Logic Necessary?

I've been looking at Skepticism, following Maddy's 2017, because it connects with the topic of this month's posts: so what if logic gives us paradox? Even highly evolved apes are unlikely to have a perfect logic. We see a tree, we know that it is a tree; that much is ordinary. We cannot rule out its being an alien quasi-stick-insect, but we never thought that we should; and now that we think about it, perhaps we could if we examined the tree more closely. Of course, we cannot ever rule out that it's an alien quasi-stick-insect of a very convincing kind, by the very definition of that kind. And were it such, it would not be a tree ... but still: so what? Nowadays we have formal logics, and modern theories of truth, much as the earliest scientists stopped relying on intuitions about geometry. What is the truth about truth? The first "truth" in that question is clearly intended to be correspondence truth, but if it does turn out to be the case that the second one cannot be correspondence, then how could the first one be? And if the first "truth" is not correspondence truth, then how satisfying could any answer to any such question be? If we have, for example, an attractive story about how truth is an attractive story, then so what? But still, we do assume that truth is correspondence truth. Even when we think about the formalities, our meta-logic is simply logic. And similarly, we simply assume that trees are not alien quasi-stick-insects of some very convincing kind. We can say that they are very probably not aliens, and then try to justify that "very probably;" and then wonder why we are doing all of that. At the end of the day, we are simply such that, for us, our logic is necessary. I see a tree, and know that it is a tree. I cannot rule out its being an alien quasi-stick-insect of a very convincing kind, and so my "know" is a sort of gamble: I assume that it isn't an alien. I don't know that it's unlikely to be one (how could I?) but I do know that it's not mad to assume that it isn't one. Since the topic is raised, I admit that it might be an alien, that I don't know that it isn't, that I don't know, in that sense, that it is a tree; but I still claim that I do know that it is a tree, in the ordinary sense. There do seem to be at least two senses of "know" in play.

Monday, February 12, 2018


It seems to be logically possible for there to be an exact copy of you, say d-you, because it seems that such a thing might exist in a parallel space-time. D-you would be physically and mentally identical to you; but it would not, of course, be you. Now, we naturally assume that none of us have been instantaneously swapped with such doppelgangers. We can never have any reason to think that any of us might have been swapped; but, that is because such swapping would be undetectable, and that is why we cannot rule out the logical possibility of such swapping.

Indeed, you cannot completely rule out the possibility that you are such a doppelganger, because you would have exactly the same memories, exactly the same sense of being yourself. There would be absolutely no empirical difference; the only difference would be semantic: reference intended to be reference to you would fail to be such reference, were it to d-you, for example (and given the falsity of Functionalism, and so forth). And of course, knowledge would be lost, e.g. if I saw d-you at a bus-stop then I would not know that you were waiting for a bus. But of course, I would know that you were waiting for a bus if I saw you at a bus-stop (and you were waiting for a bus). There is no loss of knowledge caused by not ruling out the logical possibility of d-you. We simply assume that such swapping does not happen.

Note that we do not just think it unlikely (and similarly, we do not just think it unlikely that we are brains in vats, or being fooled by demons, and so on and so forth). We do not know for sure that there are no such doppelgangers, and we do not even know for sure that there are unlikely to be any (we can have no evidence for such unlikeliness). But clearly, we are assuming that there are no such things (and nothing else of that rather wide-ranging kind). That is just an obvious empirical fact about our beliefs. (We might not notice it, because being fooled by a demon would be like being a brain in an evil scientist’s vat, and a brain in a vat is like someone having a very long vivid dream; and maybe it is only highly unlikely that you are in a coma right now.)

Thursday, February 08, 2018

Truth in Dreams

In the Cartesian argument (for Skepticism) from dreaming we are to assume that if we were dreaming, then were we to see hands in that dream, those would not be hands; but of course, they would be dream-hands in a dream-world, and so why should dream-reference to them fail? If we think of someone dreaming about hands, then clearly those are not real hands; but, were this a dream (not a dream-within-a-dream, which is what our "dreaming" would refer to), then what is meant by "real hands" within that dream would be dream-hands. You might wonder if that would be the case, had we fallen asleep having already learnt the meaning of "real hands" in the real world; but presumably we learnt the meaning of "real hands" in this world, and were this a dream then that would be a dream-world. Might we have learnt the meanings of our words in some higher realm? But, as soon as we clarify such worries, say in some Moorean way, by describing what is meant by "external thing," we tie the meanings of our words to this world: worrying about that problem resolves that problem!

Wednesday, February 07, 2018

Lots of Misprints

I've seen quite a few misprints recently, e.g. in TV text; also top of page 159, and again on page 169, in Maddy 2017 (" 'Proof on ..." instead of " 'Proof of ..."), just before she got to Moore's reason why pointing to each of his hands was a proof that there are two hands (and hence that there are external objects, and hence an external world), which was that he could similarly prove that there were three misprints on a certain page by:
taking the book, turning to the page, and pointing to three separate places on it, saying 'There's one misprint here, another here, and another here'
Maddy 2017: 164 (Moore 1939: 147) Although of course, while that proves that there are three misprints, it does not prove that there are three misprints. And while you might agree with Moore that those were misprints, that would not amount to a proof that they were. Moore, you will recall, does not have to show that there are two hands, nor even that there are two hands, he has to show the externality (so to speak) of such things as hands, given skeptical doubts, which is more like having to prove not just assume, that it is indeed a bad thing to have lots of misprints. And of course, why would we have to prove such a thing! Ask yourself what is meant by "external world" to see for yourself how it exists by definition (and note how one gestures as one does so). And yet, it is precisely that "proof" that is challenged by skeptical doubts (as the above-linked-to review of Maddy 2017 observes).

Tuesday, February 06, 2018

What do Philosophers do?

I'm half-way through Maddy's 2017 (a walk through the modern history of Skepticism), where she describes a weakness of the Argument from Dreaming: although we would not be knowing the world were we now dreaming in the ordinary way, we can rule that out in quite ordinary ways; and whereas we cannot rule out that we are dreaming in some extraordinary way (e.g. a life-long coma), why should we rule it out? Such things are unheard of! Furthermore, maybe this is a dream-world, and my hands dream-hands within it; what of it? It is far from obvious why the fact that I don't know much about the fundamental substance of my hands should get in the way of my knowing that I'm typing this with them because they exist (whether that is in a way that is to some unknown world much as dreams are to this world, or some other way).

But here's a thought: If some higher power (maybe a UFO) replaced you with a pod-person who was exactly the same as you, physically and mentally, then the people of the world would of course not know, were they to see that person before them, that you were standing there. So, if the underlying substance of the world was such that things were frequently replaced with identical copies, in such ways (and note that we cannot even know that that is unlikely), then our references would frequently fail, and we would end up knowing a lot less about the world than we assume we do. We do assume that such does not happen, but that just means that, for example, it is at best epistemic luck that people know that you are there, when they see you. At worst it is knowledge by assumption, because we do assume as much; which reminds me of Wittgenstein's "hinge propositions," which Maddy will be getting to shortly...

Perhaps we assume that things generally continue to be the same things. Or perhaps we assume that things that look the same are the same. I would not say that we know such a proposition, but maybe we do thereby know propositions that depend logically upon it, such as that I have hands. (Knowledge seems not to be some minimal amount of epistemic luck, but rather the sufficient reduction of certain kinds of epistemic luck, as required by one's context; and philosophy is a context with high standards. In philosophy we tend to accept the force of epistemic closure, because the high standard is logic (or meta-logic).)

Friday, February 02, 2018

The Essence of Cantor's Paradox

(1)    There are at least three things

Clearly there are.

(2)    Given some things, there are possible selections from them

E.g. ‘clearly’ and ‘there’ are a pair of words.

(3)    There are all the things given by reiterating (2), given (1)

Note that each possible selection is a thing.

(4)    Given some things, cardinally more selections from them are possible

That is shown by a Cantorian diagonal argument.

(5)    There are cardinally more things of kind (3) than there are things of kind (3)

That follows from (4), given (3), but is contradictory, and hence false.

My resolution begins by observing that apparently timeless possibilities could possibly become more numerous over time. It begins that way because if possible selections are becoming more numerous, then that could easily change the meaning of (3) enough to avoid (5). There is no other way of avoiding the contradiction that I can think of (and note that the main resolutions were constructivism, with its potential infinities, and going axiomatic, which means not addressing numbers of things directly); which is, after all, why this is a paradox.

Consequently this is essentially a proof by reductio ad absurdum that possible selections do become more numerous over time. And how could possible selections become more numerous, if not by a transcendent creator making definitive selections, constructing arithmetic as part of the creation of all things? There is no other way that I can think of; whereas this way is a serious possibility, because
A) mathematicians have taken constructivism surprisingly seriously, and constructivism would only be more Platonistic, and Millian, were the definitive constructions made by a transcendent creator, and
B) theologians have taken the idea of God being beyond our conception of number very seriously, e.g. the Trinity.

Thursday, February 01, 2018

Logic Needs That Hypothesis

In the Germany of the eighteen-nineties, Georg Cantor discovered the mathematical paradox that bears his name.
He put it down to the ineffability of God, even though he was only studying numbers; they were very big numbers.
But, the mathematical mainstream has since then replaced our natural conception of a collection with formal (or fictional) sets that are better behaved.
Whereas, the natural conceptions are fundamental to our actual thinking; in particular, if we cannot rely on our best thinking about formal sets, then why should formal sets be any better?
Consequently logical thinkers need to hypothesize God: only that allows those conceptions without paradox (as previously posted, and as sketched in my next post).
Over the next few posts I aim to scrutinize the elements of this, e.g. the essence of Cantor's paradox, and why we do still need logic in this democratic and scientific age.

The Problem with Prefaces

Suppose that it says, in the preface to some non-fiction book, that there is bound to be some false statement in the book, even though each statement in the book is believed to be true by the author. That is the Preface Paradox. A first analytical thought might be to regard belief as sufficiently high credence, so that we would not believe large conjunctions of our beliefs (cf. how you will probably not get a 1 or a 6 with one throw of a die, but with three dice the chance is less than 30%); but of course, we very often do (and for other reasons belief is not simply high credence, as recently posted). A more sophisticated response might discover contextualist aspects (e.g. self-reflection upon one's fallibility); but what I am interested in here is how we naturally overlook the following absurd response: Why not say that sets are simply such that sets of beliefs are like that? We could then have all that we want, and nothing that we do not want: we simply put precisely that much into the axioms of set theory! But of course, that would be too easy; what about what sets of beliefs really are? The thing is, that is essentially the mainstream response to Cantor's Paradox (which I shall be posting on in my next few posts). The Preface Paradox therefore shows how absurd the mainstream foundations of mathematics are.