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Generally we suppose that if there is any time lapse between event A and a subsequent event B, A cannot be the cause of B. But what if time were continuous, such that between any times t1 and t2, we might specify a distinct time t3? In that case, there would always be some time lapse between any two events: would that make causation as described impossible? Does conceiving of time as quantized solve the problem?

But we don't "generally" suppose that earlier events can't cause later events! Jack's earlier smoking caused his cancer, the earthquake ten minutes ago caused the tsunami now rolling across the ocean, my flick of the switch caused the light to come on a fraction of a second later, and so it goes. If anything, the "general" view is that causes precede their effects. It is not for nothing that, for example, David Hume's first attempt at a definition of a cause is as "an object, followed by another, and where all objects similar to the first are followed by objects similar to the second".

Are emotions involved in conclusions/reaching conclusions in mathematics?

Emotions are not involved in any direct way in our mathematical conclusions (for the conclusions are about numbers, or groups, or vector spaces, or sets, or whatever, not about human things like emotions.) And whether a purported conclusion is indeed a mathematical truth is an objective matter: again, emotions don't come into it. They are not involved in the proofs of the conclusions either (for proofs are deductions from more or less explicit premisses about numbers, or groups, or vector spaces, or sets, or whatever, to conclusions about such things, and still don't mention emotions). And whether a purported proof is indeed a mathematical proof is again objective matter. Does that mean emotions have no place in mathematics? Well, we do think of some proofs as beautiful or elegant or cute. And this, you might well think, is a matter of how we respond -- respond emotionally, in a broad sense -- to the proofs. And what makes a mathematician seek to prove a result in the first place (elegantly or...

People say that the more wine you drink, the more you "learn to appreciate" fine wines (we're talking about over the course of a lifetime, of course, not over the course of an evening!). Assuming this is true, is one's taste in wines actually improving over time? Or is it just changing? If the connoisseur likes dry red wine from France, and the "pleb" likes sweet white wine from Romania, what makes the connoisseur's taste superior to or more refined than the pleb's taste? Is it just the institution of wine-loving that contructs one taste as superior to the other, or do the connoisseur's taste buds literally detect marks of quality that the pleb's doesn't?

Of course there is any amount of snobbery and pseudery associated with the connoisseurship of wine. But still: it is a real phenomenon, coming over time to appreciate more of the complexities of taste and aroma and "feel" in the mouth that there can be in fine wine. That requires (enjoyable!) practice, paying attention, learning to discriminate, coming to recognize aesthetic qualities like balance and refinement. So yes, the more experienced wine enthusiast can detect differences that are really there, and which can be lost on the beginner (and, sadly, seem to some extent to get lost again as we get rather older). It isn't just a matter, then, of the connoisseur having different tastes in the sense of different preferences, but also the connoisseur will have different tastes in the sense of different and more complex experiences as he drinks. But let's not get too precious. Wine is there for civilized shared enjoyment, not for being pretentious about. And I'll add that there is nothing "pleb"...

Most foul odors we smell that give us all a shock of disgust seem to come from bacteria (at least before our mastering of chemistry). We can explain this evolutionarily as a means for making us avoid the most salient disease vectors from our humble origins (excreta, spoiled meat, putrid water, etc.). My question is this, did the selection pressures of evolution act to assign the awful olfactory sensations to the particles emitted by dangerous bacteria and their waste, OR did we evolve the response of disgust to those already-assigned sensations? In other words, does my dog experience a COMPLETELY DIFFERENT SENSATION when smelling (and rolling in) a dead animal - one that's not so bad, or does he experience the smell like I do he just LIKES IT BETTER than I do? I think this question might be about qualia, and whether there's a two-step process in how we perceive them. Do evolving organisms just shift around the few bad smells there are to the stimuli that best deserve them, or are smell sensations and...

"Different qualia or different attitudes to the same qualia?" That seems an intractable question. It should make us suspicious. Maybe the very idea of qualia (as postulated by some philosophers) is the source of the problem. For sceptical thoughts along these lines, see Dan Dennett's justly famous paper, "Quining Qualia", readable here .

Modern logicians teach us that some of the inferences embodied by the Aristotelian square of opposition (i.e., the A-E-I-O scheme) are not valid. Take the inference from the Universal Affirmative "Every man is mortal" to the Particular Affirmative "Some men are mortal": the logical form of the first proposition is a conditional ("Every x is such that if x is a man, then x is mortal") and we know that a conditional is true whenever its antecedent is false. In other words, the proposition "Every x is such that if x is a man, then x is mortal" is true even if there were no man, so the aforementioned inference is invalid. But if the universal quantifier has not ontological import, why such a logical truth as "Everything is self-idential" implies that there is something self-identical? And, above all, why the classical first order logic needs to posit a non-empty domain?

What does it mean to say that the logical form of "Every man is mortal" is "Every x is such that, if x is a man, then x is mortal"? The content of this claim is, in fact, quite obscure! However what is true is that if we are going to translate the English "Every man is mortal" into a a standard single-sorted first-order language of the kind beloved by logicians, the best we can do is along the lines of (For all x)(Fx -> Gx). And, as you say, in so doing, we don't respect the existential commitment which arguably accrues to the "Every" proposition. OK, but that's one of the prices we pay for trying to shoehorn our everyday claims involving many sorted quantifiers (as in "Every man", "no woman", "some horse", "any natural number") into an artificial language where (in any application) all the quantifiers run over a single common domain. That's a price typically worth paying in order to get other benefits (ease of logical manipulation, etc.). But if, in some context, we don't want to pay the...

I have heard that the only argument we have at the moment for the existence of free will rests on quantum mechanics, however I'm not entirely sure how this works. Could you please help me with an example of how quantum mechanics expresses our free will?

QM may give us reason for believing that determinism is false. (Actually, even this claim is problematic, but it at least has some plausibility. If we think the world behaves as QM says it does, and think that QM implies that some events are irredeemably chancy, then it seems to follow that that the state of the world at one time doesn't deterministically fix how things must be at later times.) But even if QM gives us reason for believing that determinism is false , that doesn't establish that we do have free will. For the claim that there are irredeemably chancy events plainly doesn't show anything about whether we are in control of our destiny in any interesting sense. It could still be that "free will" is an illusion, that everything that happens to us is as the result of happenings quite out of our control, and our supposedly free decisions are like the froth on the wave, doing no serious causal work. It's just that that underlying causality is chancy. Of course,...

Even if we accept Judith Jarvis Thomson's distinction between "killing" and "letting die", how can abortion be anything but horrifically unethical? Suppose I have daughter that I reluctantly take care of. I would never kill her, but I miss the disposable income and free time I had before her. Then one day I find out my daughter has rare disease and needs me to donate my kidney (or if you prefer, needs me to be tied to the machine described in violinist thought experiment). "Now's my chance!" I think. "If refuse to let her use my body, I can 'let her die' rather than 'kill' her. With my only child dead, I'll be free to live like a bachelor again. No more t-ball games for me!" Even if you grant that I have the right to let my daughter die, it still sounds like a selfish thing to do. In fact it's monstrous thing to do. Just like we can defend Fred Phelps's right to free speech while condemning the way exercises it, we can defend a woman a woman's right to bodily autonomy while condemning the way she...

I agree with everything Richard Heck says, but let me add more, recycling points I've made before in responding to other questions about abortion. Consider the following "gradualist" view: As the humanzygote/embryo/foetus slowly develops, its death slowly becomes a more serious matter.At the very beginning, its death is of little consequence; as time goeson, its death is a matter it becomes appropriate to be gradually more concernedabout. Now, note that this view seems to be the one that most of us in fact do take about the natural death of human zygotes/embryos/foetuses. After all, very few of us areworried by the fact that a very high proportion of conceptions quite spontaneouslyabort: we don't campaign for medical research to reduce that rate (and opponents of abortion don't campaign for all women to take drugs to suppress natural early abortion). Compare: we do think it is a matter for moral concern that there arehigh levels of infant mortality in some countries, and campaign and give...

Hello Philosophers Is there any axiomatized theory of arithmetic that is much stronger to be afflicted by Gödel theorems? I've read that there are axiomatized theories that are weaker than the theorem's criteria, i.e not expressive enough, and their consistency is proved within the same theory. I wondered if there would be something like that, which is stronger than the Gödel theorem's criteria for a axiomatized theory.

Gödel's first theorem, with later improvements by Rosser and others, tells us that any theory of arithmetic T that is (i) consistent, (ii) decidably axiomatized (i.e. you can mechanically check that a purported proof obeys the rules of the arithmetic) and (iii) contains Robinson Arithmetic (a very weak fragment of arithmetic) is incomplete. Strengthen T by adding more axioms and the theory will still be incomplete (unless you throw in so much it becomes inconsistent or stops being decidably axiomatized). In sum, you can't "outrun" the reach of incompleteness theorem by going to a stronger theory which is still consistent and properly axiomatized. This is explained in any standard introduction to Gödel's theorems (e.g. in the first chapter of mine).

What is the relation between logic and good reasoning? I once thought that logic was the science or study of good reasoning, but I've read a few things (mostly online, I confess) saying that logic is only a matter of "formalizing" reasoning (making it clear and unambiguous, and perhaps making possible that computers reproduce it). But whether reasoning is good should not be a concern for logic. Is that so? And if it is so, what is the current name for the study of good reasoning?

The business of logic is the evaluation of reasoning -- "do these premisses really support that conclusion"? But we want a systematic theory, not just piecemeal case studies. It is difficult to be systematic about reasoning presented in a ordinary language (think e.g. of the different ways we have of expressing generalizations in English using all/any/every/each, and the subtly different ways these behave). So ever since Aristotle, logicians have been attracted by a "divide and rule" approach. Separate the task of rendering arguments into a clear, unambiguous, tidy formalized framework, from the task of evaluating the resulting formally regimented arguments. The prime point of the formalization, though, is to aid the evaluation of arguments and develop proof-techniques for warranting complex inferences. Logicians still care whether reasoning is good!

Is it impossible that there be two recursive sets T and T* of axioms (in the same language) such that their closures under the same recursive set of recursive rules is identical and yet there is no recursive proof of this fact? It seems impossible but a simple proof of this fact would help elucidate matters!

To decide whether T and T* have the same deductive closure involves deciding whether the axioms of T* are deducible in T. That, in general, will require have a decision procedure for determining whether a given sentence is a deductive consequence of T. But theoremhood in T could be recursively undecidable.

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