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Frequently, one finds the following statment: "You cannot prove a negative." My

Frequently, one finds the following statment: "You cannot prove a negative." My question is, in this context, what is meant by the word "negative?" I understand how the word is used in mathematics and I "think" I know the meaning when used in logic. I just cannot seem to get a handle on how it is used here. Moreover, does it, perhaps, refer to a total position in the debate over the existence of God? Any comments you would make would be greatly appreciate. I enjoy your application very much and, moreso, since I am so old. Thanks. JH

This is a pretty confusing expression. What's usually meant, I think, is that a negative general proposition -- a proposition asserting that a certain kind never occurs -- requires much more by way of justification from its defender than from its opponent. Take the proposition "there are no black swans," for example. To prove it, you would have to comb through the whole universe, presumably all the way backward and forward in time, to demonstrate conclusively that nothing contained therein is a black swan. To disprove the proposition, by contrast, all you need do is produce a single black swan. Given this asymmetry, it thus makes sense to saddle the opponent, rather than the proponent, of a negative general proposition with the burden of proof.

What's confusing here is that the same sort of asymmetry is present with affirmative general propositions as well. Thus the proposition "all elks like mushrooms" requires much more by way of justification from its defender than from its opponent. To prove it, you would have to comb through the whole universe, presumably all the way backward and forward in time, to investigate all elks in regard to their fondness for mushrooms. To disprove the proposition, by contrast, all you need do is produce a single elk who doesn't like mushrooms. Given this asymmetry, it makes sense once again to saddle the opponent, rather than the proponent, of such an affirmative general proposition with the burden of proof.

In its most general formulation, the point your queried statement tries to express is then not one about negative versus affirmative propositions but one about propositions governed by different logical quantifiers. Any proposition -- negative or affirmative -- that is universally quantified requires much more by way of justification from its defender than from its opponent, who need merely produce a single counterexample, that is, prove an existentially quantified proposition.

To see this, we might state our two sample propositions in logical language. Both of them are governed by a universal quantifier ("for all x, ..."):

(1) For all x, if x is a swan then x is not black.

(2) For all x, if x is an elk then x likes mushrooms.

the negations of these two propositions, both governed by existential quantifiers ("there is an x such that...") are, respectively,

(1-) There is an x, such that x is a swan and x is black.

(2-) There is an x, such that x is an elk and x does not like mushrooms.

Clearly, much more is required of the defenders of (1) and (2) than of the defenders of (1-) and (2-). This is an interesting point, but not one well expressed in the common statement you cite.

This is a pretty confusing expression. What's usually meant, I think, is that a negative general proposition -- a proposition asserting that a certain kind never occurs -- requires much more by way of justification from its defender than from its opponent. Take the proposition "there are no black swans," for example. To prove it, you would have to comb through the whole universe, presumably all the way backward and forward in time, to demonstrate conclusively that nothing contained therein is a black swan. To disprove the proposition, by contrast, all you need do is produce a single black swan. Given this asymmetry, it thus makes sense to saddle the opponent, rather than the proponent, of a negative general proposition with the burden of proof. What's confusing here is that the same sort of asymmetry is present with affirmative general propositions as well. Thus the proposition "all elks like mushrooms" requires much more by way of justification from its defender than from its opponent. To...

Hi, I would like to ask a question about Logic. There is a formal logical

Hi, I would like to ask a question about Logic. There is a formal logical fallacy called "Circular Reasoning", are not all argument tho circular? The conclusion is always found in the premises. and then drawn from them into a conclusion.

Drawing conclusions from premises is not circular. You are going in one direction, from the premises to the conclusion. Circularity appears when you also defend your premises by appeal to the conclusion.

To illustrate with a somewhat informal but real-world typical example:

P1: The government of country A is hell-bent on territorial expansion

P2: The government of country A is expanding its military capabilities

C: The government of country A is threatening its weaker neighbors.

There's nothing circular here, the argument displays a relationship between two premises and one conclusion: the premises, if true, together support the conclusion. To explore whether they are true, and thus actually support the conclusion, we need to examine what evidence can be adduced in their support. Suppose P2 is uncontroversially true so that attention turns to P1. We ask the presenter of the argument how she can support P1. Is A's government really hell-bent on territorial expansion? Suppose she now adduces as evidence that A's government is threatening its weaker neighbors. This move involves her arguing in a circle. Even if

P2 is clearly true

P1 and P2 together are a good reason for accepting C

C is a good reason for accepting P1 --

all this gives us no good reason to accept that P1 and C are true (believing otherwise is the fallacy of circular reasoning) but rather gives us merely good reason to accept that P1 and C stand or fall together.

Upshot. Yes, conclusions are in a sense contained in, and drawn out from, premises. But this is not circular so long as these premises are not supported in a way that involves appeal to the conclusion.

Drawing conclusions from premises is not circular. You are going in one direction, from the premises to the conclusion. Circularity appears when you also defend your premises by appeal to the conclusion. To illustrate with a somewhat informal but real-world typical example: P1: The government of country A is hell-bent on territorial expansion P2: The government of country A is expanding its military capabilities C: The government of country A is threatening its weaker neighbors. There's nothing circular here, the argument displays a relationship between two premises and one conclusion: the premises, if true, together support the conclusion. To explore whether they are true, and thus actually support the conclusion, we need to examine what evidence can be adduced in their support. Suppose P2 is uncontroversially true so that attention turns to P1. We ask the presenter of the argument how she can support P1. Is A's government really hell-bent on territorial expansion? Suppose she...

Is there not something disingenuous and disrespectful in claiming that an

Is there not something disingenuous and disrespectful in claiming that an opponent's views are not sincere or belonging to themselves, but rather unconsciously motivated by psychological insecurities, social power dynamics and ideology?

There is something disrespectful about such a claim alright: one is not engaging with the opponent's expressed view on its merits but is dismissing this view as not based on conscientious, reliable reflection.

But then such a claim may be true: some people do indeed hold views that are unconsciously motivated by the kind of factors you mention. In any case, one may conscientiously reach the conclusion that another's view is so motivated, and stating such a conclusion may not then be disingenuous.

There is something disrespectful about such a claim alright: one is not engaging with the opponent's expressed view on its merits but is dismissing this view as not based on conscientious, reliable reflection. But then such a claim may be true: some people do indeed hold views that are unconsciously motivated by the kind of factors you mention. In any case, one may conscientiously reach the conclusion that another's view is so motivated, and stating such a conclusion may not then be disingenuous.

There's a logical scenario which often comes up in discussions around the

There's a logical scenario which often comes up in discussions around the question of voting. We all know the conversation... Person 1: I don't vote because my vote has no impact on the outcome of the election. Person 2: Not on it's OWN it doesn't, but if everyone thought that, no one would vote, and THEN what would happen?! Person 1: But I don't decide whether all those other people vote, I only have control of my 1 vote! My question here relates not to whether or not one should or shouldn't vote, or to the voting example alone, but rather to the logic of this situation. For this example let us assume (for the sake of the point I am interested in) that it is universally agreed that all people (including Person 1 and 2) agree that nobody voting is an outcome that everyone wishes to avoid. And also assume (despite the conversation above!) that everyone decides privately whether to vote or not, such that their decision cannot influence others decisions) Finally assume that the election involved has...

I don't think there's a named fallacy here, but I do think the principle proposed by Person 2 is unsound. If this principle were sound, then it would be impermissible to remain childless even in a world as overpopulated as ours.

The principle can be revised to be more plausible. When many people in some group are making a morally motivated effort to achieve a certain good that would not exist (or to avert a certain harm that would not be averted) without their effort, then one has moral reason to do one's fair share if one is a member of this group. This sort of principle against free-riding on the moral efforts of others can explain why one should generally vote and do so conscientiously -- at least unless one has conclusive reason to judge that enough others are already acting and that one's own effort will therefore add nothing to the outcome.

But there is also a more direct explanation of why one ought to vote. As philosopher Derek Parfit has argued, the extremely low probability of one vote affecting the outcome is compensated by the extremely large moral importance of the outcome. Thus Person 1 would likely concede that one ought to vote in "small" elections where one's vote may very well affect the outcome, e.g. in the mayoral election of one's tiny home village -- with 62 other voters, say. In this case, the probability that one will cast a deciding vote is a whopping 10 percent.

I derive this percentage as follows. With 63 people voting, there are 63!/(32!*31!) voting patterns where one side wins by one vote and an equal number of voting patterns where the other side wins by one vote. I divide this sum by the total number of voting patterns -- 2^63 -- to estimate the percentage or probability of "extremely close" outcomes. I then multiply by 32/63 to reflect the fact that only 32 out of the 63 people voting have actually cast decisive votes. More generally, for any odd-numbered electorate of 2n+1 voters, the probability that any given vote will be deciding is (2n)!/(n!^2*2^2n).

Now suppose you live in a town with four times as many other voters: 248. Do you now have less reason to vote? You'll be less likely to affect the outcome. But the outcome also matters more from a moral point of view -- nearly four times as much, I would think, because the new mayor will be seriously affecting the lives of nearly four (249/63 times) time as many people as in the first scenario.

While Parfit concluded that a morally motivated person has as much reason to vote in a larger election as in a smaller one, I can strengthen his conclusion by showing that the moral reason for voting actually becomes ever more weighty as the size of the electorate increases. This is so on the assumption that the importance of voting is proportional to the expected impact of the vote, that is, to the magnitude of the impact of the outcome multiplied by the probability of casting a deciding vote. The former of these factors is proportional to the size of the electorate. The latter of these factors is roughly proportional to squareroot (1/n). Thus, in the village of 249 voters, the probability of casting the deciding vote is still 5.06 percent, about half of 10.09 percent rather than merely a quarter. The expected impact as defined then goes up roughly with the square root of the size of the electorate. A public-spirited voter, then, who considers equally the impact of the vote upon all citizens, arguably has, other things being equal, more reason to vote the larger the election in which s/he is eligible to vote.

Here are the numbers from a quick spreadsheet calculation, showing the number of voters (yourself included), the probability of casting a deciding vote, and the expected impact (probability of casting a deciding vote multiplied by the overall number of voters):

____63_____10.092%______6.36

___249______5.061%_____12.60

___631______3.178%_____20.05

__2491______1.599%_____39.83

__6301______1.005%_____63.34

_24901______0.506%____125.91

_63001______0.318%____200.27

249001______0.160%____398.14

We see here that the probability of casting a deciding vote declines roughly with the square root of the size of the electorate. Assuming that the moral importance of the outcome increases roughly with the size of the electorate, we can conclude that the expected moral impact (and hence the strength of the moral reason in favor) of voting increases roughly with the square root of the size of the electorate.

In a US election, you might have about 120 million votes cast, so here your probability of casting a deciding vote is about 0.0073% (keeping the electorate constant, we can expect one in 6850 US elections to be decided by one vote) and the expected impact of your vote (continuous with the above calculation) is about 8740. Note that, if your vote is deciding in such an election, then a lot of other votes are also, like yours, deciding votes: in the simple case I have been discussing (assuming only two choices and leaving aside complications such as the US Electoral College), you would be one of 60,000,001 people each of whom was needed to outvote 60,000,000 others.

I don't think there's a named fallacy here, but I do think the principle proposed by Person 2 is unsound. If this principle were sound, then it would be impermissible to remain childless even in a world as overpopulated as ours. The principle can be revised to be more plausible. When many people in some group are making a morally motivated effort to achieve a certain good that would not exist (or to avert a certain harm that would not be averted) without their effort, then one has moral reason to do one's fair share if one is a member of this group. This sort of principle against free-riding on the moral efforts of others can explain why one should generally vote and do so conscientiously -- at least unless one has conclusive reason to judge that enough others are already acting and that one's own effort will therefore add nothing to the outcome. But there is also a more direct explanation of why one ought to vote. As philosopher Derek Parfit has argued, the extremely low probability of one...

No matter whether one adopts a deontological or consequentialist account of

No matter whether one adopts a deontological or consequentialist account of ethics it is apparent that there exists a moral imperative to prevent genocide. To what extent and to what cost this imperative must motivate our actions is, I suppose, a subject of serious debate, however. But how can we define genocide? Surely we can all agree that the murder of 10,000,000 people constitutes genocide. But what if we subtract one fatality? Still genocide, of course. Minus one more? The same is still true. But at some point that logic fails; when we get down to the death of one, a few, or no people we certainly no longer have a case of genocide on our hands. It seems there is a sorites paradox here. If the number of people killed is ultimately arbitrary, how is the concept of genocide meaningful? Surely we can still find moral value in the deaths of millions (or even in the death of an individual), but it seems the label in itself is ultimately kind of subjective and meaningless.

The number of victims is not the only consideration entering into the judgments of whether a genocide is taking place. Other relevant factors are the nature and size of the victim group and the motivations and intentions of the perpetrators. Still, we can hold these other factors fixed and ask your question again, for example: hypothetically lowering the number of people killed, maimed, raped, and otherwise brutalized in the Rwanda genocide, when do we reach the point at which the genocide label would no longer be applicable? Or: at what time, in those horrible months of early 1994, did the daily decision of the world's leading governments not to intervene become a decision to ignore a genocide?

You're right that there is some vagueness here. But this does not render the term meaningless. As Wittgenstein writes, there may be some unclarity about where exactly the boundary lies between two countries -- say between China and Russia -- but this does not entail that it's unclear on which side Beijing or Moscow fall. Similarly, for many terms widely used in the criminal law, terms like "negligent," "reckless," "due diligence," "reasonable person," and so on. Even the word "kill," which you seem to find unobjectionable, is subject to a Sorites problem. Suppose you hurt someone and, as a result, she dies earlier than she would otherwise have done. Have you killed her? Surely yes, if she dies within seconds of your action. Presumably no, if she lives another 80 years rather than 81. So how long exactly must she survive for you to escape the killer label?

We confront and resolve such questions all the time in legislation and jurisprudence, and the borderlines are surely arbitrary to some extent. Here "genocide" is basically in the same boat with lots of other terms and, if we tossed all those terms overboard, we'd have very little language left.

The number of victims is not the only consideration entering into the judgments of whether a genocide is taking place. Other relevant factors are the nature and size of the victim group and the motivations and intentions of the perpetrators. Still, we can hold these other factors fixed and ask your question again, for example: hypothetically lowering the number of people killed, maimed, raped, and otherwise brutalized in the Rwanda genocide, when do we reach the point at which the genocide label would no longer be applicable? Or: at what time, in those horrible months of early 1994, did the daily decision of the world's leading governments not to intervene become a decision to ignore a genocide? You're right that there is some vagueness here. But this does not render the term meaningless. As Wittgenstein writes, there may be some unclarity about where exactly the boundary lies between two countries -- say between China and Russia -- but this does not entail that it's unclear on which side Beijing or...

Theist: We should follow the Bible, and the Bible says that there is a God.

Theist: We should follow the Bible, and the Bible says that there is a God. Atheist: Why should we follow the Bible? Theist: Because the Bible says we should. Atheist: That’s circular reasoning. But then the Atheist says: We shouldn’t believe in God. Here’s logic to show he doesn't exist. Me: Why should we follow logic? Atheist: We’ve come to the conclusion that logic, and not the Bible, is right by using logic. Me: Is this not also circular reasoning? Someone please tell me why I’m wrong. Also, if I just disproved the validity of logic but used logic to disprove it, does that mean my argument is no longer valid because it’s based on logic, which is no longer valid. But if my claim is no longer valid that disproved logic, does that mean that logic is ok now. But then, that would mean that my argument is still ok, which means that… I think you get the idea. Someone please tell me why I’m wrong before my head explodes.

Fair enough, you cannot support logic by appeal to logic. But this does not disprove logic. It just shows that one attempt to justify logic is unsuccessful.

How then do we justify logic, or the Bible for that matter? You seem to think of justification as starting with nothing -- and then it's indeed hard to see how anything can ever be justified. But in real life, when we justify, or question, something we always take other things for granted: other beliefs, modes of inquiry, methods of reasoning, and so on. Each of them can be questioned too, of course, but we cannot question all of them together at once. Nothing justifies our thinking as a whole, though every part of it can be justified (or disqualified) by its fit (or incoherence) with the rest. If logic (or the Bible) makes sense to you and helps you make sense of the world, then you have a good justification for continuing to rely on it. If you find incoherence in logic, or between logic and something else you have been relying on, then you need to find a way of revising some of your commitments in order to resolve the incoherences.

So, think of justification not as something you need to start thinking, but as something you need to revise the way you think.

Fair enough, you cannot support logic by appeal to logic. But this does not disprove logic. It just shows that one attempt to justify logic is unsuccessful. How then do we justify logic, or the Bible for that matter? You seem to think of justification as starting with nothing -- and then it's indeed hard to see how anything can ever be justified. But in real life, when we justify, or question, something we always take other things for granted: other beliefs, modes of inquiry, methods of reasoning, and so on. Each of them can be questioned too, of course, but we cannot question all of them together at once. Nothing justifies our thinking as a whole, though every part of it can be justified (or disqualified) by its fit (or incoherence) with the rest. If logic (or the Bible) makes sense to you and helps you make sense of the world, then you have a good justification for continuing to rely on it. If you find incoherence in logic, or between logic and something else you have been relying on, then you need...

Are there as many true propositions as false ones? More of one than the other?

Are there as many true propositions as false ones? More of one than the other?

A few comments on the answers from Professors George and Pogge:

1. If two statements are logically equivalent, do we think of them as expressing the same proposition or two different propositions? If we think of logically equivalent statements as expressing the same proposition, and we use classical logic, in which "it is not the case that it is not the case that P" is logically equivalent to P (and if we ignore, for the moment, Prof. Pogge's worries about meaningless statements), then there's no problem with Prof. George's original pairing. In the example given by Prof. Pogge, "Bush is married" and "it is not the case that it is not the case that Bush is married" express the same proposition, and that proposition is paired with the proposition expressed by "it is not the case that Bush is married".

2. If we do not group together logically equivalent statements as suggested in 1 above, then as Prof. Pogge points out, Prof. George has paired each true statement with a false one, but some false ones are not used in the pairing. And he has also paired each false statement with a true one, with some true ones not used. In more technical mathematical language, we could say that we have a function from the true statements to the false ones that is one-to-one, but not onto, and we also have a function from the false statements to the true ones that is one-to-one but not onto. In this situation, the Cantor-Schroeder-Bernstein theorem says that there is a one-to-one correspondence between the true statements and the false statements. In fact, the proof of the theorem leads to exactly the solution that Prof. Pogge proposes: for statements that start with an even number of iterations of "it is not the case that," add an extra one, and for those that start with an odd number of iterations, remove one. (You can find more information about the Cantor-Schroeder-Bernstein theorem here.)

3. Suppose that, as suggested by Prof. Pogge, we allow for meaningless statements, and we say that if P is meaningless, then "it is not the case that P" is true. What about "it is not the case that it is not the case that P"? Is it equivalent to P, and hence meaningless? Or is it false, since "it is not the case that P" is true? If it is regarded as meaningless, then we do seem to have some "extra" true statements that are not paired with anything. But if it is false, then "it is not the case that P" can be paired with "it is not the case that it is not the case that P," and it will still be possible to make the pairing work.

4. Finally, what about the issues raised by Prof. Pogge about how many claims there are? I'm not sure exactly what is supposed to count as a "claim." Prof. Pogge suggests that we consider claims of the form "x is a real number." Well, certainly "2 is a real number," "23/57 is a real number," and "pi is a real number" are true claims. Prof. Pogge suggests that for every real number x, "x is a real number" is a claim. If that's right, then since there are uncountably many real numbers, there are uncountably many claims. But 2, 23/57, and pi are all "nameable" real numbers. Is there really any such thing as the claim "x is a real number" in the case of an "unnameable" number? If we require that a claim must be expressed by a finite sequence of letters, digits, and punctuation marks, then there are only countably many claims.

Professor George's conclusion is probably true, but the reasoning seems to me invalid. This is so, because the two "pairing" operations produce different pairs. For example, the first operation might create the pair <"Bush is married"; "it is not the case that Bush is married">. The second operation might create the pair <"it is not the case that Bush is married"; "it is not the case that it is not the case that Bush is married">. The first operation finds one unique false claim for every true one -- but some false claims are left over (for example, "3+3=9") . The second operation finds one unique true claim for every false one -- but some true claims are left over (for example, "3+3=6"). Therefore, the argument works only if it can be shown that the two sets of "left-over" claims are equal in number of members. One might try to avoid this problem by redefining Professor George's operations so that any claim that begins with an odd number of iterations of "it is not that case that" gets paired with...

Logically what is the difference between conceivable and probable or possible?

Logically what is the difference between conceivable and probable or possible?

Sometimes people use the word “possible” to mean something like, “does not entail a contradiction.” This sense of “possible” is narrower than “consistent with the laws of nature.” I gather this is what Thomas Pogge was alluding to when he said that there is a sense of “possible” that coincides with “conceivable.” Some have raised worries about whether conceivability implies possibility in this more narrow sense. Here’s an interesting example from Barry Stroud’s book on Hume (p. 50). Goldbach’s Conjecture states that every even number is the sum of two primes. At this point (as far as I know) no one has offered a proof of Goldbach’s Conjecture, and no one has disproved it either. If Goldbach’s Conjecture is false, it presumably says something that is not possible (in the narrow sense). But one might argue that I can nevertheless conceive of a state of affairs in which someone proves that Goldbach’s Conjecture is true. In that case, it seems, I would have conceived of something impossible.

Here’s another famous example from Arnauld. Arnauld is responding to Descartes’ claim in the 6th Meditation that since we can conceive of the mind existing apart from the body, it is possible that the mind can exist without the body. Arnauld objects that someone uneducated in geometry might be able to conceive of a right triangle that does not obey the Pythagorean Theorem. However this doesn’t mean that it’s possible for there to be a right triangle that fails to obey the Pythagorean Theorem.

Part of the problem is that it's not altogether clear what it is for something to be "conceivable."

The common domain these three predicates range over is that of states of affairs consisting of objects that have certain specific properties or stand in certain specific relations. Being conceivable is the easiest condition to meet. It excludes only states of affairs that we cannot think or imagine. We cannot imagine a stone that is green all over and also red all over, a stone that occupies the same space as a clump of metal, a bachelor who is married, a living horse that's not an animal, and so on. While there is a narrow sense of "possible" that coincides with "conceivable," usually being possible is a more demanding condition. To be possible, a state of affairs must not merely be conceivable, but must be consistent with what we know about this world (e.g., the laws of nature). A puddle of water turning into a human being, an animal living forever, a daytrip to another galaxy -- these are conceivable, but not possible. Being probable is more demanding still, requiring not merely that states of...

Dialetheist: "Some contradictions are true."

Dialetheist: "Some contradictions are true." My question: "Who claims (if any), that some tautologies are false?"

In colloquial speech there are some apparent tautologies that are used to make a substantive point that can be disputed. There is the famous Yogi Berra saying "it's not over till it's over" used to make the (disputable) claim that the team behind can still catch up. And there is "boys are boys" expressing the (very disputable) claim that its pointless to work toward decent behavior by men in matters sexual.

In colloquial speech there are some apparent tautologies that are used to make a substantive point that can be disputed. There is the famous Yogi Berra saying "it's not over till it's over" used to make the (disputable) claim that the team behind can still catch up. And there is "boys are boys" expressing the (very disputable) claim that its pointless to work toward decent behavior by men in matters sexual.

Referring to propositional logic conditionals, if we say that an antecedent A is

Referring to propositional logic conditionals, if we say that an antecedent A is a necessary and sufficient condition for consequent B, can we say that A caused B?

No.

That A is a necessary condition for B means that B presupposes A, that B cannot hold without A also holding.

That A is a sufficient condition for B means that A implies B, that A cannot hold without B also holding.

That A is a necessary and sufficient condition for B thus states a symmetrical relation between A and B: Neither can hold without the other, that is, both hold or neither.

If A being a necessary and sufficient condition for B indeed implied that A caused B then, given symmetry, it would likewise imply that B caused A. A and B would have caused each other -- a rather odd way for them to come about.

An example may help. In this example, A is that you are an unmarried male human adult at some given time t, and B is that you are a bachelor and this same time t.

Your being an unmarried male human adult at t is a necessary condition for your being a bachelor at t. (For you to be a bachelor at t, you must be an unmarried male human adult at t. Your being a bachelor at t presupposes your being an unmarried male human adult at t.) Your being an unmarried male human adult at t is also a sufficient condition for your being a bachelor at t. (That you are an unmarried male human adult at t implies that you are a bachelor at t.)

So, your being an unmarried male human adult at t is a necessary and sufficient condition for your being a bachelor at t.

Yet, clearly, your being an unmarried male human adult at t was not caused by your being a bachelor at t. Nor was your being a bachelor at t caused by your being an unmarried male human adult at t. In fact, for all I know, you may be female -- in which case any claims about what caused you to be a bachelor would be meaningless. Yet, even if you are female, it would remain true that your being an unmarried male human adult at t is a necessary and sufficient condition for your being a bachelor at t. This statement does not assert or entail that you are both a bachelor and an unmarried male human adult at t. Rather, it merely asserts that, at t, you are either both or neither, that you cannot be one without being the other. As it happens, it is true at any time of any X -- including men, girls, dolphins, atoms, and vacuum cleaners -- that X's being a an unmarried male human adult at this time is a necessary and sufficient condition for X's being a bachelor at this time. At any time, anything is either both (a bachelor and an unmarried male human adult) or neither.

No. That A is a necessary condition for B means that B presupposes A, that B cannot hold without A also holding. That A is a sufficient condition for B means that A implies B, that A cannot hold without B also holding. That A is a necessary and sufficient condition for B thus states a symmetrical relation between A and B: Neither can hold without the other, that is, both hold or neither. If A being a necessary and sufficient condition for B indeed implied that A caused B then, given symmetry, it would likewise imply that B caused A. A and B would have caused each other -- a rather odd way for them to come about. An example may help. In this example, A is that you are an unmarried male human adult at some given time t, and B is that you are a bachelor and this same time t. Your being an unmarried male human adult at t is a necessary condition for your being a bachelor at t. (For you to be a bachelor at t, you must be an unmarried male human adult at t. Your being a...

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