Consider the following: "If we lower the standards we lower the results, so if we raise the standards we raise the results" (in passing this is about education). I have the impression that there is a fallacy in this - even if I assume the first part of the inference, I suppose we could raise educational standards and just watch everybody fail miserably), but I cannot phrase clearly why/how this is a fallacious claim. Am I right? Is this fallacious and if so, is there a technical term for it?

Let's assume it's true that "If P, then Q". The conditional claim that you imagine being inferred from this has the structure "If not-P, then not-Q". [Not quite: I don't think the negation of "we lower standards" is "we raise standards". One way in which we might fail to lower standards is to keep them fixed.] This is indeed an incorrect inference. The first conditional claims that P is a sufficient condition for Q. While the second claims that P is a necessary condition for Q. And the latter claim simply doesn't follow from the former. For instance, it's true that if Rex is a dog, then Rex is a mammal. (Being a dog is a sufficient condition for being a mammal.) But this does not imply the false claim that if Rex is not a dog, then Rex is not a mammal. (Being a dog is not a necessary condition for being a mammal.) This fallacy is sometimes called The Fallacy of Denying the Antecedent . ("P" is called the antecedent of the first conditional claim above.)

We define the empty set as the set that contains no elements, but is there more than one empty set? So is there "an" empty set as opposed to "the" empty set? May one be able to receive values, while another is truly empty, etc.? And how is it possible to define the empty set by the absence of members or by emptiness?

The empty set is indeed defined to be that set which contains no elements. Another definition we need is that of identity of sets: we say that set A and set B are identical just in case they contain exactly the same elements, i.e., whatever is in A is also in B, and vice versa . So, with these two definitions in hand, consider empty set E 1 and empty set E 2 . Well, they are equal since any element that is in the one is in the other (for the trivial reason that neither set contains any elements). So there really is only one empty set - which is what licenses our use of the definite article "the" in " the empty set". I'm not sure I understand your last question. In set theory, you've specified a set completely when you've specified its elements. And when we say that the empty set contains nothing, we have indeed specified exactly which elements it contains (namely, none).

Is it possible to positively prove a negative?

People often say this and it can be baffling to logicians! Perhaps your use of "positively" hints at what you're getting at though. Let's assume by "prove a negative" you mean something like: establish that something of a particular kind does not exist. For instance, your "negative" statement might be: Martians do not exist. And perhaps by "positively prove" you mean: establish by pointing to a particular thing that does exist. Then an instance of your claim might be that it's not possible to establish that Martians do not exist by displaying any particular non-Martian. And that's right: just because this particular object is not a Martian it doesn't follow that there are no Martians. In general, from the fact that a particular object is not an F we cannot logically infer that there are no Fs. So, if that's what you mean, logicians will agree that it's not possible "to positively prove a negative." However, that does not mean that one cannot logically prove statements of the form: there are...

What's the difference between saying "John is fat. Mary is tall." and saying "John is fat and Mary is tall."? What does "and" mean here?

I don't see that there is much difference in terms of what you're committing yourself to regarding how the world is. In both cases your claim will be true if "John is fat" is true and "Mary is tall" is true, and false otherwise.

When we prove a statement, we show it is true. Since contradictions (statements such as "P and not-P.") are never true, we can't ever prove a contradiction. But that's precisely what we do in a proof by contradiction - we show a contradiction to be true, before declaring it absurd. This must mean we are doing something wrong. It must mean that we can't even assume a false statement to begin with. This makes sense because when we assume a statement, we pretend that it's true, but we can't pretend that a false statement is true. It's a logical impossibility. That would be like saying "1 + 1 = 4" is true. Does this mean the "proof by contradiction" method is flawed? In other words, to prove proposition P, we assume not-P and show this leads to a contradiction. But if P is true to begin with (as we want to prove it), and therefore not-P false, how can we even assume not-P is true? It's false. We can't assume it as true. It's logically impossible. For example, it's logically impossible for the square root of...

When we prove something "by contradiction", we are not proving a contradiction true. We are showing that some assumption, call it X, logically implies a contradiction, that is, logically implies a statement that is logically false. The correct lesson to draw is not that the contradiction is after all true, but rather that our assumption X is incorrect. In other words, we infer not-X. This is what it is to prove not-X by contradiction. You write that if X is false, then it's impossible to assume that it is true? Why? Is there something incoherent about supposing that Hillary Clinton is President of the U.S. and asking what would follow from that?

What is a variable and what function does it play in such quantified propositions as "There is at least a thing, x, such that x is F", or "Every x is such that it is F"? Does the variable refer to something in the world? Or does it refer only to things assigned to constants? In other words: does the variable stand for things or words? And if it stands for things, does it stand for named things or even for unnamed things?

We get confused when we assimilate variables to ordinary referring expressions like "Obama". Because, as you realize, there's no good answer to the question "What does ' x ' refer to in 'Every x is such that x is F'?", or - to put the question in colloquial English - "What does 'it' refer to in 'Everything is such that it has mass'?" The variable ' x ', or the pronoun 'it' as used above, does not stand for anything. It is used in conjunction with the quantificational expression "everything is such that" to make a claim of generality, to say that all things have a certain property. You could view it as shorthand for the claim: o 1 is F and o 2 is F and ..., where the o i 's are all the objects in the universe. Notice that in this infinite expansion of the claim "Everything is such that it is F", the "it" has disappeared. This makes it clear that "it" was never really being used to stand for something in particular. It was used, together with a quantifier (like ...

I recently graduated with a BA in philosophy. I recently applied to many Ph.D and MA program. I feel that with philosophy as competitive as it is, my record will place me in a MA program first. What can I do to distinguish myself in a MA program, for my later application to a PHD. My hope is for a top 20 school, in my area of interest, what special activities are looked for coming from an MA. The following have been suggested from a variety of sources. Please advise, did I miss anything here, are any of these wrong? Thank you for any comments you might have, 1. Maintain an good GPA 2. Publish, in both graduate and professional journals 3. Don't rely only on your own university; become involved with other nearby departments. 4. Get teaching experience. (TA, Tutoring, Teaching critical thinking) 5. Teach at a community college level (some programs allow this) 6. Gain research experience (indexing, editing etc..) 7. Directed readings in areas of study, (I'm not sure if this would help for an MA,...

My own view is that all this is incredibly wrong-headed. Singularly missing from this list is the project of immersing yourself in philosophical texts, thinking and talking to people about those texts and the issues they raise, and developing a deeper and subtler understanding of philosophical issues. I would like to think that we still live in a world in which, if you were to do this, then the rest will sort itself out appropriately. I might be wrong - but in that case, I personally would be less interested in pursuing such a career.

Is there such a thing, in philosophy, as a formula for reconciling two contradictory statements? If not, then are there guidelines or strategies to beginning the reconciliation of two contradictory statements or arguments?

People don't normally speak of contradictory arguments – unless it's shorthand for arguments that lead to contradictory conclusions. So let's focus on that. I don't think there's anything like a formula for trying to reconcile such statements. One thing philosophers sometimes do is try to find a hidden parameter that has been suppressed in the statements and then make that explicit in different ways in the two sentences thereby arriving at two statements that, though superficially contradictory, are actually perfectly consistent. To take a simple example (which has generated some discussion), we might initially be puzzled by the fact that we want to assert that Obama is young and also not young - until we realize that what we really wanted to say is that Obama is young for a President of the U.S. and that Obama is not young for a basketball player.