All chariot racers are musicians. Some chariot racers are soldiers. Therefore, some musicians are soldiers. Valid or Invalid?

Valid. Your second premise tells you that some chariot racer is a soldier. Let's call him "Alfred". So Alfred is a chariot racer and Alfred is a soldier. So Alfred is a chariot racer. This last fact, combined with the first premise, tells us that Alfred is a musician. But Alfred is also a soldier. So Alfred is both a musician and a soldier. Hence, someone is both a musician and a soldier. Which is your conclusion.

I would like to know if this can be proven I am attempting to prove G with these premises: 1. (-K and -N) > [(-P> K) and (-R> G)] 2. K> N 3. -N and B 4. -P v -R I am not sure if the premises are enough to allow the solver to prove the solution or if there should be additional premises. A response would be appreciated!

Since "-N" is true (3), then

Since "-N" is true (3), then from (2), you can infer that "-K" is true. So you know that "-K and -N" is true. Hence from premise (1) you can infer that "(-P > K) and (-R > G)" is true. Hence "-P > K" is true. Since you already showed that "-K" is true, it follows that "P" is true. But if "P" is true then it will follow from premise (4) that "-R" is true. Since you've already shown that "-R > G" is true, you can conclude that "G" is true. If you know what natural deductions are, you might find an online natural deduction proof checker and reconstruct the derivation in that.

Can you define logical validity? I'm engaged in a debate on the subject, with a friend, whom will not easily accept anyones word on the matter, so i would ask that you perhaps post your credentials? thank you for you time and effort!

If you mean "valid argument," that's typically defined as an argument such that there is no interpretation of its premises and conclusions under which all the former are true and the latter is false.

I know affirming the consequent is a fallacy, so that any argument with that pattern is invalid. But what what about analytically true premises, or causal premises? Are these not really instances of the fallacy? They seem to take its form, but they don't seem wrong. For example: 1. If John is a bachelor, he is an unmarried man. 2. John’s an unmarried man. 3. Therefore he’s a bachelor. How can 1 and 2 be true, and 3 be false? Yet it looks like affirming the consequent. 1. X is needed to cause Y. 2. We’ve got Y. 3. Therefore there must have been X. Again, it seems like the truth of 1 and 2 guarantee the truth of 3. What am I missing?

Your particular argument: 1. If John is a bachelor, he is an unmarried man. 2. John's an unmarried man. 3. Therefore he's a bachelor. isn't a valid argument. You may think it is because you believe that (3) just follows logically from (2). But it doesn't. It follows from (2) and the definition of "bachelor" as unmarried man .

Why does inconsistency entail validity?

Let me spell out the claim I think you have in mind. Any argument whose premises form an inconsistent set of sentences is a valid argument. To understand why this is so, let's be clear about what "inconsistent set of sentences" means and what "valid argument" means in this context. To say that a set of sentences is inconsistent is to say (roughly) that it is not possible that all the sentences in that set are true: they could all be false, some could be true and some false, but there is no way that all could be true. To say that an argument is valid is to say that it's not possible for all its premises to be true and its conclusion at the same time to be false; sometimes people express this by saying that the truth of the premises forces the truth of the conclusion. But now think about it: If you have an argument whose premises are inconsistent, then it's certainly not possible that all its premises be true and its conclusion false - since it's already not possible for all its...

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.)

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...

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 ...

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.