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...

A statement P about a single element in a dual or multiple set does not seem to logically exclude P applying equally to other elements in the set; yet we often talk as though "P is true of X" implies "P is not true of Y (or Z)", when X, Y, and Z all belong to some grouping. For example, take "Men work to support their families". Does this logically imply that women do not work to support their families? What about "African Americans suffer from discrimination"? Does this logically imply that Asian Americans, Hispanic Americans and white Americans (among other racial groupings) do not suffer from discrimination? Such objections are often raised in discourse. Given (x, y), "P is true of X" is thought to imply "P is not true of Y", or "Not-P is true of Y". If there is no logical exclusion above, what are these objections targeting? Is it a question of salience, rather than logic?

There is a simple reasoning. Which is better, bread or love? It seems love is better than nothing. For sure bread is better than nothing. So bread is better than love. Of course this is a wrong reasoning. But I wonder whatever logical mistake is made here?

Is it logically contradictory for a person to say that they are humble, in a broad sense? After all, humility is generally considered a desirable quality.

What books would serve as a comprehensive overview of philosophical and mathematical logic?

Hi I understand how to apply derivation rules like the rules of inference etc. My question is do we have a method of proving the rules themselves? Is there a way to prove that If P then Q; P; therefore Q? Or do we accept these rules out of intuition?

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 do philosophers mean when they describe one claim as being "stronger" or "weaker" than another?

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?

Is it possible to conceive of an irrational entity or can only rational things be conceived of? Can irrational things exist? Of course it depend on how you define rational but maybe vagueness has more creative potential for philosophical thought.