Since one's own reasoning is a basically set of rules of inference operating on a set of axiomatic beliefs, can one reliably prove one's own reasoning to be logically consistent from within one's own reasoning? Might not such reasoning itself be inconsistent? If our own reasoning were inconsistent, might not the logical consistency (validity) of such "proofs" as those of Godel's Incompleteness Theorems, be merely a mirage? How could we ever hope to prove otherwise? How could we ever trust our own "perception" of "implication" or even of "self-contradiction"?

Logic textbooks which offer a system of natural deduction containing a so called "rule of replacement" restrict this rule to logically equivalent formulae. Only these can replace each other wherever they occur. I have often wondered why this is so. It seems to me that, having e.g. p q and p&r as lines in a proof (as premisses, say), would allow one to soundly infer q&r directly from them by replacement of p by q in p&r, without requiring that p and r be logically equivalent. In less formal situations, for example, when solving a math problem, I find myself (and others) doing this all the time. I've searched the internet for this, but couldn't find any answer so far. Most grateful in advance for a reply.

Look at this inference: Premise 1: All desks have the same color. Premise 2: That desk is brown. Conclusion: All desks are brown. Now, I understand that this is a deduction. However, the conclusion is a generalization of one of the premises, and generalizations of premises are what one would expect in induction. Where did I go wrong?

I have a question about Whitehead and Russell "Principia Mathematica". Can mathematics be reduced to formal logic?

Notation: Q : formal system (logical & nonlogical axioms, etc.) of Robinson's arithmetic; wff : well formed formula; |- : proves. G1IT is always stated in the form: If Q is consistent then exists wff x: ¬(Q |- x) & ¬(Q |- ¬x) but we cannot prove it within Q (simply because there is no deduction rule to say "Q doesn't prove" (there is only modus ponens and generalization)), so it's incomplete statement, I don't see WHERE (in which formal system) IS IT STATED. (Math logic is a formal system too.) In my opinion, some correct answer is to state the theorem within a copy of Q: Q |- Con(O) |- exists x ((x is wff of O) & ¬(O |- x) & ¬(O |- ¬x)) where O is a copy of Q inside Q, e.g. ¬(O |- x) is an arithmetic formula of Q, Con(O) means contradiction isn't provable...such formulas can be constructed (see Godel's proof). But I'm confused because I haven't found such statement (or explanation) anywhere. Thank You Very Much

Following along from http://www.askphilosophers.org/question/2039: "Does the law of bivalence demand that a proposition IS either true or false today? What if the truth or falsity of this proposition is a correspondence to a future event that has yet to occur?" What's problematical about saying "yes, it's either true or false, but I don't happen to know which"? Is that substantively different from saying the same thing about an open problem in science or mathematics, to which the answer is presumably knowable but happens not yet to be known? The questioner seems to be demanding both that there be an answer, which may be a reasonable thing to want, and to be able to know what the answer is, which isn't necessarily reasonable. Is it reasonable always to expect somebody (other than deity) to know the answer to a question?

I have a small question about logic. In my text, "3 is less than or equal to pi" is translated as PvQ, where P is "3 is less than pi" and Q is "3 is equal to pi." Seems simple enough. But why isn't the statement better translated as (PvQ)&~(P&Q)? Of course, if you know what "less than" and "equal to" really mean, you'll understand that P&Q is precluded; but it bothers me that this is not explicitly stated in the translation. Someone who understands logic but not English might infer from PvQ that 3 may be simultaneously "less than" and "equal to" pi, and this strikes me as problematic.

I studied philosophy in university and I recall that one of my tutors for symbolic logic was trying to walk me through a problem by saying that if you have a large enough set of premises, two of them will inevitably contradict one another. I've always had trouble understanding (and consequently, accepting) this proposition because: if one conceives of reality as a set of claims (e.g., I am right-handed, electron X is in position Y, 2 + 2 = 4, etc.) there are an infinite number of "premises" to the "argument" that is reality and consequently reality is self-contradictory. Am I missing something here? Can you explain which of us is right about this and in which sense? I should mention that I don't necessarily have a problem with reality being self-contradictory, but that really throws symbolic logic out the window (and doesn't throw it out the window at the same time)! Thanks to all respondents for their time. -JAK

Why is question-begging considered a fallacy when it embodies a deductively valid form of reasoning?

Many claims about what is possible or logical seem to rest on what is conceivable to the human mind. But what reason do we have to believe that there's any link between the way our minds work and the way things actually are?