Is it true that anything can be concluded from a contradiction? Can you explain? It's seems like its a tautology if taken figuratively because we can indeed conclude anything if we suspend the rules of reasoning, but there is nothing especially interesting in that fact in my humble opinion.

@William Rapaport: Unless disjunctive syllogism or one of the other two rules used in the derivation fails, the "irrelevance" of the conclusion to the premise is irrelevant to whether the conclusion follows from the premise. Relevance logic has to give up at least one of those rules, none of which is easy to give up.

The topic is controversial (as I indicate below), but the inference rules of standard logic do allow you to derive any conclusion at all from any (formally) contradictory premise. Here's one way (let P and Q be any propositions at all): 1. P & Not-P [Premise: formal contradiction] 2. Therefore: P [From 1, by conjunction elimination] 3. Therefore: P or Q [From 2, by disjunction introduction] 4. Therefore: Not-P [From 1, by conjunction elimination] 5. Therefore: Q [From 3, 4, by disjunctive syllogism] Those who object to such derivations usually call themselves "paraconsistent" logicians; more at this SEP entry . They typically reject step 5 on the grounds that disjunctive syllogism "breaks down" in the presence of contradictions. I confess I've never found their line persuasive.

Is there a way to prove that logic works? It seems that the only two methods for doing this would be to use a logical proof –which would be incorporating an assumed answer into the question– or to use some system other than logic –thus proving that sometimes logic does not work.

Even asking "Is there a way to prove that logic works?" presupposes that logic does work at least at the level of its most basic laws, such as the Law of Noncontradiction, because the question itself has meaning only if the most basic laws of logic hold. To put it a bit differently: No sense at all can be attached to the notion that logic doesn't work (or even sometimes doesn't work). See also my reply to Question 4837 and Question 4884 . So we have what philosophers call a "transcendental" proof of the reliability of logic: If we can so much as ask whether logic is reliable (provably or otherwise), then it follows that the answer to our question is yes . You might say that this proof won't impress someone who doubts the most basic laws of logic in the first place. But I'd reply -- predictably -- that no sense can be attached to the notion of doubting the most basic laws of logic.

Would an omnipotent and omniscient being be bound by the laws of logic? If so, to what degree?

Yes. Completely. The tricky question is why . It's tempting to answer that necessarily everything is bound by the laws of logic because the alternative -- the claim that something isn't bound by the laws of logic -- is necessarily false. But, as I suggested in my reply to Question 4837 , no sense can be attached to the claim that something isn't bound by the laws of logic. So the claim can't be false , strictly speaking. Perhaps all we can assert is a wide-scope negation: it's not the case that something isn't bound by the laws of logic, just as it's not the case that @#$%^&*. Necessarily everything is bound by the laws of logic because the alternative is literally nonsense? I wish I had a better explanation!

Working off Kelsen, logic and rules of inference, as well as other rule based systems, are normative, "ought" based systems. If this is true, or even if it isn't, what reason do we have to take that logical rules are reasonable? In other words, why should one accept that rules of valid inference (of any system) as actually generating true responses from true premises?

To test a rule of inference, you can try to find counterexamples to it, cases in which the rule lets you derive a falsehood from true premises. Professor Vann McGee offered a well-known (and controversial) such attempt in this article . But there's no getting around rules of inference entirely. Even as you test one rule of inference you unavoidably rely on others. Because any attempt to answer the question "Why should we trust rules of inference at all?" will rely on reasoning, it will trust some rules of inference, whether or not those rules are made explicit in the reasoning. There's no way to get "outside" all rules of inference and see how they measure up against something more trustworthy than they are.

If Laws of logic are true or hold in all contexts, how can there be more one law? Do the two versions of De Mogan's laws differ? If so. how? Does the law of excluded middle differ from the law of non contradiction and from either version of De Morgans laws? Enoch

Notice that the same question arises in math, where the laws also hold no matter what. Arithmetic contains commutative laws of addition and of multiplication, associative laws of addition and multiplication, a distributive law of multiplication over addition, etc. Are those laws different? Their representations on the page certainly look different. I take it that you're asking, at bottom, how truths that hold in all possible worlds could count as distinct truths. The answer depends on how propositions are to be individuated , and here philosophers give various answers. On some theories, there's only one proposition that's true in all possible worlds, although there are indefinitely many sentences (some logical, some mathematical, some metaphysical) that express this single proposition. Other theories give a more fine-grained way of individuating propositions that allows for the existence of multiple propositions that are true in all possible worlds. You'll find more...

In predicate logic can we have valid arguments if we make an existential claim in our premises and not in the conclusion? In other words can we simply rename the existential quantifer to a "particular" quantifer or something of the sort? Does this particular quantifer always have to carry existential import?

If I understand your first question, the answer is no (unless the existential premise is superfluous). By an "existential claim," I take it you mean an existential generalization such as "There exists an x such that F x ," rather than a claim of the form "F a ," which implies an existential generalization. But you might wish to look into the rule of Existential Instantiation (or Existential Elimination in natural deduction systems); you'll find a brief summary of it here . I'm not sure I understand your second question. There are two ways of interpreting the universal and existential quantifiers: the objectual way and the substitutional way. I can't find a handy link to recommend, but if you search for discussions of those terms, you may find something relevant to your third question.

Me and my professor are disagreeing about the nature of logic. He claims that logic is prescribes norms for correct reasoning, and is thus normative. I claim that logic is governed by a few axioms (just like any in any other discipline, i.e. science) that one is asked to accept, and then follows deductively, free of any normative claims. My question is: which side is more sound? Thank you.

In this context, by "normative claims" I take it you mean claims that one ought to (or ought not to) do some particular thing. Can we get such claims out of principles of deductively valid inference? I think so. If you accept P, and you recognize that P implies Q, then there's a sense in which you ought to accept Q: you're logically and rationally committed to Q by propositions that you accept and recognize. If you accept Q, and you recognize that P implies Q, there's a sense in which you ought not to deduce P from those propositions alone: doing so would be fallacious. Now, you might say that the ought and ought not in those cases is only hypothetical: " If you want your deductive reasoning to be reliable, then you ought (or ought not)...." But I think the antecedent of that conditional (the "if" part) is easy to discharge. Plenty of people do want their deductive reasoning to be reliable, and so there's a sense in which such people really ought to use ...

what is the difference between logical necessity and metaphysical necessity?

I think of logical necessity as (predictably enough) the necessity imposed by the laws of logic. So, for example, it's logically necessary that no proposition and its negation are both true, a necessity imposed by the law of noncontradiction. But one might regard logical necessity as broader than that, since one might say that it also includes conceptual necessities such as "Whatever is red is colored." Metaphysical necessity is a bit harder to nail down. Every proposition that's logically or conceptually necessary is also metaphysically necessary, but there may be metaphysical necessities that are neither logically nor conceptually necessary, such as "Whatever is water is H2O" or "Whatever is (elemental) gold has atomic number 79." Nothing in logic or in the concepts involved makes those propositions necessary, but many philosophers say that those propositions are nevertheless "true in every possible world," which is the root idea of metaphysical necessity. Even if some proposition P isn't...

Is logic "universal"? For example, when we say that X is logically impossible, we mean to say that in no possible world is X actually possible. But doesn't this mean that we have to prove that in all possible worlds logic actually applies? In other words, don't we have to demonstrate that no world can exist in which the laws of logic don't apply or in which some other logic applies? If logic is not "universal" in this sense, that it applies in all possible words, and we've not shown that it absolutely does apply in all worlds, how can we justify saying that what is logically impossible means the not possible in any possible world, including our actual world?

I don't understand the question, because I don't understand the phrase 'a world in which the laws of logic don't apply'. I don't think any sense can be attached to that phrase. Is a world in which the laws of logic don't apply also a world in which they do apply? If no, why not? If yes, is that same world also a world in which the laws of logic neither apply nor don't apply? If no, why not? It's as if the questioner had asked, "Don't we have to demonstrate that no world can exist in which @#$%^&*?"

Is it possible for there to be a world that logic does not apply? That is, can't a "married bachelor" actually exist in some world that there is no logic or that there is a different logic that applies? And if so, then isn't it the case that we merely assume the first principles of logic (noncontradiction, identity, excluded middle, etc...) because we observe them in our actual world, which is 1 of many possible worlds? And if it is mere assumption, then can't we be wrong about them when we say they can/should apply to other possible worlds?

I don't think this question can be answered. I think no one -- including the questioner -- understands the question being asked. In asking "Is it possible for there to be a world where logic doesn't apply?" is the questioner asking (a) "Is it possible for there to be a world where logic doesn't apply?" or (b) "Is it possible for there to be a world where logic does and doesn't apply?" or (c) "Is it possible for there to be a world where logic neither applies nor doesn't apply?" or (d) "Is it possible for there to be a world where logic does apply?" If logic doesn't apply in a world, then...then what? In a world where logic doesn't apply, does logic also apply? If not, why not? Unless logic applies in every world, how can we tell which, if any, of (a)-(d) is the question that the questioner is asking?

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