## If Laws of logic are true or hold in all contexts, how can there be more one law

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 this section of an SEP article; I recommend the whole article.

## In predicate logic can we have valid arguments if we make an existential claim

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 Fx," rather than a claim of the form "Fa," 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

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.

Without disagreeing with Stephen's fine response, let me point out one other issue. You say that "logic is governed by a few axioms...and then follows deductively, without any normative claims". But there is no "following deductively" without logic: logic is about the correct norms of deductive reasoning. So this conception is flatly circular: a point made a long time ago by Quine in his paper "Truth by Convention".

I should say that there are philosophers who deny that logic is about reasoning at all. On this view, logic is about a certain relation between propositions, implication, that it aims to characterize. But then the dispute just shifts to whatever one thinks does characterize the norms of reasoning, e.g, decision theory. And, for what it's worth, my own view has always been that these philosophers have too simplistic a conception of what sorts of norms logic articulates. But that is a larger issue.

## I read once that an African tribe was asked a simple logical problem paraphrased

I read once that an African tribe was asked a simple logical problem paraphrased as follows: "Berlin is a city in Germany. There are absolutely no camels in Germany. Are there camels in Berlin?" The tribe could not provide a definitive answer, instead saying things like "I have never been to Berlin, so I cannot say whether there are camels or not" or "If Berlin is a big city, there must be camels" in other words, completely missing the logical puzzle and instead providing more pragmatic answers. Now this story may be apocryphal, since I cannot find where I read it, but it raises an interesting question. To what extent is logic universal, is it culturally biased/culturally learned, and how do we explain the answers of the tribe?

The claim that "logic is universal" is the claim that the norms of correct reasoning are universal. It is not the claim that everyone follows those norms, or that everyone reasons well.

In the story as told (apocryphal or otherwise), the tribesmen are failing to make a certain inference. That makes them poor reasoners, but it doesn't threaten the universality of logic.

## What is the truth maker for logic? In other words, why should I take logical

What is the truth maker for logic? In other words, why should I take logical truths (e.g., material implication) as true?

A few points need clarification before I can begin to answer your question.

First, logic is not concerned with truth in the way that, say, the sciences are. Logic is concerned with relationships among sentences that have truth value, not with the actual truth values of the (atomic) sentences. The only apparent exception to this might be those sentences that "must" be true (tautologies) and those that "must" be false (contradictions). But tautologies and contradictions are not atomic sentences; they are "molecular" sentences, and what makes them tautologous or contradictory are the relationships among their atomic constituents. So, for instance, "(p & ¬p)" is a contradiction because—no matter what the actual truth value of p—the truth value of "(p & ¬p)" must be false (because of the truth tables for conjunction (&) and negation (¬)). Logic isn't concerned with p's actual truth value.

Second, material implication (&rightarrow;) is not a "logical truth" nor is it even a sentence. It's a logical connective like & and ¬. Each connective has a truth table that tells you what the truth value of a molecular sentence constructed with the connective is, no matter what the actual truth values of the constituent sentences are. Again, logic isn't concerned with what those truth values are, only how they relate to each other when combined with a given connective.

So, what you may be thinking of when you said "material implication" is the rule of inference called "modus ponens":

From p

and (p &rightarrow; q),

you may infer q.

A rule of inference like this (you can also think of it as an "atomic" pattern of reasoning) is neither true nor false (any more than the number 7 is true or false). Rules of inference (patterns of reasoning, arguments, proofs—whatever you want to call them) are either "valid" or "invalid". And a rule of inference is valid if and only if it is "truth preserving", that is, if and only if, whenever the premises (in our example, these are p and "(p&rightarrow;q)") are true, then the conclusion (in our example, that's q) must be true. Logic doesn't tell you whether the premises actually are true (that's something that, say, science might be able to tell you), but it does tell you about the relationship of the premises to the conclusion, namely, that if the former are true, the latter have to be true.

So, maybe your question is this: Why should you take arguments like this as being valid? Well, consider the truth table for material implication (&rightarrow;):

p q (p&rightarrow;q)

T T T

T F F

F T T

F F T

Look at the rows that correspond to a modus ponens inference with both premises being true. There's only one such row: the first one. And in that row, q is also true. That's why modus ponens is valid.

Now, your next question might be: Why should we accept that truth table for &rightarrow;?

Here, there are two issues. First, not all logicians do accept it, or, to put it a better way, some logicians think that this connective doesn't really capture the meaning of ordinary English "if-then". One example of a different kind of logic that's been devised to deal with this is "relevance logic". Second, a more general question is: Why should we accept any truth table? Here, the answer I'll give you is that, for any combination of 1 or 2 atomic sentences with two truth values, there are only 16 possible truth tables, that is, only 16 possible logical connectives. Logic (more precisely, classical sentential or propositional logic) can be thought of as the study of them, independently of whether any of them capture the meaning of ordinary English connectives.

So, perhaps the answer to your question is: You don't have to take them as "true" or "valid". You're free to create other logics! To see some of the possible variety in logics, take a look at all the articles in the Stanford Encyclopedia of Philosophy under the heading "logic" in the index.

## what is the difference between logical necessity and metaphysical necessity?

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 logically necessary, if P is metaphysically necessary then P is true in every possible world and the negation of P is false in every possible world.

This topic is actually more controversial than my answer suggests, but that's how I'd answer your question.

## Frequently, one finds the following statment: "You cannot prove a negative." My

Frequently, one finds the following statment: "You cannot prove a negative." My question is, in this context, what is meant by the word "negative?" I understand how the word is used in mathematics and I "think" I know the meaning when used in logic. I just cannot seem to get a handle on how it is used here. Moreover, does it, perhaps, refer to a total position in the debate over the existence of God? Any comments you would make would be greatly appreciate. I enjoy your application very much and, moreso, since I am so old. Thanks. JH

This is a pretty confusing expression. What's usually meant, I think, is that a negative general proposition -- a proposition asserting that a certain kind never occurs -- requires much more by way of justification from its defender than from its opponent. Take the proposition "there are no black swans," for example. To prove it, you would have to comb through the whole universe, presumably all the way backward and forward in time, to demonstrate conclusively that nothing contained therein is a black swan. To disprove the proposition, by contrast, all you need do is produce a single black swan. Given this asymmetry, it thus makes sense to saddle the opponent, rather than the proponent, of a negative general proposition with the burden of proof.

What's confusing here is that the same sort of asymmetry is present with affirmative general propositions as well. Thus the proposition "all elks like mushrooms" requires much more by way of justification from its defender than from its opponent. To prove it, you would have to comb through the whole universe, presumably all the way backward and forward in time, to investigate all elks in regard to their fondness for mushrooms. To disprove the proposition, by contrast, all you need do is produce a single elk who doesn't like mushrooms. Given this asymmetry, it makes sense once again to saddle the opponent, rather than the proponent, of such an affirmative general proposition with the burden of proof.

In its most general formulation, the point your queried statement tries to express is then not one about negative versus affirmative propositions but one about propositions governed by different logical quantifiers. Any proposition -- negative or affirmative -- that is universally quantified requires much more by way of justification from its defender than from its opponent, who need merely produce a single counterexample, that is, prove an existentially quantified proposition.

To see this, we might state our two sample propositions in logical language. Both of them are governed by a universal quantifier ("for all x, ..."):

(1) For all x, if x is a swan then x is not black.

(2) For all x, if x is an elk then x likes mushrooms.

the negations of these two propositions, both governed by existential quantifiers ("there is an x such that...") are, respectively,

(1-) There is an x, such that x is a swan and x is black.

(2-) There is an x, such that x is an elk and x does not like mushrooms.

Clearly, much more is required of the defenders of (1) and (2) than of the defenders of (1-) and (2-). This is an interesting point, but not one well expressed in the common statement you cite.

## Is logic "universal"? For example, when we say that X is logically impossible,

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

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?

## Are first principles or the axioms of logic (such as identity, non-contradiction

Are first principles or the axioms of logic (such as identity, non-contradiction) provable? If not, then isn't just an intuitive assumption that they are true? Is it possible for example, to prove that a 4-sided triangle or a married bachelor cannot exist? Or must we stop at the point where we say "No, it is a contradiction" and end there with only the assumption that contradictions are the "end point" of our needing to support their non-existence or impossibility?

To prove a proposition is to derive it syntactically (that is, by "symbol manipulation" that is independent of the proposition's meaning). A "good" (or syntactically valid) derivation is one that begins with "first principles" (axioms) and derives other propositions from them (and from other validly derived propositions) by rules of infererence. Ideally, the rules of inference should be "truth preserving": If you start with true axioms, then all of the propositions derived from them by the rules should also be true.

So, can you prove the axioms? If so, how? The uninteresting answer is, yes, you can prove them (in a technical, but trivial, sense) just by stating them, because they don't need to be derived by any rules from anything "more basic".

So, how do you know that they are true? Well, truth and proof are two different things. Proof has to do with syntax, or valid derivation. Truth has to do with semantics, or meaning.

Ideally, truth and proof should match up: A formal system (of logic) is "sound" if all (syntactically) derived propositions are (semantically) true; that's a desirable goal, especially if you start with true axioms and follow truth-preserving rules of inference. (Of course, garbage in, garbage out: If you start with false premises, or use non-truth-preserving rules, anything can happen—you're no longer guaranteed that your end product will be true.)

And a formal system is "complete" if all semantically true propositions are (syntactically) derivable. That's also a desirable goal, and both propositional and first-order logic are complete. But if you add Peano's axioms for arithmetic to first-order logic, you get a system that is incomplete: That's (roughly) Goedel's "incompleteness" theorem, that there are true propositions of FOL+Peano that are not provable.

Returning to your question, how do you know that the axioms are true? Not by formally proving them from more basic propositions, but by using semantic methods. In the case of axioms of propositional logic, you can use truth tables. For instance, suppose that "non-contradiction" is one of your axioms. I'll assume that this can be formalized as:

~(p & ~p)

That is: It is not the case that both p is the case and that not-p is the case.

A truth-table analysis will show that this is a tautology (that it is true no matter what truth-values you assign to p).

Have you thereby "proved" the principle of non-contradiction? In the technical, syntactic sense, no: A truth-table analysis is not a formal, syntactic proof from first principles by rules of inference. In an informal, everyday sense of "proved", perhaps yes, because you've shown by semantic methods that it must be true.

So, to answer your questions: First principles are syntactically provable, but only in a trivial, technical sense. But they're not just intuitive assumptions, either, because you can give a more substantive, semantic justification for them (at least, in the propositional case).