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I have two questions about logic that have vexed me for a long time.

I have two questions about logic that have vexed me for a long time. Smith has written two great books of philosophy. Now he has come out with a third book. Therefore, that book will probably be good too. Smith has flipped a coin twice, and both times it has come up tails. Now Smith will flip the coin a third time. Therefore, that flip with probably end up 'tails' too. The logical form of inductive arguments seems to contribute nothing; the premises seem to do no logical work supporting the conclusion - is that right? Smith has written two great books of philosophy. Now he has written a third. Any author that has written two great books of philosophy, and then writes a third, has probably written a third great book. Therefore, Smith has probably written a third great book. That seems a deductive argument, because the general premise was added. And if true, the premises do seem to support with conclusion with necessity, even though the conclusion is probable; it is the knowledge of the world and not...

I think both arguments can be analyzed as inductive arguments and still distinguished in terms of their quality. The book argument is a stronger inductive argument than the coin-toss argument for a simple reason: the probability that Smith's book C is great isn't independent of whether Smith's books A and B are great.

That is, Smith's having written great books A and B makes the probability that Smith's book C is great higher than it would be had Smith not already written two great books. Important: higher than it would be otherwise, which needn't mean higher than one-half. Even though Smith's track-record raises the probability that book C is great, the track-record needn't make it more probable than not that book C is great.

By contrast, the probability of tails on any given toss of a fair coin is independent of whether the coin came up tails twice already: that history of tosses neither increases nor decreases the probability of tails on a third toss.

I think both arguments can be analyzed as inductive arguments and still distinguished in terms of their quality. The book argument is a stronger inductive argument than the coin-toss argument for a simple reason: the probability that Smith's book C is great isn't independent of whether Smith's books A and B are great. That is, Smith's having written great books A and B makes the probability that Smith's book C is great higher than it would be had Smith not already written two great books. Important: higher than it would be otherwise, which needn't mean higher than one-half. Even though Smith's track-record raises the probability that book C is great, the track-record needn't make it more probable than not that book C is great. By contrast, the probability of tails on any given toss of a fair coin is independent of whether the coin came up tails twice already: that history of tosses neither increases nor decreases the probability of tails on a third toss.

Are the laws of logic invented or are they independent of human reason? If they

Are the laws of logic invented or are they independent of human reason? If they are independent, how can they exist immaterially? What does it mean for such laws to exist in a nonphysical way?

Good question, and as fundamental a question as anyone could ask. I think that the laws of logic must be not only independent of human minds but independent of any minds, including God's mind if such exists. At any rate, I don't think anyone can see how it could be otherwise.

To say that the laws of logic depend on human or divine minds is to imply that the following conditional statement is nontrivially true:

If (1) human or divine minds had been different enough, then (2) all of the laws of logic would be different from what they are.

(By "nontrivially true," I mean that the statement is true not merely on the ground that (1), its antecedent, is logically impossible. If (1) is logically impossible, then the conditional statement is trivially true, even if (2), its consequent, is also logically impossible.)

We can't make sense of the italicized statement without presupposing that (2) is false. If the italicized statement means anything, then it doesn't mean this: If (1) human or divine minds had been different enough, then (~ 2) not all of the laws of logic would be different from what they are. But, of course, my assertion just now about the statement's meaning itself depends on holding fixed at least some of the laws of logic, i.e., it depends on presupposing (~ 2) even on the assumption that (1) is true. Therefore, we understand the italicized statement only if we presuppose that it can't be nontrivially true.

As for the nonphysical existence of the laws of logic, you might look at what I wrote in reply to Question 24874.

Good question, and as fundamental a question as anyone could ask. I think that the laws of logic must be not only independent of human minds but independent of any minds, including God's mind if such exists. At any rate, I don't think anyone can see how it could be otherwise. To say that the laws of logic depend on human or divine minds is to imply that the following conditional statement is nontrivially true: If (1) human or divine minds had been different enough, then (2) all of the laws of logic would be different from what they are . (By "nontrivially true," I mean that the statement is true not merely on the ground that (1), its antecedent, is logically impossible. If (1) is logically impossible, then the conditional statement is trivially true, even if (2), its consequent, is also logically impossible.) We can't make sense of the italicized statement without presupposing that (2) is false . If the italicized statement means anything, then it doesn't mean this: If (1) human or...

Is it possible to translate a syllogism into propositional logic? This is the

Is it possible to translate a syllogism into propositional logic? This is the example: All doctors went to medical school. Hanna is a doctor. Hanna went to medical school. Thanks a lot, Sebastiano

For any syllogism containing quantifiers such as "all," "some," and "no"/"none," you'll need predicate logic for the translation. Propositional logic alone won't suffice. But you could use propositional logic to translate a non-quantified argument that's at least similar to the syllogism: "If Hanna is a doctor, then she went to medical school. Hanna is a doctor. Therefore, Hanna went to medical school."

For any syllogism containing quantifiers such as "all," "some," and "no"/"none," you'll need predicate logic for the translation. Propositional logic alone won't suffice. But you could use propositional logic to translate a non-quantified argument that's at least similar to the syllogism: "If Hanna is a doctor, then she went to medical school. Hanna is a doctor. Therefore, Hanna went to medical school."

P1. If today is February 29th, then it is a leap year

P1. If today is February 29th, then it is a leap year P2. Today is not February 29th C. It is not a Leap Year Is this argument sound or unsound? From what I can tell it is invalid because it is possible for it to be a leap year and today not being February 29th. If it’s invalid then it should be unsound. However neither of the premises are false so it can’t be unsound? Even if it were sound, wouldn’t it technically become unsound if it happened to be February 29th in real life?

The argument is unsound because, as you say, it's invalid. It commits the well-known fallacy of denying the antecedent.

Validity is necessary (but not sufficient) for soundness. So the argument is unsound regardless of the truth or falsity of its premises.

The argument is unsound because, as you say, it's invalid. It commits the well-known fallacy of denying the antecedent . Validity is necessary (but not sufficient) for soundness. So the argument is unsound regardless of the truth or falsity of its premises.

Is there any single genuinely correct logic or so called all-purpose logic? If

Is there any single genuinely correct logic or so called all-purpose logic? If not, why should we find it?

I presume that you would dismiss out of hand the following answer to your first question: "Yes, there is a single genuinely correct, all-purpose logic, and there is no such logic, and there is more than one such logic." So I take it that your question presupposes that no correct logic could allow that answer to be true.

If you're asking whether there's any good reason to abandon the standard, two-valued, "classical" logic routinely taught to university students in favor of some non-classical logic, then I'd answer no. Some philosophers say that we ought to adopt a non-classical logic in response to such things as the Liar paradox or the Sorites paradox, but their arguments for that conclusion have never struck me as persuasive. I think that the Liar and the Sorites can be solved using only classical logic (and bivalent semantics), or at least it's too early to conclude that they can't be.

For a much more detailed answer, you might consult Susan Haack's book Deviant Logic, Fuzzy Logic: Beyond the Formalism (University of Chicago Press, 1996).

I presume that you would dismiss out of hand the following answer to your first question: "Yes, there is a single genuinely correct, all-purpose logic, and there is no such logic, and there is more than one such logic." So I take it that your question presupposes that no correct logic could allow that answer to be true. If you're asking whether there's any good reason to abandon the standard, two-valued, "classical" logic routinely taught to university students in favor of some non-classical logic, then I'd answer no . Some philosophers say that we ought to adopt a non-classical logic in response to such things as the Liar paradox or the Sorites paradox, but their arguments for that conclusion have never struck me as persuasive. I think that the Liar and the Sorites can be solved using only classical logic (and bivalent semantics), or at least it's too early to conclude that they can't be. For a much more detailed answer, you might consult Susan Haack's book Deviant Logic, Fuzzy Logic:...

Recently I asked a question about logic, and the answer directed me to an SEP

Recently I asked a question about logic, and the answer directed me to an SEP entry, which then took me to two other SEP entries, on Russell's paradox and on the Liar's paradox. Frankly, after having read through those explanations, there was a glaring omission from every cited philosopher, and I wondered if everyone was overcomplicating things: I don't see how there is any "paradox" at all. Consider the concept of a "round square" or a "six-sided pentagon." Those are nonsensical terms, because of the structural nature of the underlying grammar. They are neither logical nor illogical, they are merely grammatically inconsistent at the fundamental level of linguistic definition. The so-called "paradox" of Russell and the Liar seem to me to be exactly the same kind of nonsensical formulations: the so-called "paradox" is merely a feature of the language, these concepts also are grammatically inconsistent at the fundamental level of linguistic definition. Russell's "paradox" is just as "paradoxical" as...

If I may, I think you're being a bit too dismissive of Russell's paradox.

We start with the observation that some sets aren't members of themselves: the set of stars in the Milky Way galaxy isn't itself a star in the Milky Way galaxy; the set of regular polyhedra isn't itself a regular polyhedron; and so on. It seems that we've easily found two items that answer to the well-defined predicate

S: is a set that isn't a member of itself.

Naively, we might assume that a set exists for every well-defined predicate. (For some of those predicates, it will be the empty set.) But what about the set corresponding to the predicate S? This question doesn't seem, on the face of things, to be nonsensical or ungrammatical. But the question shows that our naive assumption implies a contradiction, and therefore our naive assumption can't possibly be true.

If I may, I think you're being a bit too dismissive of Russell's paradox. We start with the observation that some sets aren't members of themselves: the set of stars in the Milky Way galaxy isn't itself a star in the Milky Way galaxy; the set of regular polyhedra isn't itself a regular polyhedron; and so on. It seems that we've easily found two items that answer to the well-defined predicate S: is a set that isn't a member of itself . Naively, we might assume that a set exists for every well-defined predicate. (For some of those predicates, it will be the empty set.) But what about the set corresponding to the predicate S? This question doesn't seem, on the face of things, to be nonsensical or ungrammatical. But the question shows that our naive assumption implies a contradiction, and therefore our naive assumption can't possibly be true.

What is the difference between "either A is true or A is false" and "either A is

What is the difference between "either A is true or A is false" and "either A is true or ~A is true?" I have an intuitive sense that they are two very different statements but I am having a hard time putting why they are different into words. Thank you.

Perhaps I could add something here too—and perhaps it will be useful: You are right that there is a difference between the two statements that you offer, and the difference has become more significant with the rise of many-valued logics in the 20th and 21st centuries.

If one says, “A is either true or false,” then there are only two possible values that A can have—true or false. But if one says, “either A or not-A is true,” then there might be all sorts of values that A could have: true, false, indeterminate, probably true, slightly true, kind of true, true in Euclidean but not Riemannian geometry, and so on. The first formulation allows only one alternative to “true” (namely, “false”), but the second formulation allows many alternatives. The second formulation does indeed require that at least A or not-A be true, but it puts no further restrictions on what other values might substitute for “true.” (For example, perhaps A is true, and yet not-A is merely indeterminate.)

The advantage of sticking to the first formulation (often called the principle of bivalence) is that it forces us to reason from propositions that describe what is definitely so or not so, and as a result, we can actually prove things. (After all, if we were to give reasons that were neither true nor false, then our reasons would seem to end up proving nothing. Imagine, for example, someone saying, “I believe this conclusion for a good reason, but my reason is neither true nor false.” Moreover, if the conclusions we wanted to prove were also to turn out to be neither true nor false, then they would remain unprovable; what would it mean, one might ask, to “prove” the untrue?) Considerations of this sort led Aristotle to believe that scientific knowledge always depended crucially on propositions of argument that had to be true or false.

On the other hand, there are many situations in life where our ideas are so vague and indefinite that the best we can say is that a particular proposition seems somewhat true, or true to a certain degree, or true for the most part. (For example, Aristotle held that propositions of ethics were sometimes only “true for the most part.” In the Middle Ages, a number of logicians wanted to use “indeterminate” as a truth value, in addition to true and false, and in the 20th and 21st centuries, logicians have experimented increasingly with the idea that there could be many truth values, in addition to true and false. As a result, there are now various systems of many-valued logic, including so-called fuzzy logic, which assigns numerical degrees of truth to different propositions.)

All the same, the principle of bivalence still plays a fundamental role even in systems of many-valued logic, albeit at a higher level. (The second formulation that you have cited is now termed the law of excluded middle, though before the development of many-valued logics, the two formulations would have amounted to the same thing.)

Specifically, many-valued logics assign different values to various propositions and then draw conclusions from the assignments. (For example, if A is “somewhat true,” then one can conclude that A is not “entirely false.”) Nevertheless, such systems always rely on at least two crucial assumptions: (1) the propositions in question, such as A, do indeed have the assigned values or they do not, and (2) these propositions cannot both have the assigned values and not have them. The first assumption is the principle of bivalence all over again, though at the “meta” level (meaning that it applies, not to A, but to statements about A, that is, to the statements of A’s truth value). And the second assumption is the traditional law of contradiction. (For more on the law of contradiction, you might see Questions 5536 and 5777.)

In other words, the propositions treated by a system of many-valued logic are typically imprecise and indefinite, and what a many-valued logic then does is allow us to talk in a precise and definite way about the imprecise and indefinite. To achieve this result, however, the system’s own statements must be definite, and to achieve coherence, the system’s own statements must also be noncontradictory. By contrast, if one were to relax these restrictions on the system, then all one would get would be an indefinite discussion of the indefinite, or an incoherent discussion. And if this last result were all that one hoped to achieve, then there would be no need to build the system in the first place. Instead, just leap from bed in the morning, and without drinking any tea or coffee, start talking. If you are like me, you will then arrive almost instantly at the appropriate level of grogginess.

I presume that you're using the formula "~ A" to abbreviate "It is not true that A" rather than "It is false that A." If my presumption is wrong, then this response may not answer your question. Where A is some proposition , I see no difference between "It is not true that A" and "It is false that A": Every proposition that isn't true is false, and every proposition that isn't false is true. However, the same doesn't hold if A is, instead, some sentence . For a sentence can fail to be true without being false. To use an admittedly controversial example : the self-referential sentence "This sentence is not true" is neither true nor false, because the sentence fails to express any proposition in the first place (including the proposition that the sentence isn't true!). Any false sentence is not true, but a sentence can fail to be true without being false. But perhaps you meant to use the formula "~ A" to represent rejection or denial of the sentence or proposition A. Some philosophers...

If there could be a counter-argument against a premise, does that make the

If there could be a counter-argument against a premise, does that make the premise false and the argument unsound?

No. The mere possibility of a counter-argument (i.e., "there could be a counter-argument") doesn't imply that the premise is false or that an argument containing the premise is unsound. The counter-argument itself must have a true conclusion in order to guarantee that the premise against which it's a counter-argument is false.

Every sound argument has a true conclusion (although the converse doesn't hold), so if there exists a sound argument against a particular premise, then the particular premise is false. Often, however, the very soundness of that counter-argument will be a matter of controversy.

No. The mere possibility of a counter-argument (i.e., "there could be a counter-argument") doesn't imply that the premise is false or that an argument containing the premise is unsound. The counter-argument itself must have a true conclusion in order to guarantee that the premise against which it's a counter-argument is false. Every sound argument has a true conclusion (although the converse doesn't hold), so if there exists a sound argument against a particular premise, then the particular premise is false. Often, however, the very soundness of that counter-argument will be a matter of controversy.

I'm having a difficult time determining if a certain math problem should be

I'm having a difficult time determining if a certain math problem should be classified as using Formal or Informal Logic. Here it is: 1. ALL except 2 of my pets are dogs. 2. ALL except 2 of my pets are cats. 3. ALL except 2 of my pets are birds. Q: How many pets do I own? A: 2 or 3 So, while it's obvious why the answer could be 3, it's not obvious how it could be 2 as well. The reason why is because the phrase "All" could be zero, which would represent an empty set. And, of course, I could own pets other than the ones mentioned (fish / lizards). So, knowing that, we can substitute that example back into the original problem as follows: I own two, pets, which are both fish. All except 2 of my pets are dogs, which in this case, is equal to zero. So, the set of dogs can possibly be an empty set. So, anyways, I was wanting to know if the puzzle itself could be considered "formal", or is it informal because most people would mean "All" to at least equal one, and we add that assumption in there?

I interpret you as asking this: Why do we find it puzzling or counterintuitive that statements 1–3 are true in the case in which you own exactly two pets, neither of which is a dog, a cat, or a bird? Is it because we assume that "all" implies "at least one"?

Those are empirical, psychological questions whose answers I don't know. But I do think it's worth distinguishing between what "all" logically implies and what "all" conversationally implies. (You might have a look at the SEP entry on implicature.)

On the one hand, the statement "All intelligent extraterrestrials are extraterrestrials" had better be true, and its truth had better not depend on the existence of intelligent extraterrestrials. So I think there's good reason to deny that "all" logically implies "at least one."

On the other hand, someone who owns no dogs and who says "All my dogs have their shots" has said something odd or misleading, even if true. So I think there's good reason to say that, at least sometimes, "all" conversationally implies "at least one."

On the interpretation of "all," see also Question 5834.

I interpret you as asking this: Why do we find it puzzling or counterintuitive that statements 1–3 are true in the case in which you own exactly two pets, neither of which is a dog, a cat, or a bird? Is it because we assume that "all" implies "at least one"? Those are empirical, psychological questions whose answers I don't know. But I do think it's worth distinguishing between what "all" logically implies and what "all" conversationally implies. (You might have a look at the SEP entry on implicature .) On the one hand, the statement "All intelligent extraterrestrials are extraterrestrials" had better be true, and its truth had better not depend on the existence of intelligent extraterrestrials. So I think there's good reason to deny that "all" logically implies "at least one." On the other hand, someone who owns no dogs and who says "All my dogs have their shots" has said something odd or misleading, even if true. So I think there's good reason to say that, at least sometimes, "all"...

On April 10, 2014, in response to a question, Stephen Maitzen wrote: "I can't

On April 10, 2014, in response to a question, Stephen Maitzen wrote: "I can't see how there could be any law more fundamental than the law of non-contradiction (LNC)." I thought that there were entire logical systems developed in which the law of non-contradiction was assumed not to be valid, and it also seems like "real life" suggests that the law of non-contradiction does not necessarily apply to physical systems. Perhaps I am not understanding the law correctly? Is it that at most one of these statements is true? Either "P is true" or "P is not true"? or is it that at most one of theses statements is true? Either "P is true" or "~P is true"? In physics, if you take filters that polarize light, and place two at right angles to each other, no light gets through. Yet if you take a third filter at a 45 degree angle to the first two, and insert it between the two existing filters, then some light gets through. Based on this experiment, it seems like the law of non-contradiction cannot be true in...

Because the present questioner refers to my reply to Question 5536, I'll chime in here to clarify what I said there.

My point was about the fundamentality of LNC. I wrote, "I can't see how there could be any law more fundamental than the law of non-contradiction (LNC)." I gave the following reason: "Let F be any such law. If the claim 'F is more fundamental than LNC' is meaningful (and it may not be), then it conflicts with the claim 'F isn't more fundamental than LNC' -- but that reasoning, of course, depends on LNC." So that's why no law could be more fundamental than LNC, because LNC would need to be true before (in the sense of logical priority) the claim that some other law is more fundamental would even make sense.

If someone can make sense of the claim that some law is more fundamental than LNC, I'm all ears.

Because the present questioner refers to my reply to Question 5536 , I'll chime in here to clarify what I said there. My point was about the fundamentality of LNC. I wrote, "I can't see how there could be any law more fundamental than the law of non-contradiction (LNC)." I gave the following reason: "Let F be any such law. If the claim 'F is more fundamental than LNC' is meaningful (and it may not be), then it conflicts with the claim 'F isn't more fundamental than LNC' -- but that reasoning, of course, depends on LNC. " So that's why no law could be more fundamental than LNC, because LNC would need to be true before (in the sense of logical priority) the claim that some other law is more fundamental would even make sense. If someone can make sense of the claim that some law is more fundamental than LNC, I'm all ears.

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