I'm inclined to say that no tautologies are ever logically equivalent, but only because no sentences are ever logically equivalent. I take it that any tautology is a sentence in some language, as opposed to the proposition expressed by that sentence. Indeed, the etymology of the term implies that a tautology is a sentence characterized by the repetition of words: Greek tauto ("the same") + logos ("word"). An example is the English sentence "All red things are red." Unlike sentences, propositions don't contain words, so tautologies can't be propositions, strictly speaking. I interpret "logically equivalent" to mean "matching in truth-value at every possible world." Two things match in truth-value at every possible world only if both things exist at every possible world. But no sentence -- no item of any language -- exists at every possible world, because the very language of the sentence might never have existed: all language is contingent. Therefore, no two sentences are ever...

I thank William Rapaport for his comment. I'll just point out that the claim two sentences (or propositions) are logically equivalent if and only if they have the same truth values (no matter what truth values their atomic constituents, if any, have) seems to imply the following odd consequence. Take two sentences lacking atomic sentential constituents: "Snow is white" and "Obama was born in Hawaii." Both sentences are true (sorry, birthers), but isn't it odd to hold that the two sentences are logically equivalent ? Granted, they're materially equivalent, but that's just a technical way of saying that they in fact have the same truth-value. Something stronger seems required for genuine logical equivalence, which is why I prefer the definition I gave above. Fortunately, some standard textbooks do define it in that stronger way.

Hello, Joe. Except for the difference in truth-values, I see no interesting difference between your first two descriptions. "Necessarily true, but not false" is a redundant description, because any proposition that's necessarily true is ipso facto not false and in fact couldn't have been false (indeed, that's what "necessarily true" means in this context). The second description is also redundant, because any necessarily false proposition is ipso facto not true and couldn't have been true. As far as I can see, probability has nothing to do with those two descriptions. In these cases, the word "necessarily" is being used in what's often called a modal sense. The second pair of descriptions does concern probability. Any proposition that's "probably but not necessarily false" is more likely than not false but not certain to be false: the proposition has a probability greater than 0 but less than 0.5, on a scale of 0 to 1. Any proposition that's "probably but not necessarily...

I recommend starting with the SEP entry on the topic, available here . There's an article not cited by the entry that may be relevant because it takes a skeptical view of the standardly accepted solution to one of the paradoxes: "Zeno's Metrical Paradox Revisited," by David M. Sherry, Philosophy of Science 55 (1988), 58-73. Sherry argues that the standardly accepted solution "defuses" the paradox but is too ad hoc to count as a "refutation" of Zeno's reasoning.

Thank you for the argument for that claim, but your reasons for it do not particularly interest me. Wow. How very philosophical. We philosophers aren't interested in each other's reasons, after all. Now, am I supposed to be interested in the reasons you're giving for your claims? I've given a numbered-step argument for a claim about S, in particular, that you've been denying, viz. (32). You've responded by referring me to work that you say bears on a sentence that you say is "like" S. I'm not asking you to take my say-so. If (32) is false, then there's a mistake in my (1)-(8) or (24)-(32). Surely a professional logician can tell us what it is. You, Richard, claim to have established something by your (24)-(28), but your (24) and (25) both lack justification: (24) If (V) is a sentence-type, then no token of (V) expresses a proposition. (No token of a meaningless sentence expresses a proposition.) The justification you provide simply doesn't justify (24). You haven't...

Paracomplete theories, which are perhaps the most popular these days (though I do not myself incline to them) would reject (2), (10), and (16). There are many other choice points as well. Yeah, yeah. There are also theories that (claim to) reject (7), (13), and (22). Are we to think that those moves are even remotely plausible? Maybe they're plausible and not plausible, or neither? Perhaps you mean something like... No, I mean exactly this: (S) Either S is meaningless, or else S is false. (24) S is meaningless. [Repetition of (8), already established] (25) Either S is meaningless, or else S is false. [From (24) by disjunction introduction] (26) If S is a sentence-type, then S is a meaningful sentence-type or S is a meaningless sentence-type. [If P, then (P & Q) or (P & not Q).] (27) If S is a meaningful sentence-type, then (f) the token of it labeled "S" above is meaningful. (28) Not (f). [From (24)] (29) If S is a sentence-type, then S is a meaningless sentence-type. [From...

Okay, I'll defend my main claims in detail. Following Charles Parsons, you offered the following Strengthened Liar sentence: (S) Either S is meaningless, or else S is false. I claimed, and still claim, that S is meaningless. Reasoning: (1) If S is meaningful, then (a) S expresses the proposition that: Either S is meaningless or else S is false. [What else could S express if it were meaningful?] (2) If (a), then (b) S is true or S is false. [Bivalence for propositions] (3) If S is meaningful, then (b). [From (1), (2)] (4) If S is true, then S is true and not true. [Strengthened Liar reasoning] (5) If S is false, then S is true and not true. [Strengthened Liar reasoning] (6) If (b), then S is true and not true. [From (4), (5)] (7) Not (b). [From (6) by contradiction] (8) S is meaningless (i.e., not meaningful). [From (3), (7)] Before you say that (8) commits me to S by disjunction introduction, recall that I distinguish tokens from types. What (8) commits me to is the meaningful and...

Yes, those were my words. But the argument I was attributing to you...was NOT supposed to be: If (S) is meaningless, then "(S) is false" is meaningless. Then you can see why I was misled by what you actually wrote. In any case, the argument you say you meant to attribute to me contains a premise I deny, namely, that the first disjunct in (S) is "clearly meaningful." I've been claiming that (S) is meaningless, and I deny that any part of (S) says that (S) is meaningless, just as I deny that the sentence "This sentence is meaningless" (or any part of it) says of itself that it's meaningless. I understand that you wish to resist the claim that the conjunction of those two things (which happens to be (T) itself) ... You know I deny the claim in parentheses. Every conjunction has truth-conditions, but (T) has no truth-conditions. Neither do "dog" and "cat," and we don't produce a conjunction by writing an ampersand between those two inscriptions. The answer cannot just be, "Well,...

This is turning out to be an easy way of upping our response-counts! Surely I can name that sentence (S) if I so choose? I wouldn't say "surely." (After all, in bygone days we thought "Surely there's a set corresponding to any well-defined predicate we choose.") It may turn out that in this case, on pain of contradiction, the name attaches only to the token and not to the type. ...if you allow that "(S) is false" is false if (S) is meaningless... I trust that you too allow it -- indeed, that you insist on it. ...then it is hard to see why one would ever regard (S) as meaningless: It is a disjunction of meaningful disjuncts. Here you're simply repeating the claim I've been denying. (Furthermore, I don't see what how the antecedent of your conditional supports the consequent.) That, not the reasoning you took me to be attributing to you, is why I was assuming you would deny that "(S) is false" is meaningful. The reasoning I took you to be attributing to me is the...

Thanks, Richard, for your replies. Nice colloquy we're having. I hope anyone else is interested! Is there or is there not a sentence that is the disjunction of (S') and the sentence "(S) is false"? There is, and we can token it, but not by way of the sentence-token that you labeled "(S)" in your example. That's been my point all along: two type-identical sentence-tokens can be such that one is meaningful and the other isn't. The context in which a sentence-token is uttered can deprive it of propositional content. I say that's not surprising given other things we know about language. It won't do to respond "But I'm talking only about the syntax !" because you can't generate a liar paradox without assuming things about the truth-conditions of particular strings of words. Presumably you would also regard the sentence "(S) is false" as itself meaningless, on the ground that (S) is ... Goodness, no! That would be terrible reasoning. To say of a meaningless sentence-token that it's...

Me too, but that was my point: Despite appearances, (S'), which I endorse, isn't the first disjunct in (S). Similarly, despite appearances, the Epimenides sentence doesn't assert of itself that it's false. It follows that the meaningfulness of a sentence-token depends on more than the string of words it contains, but that result isn't surprising in light of other things we know about language.

PLEASE NOTE: Professor Maitzen's responses here and below were originally offered in colloquy with Professor Heck, who has since chosen to remove his contributions. [Alexander George on 6/6/2014.] But then it is a simple step of disjunction introduction to (S) itself. This simple step works only if (S) is the disjunction of (S') and "(S) is false," each of which disjuncts is meaningful. But if (S) is meaningless, then (S) isn't the disjunction of two meaningful disjuncts, and in particular it's not the disjunction of (S') and "(S) is false." I agree that this response to the Strengthened Liar implies that the meaningfulness of a sentence-token won't always be facially obvious. That implication seems less dire than the implications of some other responses.

I think you're right to suspect that the Liar (or Epimenides) sentence, "This sentence is false," is meaningless, i.e., that the sentence fails to express a proposition. But I wouldn't say that the sentence is meaningless because it's self-contradictory, like the sentence "God exists and doesn't exist." The latter sentence is surely false , in which case the sentence expresses a (false) proposition and hence isn't meaningless. If the Liar sentence is meaningless, then it doesn't assert of itself that it's false (because it doesn't assert anything), and therefore one of the premises used to generate the Liar paradox is false. Some philosophers have said that the Strengthened Liar sentence, "This sentence is not true," blocks such a solution to the paradox, on the grounds that a meaningless sentence is not true . The proper reply, I think, is to agree that a meaningless sentence is not true but to deny that the Strengthened Liar sentence asserts of itself that it's not true (again, on the...

I'd answer your three questions as follows. (1) Very important. (2) No: There are lively disagreements in logic concerning particular issues, and there may be few if any issues in logic on which everyone agrees. (3) Some philosophers say that different situations call for different kinds of logic. For what it's worth, I disagree: I'm not persuaded that there are any situations to which standard (or "classical") logic doesn't apply.

The last two of your three questions suggest this: We can't properly regard some law P as true (rather than merely as a guideline for thought) unless there's some more fundamental law Q that enables us to see that P is true. But presumably Q must also be something we properly regard as true, in which case your suggestion implies an infinite regress: there must be some more fundamental law R that enables us to see that Q is true. Likewise for R, and so on. This infinite regress may be a good reason to reject your suggestion. Why must our properly regarding P as true depend on there being some more fundamental law? In any case, I can't see how there could be any law more fundamental than the law of non-contradiction (LNC). 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.

I would caution against inferring from 'The principle of noncontradiction is an abstraction' to 'The principle of noncontradiction doesn't exist naturally or non-naturally'. A number of philosophers, and maybe an even larger number of mathematicians, think that at least some abstract objects must exist -- and exist non-naturally. It may be that the principle of noncontradiction is among those abstract objects. You may find this SEP entry on the topic helpful.

Translating the argument into symbolic terms quite literally, we get this: 'If F, then R. Not F. Therefore, not R.' That form of argument does indeed commit the fallacy of denying the antecedent: the premises don't logically imply the conclusion; the truth of the premises doesn't logically ensure the truth of the conclusion. The first premise says that F is sufficient for R; it doesn't say that F is necessary for R. In that case, R can obtain even if F fails to obtain. My hunch is that you're interpreting 'If F, then R' as 'R if and only if F': you're interpreting the conditional as a biconditional , i.e., as the claim that F is both necessary and sufficient for R. 'R if and only if F' and 'Not F' together imply 'Not R'. Your interpretation is understandable, because conversationally we often do intend to assert a biconditional when we use conditional language. A parent's 'If you clean your room, you can watch TV' usually means 'You can watch TV if and only if you clean your room...

It would be helpful to know which answers on the site you're referring to, but I'll take a stab at your question anyway. The only difference between your formulations (a) and (b) is the second disjunct in each, so I'll focus on that. I presume A is some statement. What's the difference between "Not A is true" and "A is not true"? I'm not sure there's always a difference. Depending on the system of logic or semantics, "Not A is true" can mean merely (i) "Whatever truth-value (if any) A has, it's not the value true ." Or it can mean (ii) "A is false " in systems in which every statement is true or false. The difference between (i) and (ii) is sometimes important, such as when we're dealing with the classic Liar sentence (L) "This sentence is false." One might say that L is not true and yet not false either: one might say that L is neither true nor false.

You wrote: (i) "It seems that for classical logic to make sense of a contradiction in such a way that it leads to explosion...it must define what it means to hold a contradiction in a particular way" and (ii) "[W]ouldn't it be defined in some arbitrary way that forces us into the 'explosion' scenario?" Regarding (i): If the assertion "A is true and A is false" means anything, then surely it implies that A is true and implies that A is false. I can't think of another way to construe the assertion. Are you suggesting that a conjunction doesn't imply each of its conjuncts? Regarding (ii): How is it arbitrary to infer the truth of A and the falsity of A from the assertion "A is true and A is false"? Again, I can't think of another way to understand the assertion. As far as I know, paraconsistent logicians tend to object to inferring B from (A or B) and not-A: they point out that the inference relies on the implicit assumption that not-A rules out A, an assumption they reject.