Advanced Search

Is it possible for two tautologies to not be logically equivalent?

Is it possible for two tautologies to not be logically equivalent?

The term "tautology" has no established technical usage. Indeed, most logicians would avoid it nowadays, at least in technical writing. But when the term is used informally, it usually means: sentence (or formula) that is valid in virtue of its sentential (as opposed to predicate, or modal) structure. I.e., the term tends to be restricted to sentential (or propositional) logic.

It is clear that Rapaport is assuming the sort of usage just mentioned: "a tautology is a 'molecular' sentence...that, when evaluated by truth tables, comes out true no matter what truth values are assigned to its 'atomic' constituents". Hence, on this definition, "Every man is a man" would not be a "tautology". Which is fine. It's logically valid, but not because of sentential structure.

It is all but trivial to prove, as Rapaport does, that all tautologies are logically equivalent. In fact, however, it is easy to see that Rapaport's proof does not depend upon the restriction to sentential logic. One can prove (as he of course knows) by exactly the same argument that all "valid" sentences are logically equivalent. (All logic texts prove this. It is, for example, general law (12) on p. 64 of Warren Goldfarb's Deductive Logic, which is the text we use at Brown.) The argument goes through so long as "valid" means "true in every interpretation" and "equivalent" means "have the same truth-value in every interpretation". In particular, it doesn't matter what is meant by an interpretation (at least as long as the interpretations are classical).

A couple other remarks on the discussion.

First, when Rapaport says that "two [interpreted] sentences...are logically equivalent if and only if they have the same truth values (no matter what truth values their atomic constituents, if any, have)", he is talking not about logical equivalence in general but about "truth-functional" equivalence, since only sentential constituents can have truth-values. And "Snow is white" and "Obama was born in Hawaii" are not, of course, truth-functionally equivalent. The sole atomic constituent of each sentence is that sentence itself. (Constituency is so defined as to make it a reflexive relation, trivially.) And an interpretation can perfectly well assign these two sentences different truth-values. So this is just a case of P not being equivalent to Q.

Second, as Rapaport notes, there is some controversy whether the basic notion here should apply to sentences (or formulas) or to propositions. But if it applies to syntactic items (as I would prefer), then the sentences and formulas have to be regarded as "interpreted", i.e., as having fixed meanings, or at least the "logical constants" have to be so regarded. Otherwise, indeed, no sentence will be "always true". But if the meanings are held fixed, then, for many purposes, the difference makes no difference, since the notions are inter-definable: A sentence is (logically) true if the proposition it expresses is (logically) true; a proposition is (logically) true if there is a (logically) true sentence that expresses it. The remaining philosophical issue is which notion is more fundamental.

Finally: For lots of interesting material on the word "tautology" and the history of its use in logic, see Burton Dreben and Juliet Floyd, "Tautology: How Not To Use a Word".

Someone deliberately advances a fallacious argument in an attempt to advance a

Someone deliberately advances a fallacious argument in an attempt to advance a cause she considers just. For example, she may treat contraries as if they are contradictories and thus commit a fallacy of false alternatives. Are there any living philosophers who defend the use of "noble fallacies" or "noble fallacious arguments" (and is there a better term for this kind of thing)? And are there any contemporary philosophers who criticize or condemn the practice, including when it is practiced by people who are on "their side" regarding social and political issues?

Fascinating inquiry!

I do not recall articles or books explicitly on when it is good to commit fallacies, but you might find of interest the literature on the ethics of lying. There is a great deal of philosophical work on when, if ever, it is permissible to lie, and this probably would include work on when it is permissible to deliberately engage in fallacious resigning. One primary candidate for justified deception involves paternalism in extreme cases, e.g. in a medical crisis when a parent has only five minutes to live and she asks you whether her children survived an accident, and you know that her five children were killed, is it permissible to lie by claiming, for example, you are not sure? Or, to make the case more in line with your question, would it be permissible for you to not disclose the truth about her children if it could only be done by you equivocating or begging the question or committing the fallacy of the undistributed middle? For terrific work on the ethics of lying with great attention to detailed cases and theories of meaning, see Lying and Deception by Thomas Carson.

I personally know of only one real world case in which one or more philosophers may have defended the use of fallacious reasoning. The details are a bit sketchy, but here is what I recall. In the 1980s I was at an American Philosophical Association meeting when Dan Brock then from Brown University along with some other panelists discussed the advisory role they had with the president of the USA and congress on medical decisions. The panelists spoke about the following dilemma they faced, perhaps more than once. The panelists all agreed that some policy X was optimal in terms of ethics, politics, the law etc but they believed that the persuasive*reasoning behind judging X to be optimal was highly complex and involved levels of abstract reasoning that would make the justification of X hopeless. However, the panel was aware of *or they created themselves a justification for X that was from a philosophical point of view very weak, and yet effective with the public. As I recall, the panel did not make explicit how they handled such cases this may be because the cases were recent and politically sensitive. But this is the best I can do in terms of relating a real world case of when philosophers have deliberated about the ethics of advancing arguments that are weak *possibly fallacious when they deem it the best or only option available to them.

For an interesting exchange by philosophers of when it is permissible to promote or not challenge beliefs or positions that the philosophers believe to be false, compare Iris Murdoch in her last book which more or less defends Platos notion of the golden lie with the harsh criticism of Simon Blackiburn.

What is the difference between a conclusion that is "necessarily true, but not

What is the difference between a conclusion that is "necessarily true, but not false" vs. "necessarily false, but not true"? They seem the same to me or is the answer based on probability? In the same light, what is the difference between "probably not necessarily false" and "probably but not necessarily true"? Thank you, Joe

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 true" has a probability of more than 0.5 but less than 1, on a scale of 0 to 1. In these cases, the word "necessarily" is being used in what's often called an epistemic sense, where it means "certainly".

Have Zeno's paradoxes of motion actually been satisfactorily solved? Physicists

Have Zeno's paradoxes of motion actually been satisfactorily solved? Physicists and mathematicians I've read on the matter seem to regard them as no longer important, but never explain convincingly (for my money) why they're not still important. Have philosophers said anything interesting about them recently? Could you either succinctly explain how they've been solved or point me in the direction of good recent discussions?

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.

In paradoxes such as the Epimenides 'liar' example, is it not sufficient to say

In paradoxes such as the Epimenides 'liar' example, is it not sufficient to say that all such sentences are inherently contradictory and therefore without meaning? Like Chomsky's 'the green river sleeps furiously', it's a sentence, to be sure, but that's all it is. Thanks in advance :)

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 established that V is meaningless, and I doubt it can be done. All that can be established in that regard is my (23).

(25) If (V) is a sentence-type, then no token of (V) expresses a true proposition. [~p --> q |- ~(p --> q & r)]

The justification you provide for (25) isn't even a theorem. Your reasoning wasn't "more compact" than mine; it was just sloppier.

...you are also committed to denying that (V) is a sentence-type, by the reasoning I gave.

Sloppy reasoning doesn't commit me to anything.

Next you say:

If (V) is not a meaningful type, then what else could it mean for the type (V) to be meaningless except that: No token of (V) expresses a proposition.

I suspect more sloppiness. From the claim that V isn't a meaningful type, it doesn't follow that V is a type that's meaningless. It follows merely that V is either not a type, or not meaningful, or both: i.e., my (23). Certainly "No token of V expresses a proposition" doesn't imply that V is a meaningless sentence-type. Proof: Let V be the word-type "while."

Can you define logical validity? I'm engaged in a debate on the subject, with a

Can you define logical validity? I'm engaged in a debate on the subject, with a friend, whom will not easily accept anyones word on the matter, so i would ask that you perhaps post your credentials? thank you for you time and effort!

If you mean "valid argument," that's typically defined as an argument such that there is no interpretation of its premises and conclusions under which all the former are true and the latter is false.

How important is the study of logic in philosophy, independent of any one

How important is the study of logic in philosophy, independent of any one particular philosopher or school of philosophy? Is 'logic' considered a 'neutral' subject about which 'everyone' agrees? or are there some contentious issues about what 'kind' of 'logic' applies in different kinds of situations?

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.

On what grounds are the three classic laws of thought rendered 'true'? Is there

On what grounds are the three classic laws of thought rendered 'true'? Is there a more fundamental law which enables us to see the law of identity, the law of excluded middle and the law of non-contradiction as true? If not, how can we claim that they are anything more than guidelines for thought?

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.

Hello All,

Hello All, My question is if someone makes an argument using conditional statements is the argument necessarily deductive? Basically the person claims because I am using If . . . .then clauses then that makes my argument deductive by default. I was under the impression that some conditional arguments can still be inductive based on the context of the argument. So if I claim not all conditional arguments are deductive am I correct or incorrect?

An argument using conditional statements can be an argument of any kind (it depends on what other statements are used). There is one kind of well known argument--modus ponens--that uses a conditional statement and a premise stating the antecedent of the conditional. That argument is deductive.

What is the metaphysical nature of logic itself? When we refer to a basic

What is the metaphysical nature of logic itself? When we refer to a basic principle of logic (such as non-contradiction) are we referring to something that exists which we call “non-contradiction”? Or is it simply an abstraction that doesn’t exist naturally or non-naturally?

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.

Pages