What are 'tautologies'?
Wittgenstein made a point about tautologies being what defines a statement in terms of logic. This could also be said of logic itself with well-formed formula examples. I think also to a certain degree or analogously, mathematics can be said to be tautological in terms of mathematical logic. Some proofs in mathematics are hard and non-trivial so the example isn't hard and fast.
So, what are your thoughts about tautologies apart from the standard stuff said here?
Thanks, comments welcome.
So, what are your thoughts about tautologies apart from the standard stuff said here?
Thanks, comments welcome.
Comments (90)
The evening star is the morning star. Isn't it a tautology and also contradiction, but a true statement?
My opinion (and I could be wrong) on this classic problem is that it depends on how we formalize the problem. Consider a first case:
1) ?x?y(Ex ? My ? (x=y))
Which means: There is an x, and there is a y, such that x is the evening star, and y is the morning star, and x is identical to y. In this case, "to be the evening star" is not the same predicate as "to be the morning star", because the predicate letter "E" is not identical to the predicate letter "M". So, we are not dealing with a tautology here, it's instead a contingent proposition (that happens to be true).
Now consider a second case:
2) ?x?y((x=e) ? (y=m) ? (x=y))
Which means: There is an x, and there is a y, such that x is identical to the evening star, and y is identical to the morning star, and x is identical to y. Here, the evening star is identical to the morning star, because the individual constant "e" is identical to the individual constant "m". But this is also a contingent proposition (which happens to be true), not a tautology.
In order to get a tautology, we need to consider a third case:
3) ?x((x=e) ? (x=m))
Which means: There is an x, such that if x is identical to the evening star, then x is identical to the morning star. Here, the evening star is identical to the morning star, because both of them are identical to x. This case is indeed a tautology, not a contingent proposition.
Great analysis on the points. Will go over on them again when time permits, and will get back to you if there are any points to clarify. Many thanks !!
Notice that this allows that there might be more than one evening star and more than one morning star?
?x(Ex?Mx??z(Ez?z=x)??w(Mw?w=x)) might work.
Like "The cat is on the mat or the cat is not on the mat". It says nothing.
I always found this unsatisfactorily deflationary. I think it's both interesting and significant that there are things we can know a priori. Obviously not so much in such jejune cases as John's marital status. But the principle of non-contradiction amounts to more than simply definitional truthit undergirds all reasoning. Likewise, mathematical truths (at least in a classical sense) seem to be discovered rather than invented, suggesting they reveal something real about the structure of thought or even reality, and often lead to or predict unexpected empirical discoveries. So there's something a little world-weary about using the terminology of 'taulogies'.
Yup, nice catch. We could use the uniqueness quantifier ?! (alternatively, ?[sub]=1[/sub]) to make your formula a bit easier on the eyes.
What is it exactly that we are supposed to know a priori, in this case? That bachelor means unmarried male? But that is not a priori at all its a fact about language and the world that we have to learn. In Johns case, were using a priori as a rather confusing substitute for known ahead of time or known as a background belief or something similar. Perhaps thats why it looks jejune: It doesnt really touch the issue of what genuine a priori knowledge might consist of.
Is it a tautology, though? If Wittgenstein (via @Banno) is right, then no, its not even a tautology. It doesnt follow the form of Either p or ~p. It isnt self-evident. It does tell us something about the world, both the world of language and the world of logic, of what can now be extensionally substituted.
An interesting question is, what makes logical truths (appear to be) self-evident, whereas definitional truths must be learned? And the perennial favorite: Self-evident to whom?
Then we can write (?xEx=?yMy).
That looks ugly as hell.
That step would be to invoke Quine's first dogma. Analyticity reduces to synonymy, and so is not about how the world is, but about the language we use.
I just don't like how it looks with the operator at both sides of the "=" sign, but whatever floats your boat. I'm not shaming your "logical kinks", if that's even a thing.
It's a semantic issue. The nouns have a referent. The referent could be a concept in your mind, or it could be the actual object that exists in the world.
Assume "Evening star" and "morning star" both refer to an object in the world. In that case, they are referring to the same object - so it's semantically equivalent to saying "The evening star is the evening star."
But "Evening star" and "morning star" could both just refer to your mental concepts "the point of light I see in the evening (or morning)" - the concepts refer to the context of your respective perceptions.
You could even be inconsistent, and treat one as the concept, the other as the object.
Well, no. The referent is Venus.
Here's another that spotlights semantic ambiguity:
[I]Whether they exist or not, dragons breathe fire.[/i]
I guess they are not.
But it just makes it uglier.
Obviously all of this is subjective though. Matters of taste and all that jazz.
Well, no. There is a difference between Venus and the concept of Venus. Venus is a planet. The concept of Venus, whatever it is, is not a planet. So "Venus" does not refer to the concept of Venus.
Quoting Relativist
A change of topic. From "Dragons breath fire", you can conclude that something breaths fire. You cannot conclude that there are dragons.
Here's how I would write it, if I had any say on the syntax:
?x?y((Ex & My) & (x=y))
According to ChatGpt, Venus is not a star. It is a planet. The sun is a star. Stars shine their own light. Planets don't. Planets reflect the light from the sun.
Hence, the morning star could the sun? What would the evening star be? Under this clarification is "Morning star is evening star." still a tautology? Or is it downright false?
From Witty & co, iirc, 'tautologies' are information-free, necessary repetitions (syntax) and 'logic', constituted by tautologies and rules of inference, is a consistency metric (systematicity) that is strictly applicable to grammatical (semantic) as well as mathematical (formal) expressions. Thus, I think of logic as sets of scaffoldings for excavating knowledge from nature and/or building (new) knowledge with nature that is, making explicit maps of the terrain (i.e. possibilities) which are constitutive of the terrain (i.e. actuality (e.g. Witty's "totality of facts")). Nonetheless, imo even more fundamental than tautologies, contradictions are a priori modal constraints on ontology (i.e. the instantiation of logic, ergo mathematics, semiosis & pragmatics (Spinoza, A. Meinong, U. Eco, Q. Meillassoux ...)) which entail 'impossible worlds', or necessary non-actuality.
Awesome post friend. :up:
It would be correct to say:
[I] "the sentence: 'dragons breath fire' is true whether or not dragons exist" [/i]
because:
-If dragons exist, then "dragon" refers to these existing animals.
-If dragons don't exist, then "dragon" refers to a fictional creature.
In the original sentence, "Whether they exist or not, dragons breathe fire", there's just one referrent - not 2.
Then dragons exist and are fictional creatures.
?(x)(x is a dragon and x is fictional)
or
?(x)(x is a dragon ? x is fictional)
This would imply that the set of all dragons includes all the real dragons and all the fictional creatures so-named. Some members of the set are said to breathe fire. We can't really say that "some dragons breathe fire" because fictional things don't actually breathe.
.
Why would you think fictional creatures do not breath? Or are you now saying that there are two levels of ontology, stuff that exists and stuff that is actual?
?(x)(x is a dragon & x is fictional & x breaths fire)
Looks fine to me.
Breathing is a real world activity by real world creatures. A fiction can't do this.
[Quote]are you now saying that there are two levels of ontology, stuff that exists and stuff that is actual?[/quote]
IMO there's one ontology. Dragons are either real-world creatures, or they are concepts residing in minds.
And yet it is true that dragons breath fire.
Ergo, fictional creatures can breath.
Quoting Relativist
Take a closer look at what is going on. We can set "exists' as a quantifier, ?(x)f(x), which just says that something has the property f. Then we can happily talk about dragons breathing and still say that they are fictional.
Fictional creatures are found in fiction, in the real world. Sure, you will not meet one in the street.
On this account dragons can breath fire. On your account, it is false that dragons breath fire.
The fiction of dragons includes "breathing fire". But fictions still can't engage in the real world activity.
Do you understand my objection to the original statement:
[I]
Whether they exist or not, dragons breathe fire.[/i]
I'm not saying you can't make sense of it. But strictly speaking, when a noun appears once in a sentence, it has a single referent. Fictional creature and actual creature are 2 different referents.
Sure.
Quoting Relativist
You can't say of something that does not exist, that it breaths fire. Just showing you one way to make sense of that.
What difference would they make for the statement?
"The morning star is the morning star." sound like a tautology. But it is not a tautology, when the subject means the planet Venus, and the predicate means the star Sun. Hence would it be the meaning of the words dictates on the sentence being tautology or not?
For another example, "Today is today." It sounds like tautology, but the subject means the name of a newspaper, and the predicate refers to a day in a month. Then they are not tautology.
Quoting Corvus
Is Phosphorus also the star or planet?
I don't. I don't see what's contradictory about it.
I have a name, and I also go by a nickname. If someone said "FJ is the same person as Flannel Jesus", there's no contradiction in that. Why would the evening / morning star be different? I don't see the contradiction.
It's also not necessarily a tautology, not to a person that doesn't know it's the same object they're calling both of those things.
The reason that the Morning star is morning star is because it is only visible in the mornings.
But the reason that the evening star is the evening star is because it is only visible in the evenings.
It follows,
"Morning star is evening star" is the same as "Morning star is not evening star."
Saying "Morning star is evening star" has the same meaning as
"Morning star is evening star and Morning star is not evening star."
A ^ ~A is a contradiction.
"The morning star is the evening star." is also a tautology. The morning star and evening star both refer to Venus. Hence it has the same meaning as "The morning star is the morning star.", which is a tautology.
It's easy to make a paradox out of false statements. Corvus is a human and he's not a human. If I allow myself false statements, then voila, I can produce a paradox at will.
That sounds like a strawman. You are suddenly talking about Venus, when the point of the replies was about the morning star and evening star. They may refer to Venus, but the reason they are called the morning star and evening star is the time when it is visible.
You are making up either a strawman, or you don't seem to know the point of the argument here.
How is it a strawman? You literally said "The morning star and evening star both refer to Venus."
I only highlighted it for you to let you know about the strawman. Venus was not the main point in the argument. It is mentioned to explain why the statement is a tautology i.e. they all point to the same reference viz. Venus.
Because the argument offered an explanation, calling it "making up paradoxes" was strawman. The argument didn't have to mention it, but it was just trying to be more informative.
Axiom 1: All hyenas play chess.
Axiom 2: Joe lives with two hyenas.
Statement: Joe lives with two chess players.
The statement is a tautology regardless of whether the axioms are true.
If you read the posts carefully, it is clear why it is a contradiction and why it is a tautology. All the steps of the inferences are based on the rules of logical proof.
But blatantly asserting they are a bunch of paradoxes, doesn't make sense.
I don't think anything yous aid is clear at this point.
If you read about the rules of logical proof, then maybe you would understand them? It is elementary basic rules viz. rules of elimination, assumption and addition in the textbooks. I don't think explanations on the details of the rules are the scope of the OP.
The best track record? Well, when 100s of blind men were shouting out the elephant must look like a rubber pipe standing up after feeling one of its legs accusing one normal sighted man's description of it, what could the sighted man could have done apart from saying - well good luck to youz mate? :)
Nah, I am not going to talk about logic with you again. You need to learn it yourself. I think I said enough on the tautology and contradiction. Nothing more to add to it.
Mr denying the antecedent, I think I agree.
Nothing to do with that. It was about pointing out your premise was irrelevant to the conclusion.
Obviously you forgot everything about it.
If you bring in irrational premises to the conclusion in the argument, then it doesn't get accepted in higher standard of logic. That's nothing to do with denying antecedent. You are quoting something you saw on the internet, and making your slogan for logic.
Good point. Logical truths are true in every interpretation, so they are supposedly safe from Quine's criticism. One consequence of that is the rejection of de re modality.
It might be worth taking a close look at Reference and Modality post Naming and Necessity. There is a tension here, to be sure, and Quine was correct that folk will try to smuggle Aristotelian-style essences in on Kripke's back. I'm pretty sure we can have our cake and eat it, since rigid designation and referential opacity occur in different contexts. It's a worthy topic.
Thank you for your kind comment, Arcane Sandwich. As you rightly pointed out, I am not a logician at all. I have read only a book or two on Elementary Logic books a long time ago. So I don't talk much about logic usually unless the topic requires logical explanation by its nature for clarification.
I tend to try to rely on my own reasoning rather than the formal methods on my logical reasoning in most cases. However when the topic is about something I read from the textbook, I also try to utilize them accordingly. They are all basic elementary level, of course.
OK, FJ, going to back to your initial point, you claimed my argument is made up of a bunch of paradoxes. If you could point out exactly which part of my argument are paradox and explain the reasons why they are paradoxes, then I will try to clarify them with you, if you would like me to.
I don't think I said that. Here's the quote of mine I managed to find: "It's easy to make a paradox out of false statements. Corvus is a human and he's not a human. If I allow myself false statements, then voila, I can produce a paradox at will."
I didn't say it's made of a bunch of paradoxes, I said you produced an apparent paradox, trivially, by just making false statements and claiming they're true by definition.
That is not false statements in the rule of logical proof. If I could recall it correctly, you can make up the molecular statements from atomic statements using the connectives for assumption under the rule of addition, elimination, MP and MT etc. These steps are needed to come to the required proof of conclusion.
We were talking about the Morning star and evening star. Which part is false statement in the argument?
I thought your point was the argument is from the paradoxes made up randomly with the false statements or something like that. So if contradiction is introduced for the steps of logical proof, then you claim it is a paradox, because false statement is made up and added. To my understanding, that was not a claim from someone who knew anything about logic.
Reductio ad absurdum is the most used method of logical proof from the ancient times. You call it making up false statement from paradox didn't make sense to me at all.
What does this mean? Have you lost your mind? You are so far out of touch with the English language that we literally cannot have a rational conversation.
That is a simple plain English. It means what it says.
Do you mean the essay in From a Logical Point of View?
Yep. New thread, maybe. Although given the present state of the forums it would probably turn into yet another thread about Heidegger and god.
Seems like someone doesn't understand Bunge and science.
Nice. Do you have the actual hard copy?
Oversimplifying, Quine fusses over the sentence "Cicero has six letters", and how we can get from that to "Something has six letters". The thing that has six letters is a word, not Cicero. He suggests treating modal sentences in a similar way, as substitutionally opaque. He want to do this becasue he dislikes the supposed ontological implications of other approaches.
But I don't think possible world semantics has the dire consequences he envisions, and specifically, the sort of essentialism it invokes is ontologically inert.
This not by way of an argument but an outline.
Well, a paperback reprint. But it is signed by WV Quine as well.
Quoting Banno
Understood. It'll be fun to look it over and then read some Kripke, as you suggest. I never took Kripke to be talking about essences per se; if I use the rigid designator 'that apple' I am certainly not claiming to reveal anything ontologically important about it. But it does get complicated with names, and it's fair to ask whether Kripke isn't backing into a doctrine about essences when it comes to possible-world semantics. Anyway, I'll review.
Yep. The mention of essence is a response to recent interest hereabouts, mostly amongst a small group of Thomists. A discussion of necessity here will probably the obliged to address less than clear ideas of essence.
Our concern here is that tautology is often thought of as necessity. Filling that out is a topic in itself.
I'm interested in essence, and I'm not a Thomist. Understand, Banno, that there's a holiday in Argentina called "Spanishness Day", as in, "The Day of the Spanish Essence". I do not celebrate it myself. But understand that the discussion about essences has a lot to do with who I am, and the circumstances that I live in. I don't take any of this lightly. That's why I don't agree with Analytic philosophers when they speak so lightly about essences.
Peace be with you, friend.
Then I wasn't referring to you.
Gavagai, and all of that jazz?