Ok, I'll bite. A tautology, as I understand it, is a proposition that is true, and necessarily so. A contradiction is a proposition which is necessarily false, and a contingent proposition is one that can be true or false. I see no reason to depart from the standard stuff any further than that. Why would I? Depart from the standard stuff, that is.

Wittgenstein made a point about tautologies being what defines a statement in terms of logic. — Shawn

So what did he say?

as I understand it, is a proposition that is true, and necessarily so. A contradiction is a proposition which is necessarily false, and a contingent proposition is one that can be true or false. — Arcane Sandwich

The evening star is the morning star. Isn't it a tautology and also contradiction, but a true statement?

The evening star is the morning star. Isn't it a tautology and also contradiction, but a true statement? — Corvus

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 !!

No problem, happy to help :up:

1) ∃x∃y(Ex ∧ My ∧ (x=y)) — Arcane Sandwich

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.

Wittgenstein's point (in the Tractatus) was that tautologies don't say anything about how things are. They do not tell us about the world.

Like "The cat is on the mat or the cat is not on the mat". It says nothing.

When I studied Hume as an undergrad, I was introduced to the idea of a priori truths, the standard example being that a bachelor is unmarried - because bachelors are unmarried as a matter of definition. To be told of someone, 'John is a bachelor' is also to be told that John is umarried; saying John is unmarried after having been told he is a bachelor is a tautology.

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 truth—it 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'.

1) ∃x∃y(Ex ∧ My ∧ (x=y)) — Arcane Sandwich


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. — Banno

Yup, nice catch. We could use the uniqueness quantifier ∃! (alternatively, ∃₌₁) to make your formula a bit easier on the eyes.

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. — Wayfarer

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 – it’s a fact about language and the world that we have to learn. In John’s case, we’re using “a priori” as a rather confusing substitute for “known ahead of time” or “known as a background belief” or something similar. Perhaps that’s why it looks jejune: It doesn’t 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, it’s not even a tautology. It doesn’t follow the form of “Either p or ~p”. It isn’t “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?

Yep Since your are talking about individuals, the iota operator could also be used. it is stipulated as ιxEx: The unique x such that x is E.

Then we can write (ιxEx=ιyMy).

Then we can write (ιxEx=ιyMy). — Banno

That looks ugly as hell.

Nice extension. I agree that 's "John is a bachelor" is not a tautology in the sense Wittgenstein is using.

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.

You think? It reads as "The x that is E is the y that is M". I would have said that was quite neat.

Sure. You could probably also read it as: "There is a unique x, such that x is the evening star; there is a unique y, such that y is the morning star, and x is identical to y".

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.

The evening star is the morning star. Isn't it a tautology and also contradiction, but a true statement? — Corvus

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.

The referent could be a concept in your mind — Relativist

Well, no. The referent is Venus.

Why are the parentheses necessary in this case? That's another thing that makes such a formula really ugly. Off topic, I know, but since you brought it up, I though I'd ask.

The sentence could be read either way.

Here's another that spotlights semantic ambiguity:

Whether they exist or not, dragons breathe fire.

Why are the parentheses necessary in this case? — Arcane Sandwich

I guess they are not.

Wanna hear my theory about that? When you were formulating it, you sensed on some level that the iota operator makes formulas look ugly, so you tried to "beautify" it with the parentheses.

But it just makes it uglier.

Obviously all of this is subjective though. Matters of taste and all that jazz.

The sentence could be read either way. — Relativist

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.

Whether they exist or not, dragons breathe fire. — Relativist

A change of topic. From "Dragons breath fire", you can conclude that something breaths fire. You cannot conclude that there are dragons.

I just did an x & v from another web site to get the "iota" character, changed the cap letters and didn't bother removing the already present parenthesise.

My theory has been refuted then.

Here's how I would write it, if I had any say on the syntax:

ιxιy((Ex & My) & (x=y))

"For some unique x, for some unique y, x is the evening star, y is the morning star, and x is identical to y."

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." — Relativist

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?

Well, use "Hesperus" and "Phosphorus" instead.

So, what are your thoughts about tautologies apart from the standard stuff said here? — Shawn

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: