So in what interpretation is 0=0 false?

Is Masque not saying that 0=0 is trivial and thus tautological

It's possible. I missed that interpretation.

However, "not tautological if not trivial" is logically equivalent to "trivial if tautological", and not "trivial thus tautological".

The main purpose of this thread is that I would like to understand how tautology and equality are related.

Yeah granted

Not 0 = 0.

{0...0} which really is quite a difference.

Using predicate calculus:

( (x)Fx V ~(x)Fx) is a tautology; whereas ( (x) Fx V (x) ~ Fx) is not*.

Equality or inequality in mathematics 0=0 or 0ǂ1, is related to number theory and arithmetic theorem(s). Mathematical formulae are different beasts than language or predicate calculus. But perhaps I'm misinterpreting here - sort of out of my element and others may know more or explain better.


* See Oxford Companion to Philosophy, p 866 (hardback edition). On this same page is an entry for tar-water in which the statement is made, "tar-water was probably not entirely useless." The same could be said for tautologies.

Rewriting this a few times:

1) P is a correct statement. P is not a tautology if it is not trivial.
2) P is a correct statement. If P is not trivial, then it not a tautology.
3) P is a correct statement. If P is a tautology, then P is trivial.

This is not a proper rendering of the concept. There are tautologies that are non-trivial to check. Imagine an expression of logical equivalence involving hundreds of variables.

Equality (as in, equations) are a general way of expressing a relationship between two collections of symbols that have a value (1+1=2) or are the same mathematical object (A = B as sets).

A tautology is a more restrictive concept, pertaining to the underlying logical structure. I usually don't think of logical statements being written with equalities. For example, "not(P and Q)" is logically equivalent to "(not P) or (not Q)" which makes "not(P and Q) \equiv (not P) or (not Q)" a tautology (it's true regardless of the truth values of P and Q), but I don't think I would ever write "not(P and Q) = (not Q) or (not Q)" where I really mean that to be an equal sign and not used because I don't have a different symbol to type it in its position.

Yeah, it is a form of algebraic wordplay. I came across this years ago and had not seen any real practical use except as a parlor trick.

i.e. 0 = 0, unless 0 is a multivariable algebraic symbol that can represent any number, word, phrase, or even one of Aeschylus' missing plays.

This doesn't invalidate any idea or rule out the ability to use it in practical mathematics. Especially as current capabilities push into the ability to describe coordinate systems in five-dimensional hyperspace (as an example. there are a lot of false avenues in this particular area of exploration. and some vertigo.) or really, this is something that belongs strongly in the realm of speculative geometry.

As for figuring out grocery bills, it goes back to the designation of parlor trick. Sure, it works, but you don't need it.

https://en.wikipedia.org/wiki/First-order_logic

Which appears to answer my OP. Although in that case, are there any theorems which are not tautologies?

There should be at least one. Mathematical or formal language applications of Cartesian duality perhaps?

Although I think that posit loses substance when the absence of tautology is in the expression, not the equation. Bit murky but I've found I can make do with that bit of cobbling.

The claim 0=0 is the same as the claim, "0⊆0 and 0⊆0",

which is of the form "p and p",

which is not true under all values of p (it's false when p is false), therefore it's not a tautology.

The entire point though on the original thread is that although an important identity is universally true calling it a tautology is rather nasty in a discussion of people that are not talking in terms of formal logic. Because when people claim something is a tautology they see it as trivial or repetitive/ superfluous (the fatal murder example), pretending to be important when it isnt etc.


So 0=0 is eventually equivalent to a remarkable identity like eg

https://en.wikipedia.org/wiki/Lagrange%27s_identity

But people will laugh if you write 0=0 and offer it in a discussion and they will congratulate you - say at high school but not in a PhD defense room lol- if you offer an identity eg 2^n= sum(n!/k!/(n-k)!, k from 0 to n)

A theorem is also essentially a tautology then?

https://en.wikipedia.org/wiki/Tautology_%28logic%29

What is the point of talking in these terms, to achieve what, in the context of the original thread?


Just come out and say what was the original meaning of tautology you used and why you did that there? The more honest you are the more respect you will have from me if that counts for anything. This is why i typically will refuse to horse around in threads like others do and make my jokes explicit and somehow obvious that they are jokes based on context or follow up comments.

Well, you just made a tautological statement about something you say is not a tautology.

Which is precisely my point.

I'll add that if the question were about the claim "0=0 or 0≠0", I think that would be a tautology because I think its form would be:

(p and p) or (¬p or ¬p)

Where is the tautological statement I made? In my first sentence I used the set-theoretic definition of equality (subset in both directions).

I could be wrong, I haven't studied set theory / logic beyond intro-level undergrad. Is this question less clear-cut / more open to controversy than I thought?

Edit: maybe the form isn't p & p, maybe it's p & q where q is the claim 0⊇0. But that too is not a tautology.

It's my understanding that every correct mathematical proof of a true theorem, when written in symbolic logic form, amounts to a tautology. And it's assumed that every correct proof can theoretically be reduced to symbolic logic although it's not generally done in practice.

This is why they can get algorithms to pump out new mathematical "theorems". They are tautology generating algorithms. We may have no idea what the "theorems" are saying but we know they are true.


PairTheBoard

The reason i had "issues" (lol) with it is this really; (ie your comment in the original thread is either a criticism that has no merit really based on the examples i offered to illustrate better what i meant or a joke that you never suggested it was a joke or a nitty reference that doesnt add to the discussion itself but may add to the forum this thread here that is a plus of course lol. In other words i have problems with all these possible angles so in a way i am forced to conclude i have a universal problem with its usage to describe what i said lol. Thankfully the problem i have is removed by the opportunity we have to discuss these things, eg here, in a variety of different interpretation levels and improve our encyclopedic knowledge lol)

"The word tautology was used by the ancient Greeks to describe a statement that was true merely by virtue of saying the same thing twice, a pejorative meaning that is still used for rhetorical tautologies. Between 1800 and 1940, the word gained new meaning in logic, and is currently used in mathematical logic to denote a certain type of propositional formula, without the pejorative connotations it originally possessed.

In 1800, Immanuel Kant wrote in his book Logic:

"The identity of concepts in analytical judgments can be either explicit (explicita) or non-explicit (implicita). In the former case analytic propositions are tautological."

Here analytic proposition refers to an analytic truth, a statement in natural language that is true solely because of the terms involved.

In 1884, Gottlob Frege proposed in his Grundlagen that a truth is analytic exactly if it can be derived using logic. But he maintained a distinction between analytic truths (those true based only on the meanings of their terms) and tautologies (statements devoid of content).

In 1921, in his Tractatus Logico-Philosophicus, Ludwig Wittgenstein proposed that statements that can be deduced by logical deduction are tautological (empty of meaning) as well as being analytic truths. Henri Poincaré had made similar remarks in Science and Hypothesis in 1905. Although Bertrand Russell at first argued against these remarks by Wittgenstein and Poincaré, claiming that mathematical truths were not only non-tautologous but were synthetic, he later spoke in favor of them in 1918:

"Everything that is a proposition of logic has got to be in some sense or the other like a tautology. It has got to be something that has some peculiar quality, which I do not know how to define, that belongs to logical propositions but not to others."

Here logical proposition refers to a proposition that is provable using the laws of logic.

During the 1930s, the formalization of the semantics of propositional logic in terms of truth assignments was developed. The term tautology began to be applied to those propositional formulas that are true regardless of the truth or falsity of their propositional variables. Some early books on logic (such as Symbolic Logic by C. I. Lewis and Langford, 1932) used the term for any proposition (in any formal logic) that is universally valid. It is common in presentations after this (such as Stephen Kleene 1967 and Herbert Enderton 2002) to use tautology to refer to a logically valid propositional formula, but to maintain a distinction between tautology and logically valid in the context of first-order logic (see below)."

heehaww,

Not the formal logic, but the English.

"...therefore, it's not a tautology."

Which will go a long way towards understanding why most people can't even define what a tautology is, let alone make sense of it. Of no circumstance.

I was taught that a tautology is such that when you write the truth table, the statement is true under any combination of truth values of the variables. Is that not a widely accepted definition?

Comparing the two examples I gave has given me an idea of how to think about tautologies (which may or may not be correct): a claim is a tautology if it says nothing (loosely speaking). Saying "0=0 or 0≠0" is saying nothing, whereas saying 0=0 is taking a stance and saying something (albeit not much).

Edit: I'd like to hear others weigh in on the following:

The meaning is that part in blue which I quoted in the OP, which is a purely logical definition.

Once more I think it's shades of grey, a thing much of logic, and maybe philosophy as whole, misses. Basically a "tautology" is something intuitively meaningless. Then you have the meaningless statements more difficult to spot. Then you have for example real science, with something lasting to provide. Was Einstein's theory on general relativity a tautology? After all it was only x = y = x

Tautological logic is at its most useful as an origin point or valuation peg in finding derivations and expansions using the logical statement as a rule.

In actuality, there are no rules. You as an individual will always be positioned somewhere within your location observing and expressing observations. You yourself are a tautology within your perception and without.

So, no, they aren't meaningless until you attribute to them the fluidity which a fixed position provides.

Which is essentially the philosophical definition of tautology. A correct logical expression to teach to graduate mathematical students is to recognize that both definitions have purpose in both philosophy and mathematics.

x = y = x indeed, at c but frozen lattices we are not.

You still need an axiom for this, apparently. Saying that a set is not equal to itself would surely violate:

https://en.wikipedia.org/wiki/Axiom_of_extensionality

so, in this case, interpreting p as false is not permitted.

Apparently not, although I thought it was.

Statements can be tautologies, but not proofs, I think.

Maybe worth also observing that tautology is a semantic (true/false) rather than syntactic property.

A tautology has to be true, that's a huge credit.

So much in this world is claims that x = y when it is not.

One can say that it's true that there are 180 degrees in a triangle. And they would be right. Unless someone chose a different axiomatic system, in which case they would be wrong. And then the statement "There are 180 degrees in a triangle" may not be true in the same sense that "x=x" is true. If that's the case, then there are theorems that cannot be understood as tautologies.

You could also switch the meaning of x on either side of "="

Then x could be /= x