Jesse

yguy:
I find truly awe-inspiring the extent to which reasonable people are able to deny the obvious. In this very thread, we have an intelligent poster claiming that 1+1=2 is a tautology

Jesse:
If you're making a statement about what is true within a particular axiomatic system, it is.

yguy:
tautology n.

2. Logic. An empty or vacuous statement composed of simpler statements in a fashion that makes it logically true whether the simpler statements are factually true or false; for example, the statement Either it will rain tomorrow or it will not rain tomorrow.

By this definition, it appears to me that "1+1 is either equal or to not equal to 2" is tautological, but that 1+1=2 is not. If it is, than all expressions of arithmetic addition are equally so, are they not? By extension then, all true equations are tautological.

Again, it depends whether you're viewing the equations in a purely "axiomatic" way or not. If I say "given the following axioms of peano arithmetic (insert axioms here), the theorem '1+1=2' is provably true", that would be a tautology, since it's logically impossible to be false. But again, I think that if by "1+1=2" you are referring to our mental model of what the various symbols of arithmetic "really mean", it's not a tautology.

yguy:
and another one claiming that flowers make themselves beautiful by being beautiful.

Jesse:
I haven't seen the quote you're referring to, but I suspect you're taking it out of context.

yguy:
It sounds like he never said "flowers make themselves beautiful by being beautiful", or anything like it (actually, the fact that he responded to your summary with cricket-chirping suggests he found it so ludicrous that it didn't deserve a serious response; honestly yguy, did you really interpret his response to mean 'yes, that would be an accurate summary of my views'?). Basically, I think he's saying that flowers with certain physical characteristics were more likely to attract pollinators than others, and so those characteristics were favored by natural selection. We may happen to find those characteristics "beautiful", but there's no need to assume there is some kind of objective platonic truth about whether something is "beautiful" or not (as you assume there's an objective platonic truth about whether a given act resulting in someone's death is 'murder' or not).

Jesse:
Your "independent of reality" comment is too vague. It's true that deciding whether something is necessarily true should not depend on knowing in advance whether it happens to be true in our reality.

yguy:
Where do you get that idea?

Well, for example, deciding whether or not it's necessarily true that the earth is round does not depend on knowing in advance whether it's round or flat in reality, it just depends on knowing whether it's logically possible that it could be anything other than round (if the answer is no, then we can conclude it's round in reality, but we didn't have to 'know it in advance').

Jesse:
But once you can show that something is necessarily true, that means it absolutely must be true in all possible realities, including this one.

yguy:
No, that just means you - and whoever else you can convince that you are correct - think it is.

It's possible for one's reasoning process to be unsound, I suppose. But just as you believe there is an objective truth about whether an act is murder or not, I would say there's an objective truth about whether various statements of fact like "1+1=2" and "the earth is round" are necessarily true or just contingently true.

Jesse:
So, your conclusions about whether a truth is necessary or not do have a bearing on what you should believe is true in the real world. Most theistic philosophers want God to be a necessary being that would exist in all possible realities rather than a contingent being who just happens to exist in our reality, so they obviously agree with me that this is an important distinction, even if you don't.

yguy:
I don't.

Does that mean you think there's no distinction between "things it is within God's power to do", like make a flat earth, and "things which it is not within God's power to do", like make 1+1=3? Even if you're not sure that God couldn't make 1+1=3, are you absolutely sure He could?

Anyway, if I can't convince you the distinction is meaningful than I suppose this discussion is finished. However, in future you should be aware that in the context of philosophical discussions most people do make a distinction between necessary and contingent truths, and therefore that when you compare your certainty that God exists to your certainty that 1+1=2, there's a lot of potential for confusion there unless you explain that you don't make this distinction.

yguy

I don't think that fits the definition cited, since if the axioms are false, then so, presumably, is 1+1=2. If it were a tautology, it wouldn't matter whether they are false or not.

Did they give themselves those physical charachteristics?

There is no need to assume such a thing, because it is obviously true. A beautiful woman may be a witch on the inside, but that she has physical beauty is obvious.

But you aren't answering the question: where did you get the idea that the general principle is true?

I mean, what it says to me is that deciding whether something is necessarily true is irrelevant to whether it is actually true. It's like saying you make it true by means of the logical process.

Of course. What I'm getting at is that such necessary truths are not subordinate to what we consider logical thought processes.

Yes, there is a distinction - but I don't see how that relates to the discussion.

Nope.

Jesse

yguy:
I don't think that fits the definition cited, since if the axioms are false, then so, presumably, is 1+1=2. If it were a tautology, it wouldn't matter whether they are false or not.

But if you're taking a purely axiomatic approach to anything, the question "are the axioms true or false" is meaningless. The axioms don't mean anything, they simply provide formal rules for generating new symbol-strings. We can interpret the symbols to mean something within the context of a mental model we have, but once we bring these models in we are no longer using a purely axiomatic approach (in particular, Godel's theorem shows that no matter what axiomatic system we use to try to formalize our mental model of arithmetic, there will always be statements which are true of the model but which are not provable within the axiomatic system).

Jesse:
It sounds like he never said "flowers make themselves beautiful by being beautiful", or anything like it. Basically, I think he's saying that flowers with certain physical characteristics were more likely to attract pollinators than others, and so those characteristics were favored by natural selection.

yguy:
Did they give themselves those physical charachteristics?

In this theory, there would be random variations in the flower population, and those variants which happened to be better at attracting pollinators would have a greater chance of surviving and passing on those traits to the next generation. Speaking in a sloppy way I guess you could say the flowers "figured out" how to attract pollinators better or something, but it would be more accurate to just say that the characteristic arose by a series of random variations which were selected by the external environment.

Jesse:
We may happen to find those characteristics "beautiful", but there's no need to assume there is some kind of objective platonic truth about whether something is "beautiful" or not (as you assume there's an objective platonic truth about whether a given act resulting in someone's death is 'murder' or not).

yguy:
There is no need to assume such a thing, because it is obviously true. A beautiful woman may be a witch on the inside, but that she has physical beauty is obvious.

"Obvious" to who? It might not be obvious to an alien who is not sexually attracted to humans. Do you think there is an "objective truth" about which of any pair of objects is "really" more beautiful, even if different people have different opinions about it? Is there an objective truth about whether Picasso's "Guernica" is more or less beautiful than Van Gogh's "Starry Night"?

Jesse:
Your "independent of reality" comment is too vague. It's true that deciding whether something is necessarily true should not depend on knowing in advance whether it happens to be true in our reality.

yguy:
Where do you get that idea?

Jesse:
Well, for example, <snip>

yguy:
But you aren't answering the question: where did you get the idea that the general principle is true?

Just from the meaning of "necessary" and "contingent", I guess. If something is necessary it's true in all possible realities, while if something is contingent it's true in some possible realities but not others. Assuming our reality is only one of many "possible" ones, it follows that observing whether something is true in our reality does not in itself tell you anything about whether it is necessarily true or contingently true.

Jesse:
But once you can show that something is necessarily true, that means it absolutely must be true in all possible realities, including this one.

yguy:
No, that just means you - and whoever else you can convince that you are correct - think it is.

Jesse:
It's possible for one's reasoning process to be unsound, I suppose. But just as you believe there is an objective truth about whether an act is murder or not, I would say there's an objective truth about whether various statements of fact like "1+1=2" and "the earth is round" are necessarily true or just contingently true.

yguy:
Of course. What I'm getting at is that such necessary truths are not subordinate to what we consider logical thought processes.

Yes, I never claimed that all necessary truths could be discovered to be necessary by what we consider to be logical thought processes. But plenty of them can, it seems.

Jesse:
Does that mean you think there's no distinction between "things it is within God's power to do", like make a flat earth, and "things which it is not within God's power to do", like make 1+1=3?

yguy:
Yes, there is a distinction - but I don't see how that relates to the discussion.

It seems to me that "impossible for an omnipotent being to make true" (like making 1+1=3, according to many theists) is the same as "necessarily false", while "possible for an omnipotent being to make true, but not true in the real world" (like making the earth flat) is the same as "contingently false", just by the definition of "omnipotent".

yguy

If that's the case, it appears the term "tautology" is meaningless in such cases, since the potential for falsifiability seems to be integral to the definition. IOW, given "He is either brave or not brave", the two alternatives are either true or false. You are saying the postulates are neither, therefore they cannot be tautological under the definition given previously.

The proof of 1+1=2 is based on meaningless axioms?

Most humans aren't sexually attracted to horses either, but most of us can tell a swaybacked nag from a thoroughbred.

There are perhaps men who think Molly Yard is more attractive than Nicole Kidman - just as there are people who even now think Hitler was a good guy.

I have no idea.

Jesse

Jesse:
But if you're taking a purely axiomatic approach to anything, the question "are the axioms true or false" is meaningless.

yguy:
If that's the case, it appears the term "tautology" is meaningless in such cases, since the potential for falsifiability seems to be integral to the definition. IOW, given "He is either brave or not brave", the two alternatives are either true or false. You are saying the postulates are neither, therefore they cannot be tautological under the definition given previously.

The definition you gave earlier said explicitly that it didn't matter whether the premises of a tautological statement are true or false:

For example, "if all humans are reptiles, and all reptiles can fly, then humans can fly" would be a tautology, as I understand it. Do you understand "tautology" differently?

Jesse:
The axioms don't mean anything

yguy:
The proof of 1+1=2 is based on meaningless axioms?

We can bring our own interpretation of what various terms "mean" to an axiomatic system, but the whole point of using axiomatic reasoning is that you don't have to know anything about what it means, it's a purely formal procedure for generating new symbol-strings from the axioms and the rules of inference. Douglas Hofstadter's book Godel Escher Bach, which is a good introduction to the concept of formal rules vs. meaning in mathematics and how Godel's theorem relates to both, gives a simple example of a meaningless formal system:

Axiom string: MI

Inference Rule #1: If you possess a string whose last letter is I, you can add on a U at the end.

Inference Rule #2: Suppose you have Mx. Then you may add Mxx to your collection.

Inference Rule #3: If III occurs in one of the strings in your collection, you may make a new string with U in place of III.

Inference Rule #4: if UU occurs inside one of your strings, you can drop it.

That's the whole system. From the initial axiomatic string in your collection, you can add other strings to your collection by using the rules of inference in any combination. For example, here is the "derivation" of MUIIU:

(1) MI -- axiom
(2) MII -- from (1) by rule #2
(3) MIIII -- from (2) by rule #2
(4) MIIIIU -- from (3) by rule #1
(5) MUIU -- from (4) by rule #3
(6) MUIUUIU -- from (5) by rule #2
(7) MUIIU --from (6) by rule #4

The idea of formalizing an area of math is that deriving new theorems also just becomes a matter of manipulating the appropriate symbols according to a set of rules and axioms, so there can be no ambiguity about whether a proof is valid or not, given the axioms. We may have interpretations about how to map the symbols to models in our head, but such interpretations are irrelevant to the procedure of proving new theorems within the axiomatic system.

Jesse:
"Obvious" to who? It might not be obvious to an alien who is not sexually attracted to humans.

yguy:
Most humans aren't sexually attracted to horses either, but most of us can tell a swaybacked nag from a thoroughbred.

Here you've switched from aesthetic judgements about beauty to objective issues like whether a horse is "thoroughbred" or whether it is "swaybacked". Again, the question is whether there is an objective reality about which of two people or two horses are more beautiful, or if it's a subjective taste (perhaps a taste that would nevertheless be the same in virtually all humans).

Jesse:
Do you think there is an "objective truth" about which of any pair of objects is "really" more beautiful, even if different people have different opinions about it?

yguy:
There are perhaps men who think Molly Yard is more attractive than Nicole Kidman - just as there are people who even now think Hitler was a good guy.

Yes, but you seem to be sure there is an objective truth about morality, so that people who think Hitler's actions were morally acceptable are objectively wrong. Are you equally certain that people who think Molly Yard is more beautiful than Nicole Kidman are objectively wrong, or is it just that their personal tastes are different from most other people's?

By the way, did the rest of my previous post convince you that it's meaningful to distinguish between facts that are necessarily true or false and facts which are contingently true or false? And that knowing whether a fact is true in the real world does not in itself answer this question?

yguy

Yes, but I see no room in the definition for premises which are neither true nor false.

While the truth of the syllogism in its entirety is independent of the truth of the premises, the truth of the conclusion, that humans can fly, is not. Otherwise, it appears to me that all valid syllogisms are tautologies.

If you don't know what the axioms mean, how can you possibly apply them to a proposition to test it? IOW, the axioms have to have meaning from the perspective of one operating within the system.

Is a Palamino prettier than an Appaloosa? At that point it's a matter of perference; but a swaybacked, bedraggled looking horse of any breed is physically ugly.

They are objectively wrong. The fact that there are people who think ugly is beautiful doesn't make ugly beautiful.

Within your own frame of reference I suppose it makes sense, but you failed to respond to this:

"I mean, what it says to me is that deciding whether something is necessarily true is irrelevant to whether it is actually true. It's like saying you make it true by means of the logical process."

Kimpatsu

Godwin's Law states that however mentions Hitler first loses the debate.

Jesse

yguy:
While the truth of the syllogism in its entirety is independent of the truth of the premises, the truth of the conclusion, that humans can fly, is not. Otherwise, it appears to me that all valid syllogisms are tautologies.

No, because I started it with "if humans are reptiles...". If I had just made the first premise of the syllogism, "all humans are reptiles", then the statement would be false. But since I stated it in the form of an if-then, the truth only depends on whether the conclusion would follow if the premises were true.

yguy:
If you don't know what the axioms mean, how can you possibly apply them to a proposition to test it? IOW, the axioms have to have meaning from the perspective of one operating within the system.

If your proposition is solely about whether a theorem is provable within an axiomatic system, you don't have to assign "meaning" to the theorem itself. For example, "the theorem MUIIU is provable within the axiom system I posted earlier" is a true statement. But it is pretty pointless to talk about what is provable within an axiomatic system unless you have some prior idea about what the axioms "mean" in terms of some model in your head.

Jesse:
Again, the question is whether there is an objective reality about which of two people or two horses are more beautiful, or if it's a subjective taste (perhaps a taste that would nevertheless be the same in virtually all humans).

yguy:
Is a Palamino prettier than an Appaloosa? At that point it's a matter of perference; but a swaybacked, bedraggled looking horse of any breed is physically ugly.

To all humans that may be true, but that just shows that all humans have a broadly similar set of aesthetic preferences. Is it totally inconceivable to you that another species--perhaps even an alien species--could have aesthetic preferences so different than ours that even when virtually all humans agree that one of two animals is uglier than the other, virtually all members of this species might have the opposite preference?

It also seems odd to me you could say there is an "objective truth" about whether a swaybacked, bedraggled horse is uglier than a purebred, but no objective truth about whether a Palamino is uglier than an Appaloosa. Isn't that like saying that for some moral decisions, there is an objectively right choice and an objectively wrong one, but for other moral decisions it's just a matter of personal preference?

Jesse:
Yes, but you seem to be sure there is an objective truth about morality, so that people who think Hitler's actions were morally acceptable are objectively wrong. Are you equally certain that people who think Molly Yard is more beautiful than Nicole Kidman are objectively wrong, or is it just that their personal tastes are different from most other people's?

yguy:
They are objectively wrong. The fact that there are people who think ugly is beautiful doesn't make ugly beautiful.

But how do you know that "ugly" and "beautiful" are traits that there is an objective truth about in the first place? I mean, are you a total platonist who believes that every adjective we use points to some "objective truth" in this sense? Are some people objectively cooler than others? Are some foods objectively yummier than others? Are some baby animals objectively cuter than others? Are some books objectively more boring than others?

Jesse:
By the way, did the rest of my previous post convince you that it's meaningful to distinguish between facts that are necessarily true or false and facts which are contingently true or false? And that knowing whether a fact is true in the real world does not in itself answer this question?

yguy:
Within your own frame of reference I suppose it makes sense, but you failed to respond to this:

"I mean, what it says to me is that deciding whether something is necessarily true is irrelevant to whether it is actually true. It's like saying you make it true by means of the logical process."

I think I addressed the point you were making there elsewhere:

And also:

Tell me, since you seem to believe there is an objective truth about whether it is or is not in God's power to do something, what would be wrong with identifying this with the necessary/contingent distinction like so:

truth about our reality which it would have been within God's power to make false (for example, 'the earth is round') = contingent truth

truth about our reality which it would not have been within God's power to make false (perhaps '1+1=2') = necessary truth

Does observing whether a fact is true in reality tell us anything, in itself, about whether it would have been within God's power to make it false? If we can use our own logical reasoning to deduce that, if God obeys the laws of logic, he cannot have made a statement like 1+1=2 false, does that mean we made it true by means of the logical process?

yguy

Dyslexics of the world untie!

Kimpatsu

Do you have anything worthwhile to add, Yguy?