Arithmetic and tautology

[PyramidHead]

As far as whether they are tautologous, I think so. Tautologies are dependent on an understanding of the syntax and semantics of their underlying language. Two statements that are equivalent linguistically have the same referent. So, while there may not be anything apparent about the symbols that make up "3+4" to indicate that they are equivalent to the symbol for "7", by the syntax and semantics of their underlying language (arithmetic) they have the same referent: 7. In the same way, someone who didn't speak English wouldn't know that "grass is grass" is a tautology without knowing that "is" denotes identity. Either way, a speaker of English would not learn anything new from "grass is grass," and a person who knew arithmetic like we know English wouldn't learn anything new from "3+4=7." I think both are tautologies, in that sense (trivially but necessarily true statements).

If you ask me, "what is grass," and my response is, "grass," you're gonna respond, "no shit," rightfully so. What else would something be if not the very thing it is? If you ask me, "what is meant by saying the streets were running Crimson red," and I respond, "it means the streets were running Crimson red," then you know I'm right even though you yourself might not what it means.

Even though the summation of the referents to the numerals 3 and 4 have the same referent as the numeral 7, you don't know they sum to that unless you have learned that.

But if I didn't know the English words "is" and "means", I wouldn't know that the first two statements were true either. In tautologies, the truth of the statement comes purely from the meaning of the terms, and nothing in the external world. If you don't know what the terms mean and how they are related, they won't seem true at first glance.

To illustrate my point, is the sentence "Jestem tak wysoki jak ja" a tautology?

 

[Speakpigeon]

I would guess that what was meant was: given the set of natural numbers and the definition of "+" as an internal law then 3+4=7 is a tautology.

Assuming that's the correct interpretation of the OP and assuming we use Wittgenstein's definition of "tautology" then yes 3+4=7 is a tautology.

However, we may believe and claim that "3+4=7" somehow applies acroos the board to reality, as opposed to just the purely theoretical and therefore fictional world of mathematics. If so, it should be noted that as such this is no longer a tautology, it remains to be empirically confirmed that when there are 3 apples in a box and that you then put 4 more apples in the same box there are therefore now 7 apples in that box. No longer a tautology this.

Further, if reality is not infinite then only a finite number of arithmetical tautologies could conceivably ever be empirically confirmed. If on the contrary reality is infinite it seems that even "3+4=7" could not be confirmed empirically.
EB

I was under the impression that tautologies are never empirical.

I don't seem to have said or implied otherwise.

They have to do with the definitions of terms and their conceptual relationships, not anything in the outside world. In other words, even if adding 3 apples to a basket of 4 apples resulted in there being 8 apples, it would still be true that 3+4=7 under the rules of arithmetic. Tautologies cannot be supported or refuted empirically.

I agree. So maybe you could read again what I said. I think it's clear enough. Or perhaps you could pinpoint the particular bit that's bothering you.
EB

 

[Speakpigeon]

No.

I wonder though if 3+4=3+4 is a tautology. Probably not, but pretty dang close.

Tautologies are true, but trivially so. We really don't learn anything from them. The statement, "cup boards are cup boards," is true, but trivially so--there is nothing to learn from the statement that isn't already explicit in the statement. "Grass is grass" is a tautology. The saying, "it is what it is," is profoundly tautological.

The equation 3+4=7 in sentence form reads, "three plus four equals seven." I have leaned something, or in this case reminded of what I already know. There is nothing in the part that reads, "3+4" that tells me that it's 7.

"3+4" is "3+4" is a tautology. It is what it is--after all, what else would it be!

Now, let's go back to my opening wonderment. There is a difference between X = X and X is X. The first is a statement of equivalence, but equivalence is not the same as "same." That latter is an issue of identity. "3+4" is equivalent to "7", but the number seven is a unique number in its own right and different than the others. Not even handing you two five dollar bills is identically the same as handing you a ten, even though they are equivalent monetarily speaking.

I have not come across this more restrictive definition of tautology before. Is it your opinion, then, that the classic tautology "all bachelors are unmarried" is not actually a tautology?

There are two kinds of tautologies. Originaly, it's a statement where the conclusion repeats the premise: "grass is green because grass is green". That's the sense fastis using. However, since Wittgenstein I think, there's a second understanding whereby a complex proposition is always true irrespective of the truth values of the elementary propositions of which it is made. Exemple: "p or not p". And that's the sense Juma used. Considering that the OP seems mathematical in spirit, I think only the second sense is of interest here. However, "3+4=7" is an elementary proposition, so as such it cannot be said to constitute a tautology by itself. You need to contextualise. It may be a bit complicated to express a proper tautology based on 3+4=7 but the principle seems clear enough. And "all bachelors are unmarried" is only a tautology in the second sense, as your question suggests.
EB

 

[PyramidHead]

I was under the impression that tautologies are never empirical.

I don't seem to have said or implied otherwise.

They have to do with the definitions of terms and their conceptual relationships, not anything in the outside world. In other words, even if adding 3 apples to a basket of 4 apples resulted in there being 8 apples, it would still be true that 3+4=7 under the rules of arithmetic. Tautologies cannot be supported or refuted empirically.

I agree. So maybe you could read again what I said. I think it's clear enough. Or perhaps you could pinpoint the particular bit that's bothering you.
EB

This is the part that must have confused me:

However, we may believe and claim that "3+4=7" somehow applies acroos the board to reality, as opposed to just the purely theoretical and therefore fictional world of mathematics. If so, it should be noted that as such this is no longer a tautology, it remains to be empirically confirmed that when there are 3 apples in a box and that you then put 4 more apples in the same box there are therefore now 7 apples in that box. No longer a tautology this.

I took it to mean that tautologies can be "empirically confirmed," which is not my understanding of the concept. The mapping of mathematics to reality is a separate question from whether mathematical statements are tautologies. But I see that it could mean the empirical counterparts of mathematical statements are not necessary truths. In which case, as you say, we agree.

 

[PyramidHead]

I have not come across this more restrictive definition of tautology before. Is it your opinion, then, that the classic tautology "all bachelors are unmarried" is not actually a tautology?

There are two kinds of tautologies. Originaly, it's a statement where the conclusion repeats the premise: "grass is green because grass is green". That's the sense fastis using. However, since Wittgenstein I think, there's a second understanding whereby a complex proposition is always true irrespective of the truth values of the elementary propositions of which it is made. Exemple: "p or not p". And that's the sense Juma used. Considering that the OP seems mathematical in spirit, I think only the second sense is of interest here. However, "3+4=7" is an elementary proposition, so as such it cannot be said to constitute a tautology by itself. You need to contextualise. It may be a bit complicated to express a proper tautology based on 3+4=7 but the principle seems clear enough. And "all bachelors are unmarried" is only a tautology in the second sense, as your question suggests.
EB

What do you mean by elementary versus complex propositions? It would seem that "grass is green because grass is green," when formulated logically, is always true irrespective of whether grass is actually green. Even if grass were not green, it would still be true that given the premise that grass is green, the conclusion that grass is green necessarily follows. Or did you mean something else by "the conclusion repeats the premise"?

 

[Speakpigeon]

What do you mean by elementary versus complex propositions?

"q" is elementary, "q or p" is complex. As soon as there is a logical connector involved, it's said to be complex. Of course it's debatable whether "p or not p" is really complex but that's preciselly because it's a tautology.

It would seem that "grass is green because grass is green," when formulated logically, is always true irrespective of whether grass is actually green. Even if grass were not green, it would still be true that given the premise that grass is green, the conclusion that grass is green necessarily follows. Or did you mean something else by "the conclusion repeats the premise"?

If we take "if grass is green then grass is green" to be a good approximate of "grass is green because grass is green", then its form is "p → p" and it's therefore complex, and we can see it's a tautology because it's true irrespective of the truth value of the elementary proposition "p". The difference is in the fact that "grass is green because grass is green" is not a truth-functional proposition whereas "p → p" is. So "tautology" in the second sense only applies to logical propositions (truth-functional propositions) whereas in the first sense it remains a bit vague.

The first sense is perhaps more interesting because in fact it does say something that may or may not be true if you look carefully. But that would be a derail.
EB

 

[Malintent]

"it is what it is," is profoundly tautological.

I used to think so too.. .but now I am not so sure. the expression is shorthand for "it is no more than what it appears to be". So it is not so much a tautology as it is means to give up on finding more meaning in something.

 

[fast]

"it is what it is," is profoundly tautological.

I used to think so too.. .but now I am not so sure. the expression is shorthand for "it is no more than what it appears to be". So it is not so much a tautology as it is means to give up on finding more meaning in something.

I think that would be a varient usage of the term. When a women you're vying for her affection says, "it is what it is," that can also in varient fashion express a message that brings you to the realization that feelings aren't mutual.

 

[Bronzeage]

An expression of an equality is not automatically a tautology. The fact that two certain numbers add up to seven, no matter in what order they are added is an important mathematical principle called the commutative property. This is really taxing my memory of 7th grade math.

 

[fast]

An expression of an equality is not automatically a tautology.

Agreed. A claim of equality is different than a claim of identicality. To say something is equal to something else is merely to say that there is an equivalency between the two-that they are the same in the sense they are equivalent--but not in the sense they are identical.

Ann brought me a watermelon and then brought me another while Betty brought me two. They brought me the same number of watermelons, but they didn't bring me the same watermelons. The number that each brought me is equivalent. They both brought me two. But, this is an issue of identity, not of equivalency. Each watermelon is an individual watermelon, perfectly the same as itself and different to every other, even if they are all of the same kind.

 

[Gila Guerilla]

Is the following a tautology?

3+4=7

You have provided a single statement, "three plus four equals seven".

A tautology would be something like:
3+4=7, therefore 3+4=7.

The following two are tautologous :-

There are 7 days in a week
There are (3+4) days in a week.

Here's two tautologies :-

If you have (3+4) apples, then you also have 7 apples
If you have (3+4) bananas then you have (4+3) bananas.

Isn't a tautology saying the same thing in two different ways?

3+4=7, therefore 3+4=7 is an identity.

 

[Juma]

Isn't a tautology saying the same thing in two different ways?

3+4=7, therefore 3+4=7 is an identity.

And thus a tautology.

 

[Speakpigeon]

No.

I wonder though if 3+4=3+4 is a tautology. Probably not, but pretty dang close.

Tautologies are true, but trivially so. We really don't learn anything from them. The statement, "cup boards are cup boards," is true, but trivially so--there is nothing to learn from the statement that isn't already explicit in the statement. "Grass is grass" is a tautology. The saying, "it is what it is," is profoundly tautological.

The equation 3+4=7 in sentence form reads, "three plus four equals seven." I have leaned something, or in this case reminded of what I already know. There is nothing in the part that reads, "3+4" that tells me that it's 7.

"3+4" is "3+4" is a tautology. It is what it is--after all, what else would it be!

Now, let's go back to my opening wonderment. There is a difference between X = X and X is X. The first is a statement of equivalence, but equivalence is not the same as "same." That latter is an issue of identity. "3+4" is equivalent to "7", but the number seven is a unique number in its own right and different than the others. Not even handing you two five dollar bills is identically the same as handing you a ten, even though they are equivalent monetarily speaking.

I have not come across this more restrictive definition of tautology before. Is it your opinion, then, that the classic tautology "all bachelors are unmarried" is not actually a tautology?

The sentence "all bachelors are unmarried" is formally (x)(Bx → Mx), which is not a tautology by any stretch of the imagination.

To get a tautology you need to include the definition of "bachelor" as "unmarried" and then you get: "all bachelors are unmarried therefore all bachelors are unmarried", which is obviously a tautology.

This is the same for "3 + 4 = 7". It's only a tautology if you include the definition of "3", "4", "7" and "+", or, more judiciously, the whole theory of arithmetic.

Anyway, this is a good example of the difference between formal logic and ordinary language. However, the term "tautology" in the logical sense is a term of formal logic and if you apply it outside formal logic then who knows what you mean! In the formal logic sense, "tautology" can only apply to formal expressions, such as "(p v q → p)". Expressions such as "all bachelors are unmarried" and "3 + 4 = 7 are not expressions of formal logic. Whether they are tautologies depends therefore on what you mean by them, and also by "tautology". The expresison "all bachelors are unmarried" is only acceptably a tautology, in a sense derived from formal logic, and applied to English sentences, in the sense that if you formulate the question in English I guess there had to be a presumption, and an assumption, that you take the word "bachelor" to mean "unmarried".

This also as an implication for formal logic itself. Tautologies in formal logic are only tautologies because all logicians accept the conventions of formal logic, in particular the interpretation of logical symbols and syntax. So there is always an assumption which is not always explicitly included, or even repeated each time where it would be necessary so there's no ambiguity.
EB

 

[Juma]

I have not come across this more restrictive definition of tautology before. Is it your opinion, then, that the classic tautology "all bachelors are unmarried" is not actually a tautology?

The sentence "all bachelors are unmarried" is formally (x)(Bx → Mx), which is not a tautology by any stretch of the imagination.

To get a tautology you need to include the definition of "bachelor" as "unmarried" and then you get: "all bachelors are unmarried therefore all bachelors are unmarried", which is obviously a tautology.

This is the same for "3 + 4 = 7". It's only a tautology if you include the definition of "3", "4", "7" and "+", or, more judiciously, the whole theory of arithmetic.

Anyway, this is a good example of the difference between formal logic and ordinary language. However, the term "tautology" in the logical sense is a term of formal logic and if you apply it outside formal logic then who knows what you mean! In the formal logic sense, "tautology" can only apply to formal expressions, such as "(p v q → p)". Expressions such as "all bachelors are unmarried" and "3 + 4 = 7 are not expressions of formal logic. Whether they are tautologies depends therefore on what you mean by them, and also by "tautology". The expresison "all bachelors are unmarried" is only acceptably a tautology, in a sense derived from formal logic, and applied to English sentences, in the sense that if you formulate the question in English I guess there had to be a presumption, and an assumption, that you take the word "bachelor" to mean "unmarried".

This also as an implication for formal logic itself. Tautologies in formal logic are only tautologies because all logicians accept the conventions of formal logic, in particular the interpretation of logical symbols and syntax. So there is always an assumption which is not always explicitly included, or even repeated each time where it would be necessary so there's no ambiguity.
EB

Eh. Just a waste of time. 7 is by definition = 3+4.

 

[untermensche]

Arithmetical operations like addition exist by definition.

They are not logical consequences. 4 and 3 do not logically become 7. They only do so when a specific defined operation is applied.

Tautologies involve logic not definition. At least the kind under discussion here.

 

[Juma]

Arithmetical operations like addition exist by definition.

They are not logical consequences. 4 and 3 do not logically become 7. They only do so when a specific defined operation is applied.

Tautologies involve logic not definition. At least the kind under discussion here.

Beside the point.
change the example to 1+3=5-1

 

[untermensche]

Arithmetical operations like addition exist by definition.

They are not logical consequences. 4 and 3 do not logically become 7. They only do so when a specific defined operation is applied.

Tautologies involve logic not definition. At least the kind under discussion here.

Beside the point.
change the example to 1+3=5-1

The two sides are not equal logically.

They are only equal by definition.

4 = 4 is a logical equivalence however.

A thing is logically equivalent to itself. A meaningless tautology.

 

[Juma]

Beside the point.
change the example to 1+3=5-1

The two sides are not equal logically.

They are only equal by definition.

4 = 4 is a logical equivalence however.

A thing is logically equivalent to itself. A meaningless tautology.

How do you apply these definitions without logic?

 

[Speakpigeon]

Arithmetical operations like addition exist by definition.

Then again I'm sure many people could count successfully on their fingers long before any one smart ass actually bothered to write a treaty on addition.

They are not logical consequences. 4 and 3 do not logically become 7. They only do so when a specific defined operation is applied.

Yes.

Or something else with the same effect, i.e. the "only" is somewhat misguided.

Tautologies involve logic not definition. At least the kind under discussion here.

Yes.

I couldn't bother to bring myself to make these points.
EB

 

[Speakpigeon]

The two sides are not equal logically.

They are only equal by definition.

4 = 4 is a logical equivalence however.

A thing is logically equivalent to itself. A meaningless tautology.

How do you apply these definitions without logic?

Personally, I don't understand your question.

Does anyone?
EB