ontological_realist
Member
Is the following a tautology?
3+4=7
3+4=7
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Arithmetic and tautology
- Thread starter ontological_realist
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It don't think so. That isn't true because of its logical structure, it's true because it's a conclusion reached from deductive reasoning.
A tautology would be something like "Grass is green because grass is green". Math would be more like "Grass is green because I looked at the grass and it was green". While, in one sense, the two statements are fairly equivalent, there's an important difference that the second is a truth statement because of something external brought into it, while the first is just true in and of itself.
A tautology would be something like "Grass is green because grass is green". Math would be more like "Grass is green because I looked at the grass and it was green". While, in one sense, the two statements are fairly equivalent, there's an important difference that the second is a truth statement because of something external brought into it, while the first is just true in and of itself.
You have provided a single statement, "three plus four equals seven".
A tautology would be something like:
3+4=7, therefore 3+4=7.
Random Person
Senior Member
Ha, the definition of a tautology is a tautology.
Speakpigeon
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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
PyramidHead
Contributor
I was under the impression that tautologies are never empirical. 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.
Within normal mathematics , Yes in the meaning that a tautology is a sentence that is always true.
But a tautology normally is a statemfnt made up by statement variables such as:
((P and Q) or (P and -Q)) or -P
Which is always true whatever P and Q.
fast
Contributor
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.
Shadowy Man
Contributor
tautology
[taw-TOL-uh-jee]
noun, plural tautologies.
1. see tautology
[taw-TOL-uh-jee]
noun, plural tautologies.
1. see tautology
PyramidHead
Contributor
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?
fast
Contributor
I didn't mean to cover the entire scope of the topic, just the aspect that I thought might apply to the example under question. The statement, "all bachelors are unmarried" is not just true. True, yes, but not merely so. It's a necessary truth.
Do you think equivalence should be held to the same standard as identity?
PyramidHead
Contributor
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).
fast
Contributor
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.
Eh. 3+4=7 is true by definition and thus is a tautology. That there are some steps to show it follows from the definitions of "3","+","7","=" and "4" doesnt change that.
In contrast: 3+x=7 is not a tautology.
fast
Contributor
By definition or by calculation?
By definition of course!
fast
Contributor
Lexical definition?
What? By the definition of the natural numbers. (And the decimal representation)
fast
Contributor
I assume you mean numeral. Beyond that, I could make your next argument, but I'm not rightly sure where to go after that. I guess you win for now.
This isnt about winnig or loosing. This is about finding out the truth.
Math as a whole is a big tautology.
All the constituents are defined thus the expressen is simply a view of their relations and such is always true.
To be a non-tautology there have to be a variable whose value is not fixed. And here is where empiri enters.