Is this apparently nonsensical statement logically valid?

Hi Kaiden!

You said : 

"Premise one: 2+6=9
"Conclusion: 2+6+2+6=18

Is this argument valid? Yes, it is. "

I beg to differ, because :

[Premise one : 2+6=9], whilst breaking the laws of mathematical addition, is simply a premise, so no problem there.

However, [Conclusion: 2+6+2+6=18] is false. Why? Because Premise one broke the laws of mathematical addition, but you then ignore that, and (for no good reason), applied the CORRECT laws of addition to form your conclusion.

Therefore, your conclusion is false.

Put another way : 

1 .. Premise 1 : 8=9
2 .. Conclusion : 16=18

In 1 you are ignoring a mistake in addition
In 2 you then correct the mistake to form your conclusion.

[To Jim]

Thank you for sharing your differences, Jim. In the last paragraph of my Answer’s main body, I gave an argument that 2+6=9, therefore, 2+6+2+6=18 is valid. Since you disagree that that it is valid, you will either have to prove that my argument to the contrary is itself invalid or falsify at least one of its own premises. But you have done neither. 

Presuming that I read you correctly, what you have said is that the conclusion of 2+6=9, therefore, 2+6+2+6=18 is false and so the argument is invalid. This is not right. The having of a false conclusion does not make an argument invalid. Consider the following valid argument that has a false conclusion.

1. If dogs are mammals, then sharks are mammals.

2. Dogs are mammals.

3. Therefore, sharks are mammals.

Don’t you agree that the conclusion is false? At the same time, don’t you agree that the argument is valid? It is a Modus Ponens, after all. Do you now agree with me that an argument’s having a false conclusion does not make that argument invalid?

Hello, I haven't studied proof methods yet, so I read the argument from its semantic composition. I think Jim's post makes sense because:
for 2 + 6 = 9 to be true, the sequence or the representation of the numbers must have been altered, alternatively, the meaning of "+" must have been altered.
So if 2 + 6 = 9 were true, how would I know what was "+" or any integer or 18?
It could be that 2 + 6 = 9 was from for each integer, integer = integer + 1/2, but if so then 18 was 18 +1/2. 
It could be that 2 + 6 = 9 was from "+" now stands for num1*num2 -3, but if so then 18 was 195.
Therefore, the premise 2 + 6 = 9 does not guarantee that 2 + 6 + 2 + 6 = 18. In other words, the conclusion can be false when the premise is true, so the argument is invalid.

[To LogicG]

LogicG, thank you for your thoughts. 

“In other words, the conclusion can be false when the premise is true, so the argument is invalid.”

Are you saying that it can be the case that the conclusion is false and the premise true? If that is what you are saying, then no, what you are saying is not right. It cannot be the case that the conclusion is false and the premise true. That is because it cannot be the case that the premise (2+6=9) is true.

You might say, “But grant that the premise is true, so that I can make my point. Granting that 2+6=9, my point is that the conclusion could simultaneously be false”

But in that case, we would be granting something impossible. Which is precisely to make  my point: it is impossible that the premise of the argument is true. So, it is impossible that the premise is true and the conclusion false. So, the argument is valid.

[To LogicG]

Let me say a bit more because I think your post is so important to address thoroughly. I am trying to be brief, but please ask me to clarify whenever you need me to. Strictly speaking, sentences are neither true nor false. Rather, statements are either true or false. A statement is the meaning or thought expressed by a reasonably unambiguous declarative sentence.

“for 2 + 6 = 9 to be true, the sequence or the representation of the numbers must have been altered, alternatively, the meaning of "+" must have been altered.”

A sentence whose meaning has changed is a sentence that now expresses a different statement than the one it previously expressed. When I wrote my Answer, I interpreted “2+6=9” at face value. I interpreted it as meaning just that: the quantity two and the quantity six added together is the same quantity as nine. If you alter “2+6=9” so that it does not mean that, then the argument you are evaluating is now a different argument because the statement in its premise is now a different statement (even if the sentence in both arguments is identical.)

“It could be that 2 + 6 = 9 was from "+" now stands for num1*num2 -3, but if so then 18 was 195.
Therefore, the premise 2 + 6 = 9 does not guarantee that 2 + 6 + 2 + 6 = 18. In other words, the conclusion can be false when the premise is true, so the argument is invalid.”

If “+” in premise 1 stands for “num1*num2 -3”, then what you have is a different argument from the argument that I said was valid. For the premise contains a different statement than the one contained by the premise of the argument I evaluated. The premise of the argument I evaluated was one in which 2+6=9 was interpreted at face value. The argument I evaluated is not invalid even if this other argument, whose premise is expressed by a sentence in which "+" instead stands for num1*num2 -3, is invalid.

Hi Kaiden,

I fully understand what you are trying to say. In the dogs/sharks/mammals example, I totally agree with you that as an argument, it is valid. Note the important point here -  you are using if-->then logic to move from 1 to 3, without error, or ambiguity. The concept in nonsensical, but you formed a correct conclusion using simple rules of logic.

"Since you disagree that that it is valid, you will either have to prove that my argument to the contrary is itself invalid or falsify at least one of its own premises."

You say "2+6=9, therefore, 2+6+2+6=18 is valid". Since INCORRECT arithmetic was used in the premise,  you cannot then use CORRECT arithmetic to reach your conclusion, which is what you are doing.

This is why your argument is invalid.

[To Jim]

Thank you for your kind engagement, Jim. 

“This is why your argument is invalid.”

I am not sure exactly which argument you are saying is invalid. First, there is the arithmetic argument which goes like this: 2+6=9, therefore, 2+6+2+6=18 . Second, there is the argument I gave about the arithmetic argument to the effect that the arithmetic argument is valid. Whichever argument you are referring to (even if you are referring to both), you are mistaken. The arithmetic argument is valid. Also, my argument that the arithmetic argument is valid is itself valid. 

Let’s start with my argument that the arithmetic argument is valid. Here was my argument: 

1. If an argument is valid, then it is impossible that all of its premises are true and the conclusion false, and if it is impossible that all of an argument’s premises are true and the conclusion false, then it is a valid argument.
2. The following argument is such that it is impossible that all of its premises are and the conclusion false: 2+6=9, therefore, 2+6+2+6=18. 
3. Therefore, 2+6=9, therefore, 2+6+2+6=18 is valid.

My argument has two premises and a conclusion. Premise 1 is a bi-conditional sentence that gives both the necessary and the sufficient conditions of validity. Premise 1 entails this: 

4. If it is impossible that all of an argument’s premises are true and the conclusion false, then it is a valid argument.

The reason 1 entails 4 is that 1 is a conjunction (bi-conditionals are conjunctions). A conjunction entails any one of its conjuncts. For instance, if A, then B, and if B, then A is a conjunction. It entails if A, then B, which is one of its conjuncts .  It also entails if B, then A,  which is the other of its conjuncts.   Taking a conjunction and making the inference to one of its conjuncts is a logical inference called Simplification.

Well, premise 1 of my argument is a conjunction. One of its conjuncts is that if it is impossible that all of an argument’s premises are true and the conclusion false, then it is a valid argument . So, from 1 it can be derived by Simplification that if it is impossible that all of an argument’s premises are true and the conclusion false, then it is a valid argument . That is, from 1, 4 can be validly inferred. 

Now that I have logically inferred 4 from 1 by Simplification, what more can I derive? I can take 2 and 4 and make a Modus Ponens inference that 2+6=9, therefore, 2+6+2+6=18 is valid.  

There you have it. By starting off with the two premises of my argument, I have derived the conclusion of my argument that 2+6=9, therefore, 2+6+2+6=18 is valid . I used a Simplification inference, and then a Modus Ponens inference. Since I was able to derive the conclusion of my argument from its premises, my argument is valid. 

Moving on. 

But is it true that 2+6=9, therefore, 2+6+2+6=18 is valid ? Yes, it is true. I know this because all of the premises of the valid argument that I gave for the validity of 2+6=9, therefore, 2+6+2+6=18 are true.

Let’s look back at my argument. Premise 1 gives the necessary and sufficient conditions of validity. The premise is true by definition. 

I already gave a defense of premise 2, which you have not answered. 2+6=9 is necessarily false. And it is the only premise of the arithmetic argument. Since it is necessarily false , and the only premise of the arithmetic argument, it follows that the arithmetic argument is such that it cannot be the case that all of its premises are true . And since it is impossible that all of the premises of the arithmetic argument are true, it follows that it is impossible that all of the premises are true and the conclusion false. And this is just what my own argument states in its own second premise. 

In sum, I have proven that my argument is valid. Moreover, it’s first premise is true by definition and its second premise is true by the laws of mathematics and logical inference. So, I know  that my argument’s conclusion is true, namely, that 2+6=9, therefore, 2+6+2+6=18 is a valid argument. 

Hi Kaiden,

First off, I'm not getting notifications of your replies, which is odd, as I get notified about other peoples' comments, so I have to manually go back, open up the question, then search for your replies.

Anyway, this is interesting. Let's see if we can get to the core of the problem. Now, your application of logic is fine here, except that is seems that there is something you have missed - a very crucial point, and not to do with the logic flow.

Let's look at the original post :
[1] If
[2] 2+6=9
[3] then 
[4] 2+6+2+6=18

So, on the surface it looks OK, and according to the rules of logic it is. However, there is an undercover elephant in the room, and it's this.

[2] renders the values of numbers meaningless, therefore any calculations based on [2] are also meaningless. What you have done to arrive at [4], is give numbers their correct value (thus meaning) {you have effectively multiplied by 2, 2 being the correct value of 2}, so [4] is, at the very least, an invalid conclusion.

Therefore, the argument is invalid.

To demonstrate the point, let's ask this question :

If 2+2=9 AND
4+7=31 AND
8+6=2 THEN
what is the value of 34+23 ?

This has truly rendered numbers meaningless, so the answer is impossible to find.

[To Jim]

I hope I am understanding you correctly. What I am about to say is similar to what I told LogicG.

[2] does not render the value of numbers meaningless. You are confusing numbers with numerals . “2” is not a number; it is a numeral. The numeral “2” symbolizes the quantity two. The number is the quantity two that is being symbolized . If the question you posed to me leaves me puzzled about how to answer, it is because it unclear to me what the numerals "34" and "23" mean. But a number is just the number that it is . Numbers do not lose their meaning (fifty-seven, say, doesn't lose its "fifty-sevenness"). The quantity thirty-four added to the quantity twenty-three is indeed the quantity fifty-seven, even if the numerals “34" and "23”, which themselves are merely symbols, have become ambiguous.

This is related to a distinction I explained to LogicG, as well, about the difference between a sentence and a statement. Sentences can be ambiguous, but not statements. “You are a pig” could mean different things. It might mean you are a pink barn animal that goes oink or it might mean that you are greedy. But statements just are the statements that they are. The statement you are a pink barn animal that goes oink just is that statement. When I evaluated the arithmetic argument, I was, as I said to LogicG, evaluating it according to the statements that I interpreted it as containing. And according to the numbers that I interpreted it as containing. I interpreted the argument like this:

1. The quantity two added to the quantity six is identical to the quantity nine.

2. Therefore, the quantity two added to the quantity six added to the quantity two added to the quantity six is identical to the quantity eighteen.

With those statements and numbers in place (not sentences and numerals), it is impossible that the premise is true, and it is the only premise in the argument. Therefore, it is impossible that all the premises of the argument are true. Therefore, it is impossible that all the premises of the argument are true and the conclusion false. An argument that is such that it is impossible that all its premises are true and the conclusion false is valid. Therefore, the argument is valid.

P.S. Perhaps some people have noticed, but in my posts throughout the years it has been my practice to use italics to refer to statements and use quotation marks to refer to sentences. You are greedy refers to the statement. "You are greedy" refers to the sentence. 

Hi Kaiden,

OK, let's try this another way, but with you applying your same reasoning.

[1] If
[2] 2+6=9
[3] then 
[4] 2+6+2+6=15

Is this argument valid? If not, why not? This ought to home in on the core of the the dispute.

[To Jim]

That is not an argument; that is a conditional sentence. Here is the take-away. Having a necessarily false premise is enough for an argument to be valid, as I have explained. That is the core insight to walk home with. If I keep answering your examples, I feed you that one time. By explaining the general concept of what validity is, you can take it and apply it yourself without my help. It has been a pleasure, Jim, thank you.

[To Kaiden]

OK, Kaiden. I could format it as an argument, but the outcome would be the same.

I think we'll just need to agree to disagree on this one :)

Thanks for your insights.

Hi Bo, now this is interesting. As a hypothetical, it is fine.

Dig deeper, and we find that since mathematical logic has been discarded, then any conclusion based on this discarded logic is invalid. Or so goes the argument from a mathematical logical perspective. 

I see this differently. Math (mathematical logic) is being preserved in that "+" still means addition. Numbers are simply representations of quantity, so instead of "3"=3, "2" = 3, so logic is preserved.

1 .. Premise 1 : 8=9
2 .. Conclusion : 16=18

In 1 you are ignoring a mistake in addition
In 2 you then correct the mistake to form your conclusion.

Therefore, your conclusion is invalid.