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1. The 2+2=5 thing doesn't work for this discussion because whether or not it's "true" is entirely dependent upon how we've defined it ahead of time. We've created our own arbitrary system where 2+2=4 by definition, so this is one thing we are absolutely certain of--but only because it's a result of our own theoretical constructs.

If I name my dog Rover, then I'm absolutely certain that his name is Rover, but only because I predefined the system that way. This doesn't really apply to our epistemology discussion here.

Saying that we can't know that 2+2=4 is like saying that we don't know if the word "and" really has three letters. We do know that because we decided what a letter is and what having three of them means.

2. Originally Posted by Helios
I think there are some examples of propositions we can know to be true for certain; "2+2=4", "a square has 4 sides", and, "given p&q, one can infer p (via simplification)".
Two things:

1) All of the things you mentioned are abstract concepts. They aren't directly related to anything concrete in reality.

2) Specifically,
a) "2+2 = 4" is based upon axioms, and we have to believe those axioms are true. There is still a degree of uncertainty in the axioms (although for 2+2 = 4 the degree of uncertainty is about as small as it gets).

b) "4 sides" is part of the definition of what a "square" is. It is true that things have the properties that we define them as having, but that is pointless since we arbitrarily gave them these properties to begin with.

c) "given p&q, one can infer p (via simplification)" is a property of formal logic, and we are assuming that logic can reach conclusions in a meaningful way.

Originally Posted by DigitalMethod
But you can never prove that. More importantly, there is no evidence that that is true. You can always ask "what if." You can ask questions about consciousness however you will be forever stuck in consciousness. There is no way to observe anything outside of consciousness. Therefore it is largely irrelevant to question consciousness I would say.
You certainly could prove that 2 + 2 = 5 if you changed the axioms used to reach that conclusion. Evidence is irrelevant when discussing pure mathematics, because evidence is an inferior method of reaching conclusions. Therefore pure mathematicians always ignore evidence in favor of formal logic.

3. Two things:

1) All of the things you mentioned are abstract concepts. They aren't directly related to anything concrete in reality.
Even if true, I'm not sure how this is relevant. Recall that my claim was "there are some examples of propositions we can know to be true for certain".

2) Specifically,
a) "2+2 = 4" is based upon axioms, and we have to believe those axioms are true. There is still a degree of uncertainty in the axioms (although for 2+2 = 4 the degree of uncertainty is about as small as it gets).
Which "axioms" are being referred to? As far as I am aware, the value of the addition function necessarily is 4.

b) "4 sides" is part of the definition of what a "square" is. It is true that things have the properties that we define them as having, but that is pointless since we arbitrarily gave them these properties to begin with.
Unfortunately, I do not fully understand this objection. Perhaps it ought to be asked whether, if a square, by definition, has four sides, this claim is pertinent. The proposition "A square has 4 sides" remains necessarily true; "A square has 5 sides" is, of course, necessarily false ( as we might tell a child learning elementary mathematics).

c) "given p&q, one can infer p (via simplification)" is a property of formal logic, and we are assuming that logic can reach conclusions in a meaningful way.
I've no idea what a "property of formal logic" is, or what is meant by "reach conclusions in a meaningful way". Simplification is a valid rule of inference in propositional logic; p can always be inferred from p&q, and there is no doubt whatsoever that this is the case.

4. Originally Posted by Anja
Governor Blagojovich says, "If you can't prove it, it didn't happen." See how good that works?
He didn't really get fired, of course.
(You just think he did.)

5. Originally Posted by Helios
Even if true, I'm not sure how this is relevant. Recall that my claim was "there are some examples of propositions we can know to be true for certain".
The examples are true but irrelevant. They are theoretically true, but they are entirely human constructs. You are basically saying that we can absolutely know something is true as long as it is in a system that we created entirely ourselves.

You do have an exception (sort of), but there was no actual discovery involved and the examples are not concrete in any way. My point is that one can never be absolutely certain of anything in reality.

Which "axioms" are being referred to? As far as I am aware, the value of the addition function necessarily is 4.
Every field of mathematics operates under axioms, even arithmetic. A couple of the most basic ones are
1) x = x
2) Given a = a and b = b then a + b = a + b

These axioms are both need to conclude 2 + 2 = 4. (I can't recall all of the axioms for arithmetic, but) there is also an axiom assuming that the operation addition exists.

Now these ideas may seem so basic that they don't seem like assumptions at all, but ideas which must be true. Well that is exactly the criteria for a good axiom.
I've no idea what a "property of formal logic" is, or what is meant by "reach conclusions in a meaningful way". Simplification is a valid rule of inference in propositional logic; p can always be inferred from p&q, and there is no doubt whatsoever that this is the case.
You are essentially saying the same thing that I am about your proposition. Your proposition is a property of logic. However to use logic one must assume that logic exists and can draw meaningful conclusions.

6. Originally Posted by Jennifer
He didn't really get fired, of course.
(You just think he did.)

7. The examples are true but irrelevant. They are theoretically true, but they are entirely human constructs. You are basically saying that we can absolutely know something is true as long as it is in a system that we created entirely ourselves.
Once again, my claim is: "there are some examples of propositions we can know to be true for certain". This was in response to a claim to the contrary; I did not specify of what nature these propositions are. Your task is to refute the claim that the examples adduced can be known for certain. Incidentally, all of my propositions are certain even without the existence of humans to conceive of them; 2+2=4 is certainly true irrespective of any human's existence.

Now these ideas may seem so basic that they don't seem like assumptions at all, but ideas which must be true. Well that is exactly the criteria for a good axiom.
Precisely; the axioms you cite must be true, and, consequently, 2+2=4 must also be true.

You are essentially saying the same thing that I am about your proposition. Your proposition is a property of logic. However to use logic one must assume that logic exists and can draw meaningful conclusions.
I can only repeat the foregoing: "I've no idea what a "property of formal logic" is, or what is meant by "reach conclusions in a meaningful way". I am requesting, implicitly, that you explain what is meant by these statements.

8. I thought it appropriate to at least mention that I withdrew myself from the conversation out of exhaustion. Hopefully I'll be able to gather myself and dive back in a few pages from now! Are we now arguing about the possibility of logical certainty/self-evident truths? Are we trying to refute "There are no married bachelors"?

Also, @mycroft, sense experience = true by definition? Sure. Nice.

@Costrin, I'm worn out. I'll try to come back to the debate at some point. However, for now, I'd just like to say that you rather own at this.

9. Originally Posted by silverchris9
I thought it appropriate to at least mention that I withdrew myself from the conversation out of exhaustion. Hopefully I'll be able to gather myself and dive back in a few pages from now! Are we now arguing about the possibility of logical certainty/self-evident truths? Are we trying to refute "There are no married bachelors"?
I'd rather not. I see it as pretty pointless. I apologize to the thread for bringing it up in the first place.

@Costrin, I'm worn out. I'll try to come back to the debate at some point. However, for now, I'd just like to say that you rather own at this.
Why thank you.

10. Originally Posted by Helios
Precisely; the axioms you cite must be true, and, consequently, 2+2=4 must also be true.
No we cannot be sure that "2+2=4" is true. It only seems this way. In fact "2+2=4" is based upon assumptions and in fact it can be shown that 2+2 does not always equal 4. For example in mod 3 arithmetic 2+2=1 and in mod 4 arithmetic 2+2=0.

I can only repeat the foregoing: "I've no idea what a "property of formal logic" is, or what is meant by "reach conclusions in a meaningful way". I am requesting, implicitly, that you explain what is meant by these statements.
If you haven't studied formal logic, then I'm not sure if I can explain it to you in a way that you will accept. This is my simplest attempt: There is more than one way to reason. Formal logic is one way to do so. When using any type of reasoning we assume that our method is valid in reaching useful or meaningful conclusions.

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