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## Is there a way to refute '1+1 = 2'?

Use this forum to discuss the philosophy of science. Philosophy of science deals with the assumptions, foundations, and implications of science.
Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Londoner:

No, I will know they are true not through logic but because I look at the chairs and see that there really are two of them and also that they are alike.

If I was suffering from double vision, i.e. there was really only one chair, then my description would be wrong. Logic would not come into it.

'The cat is on the mat' can be reduced to my pointing at the two objects.

But logic etc. is entirely unconnected to objects like cats and mats.

In symbolic logic we use symbols rather than words precisely because it is unspecific; any proposition would do, whether it is really true or false doesn't matter. In logic we are only interested in the connections, the symbols are simply place markers for whatever truth values we decide to give them.

It doesn't assert any fact about the world; '1 + 1 = 2' does not assert 'there are two apples'. If it did then it would be wrong! There would still be this green apple over here, and that red apple over there in the identical state to before they got 'added'. Maths only works if we move away from actual objects and deal with abstractions. 'John plus Jane' does not make 'two'. We first have to abstract them into 'people', i.e. we must abandon any interest in the truth of any fact about John or Jane, i.e. whether these people exist etc..

The question we are discussing is 'Is there a way to refute '1+1=2'?'My answer is that there is nothing to refute, because it doesn't refer to anything. We might as well be asking; 'Is there any way to refute concept'? or 'Does it exist?'
You seem to be saying that refutation is about the world of "objects". It's not. It is a logical concept that indicates a contradiction is present in a set of reasoned premises and their conclusion. You're right to say we can't refute "the world," whatever that is. Contradictions have to do with the relations between propositions and matter of their occurrence concerns the language we use to talk about the world and the logic deployed to do so. As to objects of sensual intuition, these cannot be spoken. "They" are transcendent.

Londoner
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### Re: Is there a way to refute '1+1 = 2'?

You seem to be saying that refutation is about the world of "objects". It's not. It is a logical concept that indicates a contradiction is present in a set of reasoned premises and their conclusion.
Well, refutation simply means to disprove. To disprove you have to disprove something.

In logic the premises are not reasoned, they are just symbols, the truth or falsity of which is assumed.

It is true that logic can be 'invalid'; the reason it is 'invalid' is because it breaks the rules of logic. And an argument is logical if it does follow the rules of logic. It is circular. We might equally say 'it is impossible to play football without following the FIFA rules because if you don't then it isn't football'.

Certainly we can say something like 'A and not A' is self-contradictory, but we cannot 'refute' it because it does not assert anything, or rather it asserts something and immediately takes it back.
Contradictions have to do with the relations between propositions and matter of their occurrence concerns the language we use to talk about the world and the logic deployed to do so.
Again, the distinction is between 'propositions' that appear in symbolic logic, or are represented by numbers, which are an abstraction, they are not any particular proposition. In logic we may start 'Assume A'. The question of why we should assume A is true doesn't arise. We cannot refute 'A' because all that is said about 'A' is 'let us assume it stands for a truth', not any particular truth, just 'a truth'.

But all propositions are particular and particular propositions are not like that. If proposition 'A' was 'The moon is made of cheese' (or anything else) it asserts a fact, something we might prove, disprove or be in a position of uncertainty about. And if we start of a chain of logic with 'assume A' and 'A' is false, even if we follow all the logical rules, the conclusion will be wrong, (or if we are uncertain about 'A', any conclusion will be uncertain).

So you have a choice; either the argument has content - it refers to something, in which case it may be true or false, or it doesn't refer to anything, in which case it is neither true nor false.
As to objects of sensual intuition, these cannot be spoken. "They" are transcendent.
I do not understand the phrase 'objects of sensual intuition'. I understand 'intuition' as something that appears immediately in consciousness, there being some philosophical disagreement about what sort of things that term might refer to. Is this a reference to a particular philosopher? But perhaps this is a distraction from the topic.

Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Londoner: Well, refutation simply means to disprove. To disprove you have to disprove something.

In logic the premises are not reasoned, they are just symbols, the truth or falsity of which is assumed.

It is true that logic can be 'invalid'; the reason it is 'invalid' is because it breaks the rules of logic. And an argument is logical if it does follow the rules of logic. It is circular. We might equally say 'it is impossible to play football without following the FIFA rules because if you don't then it isn't football'.

Certainly we can say something like 'A and not A' is self-contradictory, but we cannot 'refute' it because it does not assert anything, or rather it asserts something and immediately takes it back.
You need to rethink this. What you disprove is all about the logical relations between premises. I may be that you should bring an umbrella to stay dry in the rain, say, but the world of things "out there" is not what goes into an argument and its possible refutation. Arguments are all about logical relationships and nothing more, though they are pragmatically deployed.

"In logic the premises are not reasoned, they are just symbols, the truth or falsity of which is assumed "??Symbols have no truth value. Only propositions.

Yes, circular, question-begging: So there you are asking the question "why should we trust reason?" or "What is it that validates reason?" Yet the interrogative form you employ to make the question at all is itself a rational construct. You have to acknowledge that you live in logic; your thought is inherently logical. Just because you don't explicitly pronounce the logic you use doesn't mean it's absent. Just take any given thought in your head and ask, what is its logical structure? Is it an affirmation? Does it possess a tautology? A contradiction? A negation? Asserting 'a' and taking it back: 'grass is green' to 'grass is not green'. Note the copula 'is' and the negation 'not'. What are these if not logical functions?

Londoner
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### Re: Is there a way to refute '1+1 = 2'?

You need to rethink this. What you disprove is all about the logical relations between premises. I may be that you should bring an umbrella to stay dry in the rain, say, but the world of things "out there" is not what goes into an argument and its possible refutation. Arguments are all about logical relationships and nothing more, though they are pragmatically deployed.
Well, that is where we disagree.

Assume: If A then B. Assume: A. Therefore B.

There is nothing in that we can falsify because all it does is to assume something and then expounds what it has just assumed. The name for something that does that is a tautology.
"In logic the premises are not reasoned, they are just symbols, the truth or falsity of which is assumed "?? Symbols have no truth value. Only propositions.
Exactly. And as soon as we replace the symbols with any specific proposition, i.e. propositions with a meaning, then their truth or falsity is no longer something we are free to assume.
Yes, circular, question-begging: So there you are asking the question "why should we trust reason?" or "What is it that validates reason?"
What validates reason is its usefulness in context. Euclidean reasoning is useful for solving problems in Euclidean geometry. But if we are working in three dimensions then we should not trust it! Inductive reasoning is necessary and useful in science, but it is not applicable in logic. In Euclidean geometry and science we adopt certain axioms; the trustworthiness of our reasoning rests on the trustworthiness of these axioms, the reasoning isn't self-supporting.
You have to acknowledge that you live in logic; your thought is inherently logical. Just because you don't explicitly pronounce the logic you use doesn't mean it's absent. Just take any given thought in your head and ask, what is its logical structure? Is it an affirmation? Does it possess a tautology? A contradiction? A negation? Asserting 'a' and taking it back: 'grass is green' to 'grass is not green'. Note the copula 'is' and the negation 'not'. What are these if not logical functions?
'Grass is green' is a proposition. If we were doing logic, then its structure would just be a symbol like 'A'. And if we were doing logic we would have to stop thinking about grass or colour, because these are empirical matters.

The 'is' in 'grass is green' is not a logical function. We would understand its meaning in a practical context. It might serve as emphasis - that grass IS green, not brown. Or it might be saying 'grass MUST be green to count as grass'. Or we might just use it because we ought to have a verb in a sentence, although 'green grass' is quite comprehensible.

If 'is' was a logical function, what is it? It wouldn't be 'assume'. To assert 'grass is green' and say 'we will assume grass is green' mean quite different things.

Certainly words like 'is' and 'not' must have some agreed function(s) if we are going to be able to communicate, but that is syntax rather than logic.

Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Londoner: Well, that is where we disagree.

Assume: If A then B. Assume: A. Therefore B.

There is nothing in that we can falsify because all it does is to assume something and then expounds what it has just assumed. The name for something that does that is a tautology.
You can falsify anything you please. It doesn't matter to principles of reason if it actually true or not. The occasion of the particular is just one of a recapitulation of a principle which is always, already there; always, already antecedent to the occasion.
What validates reason is its usefulness in context. Euclidean reasoning is useful for solving problems in Euclidean geometry. But if we are working in three dimensions then we should not trust it! Inductive reasoning is necessary and useful in science, but it is not applicable in logic. In Euclidean geometry and science we adopt certain axioms; the trustworthiness of our reasoning rests on the trustworthiness of these axioms, the reasoning isn't self-supporting.
Confusing: Inductive reasoning is not applicable in logic?? It is rule governed thought. It is taught, examined and discussed regarding its logical features. What are you talking about?

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``````Grass is green' is a proposition. If we were doing logic, then its structure would just be a symbol like 'A'. And if we were doing logic we would have to stop thinking about grass or colour, because these are empirical matters.

The 'is' in 'grass is green' is not a logical function. We would understand its meaning in a practical context. It might serve as emphasis - that grass IS green, not brown. Or it might be saying 'grass MUST be green to count as grass'. Or we might just use it because we ought to have a verb in a sentence, although 'green grass' is quite comprehensible.

If 'is' was a logical function, what is it? It wouldn't be 'assume'. To assert 'grass is green' and say 'we will assume grass is green' mean quite different things.

Certainly words like 'is' and 'not' must have some agreed function(s) if we are going to be able to communicate, but that is syntax rather than logic.``````
You need to get your logic text and look up predicate logic.

Look: You seem to think syntax and logic are different things. They are studied differently, and they may different jargon and issues, but grammatical constructions exhibit, in their essential functions, the same intuitions as logic. Indeed: The formal study of logic is derived from an analysis of everyday language and judgment.

-- Updated October 13th, 2015, 2:08 pm to add the following --

Pressed the wrong tab and got Code: Select all. Oh well.

Spiral Out
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### Re: Is there a way to refute '1+1 = 2'?

Spiral Out wrote:Yes, I most certainly can (say we invented the very structure of logic itself), especially if that very structure of logic is 1+1=2. If the very logic itself is what is merely described by 1+1=2 then that very structure lies within the realm of the objective reality that you have previously admitted as not being within the realm of mathematical logic.

There still exists a valid refutation of mathematical logic due to the fact that the two realms cannot be reconciled, much like GR and QM.
Hereandnow wrote:Very good. But 1=1+2 does not describe its own logic. It shows it, or expresses it. But to describe logic requires a vantage point that is not logical. it would be like describing a ride on a Ferris Wheel using more Ferris Wheel rides. Description needs another medium of symbolic possibility that is alogical (which is impossible to conceive). But to ride a Ferris Wheel--this is quite different: The ride, if you will, speaks for itself; it is pure engagement. This is what the logical intuition of 1=1+2 is.

And this is why true refutation is impossible.
Are you saying that the expression of "1+1=2" is an illogical description of a logical concept?
Hereandnow wrote:Regarding the valid refutation of mathematical logic: The two realms are mathematics and logic? And I presume you have in mind some sort of reductio the two generate in some theorem. Just note that as this reductio is observed, the observer, you, are "engaged" in the logic (which is not what you have, but what you are) that generates the intuited contradiction. The best you could say is that thought and its logic is puzzlingly contradictory at times. If you think this shows that logic is just a house of cards that collapses 9as it does with such performative contradictions as "This sentence is false"), and takes 1=1+2 with it, then you would have to admit that the very logic you are employing to make this claim is duly refuted as well and your refutation is equally indictable.
The two realms are abstract concepts and objective foundations, similar to GR & QM, that cannot be comprehensively reconciled. We think we form our abstract concepts on objective foundations but such a notion is inherently not verifiable.

All things are subject to the perception thereof. In other words, things are as they appear, not as they objectively exist.

Do you think that logic is based on opinion or fact? How is fact verified? Is it a fact that the freezing point of pure water is 32 degrees fahrenheit at sea level? Isn't this statement in fact based on an abstract construct (much like mathematics) that someone created based purely on an arbitrary scale system?

The only thing that makes the statement "true" is that we have voluntarily agreed to the provisional definitions of the terms, yet those definitions still are not based in any objective reality.

Mathematics (1+1=2) is an abstract system whereby all people must agree to the provisional definitions of the structure and terms in order for that system to be "true".

The concept behind mathematics (1+1=2) is an entirely arbitrary construct which was created and (semi)voluntarily agreed upon by beings with a fundamentally limited capacity for understanding the nature of an objective reality.

It is quite easy to refute the idea that 1+1=2 simply by disagreeing with the provisional definitions of the terms.

The only thing that is irrefutable is the objective truth that an object next to another object is as it appears, which doesn't necessarily correspond to the concept of 1+1=2.

Mathematical rules are the rules we Humans created in order to effectively function within our environment and with each other, they aren't the rules of the environment itself, but merely a representation of an abstract concept. Mathematics is not a fundamental component of the environment, therefore the rules of mathematics (as arbitrary) can be refuted outside of that specific mathematical construct, especially since there is far more outside of the mathematical construct than there is within it.
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Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Spiral Out: Are you saying that the expression of "1+1=2" is an illogical description of a logical concept?
No. I didn't say that at all. I was pretty close to the exact opposite.
All things are subject to the perception thereof. In other words, things are as they appear, not as they objectively exist.
Kant said this over two hundred years ago, though in quite different terms.
Do you think that logic is based on opinion or fact? How is fact verified? Is it a fact that the freezing point of pure water is 32 degrees fahrenheit at sea level? Isn't this statement in fact based on an abstract construct (much like mathematics) that someone created based purely on an arbitrary scale system?
You're preaching to the choir. Although, were it entirely arbitrary, it wouldn't be useful. Language does show how, just not what; nor does it show the what of the how.
Mathematics (1+1=2) is an abstract system whereby all people must agree to the provisional definitions of the structure and terms in order for that system to be "true".
I don't know what you mean by a system being true, and I don't think it is so easy to call mathematics abstract. I might agree it is relatively abstract, given that empirical intuitions seem to have a stronger actual presence, a keener felt sense and by that a stronger claim to the real. But principles, rules of constructing meaning, and mathematical principles are certainly this (though they are not used like this usually), are, in empirical concepts, so vital to making an object an object, they can't be just dismissed as vacuous abstractions.
The concept behind mathematics (1+1=2) is an entirely arbitrary construct which was created and (semi)voluntarily agreed upon by beings with a fundamentally limited capacity for understanding the nature of an objective reality.
Odd. The concept is entirely arbitrary? We're not even talking about empirical concepts here. How do you deal with apodicticity? Of course, you could say, and I would agree, that our rigorous structures of thought have arisen out of evolution and its problem solving nature. Now, that would be arbitrary given that it could have been something else that won out through the ages. But I don't think so since it would entail speculating about what is not even imaginable . But on the other hand, you have to take on Husserl and his Cartesianism. Husserl puts the center of the real in the transcendental self, and at this egoic center is produced the originary principles that are the very structures of thought that makes theorizing possible at all. This is the point I am driving at. All reasoned thinking, even when you are pointing out contradictions and failings, are active in your critical thinking. You can't escape this; you can't get behind it. Objective reality?????
it is quite easy to refute the idea that 1+1=2 simply by disagreeing with the provisional definitions of the terms.
But again and again, your refutation is cast in the very logic you refute.
The only thing that is irrefutable is the objective truth that an object next to another object is as it appears, which doesn't necessarily correspond to the concept of 1+1=2.
Odd.You think one object being next to another is an objective truth? How do you know one is next to another? How do you know the principle that informs you about objects, there being next to one another, this being an objective truth? How do know what an object is? You don't consider these suspect, but they are embedded in an empirical observation, while 1+1=2 is intuitively stronger, apodictically true (to deny it entails a contradiction). You seem to say that rule "abstractions" are just things we made up. this would be true if you were talking about empirical concepts: I find a new species of flea and my addition to the lexicon is, while useful, made up. The observed regularities that warrant a new concept are singled out features; we impose the new category on the world which is made, not discovered (Rorty).

Logic is not arbitrary, but the signifiers, the noises and squiggles, are. If you think logic and math are arbitrary, then you think the thoughts that govern your thinking are arbitrary. If you think we made up apriority, you have some explaining to do.

Spiral Out
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### Re: Is there a way to refute '1+1 = 2'?

Hereandnow wrote:But again and again, your refutation is cast in the very logic you refute.
I'm not trying to refute logic, although logic is inherently disputable, I'm refuting the so-called "laws" (as they have been asserted to apply to and/or are a necessary contingent of, or representative of, objective phenomena) of an abstract construct (mathematics).
Spiral Out wrote:The only thing that is irrefutable is the objective truth that an object next to another object is as it appears, which doesn't necessarily depend upon or correspond to the concept of 1+1=2.
Hereandnow wrote:Odd. You think one object being next to another is an objective truth? How do you know one is next to another? How do you know the principle that informs you about objects, there being next to one another, this being an objective truth? How do know what an object is?
Any object, minus the conceptual descriptors, is evident through observation, and the observation of an object next to another object is a directly observable phenomenon, and does not necessarily correspond to the mathematical concept of "1+1=2".

As has been previously given as an example, other valid equations in addition to the "1"+"1"="2" paradigm (depending on the provisional definitions and rules) can include "1"+"1"="11" (literal symbolic), "1"+"1"="1" (combinative pairing), "1"+"1"="3" (reproductive pairing), etc.

If we possessed other types of senses in addition to that which we have now, we might not recognize the individuality of certain apparently separate objects, but understand them to be "one" object of a different type, one that we cannot conceive of currently.

Mathematics is a construct of a limited and incomplete perception and understanding.
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Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Spiral Out:
Any object, minus the conceptual descriptors, is evident through observation, and the observation of an object next to another object is a directly observable phenomenon, and does not necessarily correspond to the mathematical concept of "1+1=2".

As has been previously given as an example, other valid equations in addition to the "1"+"1"="2" paradigm (depending on the provisional definitions and rules) can include "1"+"1"="11" (literal symbolic), "1"+"1"="1" (combinative pairing), "1"+"1"="3" (reproductive pairing), etc.

If we possessed other types of senses in addition to that which we have now, we might not recognize the individuality of certain apparently separate objects, but understand them to be "one" object of a different type, one that we cannot conceive of currently.

Mathematics is a construct of a limited and incomplete perception and understanding.
But as I have said, but perhaps I haven't haven't been clear enough, it is the logical "sense" as Wittgenstein put it, that does not change, the apriority, the apodicticity or deep logical relations of terms that I am bringing to light vis a vis your claim that 1+1=2 is refutable. It is logicality itself. The way you demonstrate above the mutability of rules demonstrates not the conventional nature mathematics, but the arbitrariness or interchangability of the way logic is expressed. Languages may differ in their rules for concatenating words and sentences, e.g., Japanese, Hebrew, Korean, but the bare intuitions are immutable. Koreans don't Read Frege and shake their heads in incredulity!

Limited and incomplete? Who cares! Good, in fact: because logical paradoxes, which to be true paradoxes cannot be resolved, show our own impotence in getting behind the conflicting rules that cause them. We are stuck with them, as we are stuck with the principle of identity or Demorgan's theorem. You can walk away from an invented constructions, but you can't walk away from logical sense. This is impossible. You can't "think" of an illogical world (or alogical, if you choose), and this is what you would have to do to deliver a non self-refuting critique.

Londoner
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### Re: Is there a way to refute '1+1 = 2'?

Hereandnow
You can falsify anything you please.
Not unless it asserts something. Otherwise, show me how you can falsify: 'A' ('A' does not stand for anything)
Confusing: Inductive reasoning is not applicable in logic?? It is rule governed thought. It is taught, examined and discussed regarding its logical features. What are you talking about?
It is applicable - in the sort of reasoning in which it is applicable! But not in all reasoning. We do not use inductive reasoning in geometry, for example.

If you read my post again you will see that I was responding to your question 'What validates reasoning?'. I am pointing out there is no single system of reasoning which is valid in all cases, that which system we employ depends on what we are discussing.
Look: You seem to think syntax and logic are different things. They are studied differently, and they may different jargon and issues, but grammatical constructions exhibit, in their essential functions, the same intuitions as logic. Indeed: The formal study of logic is derived from an analysis of everyday language and judgment.
What do you mean by 'the same intuitions as logic'?

Logic indeed rests on axioms that might be intuited, (like the validity of induction), but such axioms are not themselves logical, just like doing Euclidean geometry requires us to accept the intuition that 'a straight line can be drawn to connect two points', but Euclidean geometry cannot prove it.

Certainly logic must reflect our thought, in the sense that we will judge its correctness by its ability to arrive at conclusions we think are correct, but that is necessarily true of everything! We can never stand outside our own heads and pass judgement on our own ability to pass judgement.

As to syntax, some grammatical constructions mirror logical ones, but others don't. Logic selects out the ones that do. In the example you gave; 'grass is green' the 'is' does not mirror a logical construction, if it had any logical function at all it would be ambiguous.

So, all 'grass is green' does is combine a subject with a predicate. Its truth depends on whether there exists a subject with that predicate. If you think this is wrong, that its truth instead depends on some matter of logic, what is it?
You need to get your logic text and look up predicate logic.
I think you should read some philosophy beyond Plato!

Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Londoner:

You can falsify anything you please.

Not unless it asserts something. Otherwise, show me how you can falsify: 'A' ('A' does not stand for anything)
not-A. 'A' is thereby falsified. The physical "stuff out there" doesn't enter into it unless you choose to apply this logical construction. THEN you will be very hard pressed to show how this relationship works. The pragmatists win the day if you ask me. Rorty, the pragmatist postmodernist, agrees that truth is made, not discovered. You've got to get it out of your head that the world is language and logic. It's not; "we" are. Read Sartre's "Nausea" where he presents the thesis of radical contingency (so-called): The world is not confined to the logic that we impose on it when we live and breathe. Rather, "stuff" can do anything and still be logically possible. "Stuff" is not logically constrained. But logic is all about constraint; it's a system of rules. Apriori rules are necessarily binding, that is, they are coercive, even as you entertain their paradoxes, and here is the central point, you employ the logic of sentence concatenation.
If you read my post again you will see that I was responding to your question 'What validates reasoning?'. I am pointing out there is no single system of reasoning which is valid in all cases, that which system we employ depends on what we are discussing.
What validates reasoning goes to logical construction. Both deductive argument and inductive arguments get their validity from their form, not from "the world". And it depends on how you want to define 'system'. I'm not interested in this. Being reasonable means to have shared, apodictic functions of cognition that make meaningful speech possible. We "fit" this on to experience and create divisions and difference among things. Some philosophers think that the differences imposed by a cognizing, synthesizing agent are not differences in the world, just differences in the way we handle the world to solve problems. They are right.
Certainly logic must reflect our thought, in the sense that we will judge its correctness by its ability to arrive at conclusions we think are correct, but that is necessarily true of everything! We can never stand outside our own heads and pass judgement on our own ability to pass judgement
Hmmmm, interesting that you would say this. I've been trying to tell you this very thing.
So, all 'grass is green' does is combine a subject with a predicate......I think you should read some philosophy beyond Plato!
I'm not being condescending when I say you should look up predicate logic. If you say that simple predication as in "the grass is green" does not possess a logical function, then you don't understand what the logic of predication is. This etic combining you speak of. First, what is it for something to be green? the "greeness" before your requires a synthetic principle that subsumes all things that have this quality. This would be characterized as an instance of a universal quantifier, something like "all observable phenomena possessing x are y's". In other words, even the the greeness, once understood as such, is an occasion of logic, the logic quantification; you are already working within the influence of a principle in the simple act of recognition, and the analysis has not even gotten to the predication of greeness.

What does Plato have to do with it? Of course, there is a point here: Kantian thinking is neoplatonic. But I am not at all saying anything about metaphysics.

-- Updated October 14th, 2015, 12:06 pm to add the following --

Ugh! Careless typing!

Londoner
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### Re: Is there a way to refute '1+1 = 2'?

Me: Not unless it asserts something. Otherwise, show me how you can falsify: 'A' ('A' does not stand for anything)

not-A. 'A' is thereby falsified.
To falsify something you have to produce an observation or argument that proves it false.

'The earth orbits the sun' is not falsified by someone saying 'The earth does not orbit the sun', a proposition is not falsified by saying 'No it's not'.

But if you disagree, then we have answered the question in the OP. Is there a way to refute '1 + 1 = 2'? Answer yes! Here goes: '1 + 1 does not equal 2.
The physical "stuff out there" doesn't enter into it unless you choose to apply this logical construction. THEN you will be very hard pressed to show how this relationship works. The pragmatists win the day if you ask me. Rorty, the pragmatist postmodernist, agrees that truth is made, not discovered. You've got to get it out of your head that the world is language and logic. It's not; "we" are. Read Sartre's "Nausea" where he presents the thesis of radical contingency (so-called):...
I have read a lot more than Nausea - I like Satre and I am deeply tempted to follow you on that diversion but I think we should stick to the point.
"Stuff" is not logically constrained. But logic is all about constraint; it's a system of rules
Quite. Nor is truth constrained by logic. Logic only expounds on truth, it does not establish that truth.
What validates reasoning goes to logical construction. Both deductive argument and inductive arguments get their validity from their form, not from "the world".
Yes, logical constructions are valid if they follow the form of logical constructions. But because a construction is valid it does not make the conclusion sound.

Horses have tails, Socrates is a horse, therefore Socrates has a tail, is valid. In logic, that the premises and thus the conclusion are false does not matter, because the premises are simply assumed to be true. We cannot falsify the conclusion that Socrates has a tail by looking at the logical construction, yet nevertheless it is false.
I'm not being condescending when I say you should look up predicate logic. If you say that simple predication as in "the grass is green" does not possess a logical function, then you don't understand what the logic of predication is. This etic combining you speak of. First, what is it for something to be green? the "greeness" before your requires a synthetic principle that subsumes all things that have this quality. This would be characterized as an instance of a universal quantifier, something like "all observable phenomena possessing x are y's". In other words, even the the greeness, once understood as such, is an occasion of logic, the logic quantification; you are already working within the influence of a principle in the simple act of recognition, and the analysis has not even gotten to the predication of greeness.
It is difficult if you introduce words like 'etic' without explaining what you mean by them, as if they were some technical philosophical term I ought to know. As far as I can see it is a term in cultural anthropology; I certainly do not 'speak of' it! Or was it just mistyped?

Likewise, I understand 'predicate logic' but not 'the logic of predication': I gather from Google the phrase occurs in a very complicated bit of Husserl that I haven't the energy to attempt to understand.

However I do know of the problem of universals like 'green' because it has been discussed by philosophers from ancient times and there is nothing simple about it - we don't 'understand it as such' - hence the problem.

But having suggested the problem you then say one particular interpretation would be 'an occasion of logic'. Well yes; if we all agree what a thing means then we can create propositions about it. If we create propositions we can draw logical inferences from them. But the inferences will still be of the same kind as Socrates' tail; their truth will rest not on the chain of logic but on the correctness of the premises.

To cut this short, I think your point now amounts to saying that because we can express a statement about the world in a symbolic form (IF we can iron out any problems and ambiguities) - the same form we would use if were expressing a purely logical relationship - then the statement becomes purely logical and its truth is purely a function of logic.

i.e. because we can write out that argument concerning Socrates, or a statement about grass, using symbols and replacing connecting words with signs, then the truth (validity) of its conclusion becomes purely a question of whether we have followed the rules of logic.

As you must gather, I do not think that is true.
What does Plato have to do with it?
Plato thought that things like numbers exist, they are not just concepts that we attach to things that do exist. Indeed, for Plato, such concepts are the only things that do exist, 'real' objects just being imperfect reflections.

With your insistence that meaning and truth reside in the heavenly realm of abstractions and not down here on earth, I think Plato would recognise you as one of his own!

Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Londoner:

To falsify something you have to produce an observation or argument that proves it false.

'The earth orbits the sun' is not falsified by someone saying 'The earth does not orbit the sun', a proposition is not falsified by saying 'No it's not'.

But if you disagree, then we have answered the question in the OP. Is there a way to refute '1 + 1 = 2'? Answer yes! Here goes: '1 + 1 does not equal 2.
You said 'A' didn't stand for anything. So, it's just a logical variable. How do you falsify a variable? Add a tilda~
I have read a lot more than Nausea - I like Satre and I am deeply tempted to follow you on that diversion but I think we should stick to the point.
Not so much a diversion as a poignant reference. You'll recall those surreal scenes in which Roquentin's tongue turns into a live centipede, and his hands into large worms (as I seem to remember), and especially that famous passage with the encounter with the chestnut tree. Especially this latter, an encounter with Being sans the particulars, sans the body of contingency that provides contextual sense in order for individual identity to announce itself. Roquentin experiences the world free of logical structure and the purpose of this bloated epiphany is to make a powerful point: the world is not logic; it has no categories, no Kantian synthetic concepts that divide and formalize. It is cognitively alien, amorphous, and entirely unspeakable.

I want make a similar point: All that is in "the world out there" (and those double inverted commas are important since language of any kind here has no place) does not provide a basis for logical engagement. (Nor does it confer Truth to any proposition because there is no such thing. Only justification. But this is, as you say, a diversion.) Sound arguments, which are supposed to be true about the world, are really about language and its rules for concatenation. I could say something "out there' causes me to say the book is on the shelf, but the equation of proposition rests with the rules and their principles of subsumption. And when I say "something causes" I am, as Wittgenstein will tell you, speaking nonsense.
Quite. Nor is truth constrained by logic. Logic only expounds on truth, it does not establish that truth
I don't really follow this. Logic expounds?

T
o cut this short, I think your point now amounts to saying that because we can express a statement about the world in a symbolic form (IF we can iron out any problems and ambiguities) - the same form we would use if were expressing a purely logical relationship - then the statement becomes purely logical and its truth is purely a function of logic.
i.e. because we can write out that argument concerning Socrates, or a statement about grass, using symbols and replacing connecting words with signs, then the truth (validity) of its conclusion becomes purely a question of whether we have followed the rules of logic.
My point has been, with some side steps, that 1+1=2 cannot be refuted because you can never get behind logic, and mathematics is a system of logic; that if you could refute the principle of addition (notwithstanding how you want to display it) you would be refuting a tautology (all mathematics is tautological. I think this is likely true.) which is apodictically coercive, and if you are fine with this, then you must be equally fine refuting any and all occasions of logicality, which would apply to the logic you employ to make your refutation and the bleeding affair is self-refuting.

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### Re: Is there a way to refute '1+1 = 2'?

Hereandnow wrote:and mathematics is a system of logic
No, it isn't. It's an abstract Human construct.

However, if we also consider logic to be an abstract Human construct then mathematics could potentially be a "system of logic" simply by association.
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Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Spiral Out:

No, it isn't. It's an abstract Human construct.

However, if we also consider logic to be an abstract Human construct then mathematics could potentially be a "system of logic" simply by association.
Human constructs are not cognitively coersive. You need to separate what is constructed and what is not. Kant did this. I certainly don't agree with everything he said, but the rules of logic are different from constructed rules of science, which are, and this is the point, all founded on the logic the language that is used to create them. This was Husserl's Cartesian argument: Empirical science issues from an originary intuitive foundation, and this is what we cannot "get behind". I may not buy wholly into his transcendental ego (but I find it tempting), but he was right about the intuitive structures that all thought issues from being unassailable.

The word "abstract" has little appeal to me. Once any and all dualisms are dismissed, where is the basis for dividing things up existentially? There are certainly differences in the manner in which they are presented in experience, but it is important to keep in mind that eidetic intuitions are powerful functions in making a thing a thing.