Are the rules of logic immutable?

[OccamsRazor]

In the past my philosophical stance was always based on a single overarching truth. This was that the laws of logic and mathematics were immutable and could not be changed for any description of reality that one may provide.

More recently I have been grappling with the question, what if the rules of logic and mathematics are not immutable but subjective or specific to our incarnation of reality?

In Michael Frayn's book The Human Touch he makes the statement:

Logic is just a system we have made up, not an inherent condition of the natural world.

Is this true? Is logic changeable?

In another thread I saw the following statement:

I don't quite know how knowing something about events inside a system from outside of the system is on the same level of impossibility as a logical impossibility. There cannot be a square circle, a rational root of a prime number or the simultaneous existence of an irresistible force and an immovable object. These are logical impossibilities..

Is this true? Could a being outside our own manisfestation of material reality not create such logical impossibilities?

I can see that here many readers of this post would begin to state that logic and mathematics were immutable. That there indeed could not exist a rational root of a prime number and these are objective truths.
This leads on to the question, how may one prove it? Bearing in mind that any proof of the immutability of logic must have its basis in logic. The question is, how can immutable logic prove istelf to objectively exist?

If we then decide that, possibly, logic is not immutable then where does this leave us? Can we ever make a metaphysical argument without firstly assuming that mathematics and logic are immutable?

[zoro]

Frege, Russell, and Wittgenstein? My condolences! Granted that Frege was brilliant; yet (tragically), he didn't see through Russell's paradox: the barber was a woman! And although Russell was equally brilliant (and I loved the man, even when I was a teenager), yet does it really take more than a massive book to "prove" that there are three feet in a yard?! As for poor Wittgenstein: his trouble, as he admitted, was that "With my full philosophical rucksack, I can only climb slowly up the mountain of mathematics" -- or dig himself out of a semantic hole.

It seems like a bottomless pit. It would be helpful to agree on definitions. My dictionary gives the following for 'logic':

1. reasoning conducted or assessed according to strict principles of validity…
• a particular system or codification of the principles of proof and inference : Aristotelian logic.
• the systematic use of symbolic and mathematical techniques to determine the forms of valid deductive argument.
• the quality of being justifiable by reason…

2. a system or set of principles underlying the arrangements of elements in a computer…

And for 'existence', it gives: "the fact or state of living or having objective reality". If you want to argue about such definitions, take it up with Webster.

As a case in point, you state: "Logic is not about proving the empirical truth of a proposition; it is about the limits and the form of that proposition." Although I certainly agree that logic has nothing to do with empiricism (who said otherwise; exactly how many straw men are in that hole with you?), yet surely only a Wittgenstein disciple would constrain logic only to "the limits and the form of that proposition"! No wonder he had such trouble climbing the mountain -- and you have such trouble digging yourself out of the hole.

Further, in contrast to your claim that I'm "(tacitly) falling back on some metaphysical ontology of what counts as identity", I'm relying on defining existence as any scientist would: an hypothesis whose predictions are to be subjected to experimental tests. That is, as I wrote in an earlier post, I maintain that all of ontology is silliness in the extreme: it's a metaphysical concoction that should be entirely subsumed by phenomenology.

Still another straw man you've dug up is: "you made a point distinguishing existence and the properties of something that exists". You're playing word games: I'm accepting the common (and Webster's) meaning for 'existence' (e.g., of a hole or an elephant), pointing out that the ideas of existence and distinctiveness are at the bases of logic and (pure) math, and that such bases can be violated.

In contrast, you're tromping into another jungle when you address other 'properties' (of a hole or an elephant), such as perimeter, area, mass, height, color, etc. The rest of us know if we have a hole in our jeans or if the elephant is dead. We also agree that the hole in our jeans can increase in size and that, when an elephant is cut in half, the mass of elephant-meat is conserved. Existence and distinctiveness, however, are different: Wittgenstein would probably still be arguing about whether we truly understand what an elephant is, but when the rest of us see a herd of elephants charging us, we adopt the hypothesis that they exist, and (even without experimental tests of that hypothesis), we come to the realization (even using true/false logic) that if we don't move, they're going to trample us into nonexistence (albeit, our body mass will be conserved).

And still another straw man! I did not state: "that [negotiating (?) the world is about probability] does not negate true/false logic". Obviously, true/false logic has great utility for closed systems (such as games, "pure math", religions, etc.) My (simple!) point was that true/false logic has severe limitations for open systems, i.e., for reality. Certainly, true/false logic is useful to examine limiting cases in which we assume, as a premiss, that some proposition is true or false, e.g., that the elephant herd will trample us. But in reality, no such proposition can be demonstrated to be valid (including that proposition). Again, we can't "prove" even that we exist!

And as for the limitations of true/false logic, I mentioned three in my original post. When you finally see the first one, when you finally dig yourself out of your semantic, Wittgenstein-inspired hole, I'd invite you to move on to the next two (dealing with changes and with probabilities).

[Furrowed Brow]

Frege, Russell, and Wittgenstein? My condolences! Granted that Frege was brilliant; yet (tragically), he didn't see through Russell's paradox: the barber was a woman! And although Russell was equally brilliant (and I loved the man, even when I was a teenager), yet does it really take more than a massive book to "prove" that there are three feet in a yard?! As for poor Wittgenstein: his trouble, as he admitted, was that "With my full philosophical rucksack, I can only climb slowly up the mountain of mathematics" -- or dig himself out of a semantic hole.

Ahhh . Of the three I’d rank Wittgenstein as numero uno.

It seems like a bottomless pit. It would be helpful to agree on definitions. My dictionary gives the following for 'logic':
1. reasoning conducted or assessed according to strict principles of validity…
• a particular system or codification of the principles of proof and inference : Aristotelian logic.
• the systematic use of symbolic and mathematical techniques to determine the forms of valid deductive argument.
• the quality of being justifiable by reason…

2. a system or set of principles underlying the arrangements of elements in a computer…

Fair enough. I think we should stick with just 1 for the time being.

And for 'existence', it gives: "the fact or state of living or having objective reality". If you want to argue about such definitions, take it up with Webster.

Nope. Fair enough again.

As a case in point, you state: "Logic is not about proving the empirical truth of a proposition; it is about the limits and the form of that proposition." Although I certainly agree that logic has nothing to do with empiricism (who said otherwise; exactly how many straw men are in that hole with you?), yet surely only a Wittgenstein disciple would constrain logic only to "the limits and the form of that proposition"! No wonder he had such trouble climbing the mountain -- and you have such trouble digging yourself out of the hole.

Ok you spotted my philosophical allegiances. I think we can widen things out for this debate, to include the form of an argument, and the form of the principles of validity. I think this covers Webster point 1, where it does not then I’ll limit myself thus.

Further, in contrast to your claim that I'm "(tacitly) falling back on some metaphysical ontology of what counts as identity", I'm relying on defining existence as any scientist would: an hypothesis whose predictions are to be subjected to experimental tests. That is, as I wrote in an earlier post, I maintain that all of ontology is silliness in the extreme: it's a metaphysical concoction that should be entirely subsumed by phenomenology.

I’ll hang back on the commitment to phenomenology for the moment, and how it subsumes metaphysics. Looks like a book in itself on just that point. Glad to hear ontology is silly. You could be a Wittgensteinian.

Still another straw man you've dug up is: "you made a point distinguishing existence and the properties of something that exists". You're playing word games: I'm accepting the common (and Webster's) meaning for 'existence' (e.g., of a hole or an elephant), pointing out that the ideas of existence and distinctiveness are at the bases of logic and (pure) math, and that such bases can be violated.

I think you are overplaying the straw man defence again. Word games? Maybe. I’d put it the other way and say I’m trying to stop you falling into ontological traps that in reality go no deeper than the sense of words.

In contrast, you're tromping into another jungle when you address other 'properties' (of a hole or an elephant), such as perimeter, area, mass, height, colour, etc. The rest of us know if we have a hole in our jeans or if the elephant is dead. We also agree that the hole in our jeans can increase in size and that, when an elephant is cut in half, the mass of elephant-meat is conserved.

Ok. Define the aspects of two holes that when added together obey rule “1 + 1 = 2’. Now defined the aspects of a hole ( that are not circular and rely on the word hole or a synonym) that obey the rule ‘1 + 1 =1’.

I’m saying hat in the latter case there is nothing on that list, other than the word hole. In fact when you use the word in the context of the latter then the word stops making sense, because it will have no context - everything that allows the word hole to make sense will appear in the list of thing that come under “1 + 1 = 2”. To persevere with the “1 + 1 = 1” argument then requires falling back on some empty metaphysics as much as you think you have avoided doing that. So what you are saying is the argument you are pooh poohing as a straw man. I’m saying that is exactly what your position amounts to if you continue down that road.

Existence and distinctiveness, however, are different: Wittgenstein would probably still be arguing about whether we truly understand what an elephant is, but when the rest of us see a herd of elephants charging us, we adopt the hypothesis that they exist, and (even without experimental tests of that hypothesis), we come to the realization (even using true/false logic) that if we don't move, they're going to trample us into non-existence (albeit, our body mass will be conserved).

How does you example of elephant in that paragraph support the assertion Existence and distinctiveness are different? True you can weigh an elephant, and measure its height etc and all these ways of understanding an elephant are distinct, yet the existence of what an elephant is - is no more than the commutation of these and a further bunch of distinct aspects. But to be guilty of boring you with the same point - sorry - the existence of an elephant is not distinct from all those aspects that define what the elephant is.

And still another straw man! I did not state: "that [negotiating (?) the world is about probability] does not negate true/false logic". Obviously, true/false logic has great utility for closed systems (such as games, "pure math", religions, etc.)

What straw man?

My (simple!) point was that true/false logic has severe limitations for open systems, i.e., for reality.

That was my point. Reality/ or open systems as you like to call them, do not negate truth false logic.

Certainly, true/false logic is useful to examine limiting cases in which we assume, as a premise, that some proposition is true or false, e.g., that the elephant herd will trample us. But in reality, no such proposition can be demonstrated to be valid (including that proposition).

But we don’t use reality to demonstrate the validity of an argument/proposition. We use logic. This is why I suspect you are conflating logic with theory of knowledge. Reality tells us whether our propositions are true or false.

Or do you mean something else here? Zoro I’m not trying to pin you to straw men arguments, but I am trying to draw out what you mean, and the implications of what you mean. You seem pretty confident you have it worked out, and I’m being a tad obtuse. From where I’m sitting you’re in muddle and don’t know it. But I really do not want to misrepresent. So if you meant something different then please advise. However, it seems to me that you’re dismissal of the semantics of the problem as word games is too hasty.

Again, we can't "prove" even that we exist!

No - because you can’t use logic to prove existence. And I’m not committed to proving we exist. I’m arguing that the rules we argue and think by are immutable.

And as for the limitations of true/false logic, I mentioned three in my original post. When you finally see the first one, when you finally dig yourself out of your semantic, Wittgenstein-inspired hole, I'd invite you to move on to the next two (dealing with changes and with probabilities).

I’ll take a peek. But as much as you retreat from Wittgenstein, the deeper you slip into the hole you are accusing me. I’m out of the hole because when I do my sums I get ‘1 + 1 = 2’.

[Furrowed Brow]

Frege, Russell, and Wittgenstein? My condolences! Granted that Frege was brilliant; yet (tragically), he didn't see through Russell's paradox: the barber was a woman! And although Russell was equally brilliant (and I loved the man, even when I was a teenager), yet does it really take more than a massive book to "prove" that there are three feet in a yard?! As for poor Wittgenstein: his trouble, as he admitted, was that "With my full philosophical rucksack, I can only climb slowly up the mountain of mathematics" -- or dig himself out of a semantic hole.

Ahhh . Of the three I’d rank Wittgenstein as numero uno.

It seems like a bottomless pit. It would be helpful to agree on definitions. My dictionary gives the following for 'logic':
1. reasoning conducted or assessed according to strict principles of validity…
• a particular system or codification of the principles of proof and inference : Aristotelian logic.
• the systematic use of symbolic and mathematical techniques to determine the forms of valid deductive argument.
• the quality of being justifiable by reason…

2. a system or set of principles underlying the arrangements of elements in a computer…

Fair enough. I think we should stick with just 1 for the time being.

And for 'existence', it gives: "the fact or state of living or having objective reality". If you want to argue about such definitions, take it up with Webster.

Nope. Fair enough again.

As a case in point, you state: "Logic is not about proving the empirical truth of a proposition; it is about the limits and the form of that proposition." Although I certainly agree that logic has nothing to do with empiricism (who said otherwise; exactly how many straw men are in that hole with you?), yet surely only a Wittgenstein disciple would constrain logic only to "the limits and the form of that proposition"! No wonder he had such trouble climbing the mountain -- and you have such trouble digging yourself out of the hole.

Ok you spotted my philosophical allegiances. I think we can widen things out for this debate, to include the form of an argument, and the form of the principles of validity. I think this covers Webster point 1, where it does not then I’ll limit myself thus.

Further, in contrast to your claim that I'm "(tacitly) falling back on some metaphysical ontology of what counts as identity", I'm relying on defining existence as any scientist would: an hypothesis whose predictions are to be subjected to experimental tests. That is, as I wrote in an earlier post, I maintain that all of ontology is silliness in the extreme: it's a metaphysical concoction that should be entirely subsumed by phenomenology.

I’ll hang back on the commitment to phenomenology for the moment, and how it subsumes metaphysics. Looks like a book in itself on just that point. Glad to hear ontology is silly. You could be a Wittgensteinian.

Still another straw man you've dug up is: "you made a point distinguishing existence and the properties of something that exists". You're playing word games: I'm accepting the common (and Webster's) meaning for 'existence' (e.g., of a hole or an elephant), pointing out that the ideas of existence and distinctiveness are at the bases of logic and (pure) math, and that such bases can be violated.

I think you are overplaying the straw man defence again. Word games? Maybe. I’d put it the other way and say I’m trying to stop you falling into ontological traps that in reality go no deeper than the sense of words.

In contrast, you're tromping into another jungle when you address other 'properties' (of a hole or an elephant), such as perimeter, area, mass, height, colour, etc. The rest of us know if we have a hole in our jeans or if the elephant is dead. We also agree that the hole in our jeans can increase in size and that, when an elephant is cut in half, the mass of elephant-meat is conserved.

Ok. Define the aspects of two holes that when added together obey rule “1 + 1 = 2’. Now defined the aspects of a hole ( that are not circular and rely on the word hole or a synonym) that obey the rule ‘1 + 1 =1’.

I’m saying hat in the latter case there is nothing on that list, other than the word hole. In fact when you use the word in the context of the latter then the word stops making sense, because it will have no context - everything that allows the word hole to make sense will appear in the list of thing that come under “1 + 1 = 2”. To persevere with the “1 + 1 = 1” argument then requires falling back on some empty metaphysics as much as you think you have avoided doing that. So what you are saying is the argument you are pooh poohing as a straw man. I’m saying that is exactly what your position amounts to if you continue down that road.

Existence and distinctiveness, however, are different: Wittgenstein would probably still be arguing about whether we truly understand what an elephant is, but when the rest of us see a herd of elephants charging us, we adopt the hypothesis that they exist, and (even without experimental tests of that hypothesis), we come to the realization (even using true/false logic) that if we don't move, they're going to trample us into non-existence (albeit, our body mass will be conserved).

How does you example of elephant in that paragraph support the assertion Existence and distinctiveness are different? True you can weigh an elephant, and measure its height etc and all these ways of understanding an elephant are distinct, yet the existence of what an elephant is - is no more than the accumulation of these and a further bunch of distinct aspects. But to be guilty of boring you with the same point - sorry - the existence of an elephant is not distinct from all those aspects that define what the elephant is.

And still another straw man! I did not state: "that [negotiating (?) the world is about probability] does not negate true/false logic". Obviously, true/false logic has great utility for closed systems (such as games, "pure math", religions, etc.)

What straw man?

My (simple!) point was that true/false logic has severe limitations for open systems, i.e., for reality.

That was my point. Reality/ or open systems as you like to call them, do not negate truth false logic.

Certainly, true/false logic is useful to examine limiting cases in which we assume, as a premise, that some proposition is true or false, e.g., that the elephant herd will trample us. But in reality, no such proposition can be demonstrated to be valid (including that proposition).

But we don’t use reality to demonstrate the validity of an argument/proposition. We use logic. This is why I suspect you are conflating logic with theory of knowledge. Reality tells us whether our propositions are true or false.

Or do you mean something else here? Zoro I’m not trying to pin you to straw men arguments, but I am trying to draw out what you mean, and the implications of what you mean. You seem pretty confident you have it worked out, and I’m being a tad obtuse. From where I’m sitting you’re in muddle and don’t know it. But I really do not want to misrepresent. So if you meant something different then please advise. However, it seems to me that you’re dismissal of the semantics of the problem as word games is too hasty.

Again, we can't "prove" even that we exist!

No - because you can’t use logic to prove existence. And I’m not committed to proving we exist. I’m arguing that the rules we argue and think by are immutable.

And as for the limitations of true/false logic, I mentioned three in my original post. When you finally see the first one, when you finally dig yourself out of your semantic, Wittgenstein-inspired hole, I'd invite you to move on to the next two (dealing with changes and with probabilities).

I’ll take a peek. But as much as you retreat from Wittgenstein, the deeper you slip into the hole you are accusing me. I’m out of the hole because when I do my sums I get ‘1 + 1 = 2’.

[zoro]

Your first challenge was "Define the aspects of two holes that when added together obey rule “1 + 1 = 2’. Now define the aspects of a hole ( that are not circular and rely on the word hole or a synonym) that obey the rule ‘1 + 1 =1’."

That's simple: in the first case, they're distinct; in the second, they're not.

That was the illustration of a part of my first point. The other part of my first point was that for simple logic to apply, the subjects must continue to exist. For that, my example was: cut an elephant in half -- an example that you seem to be studiously avoiding.

But apparently (with your next paragraph), you want still more for the definition of 'hole'. Okay. As Wittgenstein said: "the meaning of a word is its use in the language." Below, I provide three such uses -- used by people of different ages.

1. When I was a kid, other kids would taunt me with, "Look at the hole in his pants." When I'd get another hole in my pants adjacent to the first, they'd still taunt me with "Look at the hole in his pants." That experience taught me one of the limitation of Aristotelian logic: if things don't remain distinct, then 1 + 1 can equal 1.

2. For some youngsters (such as yourself), it's easy to blow a 'hole' in their semantic arguments by pulling out a dictionary -- especially if they're disciples of Wittgenstein and you show them a word's usage.

3. For an oldster such as I, well, how about this use: one would need to have a 'hole' in one's head to waste one's limited remaining time on such semantic arguments.

You next ask: "How does your example of elephant… support the assertion [that?] existence and distinctiveness are different." Hello? Go back to the original elephant. Cut it in two. Then how many elephants exist? Thus, two adjacent holes in your jeans exist (as a hole) but they're no longer distinct; two halves of elephants are distinct but no elephant continues to exist.

Again, the bases of Aristotelian logic are: 1) things exist, and 2) they are distinct. Those are two separate premisses.

You add "the existence of an elephant is not distinct from all those aspect that define what the elephant is", but I couldn't hear the rest of what you were saying -- over all the groans from the grave of poor old Wittgenstein.

You ask "what straw man". You stated "that… does not negate true/false logic." My response: nobody but you suggested that it did.

You then repeat: "Reality/ or open systems as you like to call them, do not negate [true/false] logic." Again: nobody said they did -- my point was (once again, for how many times?!): in reality, true/false logic has severe limitations -- not least of which is that we can never determine if any premiss is "true".

Ah -- finally, you revealed your point: "I'm arguing that the rules we argue and think by are immutable." Well, fine -- provided, however, we stick to the rules! For example, once again, be careful with the rules of true/false logic, because if you're not careful, then your reasoning will fail in multiple ways, e.g., 1. because you violated the conditions that things exist (e.g., two half elephants do not an elephant make) and that they are distinct (e.g., when two Black Holes merge), 2. because you violated the condition (of simple, true/false logic) that things don't change, or 3. because you assumed something is true -- which can't be validated. If you want to include such conditions in your unspecified term "immutable", then fine.

Finally, re. your final line, I'm sorry but 1 + 1 = 2 is not immutable. It's true only when the assumed conditions of logic aren't violated. I shudder to show you another example, but to help you get out of the hole that you seem to be stuck in, here goes.

Take one wad of "silly putty" (or "play dough" or similar) in one hand and another wad of the same stuff in your other hand. Now, add the two together, by smashing together the two wads of silly putty. Now, how many wads of silly putty do you have? Answer: 1 + 1 = 1.

[zoro]

Your first challenge was "Define the aspects of two holes that when added together obey rule “1 + 1 = 2’. Now define the aspects of a hole ( that are not circular and rely on the word hole or a synonym) that obey the rule ‘1 + 1 =1’."

That's simple: in the first case, they're distinct; in the second, they're not.

That was the illustration of a part of my first point. The other part of my first point was that for simple logic to apply, the subjects must continue to exist. For that, my example was: cut an elephant in half -- an example that you seem to be studiously avoiding.

But apparently (with your next paragraph), you want still more for the definition of 'hole'. Okay. As Wittgenstein said: "the meaning of a word is its use in the language." Below, I provide three such uses -- used by people of different ages.

1. When I was a kid, other kids would taunt me with, "Look at the hole in his pants." When I'd get another hole in my pants adjacent to the first, they'd still taunt me with "Look at the hole in his pants." That experience taught me one of the limitation of Aristotelian logic: if things don't remain distinct, then 1 + 1 can equal 1.

2. For some youngsters (such as yourself), it's easy to blow a 'hole' in their semantic arguments by pulling out a dictionary -- especially if they're disciples of Wittgenstein and you show them a word's usage.

3. For an oldster such as I, well, how about this use: one would need to have a 'hole' in one's head to waste one's limited remaining time on such semantic arguments.

You next ask: "How does your example of elephant… support the assertion [that?] existence and distinctiveness are different?" Hello? Go back to the original elephant. Cut it in two. Then how many elephants exist? Thus, two adjacent holes in your jeans exist (as a hole) but they're no longer distinct; two halves of elephants are distinct but no elephant continues to exist.

Again, the bases of Aristotelian logic are: 1) things exist, and 2) they are distinct. Those are two separate premisses.

You add "the existence of an elephant is not distinct from all those aspect that define what the elephant is", but I couldn't hear the rest of what you were saying -- over all the groans from the grave of poor old Wittgenstein.

You ask "what straw man?" You stated "that… does not negate true/false logic." My response: nobody but you suggested that it did.

You then repeat: "Reality/ or open systems as you like to call them, do not negate [true/false] logic." Again: nobody said they did -- my point was (once again, for how many times?!): in reality, true/false logic has severe limitations -- not least of which is that we can never determine if any premiss is "true".

Ah -- finally, you revealed your point: "I'm arguing that the rules we argue and think by are immutable." Well, fine -- provided, however, we stick to the rules! For example, once again, be careful with the rules of true/false logic, because if you're not careful, then your reasoning will fail in multiple ways, e.g., 1. because you violated the conditions that things exist (e.g., two half elephants do not an elephant make) and that they are distinct (e.g., when two Black Holes merge), 2. because you violated the condition (of simple, true/false logic) that things don't change, or 3. because you assumed something is true -- which can't be validated -- rather than work with probabilities. If you want to include such conditions in your unspecified term "immutable", then fine.

Finally, re. your final line, I'm sorry but 1 + 1 = 2 is not immutable. It's true only when the assumed conditions of logic aren't violated. I shudder to show you another example, but to help you get out of the hole that you seem to be stuck in, here goes.

Take one wad of "silly putty" (or "play dough" or similar) in one hand and another wad of the same stuff in your other hand. Now, add the two together, by smashing together the two wads of silly putty. Now, how many wads of silly putty do you have? Answer: 1 + 1 = 1.

[Furrowed Brow]

Your first challenge was "Define the aspects of two holes that when added together obey rule “1 + 1 = 2’. Now define the aspects of a hole ( that are not circular and rely on the word hole or a synonym) that obey the rule ‘1 + 1 =1’."

That's simple: in the first case, they're distinct; in the second, they're not.

That was the illustration of a part of my first point.

Woh! You’ve just whizzed by the question and repeated the same muddle. Let me start the list for you then:

“1 + 1 = 2”: stuff that accumulates includes: area, boundary, the number of exposed frayed denim threads etc

“1 + 1 = 1”: stuff that does not accumulate includes: the word hole.

My point: and I’m repeating myself again, is that all the stuff that gives meaning to the word “hole” appears on the first list. So what is appearing on the second list? Answer = nothing but a bunch of letters that when stood togther look like the word “hole“. But it can't really be the word because applied here the word hole no longer has meaning.

The other part of my first point was that for simple logic to apply, the subjects must continue to exist. For that, my example was: cut an elephant in half -- an example that you seem to be studiously avoiding.

There’s an elephant in the room eh? OK. Not avoiding. Just trying to hit one target at a time. Wittgenstein once advised Russell that philosophy should be done slowly. The sentence I put in red is a reminder of the kind of thing I think he meant. I’ll get on to elephants below.

But apparently (with your next paragraph), you want still more for the definition of 'hole'. Okay. As Wittgenstein said: "the meaning of a word is its use in the language."

First off. This quote is taken from the Philosophical Investigations. Wittgenstein’s work can be characterised into two distinct periods relay and late. The method and project of the later work being different to the more rigorous and formal early period. But let me just complete that quote for you if I may.

43. For a large class of cases-though not for all-in which we employ the word “meaning” it can be defined thus: the meaning of a word is its use in the language.

Notice the qualification of not all. We are playing one specific type of language game. How do our words connect to reality. So what gives our words physical meaning? If you can add something to the list above under the heading “1 + 1 = 1” then that give a physical meaning the word hole then we have a debate.

The muddle you in is that you at heart are an empiricist. The concept of hole seems to fail a reality check of merging holes. So you include the true/false distinction fails. Whereas what has really failed is a top down view of what counts as a category or identity. Follow through on that empiricism, then our words mean no more than what they can mean. And by mean read - refer to.

Below, I provide three such uses -- used by people of different ages.

1. When I was a kid, other kids would taunt me with, "Look at the hole in his pants." When I'd get another hole in my pants adjacent to the first, they'd still taunt me with "Look at the hole in his pants." That experience taught me one of the limitation of Aristotelian logic: if things don't remain distinct, then 1 + 1 can equal 1.

Wrong lesson to still be carrying with you into old age. Move on. It should now be teaching you to look at the nature and form of the language and how you are applying it. The limitation is that of top down abstractions -I.e. assuming a category/identity itself beyond the physical reality it refers to. In short there is not abstract category hole. Which you have proved by the hole in your pants. But that is all you have proved.

2. For some youngsters (such as yourself), it's easy to blow a 'hole' in their semantic arguments by pulling out a dictionary -- especially if they're disciples of Wittgenstein and you show them a word's usage.

What is the area of that hole? Or are you playing a different language game here. Of course you are. The point is that the early Wittgenstein was focused on working out a theory of reference. Your example of “1 + 1 = 1” relies on an implicit theory of reference. When we analyse language that way, we get into true/false logic- you can‘t avoid it. But language is mercurial and can be used other ways, but we’re playing the reference game- so we‘re into true/false logic. And I’m pointing out you’re breaking the rules of that game.

3. For an oldster such as I, well, how about this use: one would need to have a 'hole' in one's head to waste one's limited remaining time on such semantic arguments.

Well as this appears to be a metaphorical hole in the head then I shall assume you are not referring to anything in reality, and thus your words are meaningless - that is meaningless within the language game of reference and logic.

You next ask: "How does your example of elephant… support the assertion [that?] existence and distinctiveness are different?" Hello? Go back to the original elephant. Cut it in two. Then how many elephants exist?

That’s one dead elephant in two halves. I think the point you are making has its roots in the sorties paradox. How much of an elephant do you have to slice off until it stops being an elephant. So you cut off its trunk - is it still an elephant? That kind of thing. Yes?

My reply is that this is real words games. What you area accusing my criticism of playing, is your own failing. You have a huge blind spot here and I’m doing my best to draw your attention to it.

So we physically cut an elephant in half. Have we cut the concept elephant in half? Well no. But what is the natural habitat of that concept. Well - natural language. But we are playing a strange philosophical game here. By cutting the elephant in half the sum total is - one dead elephant - or is it pining for the savannah.

Thus, two adjacent holes in your jeans exist (as a hole) but they're no longer distinct;

No problem with that at all. But what makes them distinct? A concept of identity, or the contours of the denim. Your addition of distinct identities relies as an example of the failure of true false, again - repeating myself now - proves the failure of an abstract concept of identity.

two halves of elephants are distinct but no elephant continues to exist.

The bits of the elephant still exist in two halves. The elephant is dead (appropriate shift of language use), the concept elephant has not been cut in half, because the concept is a generalisation based on many elephants. (That generalisation being an empirical generalisation and not an abstraction into metaphysical classes. The nearest you get is, that thing looks like that thing looks like….we’ll call that an elephant then).

Again, the bases of Aristotelian logic are: 1) things exist, and 2) they are distinct. Those are two separate premises.

Yep. But that is why when you follow through on that kind of thinking you end up with some kind of logical atomism. Stuff exists, and is distinct, but when you start chopping reality up you find - when one is seeking to understand the logic of how were refer to the world- that our concepts are atomic, viz., built up from the world. What they are not are universal, or abstract classes imposed upon the world.

Finally, re. your final line, I'm sorry but 1 + 1 = 2 is not immutable. It's true only when the assumed conditions of logic aren't violated.

What conditions are you assuming? I have provided some clear points that lays down some conditions of logic that do not violate “1 + 1 = 2”. Who is now arguing against a straw man?

[zoro]

Amazing! You're still stuck in your metaphysical hole. You state:

“1 + 1 = 2”: stuff that accumulates includes: area, boundary, the number of exposed frayed denim threads etc

“1 + 1 = 1”: stuff that does not accumulate includes: the word hole.

My response is: try it with the two pieces of silly putty -- as in the example that you seem (conveniently?) to ignore.

You state that you're "not avoiding" (the elephant in the room) and that you're "just trying to hit one target at a time". My response: you can't seem to see the forest not only for the trees but for all the cobwebs on every single leaf and every single twig on every branch.

I'll take your word for it that "Wittgenstein once advised Russell that philosophy should be done slowly", but I'm sure thankful that Russell never took that advice: I wouldn't trade you 1000 Wittgensteins for one Russell!

You state: "If you can add something to the list above under the heading "1 + 1 = 1" then[?] that give[s?] a physical meaning [to?] the word hole, then we have a debate." Get over it! Get out of the hole that's causing you so much trouble. Surely you can find yourself some "silly putty" or "play dough" or similar -- or even, get yourself just two handfuls of dirt!

You state: "The muddle you [are?] in is that you at heart are an empiricist." Right re. the "empiricist"; wrong re. any "muddle". Look: since "hole" causes you so much difficulty, then forget about it. Never bring it up again. From here on, whenever you think you see the word "hole" read it as "a piece of silly putty" (and similar).

For example, read your next sentence "The concept of hole seems to fail a reality check of merging holes" as "The concept of a piece of silly putty seems to fail a reality check of merging pieces of silly putty." Now, to see your error, correct your sentence to read: "The concept of a piece of silly putty fails a reality check of merging pieces of silly putty" -- although it would be more informative if you wrote your sentence something similar to: "I agree with you: when two pieces of silly putty are "smooshed together" [as my grandchildren describe the process!], then using the symbol "1" to mean something exists and is distinct (not to mean mass, volume, number of atoms or any other property characterized by units or dimensions), then 1 + 1 = 1.

Re. your comments on my three examples of use of the word 'hole', I'll respond as follows.

1. You state that it was "the wrong lesson" -- obviously I disagree. You state "It should now be teaching you to look at the nature and form of the language and how you are applying it." Your silly statement brings to mind something that Popper wrote:

“In science, we take care that the statements we make should never depend upon the meaning of our terms. Even where the terms are defined, we never try to derive any information from the definition, or to base any argument upon it. This is why our terms make so little trouble. We do not overburden them. We try to attach to them as little weight as possible. We do not take their ‘meaning’ too seriously. We are always conscious that our terms are a little vague (since we have learnt to use them only in practical applications) and we reach precision not by reducing their penumbra of vaugueness, but rather by keeping well within it, by carefully phrasing our sentences in such a way that the possible shades of meaning do not matter. This is how we avoid quarrelling about words.

“Our ‘scientific knowledge’, in the sense in which this term may be properly used, remains entirely unaffected if we eliminate all definitions; the only effect is upon our language, which would lose, not precision, but merely brevity… There could hardly be a greater contrast than that between this view of the part played by definitions, and Aristotle’s view. For Aristotle’s essentialist definitions [i.e., in which a word is burdened with “capturing the essence” of some thing or process] are [imagined to be] the principles from which all our knowledge is derived; they thus [are imagined to] contain all our knowledge; and they [are imagined to] serve to substitute a long formula for a short one. As opposed to this… scientific… definitions do not contain any knowledge whatever, nor even any ‘opinion’; they do nothing but introduce new arbitrary shorthand labels; they cut a long story short.

“The problem of definitions and of the ‘meaning of terms’ is the most important source of Aristotle’s regrettably still prevailing intellectual influence, of all that verbal and empty sholasticism that haunts not only the Middle Ages, but our own contemporary philosophy; for even a philosophy as recent as that of L. Wittgenstein suffers… from this influence. The development of thought since Aristotle could, I think, be summed up by saying that every discipline, as long as it used the Aristotelian method of definition has remained arrested in a state of empty verbiage and barren scholasticism, and that the degree to which the various sciences have been able to make any progress depended on the degree to which they have been able to get rid of this essentialist method…"

2. You ask for the area of that hole. I left it as "an exercise for the reader" to see that the hole in your semantic argument could be substantially enlarged by showing a Wittgenstein disciple word usage -- but of course, it would still be one hole, as in 1 + 1 = 1.

3. Good heavens. How could a Wittgenstein disciple possibly refuse to acknowledge metaphors?!

You state (re. cutting an elephant in half): "I think the point you are making has its roots in the sorties paradox. How much of an elephant do you have to slice off until it stops being an elephant… That kind of thing. Yes?"

My response: No, not at all! I said cut the elephant in half! You then add: "My reply is that this a real word games." And then you go on with another straw man argument. So again, No! Cut the damn elephant in half!

And we're not "playing a strange philosophical game here" -- you are! I was trying to provide OccamsRazor with responses to his questions. Then, some Wittgenstein interloper by the name of Furrowed Brow leaped in, to attack with his "empty verbiage and barren scholasticism" -- and unfortunately for me, I've wasted my time trying to straighten him out. But no more -- for although, if I wasted still more of my time, then I would still have only a single hole in my head (as in 1 + 1 = 1), the hole would then be so large that I'd need to hold my head in shame.

[Confused]

Logic is driven by facts.

That is what logic is not. Logic is the rules by which we argue and draw conclusions.

Let me use the Propositional calculus as an example. This form of logic is equivalent to Boolean algebra and is reducible to truth tables. Keeping things simple the following line of argument is what goes on in the propositional calculus.

Premise 1. P then Q.
Premise 2. P
Therefore: Q

So given P then Q, and give P we can validly deduce Q. This line of reasoning is taken to be universally valid. It is a rule of thinking straight. As a set of rules it is empty of factual content. What we do is use the rule to try and think straight about the facts.

So an argument following the above form might be...

P = 'Socrates is a man'
Q = 'Socrates is mortal'.

Premise 1: If Socrates is a man then he is mortal
Premise 2: Socrates is a man.
Therefore: Socrates is mortal.

Now if someone argued "If Socrates is man then he is mortal, and Socrates is a man, therefore he is not mortal" - then you'd be saying hang on just one darn second that is illogical. You're just a crazy guy. You'd be saying that because that crazy guy would be organising the facts in an invalid way. The rules of inference determining the validity of the arguments, not the facts.

I might redefine this a bit and say that inductive logic is ruled by reasoning leading to a high probability of truth whereas deductive logic is ruled by reasoning leading to facts that constitute proof.

[Confused]

OccamsRazor: If you'd permit an analogy: while wandering, I stumbled across this loose thread on logic but was tripped up by quite a few frayed strands; with this post, perhaps I can help tie up a few loose ends. I should mention, however, that my similar undertaking (explicitly for my granddaughter) required literally hundreds of pages. A portion of the results of that effort is posted in Part 2 of my (free!) on-line book at http://zenofzero.net/ ; see especially Chapter Ib2 entitled "Some Basic Ideas about Logic" and Chapters Ih & Ii (dealing with hypotheses and probabilities); also, in Chapter 1a, I include some (derogatory!) comments about ontology (i.e., the theory of existence). Here, my plan is "just" to try to respond to questions and comments that you raised in your original post.

You quote Frayn's: "Logic is just a system we have made up, not an inherent condition of the natural world." That's nonsense! The simplest logic, Aristotelian logic, is based on two scientific results discovered by fish, monkeys, and eventually people. Subsequently, these results have been tested so many times that most people have given up trying to demonstrate that they're wrong. Now, therefore, they're called scientific "principles" (or "laws" of logic, where I put the word "law" in quotations, because nothing in science is firm enough to be called a "law"; in particular, as I'll illustrate below, there are cases where such "laws" fail).

As you no doubt know, these two scientific principles (or logical "laws") are: 1) things exist (written as A [sign unavailable showing a "three-legged equal sign"] A, read "A is identically equal to A", and called "the [logical] law of identity") and 2) things are distinct (sometime written as A [same sign with a line through it] ~A, always read as "A is not identically equal to not-A", and sometimes called "the law of contradiction" -- although it would be better described as the "principle of non-contradiction"). [I'm going to skip "the law of the excluded middle", because it has separate problems (mentioned by another poster, addressed by Bertrand Russell, and basically arising from using a member of set to define the set's boundaries).] Thus, in response to Frayn and to your question (arising from your quotation from McCulloch), "Could a being outside our own manifestation of material reality not create such logical impossibilities?", I'd say: Insofar as any "being" found itself confronted with the observations that "things exist and are distinct", then such a being will find that the "laws" of logic apply.

But there are multiple problems with your question "Can we ever make a metaphysical argument without firstly assuming that mathematics and logic are immutable?" First, realize that "basic logic" (i.e., Aristotelian logic, based on the principles mentioned above) and therefore "pure mathematics" (e.g., algebra) have severe restrictions: 1) they apply only when things exist and are distinct, 2) they apply only to "immutable" (i.e., non-changing) entities, and 3) they apply only when the only two possibilities are "true" vs. "false".

Let me (briefly!) illustrate those three limitations. 1) Suppose you have one hole in your jeans and then get a second hole adjacent to it (or suppose one Black Hole merges with another); then 1 + 1 = 1 (because the holes violate the condition that "things are distinct"). 2) She said: "They won't let people into the pub until they're 18; I'm not 18, so I can't meet you there on Thursday" -- but by Thursday, she had turned 18. (And yes, I agree that's a weak example; let me put it this way: the whole of differential calculus was developed to be able to address changes). 3) Similarly, the restriction of Aristotelian logic to "true" vs. "false" has been overcome with probability theory -- but actually, there's something here that's even more significant for any "metaphysical" argument, which I should at least mention.

Secondly, Aristotelian logic and therefore "pure mathematics" deal only with what physicists call "closed systems", such as games, "pure mathematics", comic books, and all religions! It's only for such systems that "true" vs. "false" have meaning (e.g., within such systems, it IS "true" that "three strikes and you're out", "1 + 1 = 2", "Superman is weakened by kryptonite", "Jesus is the son of God", and "Gabriel relayed Allah's messages to Muhammad"). But in reality, in what are called "open systems", we can never find a statement that can be labeled "true" -- including that statement! Thus, we can't be certain (viz., probability exactly unity) that momentum is always conserved, that E = mc^2, etc., although the associated probabilities seems to be very close to unity -- and a Nobel prize is waiting for anyone who shows that they're false! Similarly (and Descartes' silliness notwithstanding), we can't be certain even that we exist (although as I try to demonstrate in my book, the probability for the hypothesis that we exist seems to differ from unity only by about 1 part in 10^25); we may be just simulations in some god's computer game! And similarly, we can't be certain that there is no god -- although, again as I try to demonstrate in my book, the probability that there is any god (or ever was one) seems to be extremely small (somewhere in the range from about 1 chance in 10^100 to 1 chance in 10^1000).

In summary, then, I hold that we can have "meaningful" metaphysical arguments (although I'd prefer the term "philosophical arguments") -- provided that we couch our statements in terms of probabilities and that we not constrain our ideas with the inherent limitations of Aristotelian logic. In particular (and all the silly "proofs" of God's existence that have appeared during the past ~2500 years notwithstanding), one can never "prove" the existence of God (or the existence of anything!) using logic -- since one of the fundamental premisses of such logic is that "things exist". Therefore, any such "proof" contains its conclusion as a premiss. Instead, any "existence" must be treated as an hypothesis -- whose probability of validity can be determined only via the scientific method. Stated differently: to progress, ontology must be subsumed by phenomenology.

Just a few comments here. Your lengthy explaination of the simple principle of A is A and cannot be non-A is simple logic, yes. It can be reworded to say reality is reality or reality cannot be non-reality. When applying them to true mathematics, the problem I usually find is that you are expressing theoretical as opposed to empirical. Very few things can ever be true vs false except relative to the perspective viewing it.

Your example of the jeans is misleading because you fail to give all the information to explain your postulation that 1+1=1. And last I checked when using logic, one usually attempts to avoid the metaphysical and rely on empirical because in logic, inductive reasoning has many flaws (ie your Aristotlean logic being overcome by the probability theory).

In regards to your closed system I can add little because most of my information on closed systems is in regards to social evolution and genetics. But what I do know of it in regards to physics is that it is mostly applied towards entropy (S=K log W). The 2nd law of thermodynamics mainly associates that systems have an overwhelming tendency to be in a high entopy state because there exists so many ways such states can be realized. Once in a high entropy state, the system wants to stay there because it is a natural state. However, for our universe, the big bang started it off in a low entropy state mostly theorized due to gravity (Brian Greene: Fabric of the Cosmos). However in a closed system Entopy can also decrease to lead to more order with the addition of energy (ex: suns energy on universe or the energy to provide to make your bed). In application to logic, we see that using logic, entropy should increase however, by manipulation, we see lthat entopy decreases. Logic applied in a closed system can be inductive to deductive and still yeild high probabilites or proofs. In regards to an open system: you will have to clarify to me what you consider an open system. Religion? God? Superman? With inductive logic one can still show with a high probability the truth or false of each of these concepts. My way of using logic implies that deductive logic belongs in the realm of empirical while inductive in the realm of metaphysical.

In regards to your "philisophical arguments" I prefer debates, for that is what they are. And despite your final paragraph, logic isn't only applied to "proof". That would consist of deductive logic. In inductive logic we can make rational reasoning to a high degree of probability, though not as high as deductive logic, to consider something a hypothesis, rather than a theory such as with deductive logic.

Was your name Zorro in a previous life?

[Confused]

Well, until you take the time to teach yourself the difference between a "pure" number and a number with dimensions -- until you can differentiate between existence and properties of something that exists -- then it seems that my best choice is to leave you in the hole, digging yourself deeper and deeper.

Are you sure you weren't Zorro. Your arguements seem to be taking on the same demeaning and rude tone with the same irrationality. The properties of something that exists is usually what is used to define its existence. However, having properties of something hypothesized to exist doesn't necessarily constitute proof of its existence. For example: God. We have mutliple books from multiple authors to attest to the existence of God. However, that proof may define the existence of God, it doesn't define Gods existence.