Thread: Is Logic Necessarily Infallible?

Originally Posted by thegnome54

"IF
p is q
AND
q is a
THEN
p is a"

If that's what you mean, then that is a logical process.

So what is this process aimed at then? You said logic/math is not a process but then you say it is here.
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Originally Posted by O'nus

So what is this process aimed at then? You said logic/math is not a process but then you say it is here.
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That is not 'math', it is 'a process using math'. The aim of the process is to prove that p is a. It uses math, but it's not 'math', it's an application of it.

Originally Posted by thegnome54

That is not 'math', it is 'a process using math'. The aim of the process is to prove that p is a. It uses math, but it's not 'math', it's an application of it.

So, the following is a process using math:
2+2=4
(2+2=4 is = to if p and q, then y)

And you are saying that it is not an application of math and that it is not math.

So, how can I have math all on its own? Can you demonstrate?
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Originally Posted by O'nus

So, how can I have math all on its own? Can you demonstrate?
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Math is the language - the numbers, symbols, and rules.

It's like English versus a sentence. The sentence is written in English, but the sentence has a purpose, while English itself is simply a system of communication.

These rules are infallible...

http://lc.brooklyn.cuny.edu/smarttutor/logic/rules.html

Originally Posted by thegnome54

Math is the language - the numbers, symbols, and rules.

It's like English versus a sentence. The sentence is written in English, but the sentence has a purpose, while English itself is simply a system of communication.

I totally agree. Humans made up the symbols, but the mathematical/logical rules the symbols are used to represent are facts.

Originally Posted by Universal Mind

These rules are infallible...

Yes, but only within the framework of logic - within the mindset which allows for 'statements' to be 'true' or 'false' - these are human constructs which are not necessarily reflective of the way the universe really works.

Originally Posted by thegnome54

Math is the language - the numbers, symbols, and rules.

It's like English versus a sentence. The sentence is written in English, but the sentence has a purpose, while English itself is simply a system of communication.

So math is the symbols 2 and 2 and that adding them together would be called addition. The process "2 and 2 is 4" is called..?
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Originally Posted by O'nus

So math is the symbols 2 and 2 and that adding them together would be called addition. The process "2 and 2 is 4" is called..?
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That's known as an 'expression', which uses math to express a logical truth.

Originally Posted by thegnome54

Yes, but only within the framework of logic - within the mindset which allows for 'statements' to be 'true' or 'false' - these are human constructs which are not necessarily reflective of the way the universe really works.

Are you talking about the language symbols or what the symbols are used to represent? What are you saying are human constructs? I think those rules were true long before there was life on Earth.

Originally Posted by thegnome54

That's known as an 'expression', which uses math to express a logical truth.

But math and logic are the same..?
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Originally Posted by O'nus

But math and logic are the same..?
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I'm flabbergasted that this many posts have taken place to try to define for you what "math" means. Do you really not know, or are you trying to make some obscure point we aren't getting?

Perhaps you should take a few days to do a little reading and come back when the conversation can continue on topic.

Originally Posted by skysaw

I'm flabbergasted that this many posts have taken place to try to define for you what "math" means. Do you really not know, or are you trying to make some obscure point we aren't getting?

Perhaps you should take a few days to do a little reading and come back when the conversation can continue on topic.

I obviously have an idea of what math, logic, and science are. But I appreciate the facetious implications.

Further, math and logic and analguous so I was under the impression that it was on topic considering they are paralell.

My questioning is to show the vague definitions being used for magic and logic. I do not think logic is being properly represented in this thread and the preceeding posts have demonstrated that.

When considering an investigation to the fallibility of something, consider that your semantic definitions may be the very culprit for the implausabile fallibility.
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Originally Posted by O'nus

I obviously have an idea of what math, logic, and science are. But I appreciate the facetious implications.

Originally Posted by O'nus

But math and logic are the same..?

The answer to this question is "no." This answer appears many times throughout this thread, both implicitely and explicitely. The semantic definitions have been nailed down and beaten to death with an oversized mallet. Nobody thinks math and logic are the same thing, and no one has implied it, save possibly you.

Originally Posted by skysaw

The answer to this question is "no." This answer appears many times throughout this thread, both implicitely and explicitely. The semantic definitions have been nailed down and beaten to death with an oversized mallet. Nobody thinks math and logic are the same thing, and no one has implied it, save possibly you.

Math and logic are fundamentally the samething. Let me demonstrate:

2+2=4
If P and P then Y
2+3=5
If P and R then L

I will now have to fully illustrate the problems held within the thread. Expect my response shortly.
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Math
Logic
Science
Math/Logic

Science is not numbers and graphs. That's math. Science is the gathering of empirical knowledge.

Math is now numbers and graphs.
Science is the gathering of empirical knowledge. (Which inclines us to ask - how?)

Math is a way of writing shorthand logic, while science is the acquisition of knowledge through logic and experimentation. Though they are essentially inseparable, that does not make them one and the same.

Math is now writing shorthand logic.
Science is gathering empirical knowledge through logic(which, as above stated, is math) and experimentation.
Syllogism here:
Math is Logic
Science is Logic
Science is math - fallacious but we're getting to the main point.

Math is nothing but logic, but I don't think math is science. I don't care who says so, it doesn't make sense to me. Are you sure you don't just mean that math is equal to logic

Math is logic. Math is not science.

Math and science ARE related, they're just not the exact same thing, that's all.

We can say here that math and science are correlated because of logic.

Math is a way of expressing logic.
Science is the process of gathering knowledge through empirical means.

Math is logic. Science is gathering of knowledge (again, how?)

Math/logic is not a process, it is a tool. A shovel is not the same as digging, though they are mostly inseparable.

Math/logic is now not a process but a tool (what is the difference?).
Diggin here is supposed to be analguous to science, and, at this moment, that is not what I am questioning.

"IF
p is q
AND
q is a
THEN
p is a"
If that's what you mean, then that is a logical process

Math/logic (which we have established are analguous) are now a process as opposed to previous statements.

Neither is a process. Both have processes.

This seems to imply that science and math both have processes and broad definitions. ie:
Math = algebra. Science = logic
This is contrary to what we have established above. Or, I am confused to the definitions (which is why I asked)

I am saying that science is a process - the process of gathering knowledge through empirical means.
Math is not a process, it is more of a language, a shorthand way of expressing logic so others may understand it.
The difference, the way I see it, is that science is a process with a goal, whereas math is just a tool, a way of communicating

Science is a process to gathering knowledge through empirical means.
Now math is not a process but logical representation.
Earlier we established that it was a process and that it was analguous.
Furthermore, math is aimed at knowledge through empirical means. So, I asked, what is the difference then?

Math is the language - the numbers, symbols, and rules.

Math is only the language. But what is it then when we use them? Logic? This has been equivocated too often now.

That's known as an 'expression', which uses math to express a logical truth.

Now math is an expression of logic. Which is very close to what I was trying to make a point of. That logic IS math. Bertrand Russell and Wittgenstein investigated the symbolism of such and if you ever see the symbols used in logic, they are intend to be analguous to the mathematical process - because they are. The implications of this is what I bring up in my other thread - the Tractatus.

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After reviewing this entirely, I have noticed the main problem.

I am saying that logic IS math. (When I say logic, I do not mean simply 'thinking about things' but the logical processes as set forth by Bertrand Russell - http://lc.brooklyn.cuny.edu/smarttutor/logic/rules.html [thanks Universal Mind]) They are the samething with different symbols. That is the only difference.

I think there is an inclination to separate the two and that is what I was trying to extrapolate.
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Originally Posted by O'nus

After reviewing this entirely, I have noticed the main problem.

I am saying that logic IS math. (When I say logic, I do not mean simply 'thinking about things' but the logical processes as set forth by Bertrand Russell - http://lc.brooklyn.cuny.edu/smarttutor/logic/rules.html [thanks Universal Mind]) They are the samething with different symbols. That is the only difference.

I think there is an inclination to separate the two and that is what I was trying to extrapolate.
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Math is NOT logic. You are implying that they are entirely the same thing - if they were, we wouldn't have two words for them.

Earlier I said that math was entirely logic - but I didn't mean that they were one and the same.

My definitions have never changed throughout this, you have simply misinterpreted some of my statements. I've always held that:

Math is a language which is used to express logic - it is not a process, it is a tool.

Science is a process by which knowledge is gathered using logic and empirical means.

Math is not the same as logic, it is simply a popular way of expressing it.

Originally Posted by thegnome54

Math is NOT logic. You are implying that they are entirely the same thing - if they were, we wouldn't have two words for them.

Earlier I said that math was entirely logic - but I didn't mean that they were one and the same.

And here we expose the problem. Only until this recent century have we analyzed the similarities between logic and math. They are fundamentally the same thing. Here are some sources from the Stanford Ecyclopedia of Philosophy that thoroughly illustrate the similarities:

Constructive Mathematics: http://plato.stanford.edu/entries/ma...-constructive/
Type Theory: http://plato.stanford.edu/entries/type-theory/
Provability Logic: http://plato.stanford.edu/entries/logic-provability/
Fuzzy Logic: http://plato.stanford.edu/entries/logic-fuzzy/

To help the reader not familiar with the basic notions of higher mathematics I comment here on two notions used:

Continuous t-norm. A t-norm is a particular operation x*y with arguments and values in the real unit interval [0,1]. Such an operation is continuous, intuitively speaking, if small changes of the arguments lead only to small changes of the result of the operation. Precisely, for each ε > 0 there is a δ > 0 such that wherever |x1 − x2| < δ and |y1 − y2| < δ then |(x1*y1) − (x2*y2)| < ε.

Infimum and supremum of a subset of the real unit interval [0,1]. Let A be a set of truth values, hence a subset of [0,1]. A truth value x is a lower bound of A if x ≤ y for each element y of A; it is the infimum of A if it is the largest lower bound (notation: x = inf(A)). Clearly, if A has a least element then this element is its infimum; but if A has no least element then its infimum is not its element. For example if A is the set of all positive truth values (x > 0) then inf(A)=0. Dually, x is an upper bound of A if x ≥ y for all y in A; the supremum of A is its least upper bound

Before mathematicians assert something (other than an axiom) they are supposed to have proved it true. What, then, do mathematicians mean when they assert a disjunction P Q, where P and Q are syntactically correct statements in some (formal or informal) language that a mathematician can use? A natural — although, as we shall see, not the unique — interpretation of this disjunction is that not only does (at least) one of the statements P, Q hold, but also we can decide which one holds. Thus just as mathematicians will assert that P only when they have decided that P by proving it, they may assert P Q only when they either can decide — that is, prove — that P or decide (prove) that Q.

With this interpretation, however, mathematicians run into a serious problem in the special case where Q is the negation, ¬P, of P. To decide that ¬P is to show that P implies a contradiction (such as 0=1). But it will often be that mathematicians have neither decided that P nor decided that ¬P. To see this, we need only reflect on the following:

Goldbach Conjecture:
Every even integer > 2 can be written as a sum of two primes,

which remains neither proved nor disproved despite the best efforts of many of the leading mathematicians since it was first raised in a letter from Goldbach to Euler in 1742. We are forced to conclude that, under the very natural interpretation of P Q just canvassed, only an optimist can retain a belief in the law of excluded middle, P ¬P.

Traditional, or classical, mathematics gets round this by widening the interpretation of disjunction: it interprets P Q as ¬(¬P¬Q), or in other words, “it is contradictory that both P and Q be false”. In turn, this leads to the idealistic interpretation of existence, in which xP(x) means ¬x¬P(x) (“it is contradictory that P(x) be false for every x”). It is on these interpretations of disjunction and existence that mathematicians have built the grand, and apparently impregnable, edifice of classical mathematics which serves a foundation for the physical, the social, and (increasingly) the biological sciences. However, the wider interpretations come at a cost: for example, when we pass from our initial, natural interpretation of P Q to the unrestricted use of the idealistic one, ¬(¬P¬Q), the resulting mathematics cannot generally be interpreted within computational models such as recursive function theory

I hope this elucidates my point.
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Originally Posted by thegnome54

Math is not the same as logic, it is simply a popular way of expressing it.

I think math is more than a process of using symbols to express a type of logic. Math encompasses also the truths behind the symbols. But I would not say that math and logic are synonymous. True math is 100% logical, and I would say math is a form of logic, but there is a lot of logic that is not mathematical.

Originally Posted by Universal Mind

I think math is more than a process of using symbols to express a type of logic. Math encompasses also the truths behind the symbols. But I would not say that math and logic are synonymous. True math is 100% logical, and I would say math is a form of logic, but there is a lot of logic that is not mathematical.

It seems to rely on how you define logic. In one perspective, as stated above, you can see the similarities between the two. However, if you define logic as common sense or critical thinking, then it is no longer fundamentally paralell to mathematics.
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Originally Posted by O'nus

It seems to rely on how you define logic. In one perspective, as stated above, you can see the similarities between the two. However, if you define logic as common sense or critical thinking, then it is no longer fundamentally paralell to mathematics.
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I don't know - common sense and critical thinking can be expressed using math more often than you'd think.

I was able to do algebra in fourth and fifth grade, but I was really just using the logic - I didn't know what 'x' meant, or any of the mathematical notation (or the rules for that matter, like PEMDAS, etc.) When I finally did learn algebra, I realized that it was just a way of writing down the logic you use to solve these problems. The same holds for advanced math - math and logic are 'similar' because math is just an easy way of expressing logic.

I don't know of any logical truths which can't be expressed using math, really.

Originally Posted by thegnome54

I don't know - common sense and critical thinking can be expressed using math more often than you'd think.

I was able to do algebra in fourth and fifth grade, but I was really just using the logic - I didn't know what 'x' meant, or any of the mathematical notation (or the rules for that matter, like PEMDAS, etc.) When I finally did learn algebra, I realized that it was just a way of writing down the logic you use to solve these problems. The same holds for advanced math - math and logic are 'similar' because math is just an easy way of expressing logic.

I don't know of any logical truths which can't be expressed using math, really.

Yeah, exactly. Logic also has its own individual notation and symbols for expression which are really just mathematical forumla's.

Have you ever read anything regarding Chaos Theory? I think what your original post asks would be related to the idea that, if we had all the variables we needed in the world (if.. although it is nearly impossible), we could predict every event in all times.

If we got all the variables..
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Originally Posted by O'nus

Have you ever read anything regarding Chaos Theory? I think what your original post asks would be related to the idea that, if we had all the variables we needed in the world (if.. although it is nearly impossible), we could predict every event in all times.

If we got all the variables..
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Yeah, but causal determinism only applies if we assume that there is no such thing as a truly random event - quantum mechanics casts doubt on that assumption.

Originally Posted by thegnome54

Yeah, but causal determinism only applies if we assume that there is no such thing as a truly random event - quantum mechanics casts doubt on that assumption.

I don't see how an event could be truly random. It if it is truly random, it has a completely uncaused component, and that is something I would consider magic.