An introduction to formal logic

Syllogism is a linear phenomena!

All humans are mortals, all philosophers are humans, all philosopher are mortals!
This syllogism works because all members in the set have same properties, called group properties, all these members have thus these properties; humans, mortals, philosophers.

But the members in the set of energy are fuzzy, because they have different properties, hence some is made up of mass, some is made up of photons, a car simply doesn't belong to the set of light. Fuzziness cannot be set in a syllogistic proposition. For instance, all mass is made up of energy, photons are energy, but are not made up of mass, therefore; nonlinear phenomena doesn't work in a syllogistic way! :)
 
That being the reason why the syllogism is not an adequate system of reasoning for much of what's been found in the last 2000 years.

I'd even go so far as to say that logic alone will get you nowhere, but it will help to identify formally invalid reasoning. It's a tool of limited usefulness, when pushed beyond its limits, "results may be unpredictable".
 
MASS = ENERGY
Mass is made up of energy, photons are made up of energy, photons are made up of mass.
Invalid syllogism. :(
(On top of the fact that mass=energy is only a loose statement, and in the context under which you are trying to use it here, it is definitely an incorrect statement.)

Photons can only travel at light speed, because they have no mass, so I am wondering about, if Aristotelian logic can deal with asymmetrical relations?
The `apparant' contradiction here is not one of Aristotelian logic, but of incorrect premises and invalid syllogism, I think.

This is not to say that Aristotelian logic has no limitations, just that the above is not an example of that.

all mass is made up of energy, photons are energy, but are not made up of mass, therefore; nonlinear phenomena doesn't work in a syllogistic way!
I am not sure what you mean by `nonlinear phenomena' here. As I expressed above, I do not think this is a failure of the `syllogistic way' at all.
 
TO DORMAN

1 I define linear as commutative!
A is married with B, B is always married with A
A philosopher is a mortal human.

2 I define non-symmetrical as sometimes commutative!
A is brother to B, B is sometimes brother to A, and B is sometimes sister to A.
A philosopher is a mortal human, but a mortal human is sometimes a philosopher!

3 I define asymmetrical as never commutative!
A is father to B; B is never father to A.
Aristotelian logic has no concepts of asymmetrical relations!

My borderline between 2, and 3 is fuzzy, hence 2 and 3 is included in my usage of the word nonlinear, but my discrimination between linear and nonlinear is sharp! Binary syllogism is proper to use in connection with linear phenomena, but have no merit outside its realm. A parallel can be drawn to Euclidian geometry, and its improperness to compute non-Euclidian space, hence non-Euclidian Geometry is computable there! Binary logic can only compute members, which only belongs to the same set, but fuzzy logic can compute members, which belongs both to the same set, or many different sets, because binary logic is a special case of fuzzy logic!

Fuzzy logic, and/or nonlinear logic!
The binary logic of modern computers often falls short when describing the vagueness of the real world. Fuzzy logic offers more graceful alternatives.
http://www.fortunecity.com/emachines/e11/86/fuzzylog.html

To make fools of our selves in public is the best way to remove our misunderstandings! :D
 
Peter Soderqvist said:
[1 I define linear as commutative!
A is married with B, B is always married with A
A philosopher is a mortal human.

I have often trouble reading your posts because you use terms in very nonstandard ways, like in this case.

Normally, linear is used to describe a thing where there are no exponents. For example, x = 2y is a linear equatiation but x^2 = 1 is not, since x is raised to the power of two. Similarily, f(x) = 4x+5 is a linear function.

The term commutative is usually used on binary operators. For example, for natural numbers + is a commutative operator since x + y = y + x, but - is not since x-y != y-x in general. A binary operator is a function that takes two arguments and produces a value out of them.

In your first example sentence you are describing a binary relation over the set of persons. A binary relation is a collection of pairs that are somehow, well, related to each other. In your example the pairs consist of two persons, A and B , that are married to each other. Relations and functions are very similar, but not equivalent. An unary function (like the f(x) above) is a binary relation of a specific form and binary functions
are similarily special cases of ternary relations (relations between three elements). This is why the term commutative is strange here.

In case of binary relations the correct term is symmetric. A relation is symmetric if A and B are in relation always when B and A are.

I have no idea what you are trying to say with your second example, since in my eyes it states that the set of philosophers is a subset of the set of mortal humans and I can't see how linearity, commutativity, or symmetry applies here.

2 I define non-symmetrical as sometimes commutative!
A is brother to B, B is sometimes brother to A, and B is sometimes sister to A.
A philosopher is a mortal human, but a mortal human is sometimes a philosopher!

While technically not wrong (with the exception of using "commutative"), this is a little dangerous definition since nonsymmetric could be confused with antisymmetric. I would prefer writing "not symmetric" as two separate words to minimize the danger of misconceptions.

3 I define asymmetrical as never commutative!
A is father to B; B is never father to A.
Aristotelian logic has no concepts of asymmetrical relations!

To tell the truth, I've never understood why anybody still wants to use Aristotelian logic. It was created over 2300 years ago and it has clear limitations. Why not use modern formal logic and its notations, either propositional, predicate, or some altogether different one depending on the problem to model? Sure, there are examples where even predicate logic doesn't have enough expressive power, but all the examples in your post are nicely handled using it.

Binary logic can only compute members, which only belongs to the same set, but fuzzy logic can compute members, which belongs both to the same set, or many different sets, because binary logic is a special case of fuzzy logic!

Fuzzy logic is an important tool in the domains where it can be used (controllers, game AI, etc.), but it too has its limitations. If you create a large model using it, it is very easy to end up with a system that nobody really knows what it is doing. Then there are also cases where the fuzzy interpretation of "or" as "max" and "and" as "min" is not desirable (e.g., when working with probabilities).
 
Do you understand this?

TO LW

You wrote I have no idea what you are trying to say with your second example, since in my eyes it states that the set of philosophers is a subset of the set of mortal humans and I can't see how linearity, commutativity, or symmetry applies here.

I wrote on page 4, 11-19-2002 10:42 AM: All humans are mortals, all philosophers are humans, all philosophers are mortals! This syllogism works because all members in the set have same properties, called group properties, all these members have thus these properties; humans, mortals, philosophers.

Soderqvist1: They are commutative, because they belong to the same set, because all humans are mortals, all humans are philosophers there, but outside of the set; all humans are philosophers are false to facts! Because, if we fond some mortal humans who are not a philosopher he doesn't belong to the set either, because members in a subset or other sets are not philosophers, all humans are simply not philosophers!
 
Re: Do you understand this?

What!?

Please rephrase all that somehow. It sounds like you're saying that if we have only humans and philosophers, that they're the same?

You really lost me.

In the universe of rectangles, there are squares and non-squares.
 
Regarding rephrase!

TO WHITEFORK

White fork wrote 11-20-2002 05:36 PM: Please rephrase all that somehow. It sounds like you're saying that if we have only humans and philosophers, that they're the same? You really lost me.

Soderqvist1: Syllogism can only compute members, which only belong to one set, because syllogism is not a fuzzy phenomenon! It is called the Aristotelian law of Identity; A is A. The law of contradiction; something cannot be both A and not A; and the law of the excluded middle, something must be either A or not A. Any set can have many characteristics, but any characteristic it has, is part of its identity.

You wrote: In the universe of rectangles, there are squares and non-squares.

Soderqvist1: In the set of rectangles, their members are squares, and non-squares,
A chessboard is a rectangle; a chessboard is either a square or a non-square.
A poker card is a rectangle; a poker card is either a square or a non-square.

A is A: Aristotle's Law of Identity
http://www.importanceofphilosophy.com/Metaphysics_Identity.html

Soderqvist1: I define symmetrical relation as one to one correspondence.
An apple cut from top to down, into two halves are symmetrical, and something similar can be said about our mirror picture when we look into the mirror. Therefore A is always brother to B, and B is sometimes brother to A, and B is sometimes sister to A, this is a non-symmetrical relation, because all A's are made up of males, but all B's are made up of both males and females. In special cases, let's say, Adam is brother to Bill, and Bill is brother to Adam is a symmetrical relation, Adam is brother to Barbara, and Barbara is a sister to Adam is a asymmetrical relation, therefore, this phenomenon-as-a-whole is a non-symmetrical relation! :)
 
We're dealing with two different species of "to be" here (apologies to Bill Clinton, but it depends on what your definition of "is" is).

The "IS" in A is A (Aristotlean identity, "I am the king of Spain") is not the same as the "is" in "I am one of the crowned heads of Europe"). The first is not only symmetric, it is reflexive. The second assigns a member to a set, and is generally not either reflexive or symmetrical, unless the set contains only one member (and then it's an identity relationship). I suspect we may attempting to say the same thing, however.
 
Soderqvist1: It appears to me that the Heads of Europe is the set or identity in general, and the Spaniard king is one particular identity, and identity means sameness in all respects, and the identity is invalid when one of the characteristics is left out! But the question is; shall we define by intension, or extension?
Do you know the difference between definition by intension, and extension?

I am short of time here but I will be back tomorrow! :)
 
I know the difference. I think I may need to learn Swedish, though. We seem to have a language barrier going here. Sorry for my denseness.
 
To-be related or not?

TO WHITEFORK

White fork wrote on page4, 11-21-2002 12:38 PM: We're dealing with two different species of "to be" here (apologies to Bill Clinton, but it depends on what your definition of "is" is).

The "IS" in A is A (Aristotelian identity, "I am the king of Spain") is not the same as the "is" in "I am one of the crowned heads of Europe"). The first is not only symmetric, it is reflexive. The second assigns a member to a set, and is generally not either reflexive or symmetrical, unless the set contains only one member (and then it's an identity relationship). I suspect we may attempting to say the same thing, however.

Soderqvist1: "I am the king of Spain" is the "is of identity" and the "I am one of the crowned heads of Europe" is a relational term, the "is of predication" assigning something to a class, in this case the class of crowned heads of Europe, there the set, or class has at least one member, namely; the king of Spain!

Syllogism cannot deal with Bertrand Russell's paradox; All Cretans are liars, I am a Cretan, therefore I am a liar? I have tell you the truth that; I am a Cretan, thus I doesn't belong to the set of liars, at least one Cretan isn't a liar, namely me! ;)

The modern study of fuzzy logic and partial contradictions had its origins early in this century, when Bertrand Russell found the ancient Greek paradox at the core of modern set theory and logic. According to the old riddle, a Cretan asserts that all Cretans lie. So, is he lying? If he lies, then he tells the truth and does not lie. If he does not lie, then he tells the truth and so lies. Both cases lead to a contradiction because the statement is both true and false.

Russell found the same paradox in set theory. The set of all sets is a set, and so it is a member of itself. Yet the set of all apples is not a member of itself because its members are apples and not sets. Perceiving the underlying contradiction, Russell then asked, "Is the set of all sets that are not members of themselves a member of itself ?" If it is, it isn't; if it isn't, it is.

Faced with such a conundrum, classical logic surrenders. But fuzzy logic says that the answer is half true and half false, a 50-50 divide. Fifty percent of the Cretan's statements are true, and 50 percent are false. The Cretan lies 50 percent of the time and does not lie the other half. When membership is less than total, a bivalent system might simplify the problem by rounding it down to zero or up to 100 percent. Yet 50 percent does not round up or down.

In the 1920s, independent of Russell, the Polish logician Jan Lukasiewicz worked out the principles of multivalued logic, in which statements can take on fractional truth values between the ones and zeros of binary logic. In a 1937 article in Philosophy of Science, quantum philosopher Max Black applied multivalued logic to lists, or sets of objects, and in so doing drew the first fuzzy set curves. Following Russell's lead, Black called the sets "vague."
http://www.fortunecity.com/emachines/e11/86/fuzzylog.html

I hope the language barrier is removed? :)

I will be back at Monday!
 
I understand. Thank you.

(The comment about learning Swedish was not meant to be sarcastic, in any case. It's getting near the time for me to study an new language and I don't know any of the Scandinavian ones. Maybe for 2003).
 
I hope you will select Swedish in the year 2003, a first lesson, or introduction here; we don't say – I am right, instead of that we say; I have right converted into Swedish; Jag har rätt, and I am wrong, corresponds to; I have wrong converted into Swedish; Jag har fel! :)
 
What is logical existence?

Logical existence is freedom from self-contradiction, a circle is a circumference with points in equal distance from its center; therefore a square circle has no logical existence!
 
Do you take a Platonic stand on the existence of mathematical entities? That topic is truly a fruitful one for discussion.
 
My stance is non-Aristotelian! :)

TO WHITE FORK

Soderqvist1: I assume you mean that platonic mental objects are eternal verities, and are independent from our minds! I consider this Platonic stance as meaningless, because we have no means to verify its independency! But my stance is that; words are not the objects we are talking about, and words are hopelessly over defined by intension, and hopelessly under defined by extension, because a map cannot map all its territory! For instance, a circle is a static concept, which will never be found in our dynamic world of physicochemical processes, hence all what can be found there is a ring, with some color, thickness. The word "circle" is over defined by intension, because externally, its points are not at equal distance from its center, and the perfect concept in our minds, are under defined by extension, because it lacks the external characteristics (color, thickness)! Therefore Aristotelian A = A is not externally truth!

Structural differential (a picture) can show you what cannot be said!
http://userwww.sfsu.edu/~rsauzier/Korzybski.html

Soderqvist1: Emphasis in bold type by me!

Richard Dawkins Charles Simonyi Professor In the Public Understanding of Science Oxford University, Oxford, England From his book The Selfish Gene, Chapter 2 the Replicators page 18: Should we then call the original replicator molecules "living"? Who cares? I might say to you " Darwin was the greatest man who has ever lived", and you might say No, Newton was, but I hope we would not prolong the argument! The point is that no conclusion of substance would be affected whichever way our argument was resolved. The facts of the lives and achievement of Newton, and Darwin remain totally unchanged whether we label them great or not.

Similarly, the story of the replicator molecules probably happened something like the way I am telling it, regardless of whether we choose to call them "living". Human suffering has been caused because too many of us cannot grasp that words are only tools for our use, and that the mere presence in the dictionary of a word like "living" does not mean, it necessarily has to refer to something definite in the real world. Whether we call the early replicators living or not, they were the ancestors of life; they were our founding fathers

Living or non-living?
Were these early self-replicating molecules living or non-living? Dawkins: that's an idle question. 'No conclusion of substance would be affected by whichever way our argument was resolved.'
http://www.mdx.ac.uk/www/psychology/cog/psy1100/dawkins.htm#sect5

Alfred Korzybski: Whatever we may say something is, obviously is not the 'something' on the silent levels. Indeed, as Wittgenstein wrote, 'What can be shown, cannot be said.' In my experience I found that it is practically impossible to convey the differentiation of silent (unspeakable) levels from the verbal without having the reader or the hearer pinch with one hand the finger of the other hand. He would then realize organismally that the first-order psycho-logical direct experiences are not verbal. The simplicity of this statement is misleading, unless we become aware of its implications, as in our living reactions most of us identify in value the two entirely different levels, with often disastrous consequences. Note the sadness of the beautiful passage of Eddington on page. He seems to be unhappy that the silent levels can never be the verbal levels. Is this not an example of unjustified 'maximum expectation'?

I firmly believe that the consciousness of the differences between these levels of abstractions; i.e., the silent and the verbal levels is the key and perhaps the first step for the solution of human problems. This belief is based on my own observations, and studies of the endless observations of other investigators. There is a tremendous difference between 'thinking' in verbal terms, and 'contemplating', inwardly silent, on non-verbal levels, and then searching for the proper structure of language to fit the supposedly discovered structure of the silent processes that modern science tries to find.

If we 'think' verbally, we act as biased observers and project onto the silent levels the structure of the language we use, and so remain in our rut of old orientations, making keen, unbiased, observations and creative work well-nigh impossible. In contrast, when we 'think' without words, or in pictures (which involve structure and therefore relations), we may discover new aspects and relations on silent levels, and so may produce important theoretical results in the general search for a similarity of structure between the two levels, silent and verbal. Practically all important advances are made that way.
http://www.esgs.org/uk/art/ak2.htm

Albert Einstein: These thoughts did not come in any verbal formulation. I rarely think in words at all. A thought comes, and I may try to express it in words afterward. Quoted in H Eves Mathematical Circles Adieu (Boston 1977).
http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Einstein.html
 
whitefork said:
That being the reason why the syllogism is not an adequate system of reasoning for much of what's been found in the last 2000 years.

I'd even go so far as to say that logic alone will get you nowhere, but it will help to identify formally invalid reasoning. It's a tool of limited usefulness, when pushed beyond its limits, "results may be unpredictable".

Oh my... does this mean that the existence of God and an afterlife cannot be proven by means of logic alone?!? :eek:
 
Regarding White Fork's syllogism is not an adequate system of reasoning!

I think he mean that Aristotelian syllogism has made us too black or white orientated!
Not because syllogism is invalid, but because of it's over emphasis on either it is, or not? For instance, Daniel Dennett in his book, Darwin's Dangerous Idea, has rejected Aristotle's essentialism, because objects has no inherent "essence" because we cannot say the exact moment in time when a cell is split in two, but it is an easy task after a while to say; now they are two there, and something similar can be said about speciation, we cannot point out the exact borderline between the old one, and the new species, because a species have no essence, and for the same reason variations in a species, or speciation, is not an clear-cut question, the borderline there is simply fuzzy. But the creationists believe in Aristotle's essence, because Aristotle's philosophy is part of the Christian doctrine, and because of that, has rejected the Darwinian evolution! :)
 

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