Archive | October, 2012

Defining Vision – On First-Order Logic’s Syllogisms

4 Oct

To a question that stated

There is an old joke that goes something like this: “If God is love, love is blind, and Ray Charles is blind, then Ray Charles is God.” Explain, in the terms of first-order logic and predicate calculus, why this reasoning is incorrect. 

I answered…

 

“Smokin’ Boolean Operators, Batman!… we are getting closer to The Core!… careful!”

Yeah, some say I’m 5 – going on 50.

Yet, having to take this ‘walk down memory lane’ (pun, of course, INTENDED!) does help me ‘jog’ some otherwise ‘scary’ terms – into sheer daylight, when the pressures of the job market make Scripting/Programming a must – and there are all those Java classes, a’brewin’!

… Google? First Order Logic = George Boole, then? … and all this breaking things down to their simplest terms? eliminating them “Ambi- Sisters” as to avoid disaster (yeah, ambi-guity and ambi-valence… Left Hemisphere? relax!) anyway, one of the top results (besides Wiki, of course!) on that ‘Philosopher’s Stone’ that Larry and Sergei put together* states:

“In order to know the truth value of the proposition which results from applying an operator to propositions, all that need be known is the definition of the operator and the truth value of the propositions used.” (Lander.Edu, 2012)

and before that…

  1. “We are going to set up an artificial “language” to avoid the difficulties of vagueness, equivocation, amphiboly, and confusion from emotive significance.

    1. The first thing we are going to do is to learn the elements of this “new language.”
    2. The second is to learn to translate ordinary language grammar into symbolic notation.
    3. The third thing is to consider arguments in this “new language.”

  2. Symbolic logic is by far the simplest kind of logic—it is a great time-saver in argumentation. Additionally, it helps prevent logical confusion when dealing with complex arguments.. (ibid)

(… and thanks to the daring classmate who helped brought forth the ‘symbolic logic’ nugget… Golden!)

Anyway, echoes of the Technical Writing class, with its ‘Three C’s: Clear Concise AND Complete’ abound; and yes, the article mentions propositions serving as ‘bindings’ or ‘connectors’ to all of these… therefore…

… is that the EXACT Same Lewis Carroll?… WHOA!… first Bertrand, now Lewis… pass the Hookah – of Knowledge!

OMG!… think of all the ‘skip-to-the-ends’ committed every single ‘millisecond’… WHOA!

“The Learner, who wishes to try the question fairly, whether this little book does, or does not, supply the materials for a most interesting mental recreation, is earnestly advised to adopt the following Rules:—

(1) Begin at the beginning, and do not allow yourself to gratify a mere idle curiosity by dipping into the book, here and there. This would very likely lead to your throwing it aside, with the remark “This is much too hard for me!”, and thus losing the chance of adding a very large item to your stock of mental delights. This Rule (of not dipping) is very desirable with other kinds of books——such as novels, for instance, where you may easily spoil much of the enjoyment you would otherwise get from the story, by dipping into it further on, so that what the author meant to be a pleasant surprise comes to you as a matter of course. Some people, I know, make a practice of looking into Vol. III first, just to see how the story ends: and perhaps it is as well just to know that all ends happily——that the much-persecuted lovers do marry after all, that he is proved to be quite innocent of the murder, that the wicked cousin is completely foiled in his plot and gets the punishment he deserves, and that the rich uncle in India (Qu. Why in India? Ans. Because, somehow, uncles never can get rich anywhere else) dies at exactly the right moment——before taking the trouble to read Vol. I his, I say, is just permissible with a novel, where Vol. III has a meaning, even for those who have not read the earlier part of the story; but, with a scientific book, it is sheer insanity: you will find the latter part hopelessly unintelligible, if you read it before reaching it in regular course…

…Mental recreation is a thing that we all of us need for our mental health; and you may get much healthy enjoyment, no doubt, from Games, such as Back-gammon, Chess, and the new Game “Halma”. But, after all, when you have made yourself a first-rate player at any one of these Games, you have nothing real to show for it, as a result! You enjoyed the Game, and the victory, no doubt, at the time: but you have no result that you can treasure up and get real good out of. And, all the while, you have been leaving unexplored a perfect mine of wealth. Once master the machinery of Symbolic Logic, and you have a mental occupation always at hand, of absorbing interest, and one that will be of real use to you in any subject you may take up. It will give you clearness of thought——the ability to see your way through a puzzle——the habit of arranging your ideas in an orderly and get-at-able form——and, more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will so continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art. Try it. That is all I ask of you! (Carroll, Lewis, 1897)

Anyway… it’s a Math class, right?… so no skipping, indeed!

  • [INSERT ‘SMILING CHESHIRE’ DISAPPEARING INTO THIN AIR HERE]

Bingo!… (or should I say “Eureka!”)(no, no bathtubs were harmed in the making of this DQ… yet… I’m kinda floatin’ in a pool of fresh – yet eternal – wisdom here!)

Syllogisms? that’s the crux of the DQ?

(First-Order Logic, believe Lewis took care of it, as follows…)

“When a Trio of Biliteral Propositions of Relation is such that

(1) all their six Terms are Species of the same Genus,
(2) every two of them contain between them a Pair of codivisional Classes,
(3) the three Propositions are so related that, if the first two were true, the third would be true,

the Trio is called a ‘Syllogism’; the Genus, of which each of the six Terms is a Species, is called its ‘Universe of Discourse’, or, more briefly, its ‘Univ.’; the first two Propositions are called its ‘Premisses’, and the third its ‘Conclusion’; also the Pair of codivisional Terms in the Premisses are called its ‘Eliminands’, and the other two its ‘Retinends’.
The Conclusion of a Syllogism is said to be ‘consequent’ from its Premisses: hence it is usual to prefix to it the word “Therefore” (or the Symbol “∴”).
[Note that the ‘Eliminands’ are so called because they are eliminated, and do not appear in the Conclusion; and that the ‘Retinends’ are so called because they are retained, and do appear in the Conclusion.

Note also that the question, whether the Conclusion is or is not consequent from the Premisses, is not affected by the actual truth or falsity of any of the Trio, but depends entirely on their relationship to each other.

As a specimen-Syllogism, let us take the Trio

“No x-Things are m-Things;
No y-Things are m′-Things.
No x-Things are y-Things.”

which we may write, as explained at [p. 26], thus:—

“No x are m;
No y are m′.
No x are y”.

Here the first and second contain the Pair of codivisional Classes m and m′; the first and third contain the Pair x and x; and the second and third contain the Pair y and y.

Also the three Propositions are (as we shall see hereafter) so related that, if the first two were true, the third would also be true.

Hence the Trio is a Syllogism; the two Propositions, “No x are m” and “No y are m′”, are its Premisses; the Proposition “No x are y” is its Conclusion; the Terms m and m′ are its Eliminands; and the Terms x and y are its Retinends” (ibid)

Hmmmmm….

Bruce said “Predicate Calculus” though… and yes, above is actually EXCLUDING x = y (Deflating? Defeating? Nullifying?… how do you say “I Won” In Math?)… so are we almost there?

Found a great lecture… had to skip, 96 slides?… found this nugget?

“English to Predicate Logic

In would be difficult to reason about such statements using only the propositional calculus. With the predicate calculus we can reason over collections of objects.

  • Formalizing English or mathematical expressions in predicate logic is useful:
  • Firstly, it enforces precision and exposes ambiguities. Does “between 80 an n” include 80 and n? This question must be answered during formalization.
  • Secondly, once formalized we can use inference rules to reason formally about the objects under consideration.”

Author then goes on to ‘translate’ English predicates into mathematical notation… a Keeper? or time to hit the shelves (hoped to say the ‘sheets’ but this one? ready to ‘defrost’ by Saturday!)

“A logical theory is said to be strongly complete iff all of its models are isomorphic. To query such a theory – i.e. such a world representation – is thus an easy task: a query can be translated
into a formula p, which has the same value a in all these models. So 01 is said to be the answer to p. However, with an incomplete theory, such a formula q will have distinct values
in models, so no such a definite answer can be defined.

Lipski takes a more general approach, defining two “answers” to a query, namely the upper bound and the lower bound. The lower (respectively upper) bound of a query pl in a theory
Tis defined as “true” iff Q, is true in all models (at least one model) of I: Intuitively, the lower bound of p could be called p’s “necessity”, while upper bound would be seen as its
“possibility”. Definitions of lower and upper bounds are clearer if open queries (“give all x such that p”) are considered. Then, lower bound is the set of sure answers and upper bound
the set of possible (i.e. consistent with the theory) answers, and lower bound is obviously included in upper bound. Lipski also defines an infromation @re)order, which relates two theories
if models of the first one (considered as “more informative”) also are models of the second one. The set of the theories, preordered by this information order, would be a Kripke’s
model – where the value of 00p respectively Ooq) is the lower (upper) bound of q – if these theories would be complete!

As found in a way (perhaps not the only one) to make such theories complete is to translate them into a many-valued logic framework.”

(Whoa!… is this why Mathematicians cum Philosophers may also end up – or start off – as Theologians? – as again, in terms of not only its universality, but its closeness to “The Divine”, well… (thinking St. Thomas Aquinas! – and Bertrand’s Rebuttals, too?!)

“Upper and Lower Bounds”? “Sure Answers”…  “The Set of Possible Answers – All Obviously Included”… so can one actually say “I’m a Holisticist”?

… Anyway, think boundaries may have been breached by now? (or do Fractals apply to DQ’s?)…

Frank/cisco Palacio
CDT/CST Time Zone (Central Illinois)
Francisco.J.Palacio@GMail.Com

References

Introduction to Symbolic Logic: Philosophy.Lander.Edu (2012) Retrieved from: http://philosophy.lander.edu/logic/symbolic.html

Caroll, Lewis: Symbolic Logic (1897) Retrieved from: http://www.gutenberg.org/files/28696/28696-h/28696-h.htm

Browne, Patrick: Predicate Calculus Lecture Comp.Dit.Ie (2012) Retrieved from: http://www.comp.dit.ie/pbrowne/compfund1/lecture5.pdf

Ostermann, P. (1990). Many-valued modal logics: Uses and predicate calculus. Mathematical Logic Quarterly, 36(4), 367-376. doi:10.1002/malq.19900360411 Retrieved from: https://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=62674716&site=eds-live

Dekkers, W., Bunder, M., & Barendregt, H. (1998). Completeness of two systems of illative combinatory logic for first-order propositional and predicate calculus. Archive For Mathematical Logic, 37(5/6), 327.Retrieved from: https://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=4684426&site=eds-live

*Turning disparate shreds of information into Shareholder Value?… reeks of Alchemy, does it not?

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Fuzzy Bio (Intro to the Discrete Math Class)

4 Oct

So how does one intro oneself when the Mathematics Professor is THE real deal?

Like this!… “Tongue in Cheek”!

Hi.

Almost hitting the half century mark, so there’s risk of TMI (Too Much Information) involved.

  • Saw the LinkedIN? Feel free to ‘connect’ via the e-mail below; I guess by the time we’re done with the class, we’d have learned enough from each other, right?
  • Saw the ‘IceBreaker? Toastmasters, been doing it since 2008, absolutely enjoy the program – which in my case, has helped me deal with my profligate proclivity towards loquaciousness, aye!
  • Saw the ‘Hochdeutsch’ Exchange with Herrn Doktor Stephan?… fancy myself a wannabe polyglot, as though there are two missing (ought to be five, last I hear – yet many encourage me as I will push variants and dialects, given my overall enthusiasm for linguistics prevailing) … the main ones?.. Spanish, mother tongue, recently fortified with some ‘Cubanisms’ after living in Hialeah (yeah, Calle Ocho? heard of it?… Great iteration of Mardi Gras, according to some!) for close to three years; English, been mangling it since 1980 when I arrived in a small town in the midwest for a two-year stay at an Army JROTC school, as to complete Junior and Senior Years of High School, correct-o!… Also tried Japanese for a year or so; gave up after realizing it DOES help if you learn your Kanji (and Katakana/Hiragana ‘Romajis’) as early as possible… 5,000 to get out of High School? Oy Vey!… German? As I mentioned earlier, spent about six years at a local branch of Goethe Institut, were it not for the fact that I was the proverbial “Last Man Standing” they would have been able to test me on the “Oberstufe Eins” – their equivalent of the American TOEFL (Test of English as a Foreign Language)… Finally? Spanglish, if we ever get the dialect to be properly recognized, maybe counts…. (and yes, even been thought of as being fluent in French, Italian and Portuguese and Galician – only because we used to kid around and practice their accentuation, what with the great immigrant influxes in the fifties available – back when Venezuela was young and rich and underpopulated – not the current shell of its former self, having returned to the rule of the “Caudillo” (Strongman) recently.)

Otherwise, been on/near/around/inside I.T. Systems most of my life… yeah, had a Macintosh when they were cool, hold on dearly to my PIII/384 Mb RAM Gateway, and don’t mind tinkering with a couple of other carcasses (including my current Droid, believe second or first gen one? runs good – and HAS a keyboard!) – while I squirrel away some moolah to get me a ‘Phablet’… or should it be an iPad?… Choices, choices, choices!

Bottom line? Praying and hoping to get this one out of the way, as yes, Math is NOT my forte – even though many think there’s hope, as after all, it’s the most Universal of ‘languages’ out there, right?… so there’s that linguistics angle, to the rescue, yet again!

Saludos!