July 31, 2017 Monday

Bedtime Story 

Understanding Predicate Grammatically and Syntactically

Last night I left you with fascinating subject of predicate which can be studied at the least under three different fields: mathematical logic as we have already been through, grammar and linguistics. 

The traditional grammar perspective of predicate is perhaps well known and understood whereas the second one, viz. semantics, which is more modern, takes its inspiration from logic founded by Gottlob Frege.

By the way, the name of this classic nineteenth century logician and mathematician will keeps popping up whenever one goes into mathematical logic or foundations of mathematics.

Seen from the old grammatical perspective, a predicate is a property that an object has.

In other words, a predicate is associated with a true value of certain object.

Of course, the way we as children were taught was slightly different though it meant the same.

We are taught that the predicate is one of the two pain parts of a sentence, the other being the subject.

The predicate provided information about the subject and in a way, tells something true about the subject or at least, assumed to do so.

Consider the sentence:

The ape is enjoying listening to music.

So here we have the declarative sentence that is linking the subject (the ape) to a verb along with an adverb, the whole of second part being the verb phrase.

The whole of the second part of the sentence, which is essentially a verb phrase, is the predicate.

The more modern view of predicate that comes from the influence of Gottlob Frege where predicate is seen as assigning a property to a single argument or something that relates two or more arguments to each other.

With this much in your armory, let us return back to the concept of well-formedness in language.

In linguistics, well-formedness is that attribute of a sentence or a clause that adheres to all the grammar of that language.

But even here there can be a problem.

Consider the sentence below that confirms to all the grammatical rules of English language.

Colorless green ideas sleep furiously.  

This is a famous sentence that was constructed by the most famous linguistic ever - Noam Chomsky in his 1957 book “Syntactic structures”.

This, by the way, was his first book which actually was more like a monograph of about a hundred pages targeted particularly for the students of his field.

What he wanted to prove was that a sentence can be perfectly grammatically well-formed (no syntax errors) and yet can be semantically a nonsense.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

July 30, 2017 Sunday

Bedtime Story 

The Concept of "Well-Formedness" in Natural Languages

You may scoff at the banality of the Convention-T and it really does look overdone.

This is so because both the object language and the metalanguage are the same; same as in English language.

Now have a look at this sentence.

“Roza krasnaya” is true if, and only if, rose is red.

In the above sentence where the object language is Russian and the metalanguage is English perhaps it appears to our mind more sensible and meaningful.

Now it must be made clear that when Tarski originally conceived of this idea, he had meant to be applicable only to formal languages and not to natural languages.

After having discussed so much about formal calculus, you can surely guess what a formal language would look like.

Yes, exactly like the Principia Mathematica.

A formal language is a set of strong of symbols along with very specific set of rules.

It needs to be emphasized that all such ideas of formalizing mathematics and language has its origins in the Begriffsschrift, the book on logic penned down by Gottlob Frege way back in 1879.

Though the title of the book literally translates as “concept-script”, its more descriptive name would be “a formula language, modeled that on arithmetic and pure thought.”

Why do you think Tarski thought his idea of Convention T was not applicable to the natural languages?

Well, for two very important reasons at least.

First, there is no way of deciding if a sentence of a natural language is well formed.

You may wonder what on earth is meant by “well formed” or rather what is well-formedness.

Well, in a natural language well-formedness refers to that quality when a clause confirms to the grammar.

Clause as you may know, but have forgotten, is the smallest grammatical unit that can express a complete proposition.

And what you may ask is a proposition?

A proposition is a statement that expresses something that is either true or false.

A clause consists of a subject and a predicate.

Now predicate is a complex term with multiple meanings.

It is something that seems very familiar and yet strangely inexplicable.

We have discussed time and again the meaning of predicate from the perspective of mathematical logic.

Now let us examine from the point of linguistics and grammar.

Within linguistics and grammar, there are two competing views.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

July 29, 2017 Saturday

Bedtime Story 

When Philosophy Got Mathematized

So let’s read what Bertrand Russell had to say of analytical philosophy.

“Modern analytical empiricism differs from that of Locke, Berkeley and Hume by its incorporation of mathematics and its development of a powerful logical technique.

It is thus able, in regard to certain problems, to achieve definite answers, which have the quality of science rather than philosophy.

It has the advantage, in comparison with the philosophies of the system-builders, of being able to tackle its problems one at a time, instead of having to invent at one stroke a block theory of the whole universe.

Its methods, in this respect, resemble those of science.”

Tarski’s 1933 landmark paper was essentially an attempt to resolve the Liar Paradox, though in this endeavor he made great many metamathematical discoveries.  

The paper is long and I shall not go very deep into it.

I shall squeeze it into a short bedtime story by extracting the general essence of it.        

As discussed earlier in the Liar Paradox, Tarski considered it vital that a language needs to be distinguished and separated into two parts, the object language and the metalanguage.

Metalanguage just like metamathematics is when language or its symbols is being used to discuss language itself.

In this case the language that is spoken about or examined becomes the object language.

These two are separated from each other by the use of either quotation marks or putting the object language in italics or separating them apart in different lines.

A very simple example of metalanguage that is known as embedded metalanguage is found in our ordinary or natural languages.

The words such as verb, noun, adjective that describe features of their language are in fact metalanguage.

Tarski encouraged that whenever a sentence spoke about the other sentence say P, the sentence P should be rendered in quotes.

With this Tarski had introduced a condition that came to be known as Convention T (I fancy that letter T stands for Tarski).

Any viable truth for every sentence “P” must have the following form:

“P” is true if, and only if, P.  

In the field of logic, if and only if (often abbreviated to iff), is a biconditional logical connective between statements.

An example of this would be,

“Rose is red” is true if, and only if, rose is red.

Such kind of statements are called ‘T-sentences’.

The part then within the quotation marks is the object language and the rest that follows it is the meta-language.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

July 28, 2017 Friday

Bedtime Story 

Liar Paradox and Gödel 

To resolve the Liar Paradox, Tarski had proposed a hierarchy of languages wherein the key feature of the hierarchy would be that it would be permissible for sentences in the higher semantic hierarchy to assign truth values to sentences lower in the semantic hierarchy, but not the other way around.

Yet such a system of “languages” has its own deficiency or limitation.

The major limitation is that such a system of “languages” is incomplete.

Consider the following statement regarding the system of “languages” that Tarski has envisaged:

‘For every statement in the level x of the hierarchy, there is statement at level x + 1 which asserts that the first statement is false.’

It seems to be a fair, true and meaningful statement about the hierarchy that Tarski talks about.

But then it is applicable to statements at every level of hierarchy.

For that to happen, it should be above and beyond every level of hierarchy which only means that it cannot be within the hierarchy.

The Liar Paradox is immensely intriguing and fascinating because not only it poses a conundrum for the logicians but also has been exploited by them in proving their unintuitive theorems.

  

Gödel in his 1931 theorems had used a modified version of the Liar Paradox.

He had replaced the sentence “this sentence is false” with “this sentence is not provable” with that ingenious formula G.

In a way, when Gödel explored the truth and the provability of his Formula G, he was analyzing in a formal way the truth of the liar sentence.

I shall end my discussion of the Liar Paradox with this much.

Now I would like to move on to a very long paper that Tarski published in 1933, very soon after Gödel’s 1931 landmark incompleteness papers.

In this paper he attempted to give a mathematical definition of truth for formalized languages.

In 1935 it was translated to German and it was only in 1956 that it managed to get itself translated into English.

It was all done by Tarski himself and he is the author of the book:

Logic, Semantics and Metamathematics: Papers from 1923 to 1938.

The book which is a collection of seventeen papers is considered to be a landmark in the history of Analytical Philosophy.

In case if you are wondering what the hell is this analytical philosophy, let us read what Bertrand Russell had to say of it which was then a new and emerging subject.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

July 27, 2017 Thursday

Bedtime Story 

Tarski Offers a Solution to the Liar Paradox

If you ask me from where I got all this material that dates back centuries, that would be a very valid question and I would be obliged to answer.

Since it is generally a one way correspondence, I will assume that the question was raised and hence I will force myself to reveal my sources. 

I got the material for my recent bedtime stories from a paper published in 2009 by two scholars, Ahmed Alwishah of Stanford University and David Sanson of Ohio State University in the journal Vivarium.

I think it is important to give some reference even in writing something as banal and as unimportant as bedtime stories.

Now we shall come back to Alfred Tarski and see what he had to say on the centuries old Liar Paradox.

Tarski saw the whole paradox as an outcome of the structure of language.

While Tusi too had found the origin of the Liar Paradox in the self-referential structure of the language, he really could come out with no solution to its resolution.

Tusi got over the problem by simply pronouncing that such self-referential declarative sentences are not deserving of being assigned truth values.

Tarski, on the other hand, offered a solution to this long-standing vexing conundrum.  

He recommended that whenever one needs to assign truth-value to a sentence, one needs to consider language to be made up of different levels or hierarchy.

He says that the Liar Paradox arises because our language is “semantically closed.”

In a closed language like ours it is possible for one sentence to state the truth or falsity of another sentence or even itself.

This can be avoided only if we assign the idea of hierarchy in languages.

Then within it, only the sentences of languages at higher levels be allowed to predicate truth level to sentences in lower levels of language.     

In such a case, the sentence to which the truth (or false) value is being assigned would be the “object language” whereas the sentence which is referring or assigning these values would be a part of a “meta-language”.

In such a system of “languages”, it would be permissible for sentences in the higher semantic hierarchy to assign truth values to sentences lower in the semantic hierarchy, but not the other way around.

This will prevent the paradox from arising as it will prevent the system from becoming self-referential.

This is a brilliant thought mon ami, isn’t it no matter how unrealistic may be its applicability.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

July 26, 2017 Wednesday

Bedtime Story 

Liar Paradox and the End of Islamic Golden Age

Tusi did not stop merely at his radical analysis of the Liar Paradox.

He went further in his writings and gave his clarification on the definitions of “true” and “false”.

“…its being false, insofar as it is a declarative sentence, does not necessitate its being true.

Instead, its being false necessitates the denial of its being false, insofar as it is that-about-which-something-is declared, and [necessitates] its being false, insofar as it is a declarative sentence.

Hence we should not concede that, in this way, the denial of its being false necessitates its being true.”

Here is crucial difference between Abhari and Tusi on the very definitions of ‘true’ and ‘false’.

Abhari insisted that a statement is true when it is in agreement with its subject and false when it is opposite of that.

Tusi disagrees strongly with this reasoning.

Not in general, of course, but specifically when it comes to self-referring declarative sentences.

This kind of reasoning, he insists, works for most other statements but would not be applicable to a declarative sentence that declares its own subject to be false.

In such a specific case, two opposite parts end up in disagreement with each other.

The same subject cannot be in disagreement with itself.

So what is Tusi’s take finally on the Liar Paradox?

Tusi says that a self-referenced declarative sentence that declares itself to be false, like the case in the Liar Paradox, can neither be false nor true.

To such sentences, the definition of true or false is simply not applicable.

With this much I am going to leave behind the glorious Islamic Golden Age that was brought to end probably in 1258 with the Siege of Baghdad.

Baghdad was then the capital of Abbasid Caliphate and the seat of science, philosophy, invention and culture.

Though it is often repeated that religion was the cause of the end of Islamic Golden Age, it may not necessarily be true.

What destroyed the Caliphate and with it the Islamic Golden Age was the barbaric and brutal hoards, the likes of which the world had never seen before.

The fall of the Islamic Golden Age coincides exactly with the rise and the expansion of the Mongol Empire that had unleashed upon the world destruction the likes of which was never seen before.

Twentieth century, of course, would take warfare, bloodshed and brutality to totally new heights mostly by the great European Powers.

The Turco-Mongol invasions, it is estimated, killed approximately 5% of the world population of that time.  

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

July 25, 2017 Tuesday

Bedtime Story 

Tusi Gets the Crux of Liar Paradox

Tusi after thinking deeply on the Liar Paradox came to the conclusion that a paradox is generated when a declarative sentence declares something about itself.

This does not always happen in all instances of declarative sentences declaring something about themselves but in special circumstances as in the Liar Paradox.

Consider the example of the last night.

When the D1 declared something about D2, there was no paradox even if the statement D2 had been something about falsity.

For better illustration consider these statements:

[S1] S2 is false

[S2] This statement is false

Notice here that if you consider the statement [S2] by itself, then the Liar Paradox arises.

But when both [S1] and [S2] are taken together, the paradox disappears.    

If the statement D1 declares itself falsely to be not D1, then this false declaration self-referring itself to be ‘false’ generates a paradox.

The key point is the self-reference and Tusi’s genius lay in grasping that pith of the entire paradox.

Let us see how Tusi stated it some 800 years ago:

“Moreover, if the first declarative statement declares itself to be false, then [both] its being true, insofar as it is a declarative sentence, and its being false, insofar as it is that-about-which-something-is-declared, are concomitant.

Thus the following paradox can be generated: The first declarative sentence, which is a declaration about itself, namely that is false, is either false or true.

If it is true, then it must be false, because it declares itself to be false.

If it is false, then it must be true, because if it is said falsely, then it will become true, which is absurd.”

These lines are astounding brilliant and the kind of thinking displayed in them is a landmark in the history of the Liar Paradox.

Tusi makes three brilliant analytical points that were totally novel at that time.

[1] The liar sentence is singular wherein the subject is itself.

[2] He clearly invokes the idea of self-reference.

[3] He has pin pointed out the crucial assumption that is guilty of creating the whole problem – the assumption that a declarative sentence, by its very own nature, can declare something about anything.

 Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg

July 24, 2017 Monday

Bedtime Story 

Tusi is not Impressed with Abhari's Reasoning

Abhari in his solution to the Liar Paradox is invoking the negation of conjunction which you will get now if you read back his lines:

“To solve the paradox we should not concede that if it is false then one of his sentences is true.

For its being true is taken to be the conjugation of its being true and being false.

Therefore its being false necessitates the non conjunction of its being true and being false.

And the non-conjunction of its being true and being false does not necessitate its being true.”

The only problem in his argument is that in some cases negation of a conjunction does merit negation of conjunct.

This is so when the conjuncts are logically equivalent.

Logical equivalence means when one follows from the other or vice versa.

Since the two statements:

“The Liar sentence is true” and

“The Liar sentence is false” are logically equivalent, negation of a conjunction can allow negation of conjunct.

So Abhari’s solution though very painfully thoughtful does fall apart under detailed scrutiny of modern logic.

Tusi himself was hardly convinced with Abhari’s reasoning of the liar paradox.

Tusi argued that no matter what truth condition (conjunction or conditional) Abhari wishes to associate with the Liar Paradox, it does not matter.

It can be argued directly that its being false entails the negation of its being false, and so entails its being true.

Tusi then proposed his own solution to this great paradox.  

He wrote, “If a declarative sentence, by its nature, can declare-something-about anything, then it is possible that it itself can declare-something-about another declarative sentence.”

Tusi is saying that nothing can prevent one declarative sentence to declare something about another declarative sentence.

Consider the following two declarative sentences D1 and D2.

D1. “It is false”

D2. “Abhari is fasting”

D1 can declare D2 to be false.

Meaning “It is false that Abhari is fasting”

There is no paradox being generated here simple because both the declarative sentences have two different subjects.

A paradox is generated when a declarative sentence declares something about itself.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

Advertisements

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg