HIJRAH 1430 Ramadhan 07?
FREEDAY, August 28, 2009
UP DILIMAN CAMPUS
I shall rewrite the posting "MUTUALLY EXCLUSIVE & JOINTLY EXHAUSTIVE" because it is more about PHILO OF RELIGION rather than LOGIC. I shall probably delete this last 2 postings of mine... The more fitting title should have been "LOGIC DIAGRAMS". I am reminded of MARTIN GARDNER'S book of the same title which introduced me to the matter. I read this book when I was a college freshman. I remember very well that section of the UP MAIN LIB. The PHILO books were all found in front at the left side of what is now the Reference section. Other excellent books on Literature etcetc were found at the middle & back part. I devoured all the good ones - THE COMPLETE WORKS of H.G. WELLS, COMPLETE SHERLOCK HOLMES, SCI-FI, NIETZSCHE, etcetc ... I was an ELECTRICAL ENGINEERING major then. One of my engineeering classmates who saw the books I had been borrowing warned me to stop reading these & other pocketbooks... He knew better... Guess what? I failed to finish my original course - precisely because I was studying - different subjects!! Who told you studying hard cannot lead to failure/flunking?
LOGIC DIAGRAMS:
Euler Circles & Venn Diagrams
One may claim that these spatial representations of logical notions are some of the very first
CONCEPTUAL MAPS. Conceptual maps are another important topic I wanted to tackle in this BLOG but I shall just insert URLs of excellent websites devoted to this topic. And good book titles too. Conceptual mapping has been associated with such writers as TONY BUZAN et al but actually these things are as old as the MEDIEVAL period.
I have always considered PATTERN & ORDERLINESS as hallmarks of a RATIONAL MIND; that intelligent people practice ORDERED/ORGANIZED/SYSTEMATIC thinking. [BUT I once had a very intelligent MAGNA CUM LAUDE engineering graduate teacher who taught very well but had terrible handwriting & was very topsy-turvy/disorganized! I guess to every rule there is an exception!] Educators have determined that drawing diagrams & patterned maps help in thinking. Apparently, the brain imitates the map & reorganizes itself. BUT, again, the brain actually does not seem to be organized when it comes to storing ideas in memory - It seems that
there is no single spot or area in which all our ideas are stored, clumped together. Like in a computer diskette wherein the data are NOT stored in ordered, contiguous areas but are scattered all over the disk sectors. (DEFRAGMENTING one's hard drive tries to join together
these fragmented data, thus saving space & time.)
Compare a field or garden wherein the plants/trees are scattered randomly with another one wherein these are patterned as in rows and columns, circular hedges etc See the difference? Yet patterns do not seem to be common in NATURE. When we find such patterns in the outside world we automatically assume there is a pattern-maker, a designer, that human beings must have created these... Is NATURE DISORDERLY & ARE HUMANS DESIGNERS/PATTERN-MAKERS?
Circles, Squares, Triangles...
CONSPIRATORIAL GEOMETRY...
JUST DELETED IN THIS EXACT SPOT THREE IMAGES of CIRCLE & TWO EQUILATERAL TRIANGLES WHICH TURNED OUT DISORDERED!... I CANNOT MASTER THIS BLOGGER ENTRY FORM. MY TXT FILE ENDS UP DISORDERED...
-SQUARING THE CIRCLE
-RULER & COMPASSS GEOMETRIC CONSTRUCTIONS
-HOW TO DRAW A PERFECT PENTAGRAM STAR, PENTAGON
-GEOMETRIC ORIGAMI PAPER FOLDINGS
-THE NUMBER pi
-THE NUMBER e
-THE GOLDEN PROPORTION/RATIO
BUT THESE ARE ALL ABOUT GEOMETRY & MATHEMATICS...
I am writing only about LOGIC...
=======================
jamaal_the_jobless@yahoo.com
=======================
Thursday, August 27, 2009
Monday, August 17, 2009
POST 6 MUTUALLY EXCLUSIVE & JOINTLY EXHAUSTIVE
UPLOADED ON
HIJRAH 1430 Sha'ban 25
Monday, August 17, 2009
UP DILIMAN CAMPUS
GEOMETRY & MATHEMATICS OF RELIGION
PART 1 OF 2
Geometrical/Mathematical Tool for aiding one's thinking about concepts
WARNING: GROSSLY OVERSIMPLIFIED
Consider the three circles intersecting at their centers... I will not go into the geometry of the plane figures but those who studied high school geometry know that a lot of theorems can be derived from such a configuration... I will use the circles a la Venn Diagram or Euler Circle... I shall try to categorize & classify human beings according to their professed religions & simplify the case to just the three major ABRAHAMIC faiths: Judaism, Islam & Christianity... [PORTION DELETED]
Now consider the mathematical model of the intersecting circles wherein masses of human beings are represented as plane areas This is an extremely simplified model because other individuals or groups of human beings are not represented - where are the ATHEISTS or AGNOSTICS, the HINDUS, the BUDDHISTS, the CONFUCIANS the ANIMISTS, the NEW AGERS, the WICCANS, the DEVIL-WORSHIPPERS etcetcetc... But this is precisely one purpose of (mathematical) modeling, to simplify a problem so that a pattern may be discerned... Of course the modeller should be aware of the limitations of the model she is using... And what conclusions are justified or may be derived from the limited model... In real science, + or - approximations and margins for error are provided... SCIENCE WAS, IS AND ALWAYS WILL BE A "THEORY" AS IN AN UNFINISHED ENDEAVOR...
The three circles divide the plane into seven(7) contiguous (close together) sectors. These sectors are similar because the image has some symmetry... We let the plane represent the human beings who adopt/profess even NOMINALLY (just in name or in public documents though they never follow the religion's commandments, injunctions or teachings... So all Christians are found within the circle of CHRISTIANS, all Muslims are within the ISLAM circle & the Jews are found within the JEW circle.
OH BY THE WAY NOT ALL THINKING INDIVIDUALS AGREE WITH SUCH MATHEMATICAL MODELLING OF HUMAN BEINGS (ONE OF THEM IS LYNDON LAROUCHE WHO CRITICIZES THE WAY THINKERS SINCE NEWTON INCLUDING SUCH FIGURES AS JOHN VON NEUMAN ETCETC HAVE GROSSLY OVERSIMPLIFIED HUMAN BEINGS... The crucial element, the HUMAN FACTOR, the HUMANITY of the INDIVIDUAL HUMAN is not presented... Nevertheless, one can see the possible practical uses of such models
in crowd study, crowd control, handling masses of people, mass classification etctetc MARXISTS I think adopt such broad generalizations with their conceptions of CLASS, PROLETARIAT , etctetc Of course the PHILOSOPHY OF MARXISM IS DIALECTICAL MATERIALISM...
OF COURSE, I AM AWARE THAT I MYSELF CAN BE TREATED AS A MERE POINT IN THIS GEOMETRIC MODEL BUT MY EXISTENTIAL-RELIGIOUS POSITION CAN COMPLETELY
GO AGAINST THIS POINT OF VIEW... I CAN CLAIM I HAVE MY OWN CIRCLE OF ONE POINT & IT DOES NOT AT ALL APPEAR IN THE MODEL!... This is one nice thing about ISLAM that I like: it does not destroy one's INDIVIDUALITY... Even if one goes to pray with the JAMAAH still after that one can pray alone by oneself... Even the PROPHET practiced such silent aloneness (not loneliness - one may be alone but one is not necessarily lonely & one may be in a crowd of laughing humans & yet one is sadly alone!!!) THus, there are SUFIs and other
interpreters of ISLAM who believe there is an HIDDEN ESOTERIC side to the QuR'AN & the DEEN etcetctetc
PART 2 OF 2
HIJRAH 1430 Sha'ban 24
Sunday, August 16, 2009 GC
At Home
Continuation
=========================================
"MUTUALLY-EXCLUSIVE & JOINTLY-EXHAUSTIVE"
=========================================
THE CIRCLE OF THREE RELIGIONS [ANOTHER JAMAAL ORIGINAL TADAAHHHH!!!]
Continuing my grade school diagram of the three major ABRAHAMIC faiths... One who identifies himself with only one or just two or all three of the three major world religions may view human beings as being classified or falling into one of three areas of the seven (7) sectors in the diagram. Three sectors are colored and the other four I simply label with the letters "A", "B" & "C" and the center sector with the numeral "1". If the three circles represent all human beings belonging
to the three religions of ISLAM, JUDAISM & CHRISTIANITY, then we may consider the RED sector as those individuals who are exclusively JEWS, the BLUE sector as "pure" CHRISTANS & the GREEN as exclusively MUSLIMS. There are no half-cooked, half-breeds, 50-50, so-so in these sectors. [ALWAYS REMEMBER or NEVER FORGET: this is just a model & overly-simplified...] Now, what or who are those found/located/represented in sectors A, B & C? And those in the middle sector 1? I would say that those in sectors A would form the intersection of JEWS & CHRISTIANS, meaning those whose father might be a JEW & mother a CHRISTIAN & professes either religion [But I think one can be a real Jew only if one's mother is a JEWESS...?] These can also be the sector of the ALLIANCE of JEWS & CHRISTIANS (Those who believe JEWS & CHRISTIANS are not antagonistic to each other & may even be allied...) Similarly
sector B would be those JEWS & MUSLIMS who are allied or do not consider the other as enemy & sector C would form the sector of MUSLIM-CHRISTIAN alliance... There is a saying that "your enemies enemy is your ally" or something similar but I don't fully agree with this...
Anyway, the point in this exercise is the diagram shows us that there is the possibility (We are not claiming that indeed "there actually exists" out there in the real world) of alliances between MUSLIMS & CHRISTIANS, JEWS & MUSLIMS, & CHRISTIAN & JEWS. IF (Those who are not familiar with COUNTERFACTUALS may research/read a little in popular books on Philosophy or the Internet.) - IF the colored sectors represent "purists" who reject the "unpure" then we may say that the colored sectors represent mutually exclusive sectors of MUSLIMS, CHRISTIANS & JEWS... And the intersecting sectors would be the less antagonistic as in accomodating sectors of each RELIGION... Sector A may represent the MUSLIM-JEW ALLIANCE AGAINST CHRISTIANS, sector B may represent the CHRISTIAN-JEW ALLIANCE AGAINST MUSLIMS, and sector C may be the MUSLIM-CHRISTIAN ALLIANCE AGAINST JEWS! And the central sector would be the universalists as in those who find all three religions as all acceptable & NOT EXCLUSIVIST. Does this mean that those in sector 1 are AGAINST the PURISTS in all three religions? Or does this mean that sector 1 represents those who can go along peacefully with any member of any of the three religions?... I PREFER TO VIEW SECTOR 1 AS THE TRUE MUSLIM WHO IS ALSO THE TRUE CHRISTIAN & THE TRUE JEW... THE TRUE MUSLIM IS NON-DIFFERENT FROM THE TRUE CHRISTIAN & THE TRUE JEW.
DEAR READER, What do your think?
Exercise (in futility:) or Homework:
How about well-known or popular individual personalities whom we have come to identify with this or that religion like the present Roman Catholic Pope Benedict or Pope John Paul II or Usamah Binladin or Saudi King Abdullah or ex-Pres. George Bush or Pres. Obama or Yasser Arafat or Ayatullah Khomeini or ETCETCETCETCETCETC...
WHERE WILL YOU LOCATE EACH PERSONALITY? WHAT DOES IT MEAN FOR THEM TO BE LOCATED WITHIN THAT SECTOR OF OUR GEOMETRIC MODEL?
If Mr. X hates Jews & believes in an alliance of Muslims & Christians against the third party, then Mr. X would fall in sector C. But is this interpretation correct or true?
WHAT IS THE TRUE?
SPECIAL BONUS PROBLEM:
USING THE SAME MODEL ANALYZE THE MAJOR RELIGIONS OF THE WORLD
INCLUDING HINDUISM, BUDDHISM, SHINTOISM, CONFUCIANISM, CONFUSIONISM ETC ETC TO BE SUBMITTED "PASA" ASAP
HIJRAH 1430 Sha'ban 25
Monday, August 17, 2009
UP DILIMAN CAMPUS
GEOMETRY & MATHEMATICS OF RELIGION
PART 1 OF 2
Geometrical/Mathematical Tool for aiding one's thinking about concepts
WARNING: GROSSLY OVERSIMPLIFIED
Consider the three circles intersecting at their centers... I will not go into the geometry of the plane figures but those who studied high school geometry know that a lot of theorems can be derived from such a configuration... I will use the circles a la Venn Diagram or Euler Circle... I shall try to categorize & classify human beings according to their professed religions & simplify the case to just the three major ABRAHAMIC faiths: Judaism, Islam & Christianity... [PORTION DELETED]
Now consider the mathematical model of the intersecting circles wherein masses of human beings are represented as plane areas This is an extremely simplified model because other individuals or groups of human beings are not represented - where are the ATHEISTS or AGNOSTICS, the HINDUS, the BUDDHISTS, the CONFUCIANS the ANIMISTS, the NEW AGERS, the WICCANS, the DEVIL-WORSHIPPERS etcetcetc... But this is precisely one purpose of (mathematical) modeling, to simplify a problem so that a pattern may be discerned... Of course the modeller should be aware of the limitations of the model she is using... And what conclusions are justified or may be derived from the limited model... In real science, + or - approximations and margins for error are provided... SCIENCE WAS, IS AND ALWAYS WILL BE A "THEORY" AS IN AN UNFINISHED ENDEAVOR...
The three circles divide the plane into seven(7) contiguous (close together) sectors. These sectors are similar because the image has some symmetry... We let the plane represent the human beings who adopt/profess even NOMINALLY (just in name or in public documents though they never follow the religion's commandments, injunctions or teachings... So all Christians are found within the circle of CHRISTIANS, all Muslims are within the ISLAM circle & the Jews are found within the JEW circle.
OH BY THE WAY NOT ALL THINKING INDIVIDUALS AGREE WITH SUCH MATHEMATICAL MODELLING OF HUMAN BEINGS (ONE OF THEM IS LYNDON LAROUCHE WHO CRITICIZES THE WAY THINKERS SINCE NEWTON INCLUDING SUCH FIGURES AS JOHN VON NEUMAN ETCETC HAVE GROSSLY OVERSIMPLIFIED HUMAN BEINGS... The crucial element, the HUMAN FACTOR, the HUMANITY of the INDIVIDUAL HUMAN is not presented... Nevertheless, one can see the possible practical uses of such models
in crowd study, crowd control, handling masses of people, mass classification etctetc MARXISTS I think adopt such broad generalizations with their conceptions of CLASS, PROLETARIAT , etctetc Of course the PHILOSOPHY OF MARXISM IS DIALECTICAL MATERIALISM...
OF COURSE, I AM AWARE THAT I MYSELF CAN BE TREATED AS A MERE POINT IN THIS GEOMETRIC MODEL BUT MY EXISTENTIAL-RELIGIOUS POSITION CAN COMPLETELY
GO AGAINST THIS POINT OF VIEW... I CAN CLAIM I HAVE MY OWN CIRCLE OF ONE POINT & IT DOES NOT AT ALL APPEAR IN THE MODEL!... This is one nice thing about ISLAM that I like: it does not destroy one's INDIVIDUALITY... Even if one goes to pray with the JAMAAH still after that one can pray alone by oneself... Even the PROPHET practiced such silent aloneness (not loneliness - one may be alone but one is not necessarily lonely & one may be in a crowd of laughing humans & yet one is sadly alone!!!) THus, there are SUFIs and other
interpreters of ISLAM who believe there is an HIDDEN ESOTERIC side to the QuR'AN & the DEEN etcetctetc
PART 2 OF 2
HIJRAH 1430 Sha'ban 24
Sunday, August 16, 2009 GC
At Home
Continuation
=========================================
"MUTUALLY-EXCLUSIVE & JOINTLY-EXHAUSTIVE"
=========================================
THE CIRCLE OF THREE RELIGIONS [ANOTHER JAMAAL ORIGINAL TADAAHHHH!!!]
Continuing my grade school diagram of the three major ABRAHAMIC faiths... One who identifies himself with only one or just two or all three of the three major world religions may view human beings as being classified or falling into one of three areas of the seven (7) sectors in the diagram. Three sectors are colored and the other four I simply label with the letters "A", "B" & "C" and the center sector with the numeral "1". If the three circles represent all human beings belonging
to the three religions of ISLAM, JUDAISM & CHRISTIANITY, then we may consider the RED sector as those individuals who are exclusively JEWS, the BLUE sector as "pure" CHRISTANS & the GREEN as exclusively MUSLIMS. There are no half-cooked, half-breeds, 50-50, so-so in these sectors. [ALWAYS REMEMBER or NEVER FORGET: this is just a model & overly-simplified...] Now, what or who are those found/located/represented in sectors A, B & C? And those in the middle sector 1? I would say that those in sectors A would form the intersection of JEWS & CHRISTIANS, meaning those whose father might be a JEW & mother a CHRISTIAN & professes either religion [But I think one can be a real Jew only if one's mother is a JEWESS...?] These can also be the sector of the ALLIANCE of JEWS & CHRISTIANS (Those who believe JEWS & CHRISTIANS are not antagonistic to each other & may even be allied...) Similarly
sector B would be those JEWS & MUSLIMS who are allied or do not consider the other as enemy & sector C would form the sector of MUSLIM-CHRISTIAN alliance... There is a saying that "your enemies enemy is your ally" or something similar but I don't fully agree with this...
Anyway, the point in this exercise is the diagram shows us that there is the possibility (We are not claiming that indeed "there actually exists" out there in the real world) of alliances between MUSLIMS & CHRISTIANS, JEWS & MUSLIMS, & CHRISTIAN & JEWS. IF (Those who are not familiar with COUNTERFACTUALS may research/read a little in popular books on Philosophy or the Internet.) - IF the colored sectors represent "purists" who reject the "unpure" then we may say that the colored sectors represent mutually exclusive sectors of MUSLIMS, CHRISTIANS & JEWS... And the intersecting sectors would be the less antagonistic as in accomodating sectors of each RELIGION... Sector A may represent the MUSLIM-JEW ALLIANCE AGAINST CHRISTIANS, sector B may represent the CHRISTIAN-JEW ALLIANCE AGAINST MUSLIMS, and sector C may be the MUSLIM-CHRISTIAN ALLIANCE AGAINST JEWS! And the central sector would be the universalists as in those who find all three religions as all acceptable & NOT EXCLUSIVIST. Does this mean that those in sector 1 are AGAINST the PURISTS in all three religions? Or does this mean that sector 1 represents those who can go along peacefully with any member of any of the three religions?... I PREFER TO VIEW SECTOR 1 AS THE TRUE MUSLIM WHO IS ALSO THE TRUE CHRISTIAN & THE TRUE JEW... THE TRUE MUSLIM IS NON-DIFFERENT FROM THE TRUE CHRISTIAN & THE TRUE JEW.
DEAR READER, What do your think?
Exercise (in futility:) or Homework:
How about well-known or popular individual personalities whom we have come to identify with this or that religion like the present Roman Catholic Pope Benedict or Pope John Paul II or Usamah Binladin or Saudi King Abdullah or ex-Pres. George Bush or Pres. Obama or Yasser Arafat or Ayatullah Khomeini or ETCETCETCETCETCETC...
WHERE WILL YOU LOCATE EACH PERSONALITY? WHAT DOES IT MEAN FOR THEM TO BE LOCATED WITHIN THAT SECTOR OF OUR GEOMETRIC MODEL?
If Mr. X hates Jews & believes in an alliance of Muslims & Christians against the third party, then Mr. X would fall in sector C. But is this interpretation correct or true?
WHAT IS THE TRUE?
SPECIAL BONUS PROBLEM:
USING THE SAME MODEL ANALYZE THE MAJOR RELIGIONS OF THE WORLD
INCLUDING HINDUISM, BUDDHISM, SHINTOISM, CONFUCIANISM, CONFUSIONISM ETC ETC TO BE SUBMITTED "PASA" ASAP
Saturday, July 18, 2009
POST 5 NECESSITY & SUFFICIENCY
6:15 A.M. Manila time
Sunday, July 20, 2009 Gregorian Calendar
Hijrah 1430 Rajab 25
At Home
=================================
NECESSARY & SUFFICIENT CONDITIONS
=================================
by: Jam bin Javi
Let us examine the following statements:
To be Malay is to be Muslim.
IF one is (a) Malay, THEN one is (a) Muslim.
If one is Malay, then, NECESSARILY, one is Muslim.
To be Muslim is a NECESSARY CONDITION to be Malay.
To be Malay is a SUFFICIENT CONDITION to be Muslim.
All Malays are Muslims. But not all Muslims are Malays.
Important notions or concepts that must be noted in the above consideration are:
- CONDITION (to be Malay = "Malay-ness" & to be Muslim = "Muslim-ness")
- The terms IF & THEN and the whole CONDITIONAL statement "If _, then _."
- NECESSITY/NECESSARY CONDITION
- SUFFICIENCY/SUFFICIENT CONDITION
A graphical tool taught in grade school math may help one's understanding.
Using a modified Venn Diagram:
For simplicity, the diagram does not include a so-called UNIVERSAL SET, a third bigger square that should enclose both squares. The MEMBERS of our sets are portrayed as individual points or items rather than as plane areas. We are dealing with INDIVIDUAL HUMANS and the UNIVERSAL SET should be the set of HUMANS. From the diagram, it is easy to see that all INDIVIDUALS who are (found or located) within the square for Malays are also within the square for Muslims, meaning, ALL Malays are Muslims. If one is Malay, then one is Muslim.
But not all Muslims are Malays because there are INDIVIDUALS that are outside the square for Malays and yet are within the square for Muslims, meaning, there are Muslims who are not Malay or, EQUIVALENTLY, NOT ALL Muslims are Malays.
Actually, the presentation is an oversimplified account since the notion of an individual representing or exemplifying an attribute/property or a condition is more complicated than as given. In this case, a single or lone individual exemplifies the attribute or condition of Malayness or Muslimness or both. But, how does Muslimness or Malayness relate to one's being an individual? Can't an individual be a half or just 1/4 Muslim or Malay? And what does it mean to be BOTH Muslim AND Malay? In the diagram, an individual stick-figure represents a unit Malay or a unit Muslim and this is the smallest item which cannot be broken down or analyzed further.
In the mathematical language of SET THEORY, one says that "the SET of Malays forms a SUBSET of the SET of Muslims". A set is a mathematical notion - a group of objects or individuals all together conceived as a single entity - a COLLECTION with each member exemplifying a common attribute of all the other members of the set. Another subset of the set of MUSLIMS may be the set of ARABS, or the set of MOORS.
To satisfy the condition of Malayness is to belong to the class or set of Malays
To satisfy the condition of Muslimness is to belong to the class or set of Muslims
CONDITION of MALAYness = To BE MALAY = To be a MEMBER of/To BELONG to the CLASS or SET of MALAYS
CONDITION of MUSLIMness = To BE MUSLIM = To be a MEMBER of/To BELONG to the CLASS or SET of MUSLIMS
To be Malay is a sufficient condition to be Muslim and to be Muslim is a necessary condition to be Malay. Being Malay guarantees being Muslim. Alternatively,
One cannot be Malay if one is not Muslim.
" " " " unless one is Muslim.
ADDED: 081209
Condition A is necessary for condition B if its FALSITY or NON-OCCURENCE guarantees the FALSITY or NON-OCCURENCE of condition B.
Condition A is sufficient for condition B if its TRUTH or OCCURENCE guarantees the TRUTH or OCCURENCE of condition B.
Necessity and sufficiency are CONVERSE. That a condition A suffices for another condition B
also means that condition B is necessary for condition A.
EXERCISE: Contemplate the case for ARABS or MOORS instead of MALAYS.
O
_|_
/ \
TO BE CONTINUED/EMENDED
Sunday, July 20, 2009 Gregorian Calendar
Hijrah 1430 Rajab 25
At Home
=================================
NECESSARY & SUFFICIENT CONDITIONS
=================================
by: Jam bin Javi
Let us examine the following statements:
To be Malay is to be Muslim.
IF one is (a) Malay, THEN one is (a) Muslim.
If one is Malay, then, NECESSARILY, one is Muslim.
To be Muslim is a NECESSARY CONDITION to be Malay.
To be Malay is a SUFFICIENT CONDITION to be Muslim.
All Malays are Muslims. But not all Muslims are Malays.
Important notions or concepts that must be noted in the above consideration are:
- CONDITION (to be Malay = "Malay-ness" & to be Muslim = "Muslim-ness")
- The terms IF & THEN and the whole CONDITIONAL statement "If _, then _."
- NECESSITY/NECESSARY CONDITION
- SUFFICIENCY/SUFFICIENT CONDITION
A graphical tool taught in grade school math may help one's understanding.
Using a modified Venn Diagram:
For simplicity, the diagram does not include a so-called UNIVERSAL SET, a third bigger square that should enclose both squares. The MEMBERS of our sets are portrayed as individual points or items rather than as plane areas. We are dealing with INDIVIDUAL HUMANS and the UNIVERSAL SET should be the set of HUMANS. From the diagram, it is easy to see that all INDIVIDUALS who are (found or located) within the square for Malays are also within the square for Muslims, meaning, ALL Malays are Muslims. If one is Malay, then one is Muslim.But not all Muslims are Malays because there are INDIVIDUALS that are outside the square for Malays and yet are within the square for Muslims, meaning, there are Muslims who are not Malay or, EQUIVALENTLY, NOT ALL Muslims are Malays.
Actually, the presentation is an oversimplified account since the notion of an individual representing or exemplifying an attribute/property or a condition is more complicated than as given. In this case, a single or lone individual exemplifies the attribute or condition of Malayness or Muslimness or both. But, how does Muslimness or Malayness relate to one's being an individual? Can't an individual be a half or just 1/4 Muslim or Malay? And what does it mean to be BOTH Muslim AND Malay? In the diagram, an individual stick-figure represents a unit Malay or a unit Muslim and this is the smallest item which cannot be broken down or analyzed further.
In the mathematical language of SET THEORY, one says that "the SET of Malays forms a SUBSET of the SET of Muslims". A set is a mathematical notion - a group of objects or individuals all together conceived as a single entity - a COLLECTION with each member exemplifying a common attribute of all the other members of the set. Another subset of the set of MUSLIMS may be the set of ARABS, or the set of MOORS.
To satisfy the condition of Malayness is to belong to the class or set of Malays
To satisfy the condition of Muslimness is to belong to the class or set of Muslims
CONDITION of MALAYness = To BE MALAY = To be a MEMBER of/To BELONG to the CLASS or SET of MALAYS
CONDITION of MUSLIMness = To BE MUSLIM = To be a MEMBER of/To BELONG to the CLASS or SET of MUSLIMS
To be Malay is a sufficient condition to be Muslim and to be Muslim is a necessary condition to be Malay. Being Malay guarantees being Muslim. Alternatively,
One cannot be Malay if one is not Muslim.
" " " " unless one is Muslim.
ADDED: 081209
Condition A is necessary for condition B if its FALSITY or NON-OCCURENCE guarantees the FALSITY or NON-OCCURENCE of condition B.
Condition A is sufficient for condition B if its TRUTH or OCCURENCE guarantees the TRUTH or OCCURENCE of condition B.
Necessity and sufficiency are CONVERSE. That a condition A suffices for another condition B
also means that condition B is necessary for condition A.
EXERCISE: Contemplate the case for ARABS or MOORS instead of MALAYS.
O
_|_
/ \
TO BE CONTINUED/EMENDED
Sunday, April 19, 2009
POST 4 LOGIC CARTOON
April 20, 20009
I enjoy cartoons very much because they carry so much MEANING/message in such a short/small space... Am an amateur cartoonist & these EGGheads are my original conceptions... My thesis focuses on the work of the German philosopher-mathematician-logician GOTTLOB FREGE (1848-1925), the generally-acknowledged originator-founder of modern SYMBOLIC LOGIC.
S - Stolen from jam-bin-javi
NEXT: NECESSARY & SUFFICIENT CONDITIONS
I enjoy cartoons very much because they carry so much MEANING/message in such a short/small space... Am an amateur cartoonist & these EGGheads are my original conceptions... My thesis focuses on the work of the German philosopher-mathematician-logician GOTTLOB FREGE (1848-1925), the generally-acknowledged originator-founder of modern SYMBOLIC LOGIC.
S - Stolen from jam-bin-javiNEXT: NECESSARY & SUFFICIENT CONDITIONS
Monday, March 2, 2009
POST 3 LOGIC FOR BELIEVERS
Moonday, March 02, 2009
@Home
==============
LOGIC FOR BELIEVERS
==============
by jam_bin_javi
Let us examine the following STATEMENT (which upon INTERPRETATION may be characterized as a RELIGIOUS one) and ANALYZE it using the principles of SYMBOLIC LOGIC,
"IF YOU LOVE THIS LIFE, THEN YOU ARE NOT A TRUE BELIEVER."
Logicians would classify this as a compound CONDITIONAL or HYPOTHETICAL statement consisting of two component (sub-)statements, namely, the ANTECEDENT which follows the term "IF" and the CONSEQUENT which follows the term "THEN."
ANTECEDENT: "YOU LOVE THIS LIFE."
CONSEQUENT: "YOU ARE NOT A TRUE BELIEVER."
Symbolically, we can make the following representations:
L = "YOU LOVE THIS LIFE."
T = "YOU ARE A TRUE BELIEVER."
The DENIAL or NEGATION of the second statement is:
"IT IS NOT THE CASE THAT YOU ARE A TRUE BELIEVER."
This is symbolized by:
~T
From the logical viewpoint, the last statement is EXACTLY THE SAME AS or
IDENTICAL WITH [the logically proper word is EQUIVALENT TO/WITH]
"YOU ARE NOT A TRUE BELIEVER."
Thus, the original compound statement when symbolized becomes,
L [IMPLIES OR IMPLICATION SIGN] ~T
According to the RULE OF REPLACEMENT called MATERIAL IMPLICATION,
L IMPLICATION SIGN ~T IFF ~L V ~T
And by another RULE OF REPLACEMENT called DE MORGAN'S THEOREM,
~L V ~T IFF ~(L AND T)
Thus, our original statement can be expressed symbolically in another form,
~(L AND T)
When translated into English, this is read as,
"IT IS NOT THE CASE THAT YOU LOVE THIS LIFE AND YOU ARE A TRUE BELIEVER."
Or equivalently,
"YOU CANNOT BOTH LOVE THIS LIFE AND BE A TRUE BELIEVER."
Here, we have two cases each of which completely excludes the other. This is the essential meaning or content of the original conditional statement: that one case totally excludes the other. They cannot both be true at the same time. Note also that in the original conditional statement, we are NOT ASSERTING that "YOU LOVE THIS LIFE."
Moreover, it is extremely important to note that the denial or negation of the statement,
"YOU LOVE THIS LIFE."
is NOT,
"YOU HATE THIS LIFE."
but rather,
"YOU DO NOT LOVE THIS LIFE."
And that "YOU DO NOT LOVE THIS LIFE." is NOT EQUIVALENT to "YOU HATE THIS LIFE." To claim that these two are equivalent is illogical or bad reasoning.
IF YOU HATE THIS LIFE, THEN YOU DO NOT LOVE THIS LIFE.
[TRUE (On INTERPRETATION)]
IF YOU DO NOT LOVE THIS LIFE, THEN YOU HATE THIS LIFE.
[NOT QUITE. What if you couldn't care less or you are detached or simply disinterested? "Apathetic" is a LOADED word...]
In symbolic logic, the statements,
"YOU DO NOT LOVE THIS LIFE." and "YOU HATE THIS LIFE." would be symbolized or represented differently. The analysis of the underlying MEANINGS of the component terms and the actual DETERMINATION of the TRUTH-VALUES of these statements lie outside the purview or scope of logic.
Lastly, our original statement has its equivalent TRANSPOSITION form,
"IF YOU ARE A TRUE BELIEVER, THEN YOU DO NOT LOVE THIS LIFE."
From the logical viewpoint, this is the same statement as the original.
(S)Written in 2006 ["another CONSPIRATORIAL pasiklab" the original states...]
MODIFIED/ABRIDGED BY THE AUTHOR to suit this AL-MANTIQ BLOG which is intended for everyone...
POSTED March 02, 2009
(S) = STOLEN FROM Jamaal bin Javier
***COPY AT YOUR OWN RISK***
@Home
==============
LOGIC FOR BELIEVERS
==============
by jam_bin_javi
Let us examine the following STATEMENT (which upon INTERPRETATION may be characterized as a RELIGIOUS one) and ANALYZE it using the principles of SYMBOLIC LOGIC,
"IF YOU LOVE THIS LIFE, THEN YOU ARE NOT A TRUE BELIEVER."
Logicians would classify this as a compound CONDITIONAL or HYPOTHETICAL statement consisting of two component (sub-)statements, namely, the ANTECEDENT which follows the term "IF" and the CONSEQUENT which follows the term "THEN."
ANTECEDENT: "YOU LOVE THIS LIFE."
CONSEQUENT: "YOU ARE NOT A TRUE BELIEVER."
Symbolically, we can make the following representations:
L = "YOU LOVE THIS LIFE."
T = "YOU ARE A TRUE BELIEVER."
The DENIAL or NEGATION of the second statement is:
"IT IS NOT THE CASE THAT YOU ARE A TRUE BELIEVER."
This is symbolized by:
~T
From the logical viewpoint, the last statement is EXACTLY THE SAME AS or
IDENTICAL WITH [the logically proper word is EQUIVALENT TO/WITH]
"YOU ARE NOT A TRUE BELIEVER."
Thus, the original compound statement when symbolized becomes,
L [IMPLIES OR IMPLICATION SIGN] ~T
According to the RULE OF REPLACEMENT called MATERIAL IMPLICATION,
L IMPLICATION SIGN ~T IFF ~L V ~T
And by another RULE OF REPLACEMENT called DE MORGAN'S THEOREM,
~L V ~T IFF ~(L AND T)
Thus, our original statement can be expressed symbolically in another form,
~(L AND T)
When translated into English, this is read as,
"IT IS NOT THE CASE THAT YOU LOVE THIS LIFE AND YOU ARE A TRUE BELIEVER."
Or equivalently,
"YOU CANNOT BOTH LOVE THIS LIFE AND BE A TRUE BELIEVER."
Here, we have two cases each of which completely excludes the other. This is the essential meaning or content of the original conditional statement: that one case totally excludes the other. They cannot both be true at the same time. Note also that in the original conditional statement, we are NOT ASSERTING that "YOU LOVE THIS LIFE."
Moreover, it is extremely important to note that the denial or negation of the statement,
"YOU LOVE THIS LIFE."
is NOT,
"YOU HATE THIS LIFE."
but rather,
"YOU DO NOT LOVE THIS LIFE."
And that "YOU DO NOT LOVE THIS LIFE." is NOT EQUIVALENT to "YOU HATE THIS LIFE." To claim that these two are equivalent is illogical or bad reasoning.
IF YOU HATE THIS LIFE, THEN YOU DO NOT LOVE THIS LIFE.
[TRUE (On INTERPRETATION)]
IF YOU DO NOT LOVE THIS LIFE, THEN YOU HATE THIS LIFE.
[NOT QUITE. What if you couldn't care less or you are detached or simply disinterested? "Apathetic" is a LOADED word...]
In symbolic logic, the statements,
"YOU DO NOT LOVE THIS LIFE." and "YOU HATE THIS LIFE." would be symbolized or represented differently. The analysis of the underlying MEANINGS of the component terms and the actual DETERMINATION of the TRUTH-VALUES of these statements lie outside the purview or scope of logic.
Lastly, our original statement has its equivalent TRANSPOSITION form,
"IF YOU ARE A TRUE BELIEVER, THEN YOU DO NOT LOVE THIS LIFE."
From the logical viewpoint, this is the same statement as the original.
(S)Written in 2006 ["another CONSPIRATORIAL pasiklab" the original states...]
MODIFIED/ABRIDGED BY THE AUTHOR to suit this AL-MANTIQ BLOG which is intended for everyone...
POSTED March 02, 2009
(S) = STOLEN FROM Jamaal bin Javier
***COPY AT YOUR OWN RISK***
Wednesday, February 4, 2009
POST 2 LOGIC PAPER 1
Tuesday
February, 03, 2009
Manila
THE FOLLOWING PAPER CAN ONLY BE APPRECIATED BY SOMEONE WHO HAS TAKEN PREDICATE LOGIC (& NOT JUST ARISTOTELIAN OR PROPOSITIONAL LOGIC) PLKUS A LITTLE BACKGROUND IN MODERN METAPHYSICS. THE U.P. DILIMAN UNDERGRAD COURSES PHILO 11 & PHILO 12 ARE SUFFICIENT (PRE)REQUISITES... BUT THE PAPER WITH ITS BROAD STROKES CAN SERVE AS A USEFUL GUIDE FOR INVESTIGATORS INTO LOGIC & METAPHYSICS.
==========================
ON ONTOLOGICAL COMMITMENT
==========================
by JAMAAL bin JAVIER, A.B. Philosophy, on-going M.A. in Philosophy
University of the Philippines at Diliman
ABSTRACT
This brief but comprehensive paper was submitted in one of the author's graduate courses in logic. Here, he seeks to answer the given guide question posed in the first line of the INTRODUCTION and tries to develop a rough outline, a skeleton for a more expanded and detailed work that serves as a chapter or section in his proposed Master of Arts in Philosophy THESIS on logic.
INTRODUCTION
Where do metaphysical or ontological presuppositions enter into our formal logical systems?
This question may be answered by considering the connection, if there is any, between the areas of logic and metaphysics, both broadly construed as systems in philosophy. There are three notions involved in this consideration, namely,
1) ontology,
defined as
(sense 2) "the assumptions about existence underlying any conceptual scheme or any theory or system of ideas" (FLEW 1984p),
2) logic,
as a symbolic system of formal language, and
3) a third relating term or item.
This third element (which is actually a system of logic according to the Polish logician Alfred Tarski) is semantics. Semantics is, roughly, the study of the relations obtaining between logico-linguistic objects and what they express. By the term "logico-linguistic", I am referring to language in its broadest, most general construal. And this sense includes both natural languages such as English or Bahasa and artificial ones such as computer languages and formal languages such as the various calculuses/calculi of symbolic logic. Now according to...
A formal logical system admits of interpretation of its syntactical elements through or via semantics, and ontologic(al) notions such as existence arise when logical symbols are interpreted or given meaning. In particular, in first-order predicate logic (or equivalenly, quantificational theory or predicate calculus), the philosophical notion of existence is formally expressed using the notion of the quantifier and symbolized by "E". One component of a typical well-formed formula in quantificational logic is the symbol "Ex" which is translated into colloquial English as, "There exists an x." The variable "x" stands for any individual and can take on different values depending on the domain of discourse which, in turn, depends on the intended interpretation, that is, on what is being talked about. A group of statements, a language, or a theory, can be translated into the idioms of predicate logic thereby producing quantified statements that make implicit reference to a totality of individuals. It is relative to such a totality that ontological notions - such as presuppositions about existence and particular entities, come into the picture. Existential presuppositions, i.e., claims about the existence of certain entities or kinds of entities may lie hidden in a theory (i.e. implicitly) and covertly enter its corresponding formal logical system. But these can be laid bare and made explicit using the techniques of higher formal logic.
QUINEAN ONTOLOGY
The American logician WIllard Van Orman Quine (1800-1985) was one of the pioneering theorists who devoted much effort in trying to come up with a workable account of the relation between logic and ontology. In particular, he is noted for his contribution, the idea of (the criterion of) ontological commitment [In his work WORD & OBJECT(19), Quine used the phrase "ontic commitment" instead] which he expounded and developed in his various works in logic. Ontological commitment refers to the metaphysical presuppositions of a theory and Quine sought an answer to the general question, "What are the metaphysical presuppostions of a theory?" He was endeavoring to determine what a given theory (implicitly) says there is. Working within the technical ambit of quantificational logic, Quine tried to solve the problem of how to lay bare the ontological assumptions implicit in a theory. And he came up with his proposed notion of "ontological commitment", articulated in a so-called "criterion". The core idea is expressed in his famous formula or dictum,
"To be is to be the value of a (bound individual) variable." (QUINE, p.)
By "bound", it is meant here that the variable lies within the scope of a quantifier, e.g., (x) or Ey, where the variables x and y range over individuals. Otherwise, it is "free". An expression in the predicate calculus is a statement if and only if all occurrences of its variables are bound. If any occurrence is free, then the expression is not a statement but rather a propositional function and ,therefore, is neither true nor false. It has no truth-value. The predicate calculus takes into account not only statements or propositions but propositional functions as well and its expressions may make implicit reference to (an) individual or to a totality of individuals. This happens whenever there is an expression that contains a quantifier. In a quantified expression, variables take on values or are said to be "instantiated". Instantiation is accomplished by replacing the variables by constant terms that stand for particular individuals. All these individuals that may be substituted as constant terms within a particular theory constitute a
totality, a set of individuals which is called the domain or universe of discourse. This totality, in effect, contains everything one wants to talk about. For example, in mathematics, one normally talks about numbers and in linguistics, words or phrases. It is with reference to such domain of individuals that the notion of ontological commitment applies or arises.
EXISTENCE IS NOT A PREDICATE
An important distinction must be made between the usual or commonsensical existence statements made in ordinary, colloquial language (e.g., "There are stones.") and the logical notion of existence as treated within the formal language of first-order predicate calculus. Formal languages are more regimented and, unlike ordinary language, do not readily admit of changes, variations or nuances.They are subject to strict rules of syntax and are more precise and rigid. Lest we forget, here, under present consideration and subject to scrutiny is the quantificational-logical notion of existence or the notion of existence as expressed in the language of first-order predicate calculus. This is the sense of "existence" used or appealed to by philosophers, paricularly by the logicians, in their formal language of logic whenever they are deliberating about existence. Existence is a fundamental concept in philosophy that lies at the core of most philosophical issues. In higher-order logic, existence is not a first-level [I use "level" instead of "order".] predicate, attribute, or property unlike those expressed in the simple
propositons of propositional logic. Existence is rather a second-level predicate, a "property of properties." In predicate logic, a statement is further broken up (or "analyzed") into an attribute which applies to an individual. [Thus the adjective "predicate"] Formally, this action is captured by using the generic symbol PHI(x) where "PHI" is a first-level predicate and the variable x ranges over individuals of the domain of discourse. The phrase "existence is not a
(first-level) predicate" [cf. Kant & other philosphical writings] means that the notion of existence is NOT captured by simply rendering it (in the formal apparatus of logic) as "E" applied to x as above-mentioned or as it is usually done in the ordinary (lower order) logic of propositions. Here, one may assert that an entity m exists and this is expressed as a simple statement symbolized by a single letter M. Predicate logic allows a more detailed treatment of proofs and arguments and a finer-grained analysis of statements especially those involving the idea of multiple generality. And in first-order predicate logic, the notion of existence is captured and formally expressed by the symbol "E". A well-formed formula of quantificational logic is ExPHI(x) roughly translatable as "There exists an x with the attribute PHI. This clearly shows that existence is not a mere first-level predicate.
The existential quantifier symbolized by "E" carries existential import or is said to assert existence. It is a "logically regimented way" of saying that "There is" or "There exists". Existence is what existential quantification expresses. There are things of kind F if and only if ExFx. For Quine, it is "unreasonable to ask for an explication of existence in simpler terms." QUINE 1969
For Quine, "the question of the ontological commitment of a theory does not properly arise except as that theory is expressed in classical quantificational form." (QUINE p.) This means that any assertion about existence can be made only within a conceptual scheme or a theoretical framework which can be expressed in the language of predicate logic prior to the further application of the criterion of ontological commitment. "Quine's critieerion is framed within that context of the logic of quantification." (SEVERENS [.) [NOTE VERY WELL that the primary concern here is NOT with existence per se but with "Imputations of existence, on what a theory says exists." [QUINE]
ONTOLOGY VS ONTOLOGICAL COMMITMENT OF A THEORY
In his work, Quine was compelled to make a finer distinction between the ontology of a theory and its ontological commitment.[Roughly, between what there is in a theory and what the theory says there is] and this is briefly tackled here in passing.
According to James Oliver,
"The ontology of a theory is the range of values of the variables of the theory whereas
its ontological commitment is often only a part of its ontology. The ontology or range
of values is a matter of the conceptual scheme or language system in which the theory
is stated." (OLIVER, 1974)
Different ontologies, that is, different ranges of values may satisfy a given theory. For example, again in mathematics, a theorem expressed as a statement to the effect that, "For all whole numbers x, x possesses the numerical property P [Formally: (x)(Nx IMPLIES Px)] may be satisfied by the set of even numbers and at the same time also by the set of odd numbers with the two mutually exclusive and jointly exhaustive sets forming the the set of natural numbers N. One may treat as a mathematical theory a subset of the mathematical statements concerned with the properties of whole numbers and thus one may have two ontologies - one containing only the set of even numbers and the other containing only the set of odd numbers wuth both fulfilling the theory that yields the quantified statement Ex(x is a natural number) [or identically, Ex(xeN). Hence, it is shown that the ontology of a theory need not be identical with the ontological commitment of the theory. The objects that the theory assumes may not be those to which it is ontologically committed.
Robert Audi avers,
"A theory is ontologically committed to a given ogbject only if that object is common to all of
the ontologies fulfilling the theory. And the theory is ontologically committed to objects
of a given class provided that class is not empty according to each of the ontologies fulfillling
the theory." (AUDI 1995)
CRITICISMS
At the basic level, a common attack against Quine's work is the apparent circularity in his
conception of existence and this a criticism that applies not only to the notion of logical existence but to logic itself. Various non-academic discussions in the Internet (2004) revolve around this point or issue, what seems to be the assumption of an entity's existence by the very act of one's making a statement about it. This is expressed formally in the formula
a exists = def Ex(x=a)
As it stands, this is an "explication of singular existence". But for Quine, "explication of the existential quantifier itself and of general existence" is "a forlorn cause." (QUINE 1969)
Apparently, it leads nowhere. [The underlying idea is somewhat vaguely similar or analogous to the notion of "bootstrapping" in computers and electronics, or in common experience, "carrying oneself through one's boots or shoelaces."] On a more scholarly vein, Quine's overall postion regarding his doctrine of ontological commitment essentially adopts the so-called objectual interpretation of the quantifier ahd those who subscribe to the substitutional interpretation have a basis or a good reason to reject his criterion. This despite Quine's own objections discussed in his Existence and Quantification (QUINE 1969)
On a still higher level of scholarship, the Japanese-American philosopher Charles Chihara has
vigorously criticized Quinean criterion of ontological commitment. Some of these criticisms shall be mentioned minus any further reaction or detailed discussion. According to Chihara, Quine's use of the term "theory" is ambiguous and equivocates between the notion of a theory as a deductively closed (i.e., covers all its logical consequences) set of fully interpreted sentences and that as otherwise. Moreover, Quine has applied his criterion to single sentences and it would appear that, since his criterion is intended to determine the ontological comittment of theories, he is imputing that "a single sentence is a kind of degenerate theory." (CHIHARA 1974)
Also, Chihara, quoting the works of the American philosopher Donald Davidson and the Finnish logician Jaakko Hintikka, points to a possible risk of misreading Quine, leading to the misconstrual that there is no need to specify a particular universe of discourse for the theory (CHIHARA 1974) Furthermore, Chihara has shared a criticism by John Searle to the effect that Quine's criterion is vulnerable to a reductio for apparently it provides a mere condition of sufficiency and not of necessity and sufficiency as it claims to have. (CHIHARA 1974) Lastly, it is usually assumed that the notion of ontological commitment is a semantic one as Quine himself has admitted that his criterion is a semantical formula. (QUINE 1953) But in his paper, James Oliver considers the question (his seventh) of the pragmatic basis of ontological commitment (OLIVER 1974)
ALTERNATIVES
In his paper, Chihara has provided other alternative criteria of ontological commitment and their formulations. He presented four criteria in all: Quine's, Chateaubriands's, Church's and Chomsky & Scheffler's. Chateaubriand's criterion which was originally presented in that writer's Ph.D. dissertation is extensively if not tortuously discussed by Chihara and involves the "notions of theorems of a theory" to "avoid talk about necessary and/or contingent truths." CHIHARA, 1974 p) On the other hand, Church's critierion is formulated as follows:
"The assertion of (Ex)(M) carries ontological commitment to entities x such that M," where the letter 'x' may be replaced by any variable, the italicized letter 'x' may be replaced by any name of a the same variable, the letter 'M' may be replaced by any open sentence containing only the above variable, and the italicized letter 'M' may be replaced by any name of this open sentence."
Noam Chomsky and Israel Scheffler state a criterion similar to Church's but in the form,
"A theory T makes a _____-assumption if and only if it yields a statement of the form "(Ex)(x is (a) _____)" (CHIHARA 1974)
According to Chihara's analsis, Quine's later writings aparently reflected a move towards a Chateaubriand-type of criterion. Of the four criteria, Chateaubriands's is deemed by Chihara as "the most adequate."
SIGNIFICANCE & APPLICATIONS
The notion of ontological commitment has important applications in philosophy particularly in the philosophy of language, philosophy of mathematics, philosophy of science, and in metaphysics. A specific application is the drawing of the distinction between nominalistic and realistic languages as expressions/manifestations of the metaphysical positions of nominalism and realism, where, in realistic languages, words of abstract or general nature" are "substituents for variables" unlike in the case of nominalistic languages. (CHIHARA 1974) Other important philosophical topics closely related with ontological commitment are:
-the nature and relation of intensionality & extensionality;
-the interpretation of the quantifier;
-the nature of theories and their interpretations;
-the nature of kinds;
-the question of existence of nearly possible entities; and
-the significance of Ockham's Principle (CONCISE ROUTLEDGE ENCYBLOPEDIA OF PHILO)
CONCLUSiON
THe foregoing brief considerations show that the logical system of first-order predicate calculus is closely intertwined with ontology via the semantical interpretation of the syntactical elements of the quantifier and its bound variables. Such is the case whether the intended universe of discourse involves individuals as numbers, words, or the entities of natural science, that is, whether the theory under study is mathematical, linguistic or scientific. Quine's criterion of ontological commitment may be applied by translating the given theory into the quantificational language of the predicate calculus. The notion of ontological commitment lays bare a fertile area of research of sufficient depth and breadth for aspiring researchers in philosophy especially in logic and metaphysics. Its usefulness and applicability in wide ranges
of philosophical topics provides enough warrant for a closer investigation of the deeper implications of the concept and the issues involved. Despite its origins from the highly- technical, forbidding realm of quantificational theory, the concept of ontological commitment, treated qualitatively, can serve as a unifying principle in tackling other less abstruse areas in philosophy. As with Tarski's Conception of Truth which evolved from scientific semantics and the calculus of classes, the concept of ontological commitment as propounded by Quine will continue to be a legitimate topic for scholarly philosophical papers, polemical or otherwise.
REFERENCES:
AUDI
CHIHARA
FLEW
MARTIN
OLIVER
PRESLEY
QUINE
QUINE
QUINE
====
PAPER AS IT STANDS MAY BE CHARGED WITH PLAGIARISM...
STILL UNFINISHED ACTUALLY...
UPDATED Friday Feb 13, 2009
WARNING: COPY AT YOUR OWN RISK!
February, 03, 2009
Manila
THE FOLLOWING PAPER CAN ONLY BE APPRECIATED BY SOMEONE WHO HAS TAKEN PREDICATE LOGIC (& NOT JUST ARISTOTELIAN OR PROPOSITIONAL LOGIC) PLKUS A LITTLE BACKGROUND IN MODERN METAPHYSICS. THE U.P. DILIMAN UNDERGRAD COURSES PHILO 11 & PHILO 12 ARE SUFFICIENT (PRE)REQUISITES... BUT THE PAPER WITH ITS BROAD STROKES CAN SERVE AS A USEFUL GUIDE FOR INVESTIGATORS INTO LOGIC & METAPHYSICS.
==========================
ON ONTOLOGICAL COMMITMENT
==========================
by JAMAAL bin JAVIER, A.B. Philosophy, on-going M.A. in Philosophy
University of the Philippines at Diliman
ABSTRACT
This brief but comprehensive paper was submitted in one of the author's graduate courses in logic. Here, he seeks to answer the given guide question posed in the first line of the INTRODUCTION and tries to develop a rough outline, a skeleton for a more expanded and detailed work that serves as a chapter or section in his proposed Master of Arts in Philosophy THESIS on logic.
INTRODUCTION
Where do metaphysical or ontological presuppositions enter into our formal logical systems?
This question may be answered by considering the connection, if there is any, between the areas of logic and metaphysics, both broadly construed as systems in philosophy. There are three notions involved in this consideration, namely,
1) ontology,
defined as
(sense 2) "the assumptions about existence underlying any conceptual scheme or any theory or system of ideas" (FLEW 1984p),
2) logic,
as a symbolic system of formal language, and
3) a third relating term or item.
This third element (which is actually a system of logic according to the Polish logician Alfred Tarski) is semantics. Semantics is, roughly, the study of the relations obtaining between logico-linguistic objects and what they express. By the term "logico-linguistic", I am referring to language in its broadest, most general construal. And this sense includes both natural languages such as English or Bahasa and artificial ones such as computer languages and formal languages such as the various calculuses/calculi of symbolic logic. Now according to...
A formal logical system admits of interpretation of its syntactical elements through or via semantics, and ontologic(al) notions such as existence arise when logical symbols are interpreted or given meaning. In particular, in first-order predicate logic (or equivalenly, quantificational theory or predicate calculus), the philosophical notion of existence is formally expressed using the notion of the quantifier and symbolized by "E". One component of a typical well-formed formula in quantificational logic is the symbol "Ex" which is translated into colloquial English as, "There exists an x." The variable "x" stands for any individual and can take on different values depending on the domain of discourse which, in turn, depends on the intended interpretation, that is, on what is being talked about. A group of statements, a language, or a theory, can be translated into the idioms of predicate logic thereby producing quantified statements that make implicit reference to a totality of individuals. It is relative to such a totality that ontological notions - such as presuppositions about existence and particular entities, come into the picture. Existential presuppositions, i.e., claims about the existence of certain entities or kinds of entities may lie hidden in a theory (i.e. implicitly) and covertly enter its corresponding formal logical system. But these can be laid bare and made explicit using the techniques of higher formal logic.
QUINEAN ONTOLOGY
The American logician WIllard Van Orman Quine (1800-1985) was one of the pioneering theorists who devoted much effort in trying to come up with a workable account of the relation between logic and ontology. In particular, he is noted for his contribution, the idea of (the criterion of) ontological commitment [In his work WORD & OBJECT(19), Quine used the phrase "ontic commitment" instead] which he expounded and developed in his various works in logic. Ontological commitment refers to the metaphysical presuppositions of a theory and Quine sought an answer to the general question, "What are the metaphysical presuppostions of a theory?" He was endeavoring to determine what a given theory (implicitly) says there is. Working within the technical ambit of quantificational logic, Quine tried to solve the problem of how to lay bare the ontological assumptions implicit in a theory. And he came up with his proposed notion of "ontological commitment", articulated in a so-called "criterion". The core idea is expressed in his famous formula or dictum,
"To be is to be the value of a (bound individual) variable." (QUINE, p.)
By "bound", it is meant here that the variable lies within the scope of a quantifier, e.g., (x) or Ey, where the variables x and y range over individuals. Otherwise, it is "free". An expression in the predicate calculus is a statement if and only if all occurrences of its variables are bound. If any occurrence is free, then the expression is not a statement but rather a propositional function and ,therefore, is neither true nor false. It has no truth-value. The predicate calculus takes into account not only statements or propositions but propositional functions as well and its expressions may make implicit reference to (an) individual or to a totality of individuals. This happens whenever there is an expression that contains a quantifier. In a quantified expression, variables take on values or are said to be "instantiated". Instantiation is accomplished by replacing the variables by constant terms that stand for particular individuals. All these individuals that may be substituted as constant terms within a particular theory constitute a
totality, a set of individuals which is called the domain or universe of discourse. This totality, in effect, contains everything one wants to talk about. For example, in mathematics, one normally talks about numbers and in linguistics, words or phrases. It is with reference to such domain of individuals that the notion of ontological commitment applies or arises.
EXISTENCE IS NOT A PREDICATE
An important distinction must be made between the usual or commonsensical existence statements made in ordinary, colloquial language (e.g., "There are stones.") and the logical notion of existence as treated within the formal language of first-order predicate calculus. Formal languages are more regimented and, unlike ordinary language, do not readily admit of changes, variations or nuances.They are subject to strict rules of syntax and are more precise and rigid. Lest we forget, here, under present consideration and subject to scrutiny is the quantificational-logical notion of existence or the notion of existence as expressed in the language of first-order predicate calculus. This is the sense of "existence" used or appealed to by philosophers, paricularly by the logicians, in their formal language of logic whenever they are deliberating about existence. Existence is a fundamental concept in philosophy that lies at the core of most philosophical issues. In higher-order logic, existence is not a first-level [I use "level" instead of "order".] predicate, attribute, or property unlike those expressed in the simple
propositons of propositional logic. Existence is rather a second-level predicate, a "property of properties." In predicate logic, a statement is further broken up (or "analyzed") into an attribute which applies to an individual. [Thus the adjective "predicate"] Formally, this action is captured by using the generic symbol PHI(x) where "PHI" is a first-level predicate and the variable x ranges over individuals of the domain of discourse. The phrase "existence is not a
(first-level) predicate" [cf. Kant & other philosphical writings] means that the notion of existence is NOT captured by simply rendering it (in the formal apparatus of logic) as "E" applied to x as above-mentioned or as it is usually done in the ordinary (lower order) logic of propositions. Here, one may assert that an entity m exists and this is expressed as a simple statement symbolized by a single letter M. Predicate logic allows a more detailed treatment of proofs and arguments and a finer-grained analysis of statements especially those involving the idea of multiple generality. And in first-order predicate logic, the notion of existence is captured and formally expressed by the symbol "E". A well-formed formula of quantificational logic is ExPHI(x) roughly translatable as "There exists an x with the attribute PHI. This clearly shows that existence is not a mere first-level predicate.
The existential quantifier symbolized by "E" carries existential import or is said to assert existence. It is a "logically regimented way" of saying that "There is" or "There exists". Existence is what existential quantification expresses. There are things of kind F if and only if ExFx. For Quine, it is "unreasonable to ask for an explication of existence in simpler terms." QUINE 1969
For Quine, "the question of the ontological commitment of a theory does not properly arise except as that theory is expressed in classical quantificational form." (QUINE p.) This means that any assertion about existence can be made only within a conceptual scheme or a theoretical framework which can be expressed in the language of predicate logic prior to the further application of the criterion of ontological commitment. "Quine's critieerion is framed within that context of the logic of quantification." (SEVERENS [.) [NOTE VERY WELL that the primary concern here is NOT with existence per se but with "Imputations of existence, on what a theory says exists." [QUINE]
ONTOLOGY VS ONTOLOGICAL COMMITMENT OF A THEORY
In his work, Quine was compelled to make a finer distinction between the ontology of a theory and its ontological commitment.[Roughly, between what there is in a theory and what the theory says there is] and this is briefly tackled here in passing.
According to James Oliver,
"The ontology of a theory is the range of values of the variables of the theory whereas
its ontological commitment is often only a part of its ontology. The ontology or range
of values is a matter of the conceptual scheme or language system in which the theory
is stated." (OLIVER, 1974)
Different ontologies, that is, different ranges of values may satisfy a given theory. For example, again in mathematics, a theorem expressed as a statement to the effect that, "For all whole numbers x, x possesses the numerical property P [Formally: (x)(Nx IMPLIES Px)] may be satisfied by the set of even numbers and at the same time also by the set of odd numbers with the two mutually exclusive and jointly exhaustive sets forming the the set of natural numbers N. One may treat as a mathematical theory a subset of the mathematical statements concerned with the properties of whole numbers and thus one may have two ontologies - one containing only the set of even numbers and the other containing only the set of odd numbers wuth both fulfilling the theory that yields the quantified statement Ex(x is a natural number) [or identically, Ex(xeN). Hence, it is shown that the ontology of a theory need not be identical with the ontological commitment of the theory. The objects that the theory assumes may not be those to which it is ontologically committed.
Robert Audi avers,
"A theory is ontologically committed to a given ogbject only if that object is common to all of
the ontologies fulfilling the theory. And the theory is ontologically committed to objects
of a given class provided that class is not empty according to each of the ontologies fulfillling
the theory." (AUDI 1995)
CRITICISMS
At the basic level, a common attack against Quine's work is the apparent circularity in his
conception of existence and this a criticism that applies not only to the notion of logical existence but to logic itself. Various non-academic discussions in the Internet (2004) revolve around this point or issue, what seems to be the assumption of an entity's existence by the very act of one's making a statement about it. This is expressed formally in the formula
a exists = def Ex(x=a)
As it stands, this is an "explication of singular existence". But for Quine, "explication of the existential quantifier itself and of general existence" is "a forlorn cause." (QUINE 1969)
Apparently, it leads nowhere. [The underlying idea is somewhat vaguely similar or analogous to the notion of "bootstrapping" in computers and electronics, or in common experience, "carrying oneself through one's boots or shoelaces."] On a more scholarly vein, Quine's overall postion regarding his doctrine of ontological commitment essentially adopts the so-called objectual interpretation of the quantifier ahd those who subscribe to the substitutional interpretation have a basis or a good reason to reject his criterion. This despite Quine's own objections discussed in his Existence and Quantification (QUINE 1969)
On a still higher level of scholarship, the Japanese-American philosopher Charles Chihara has
vigorously criticized Quinean criterion of ontological commitment. Some of these criticisms shall be mentioned minus any further reaction or detailed discussion. According to Chihara, Quine's use of the term "theory" is ambiguous and equivocates between the notion of a theory as a deductively closed (i.e., covers all its logical consequences) set of fully interpreted sentences and that as otherwise. Moreover, Quine has applied his criterion to single sentences and it would appear that, since his criterion is intended to determine the ontological comittment of theories, he is imputing that "a single sentence is a kind of degenerate theory." (CHIHARA 1974)
Also, Chihara, quoting the works of the American philosopher Donald Davidson and the Finnish logician Jaakko Hintikka, points to a possible risk of misreading Quine, leading to the misconstrual that there is no need to specify a particular universe of discourse for the theory (CHIHARA 1974) Furthermore, Chihara has shared a criticism by John Searle to the effect that Quine's criterion is vulnerable to a reductio for apparently it provides a mere condition of sufficiency and not of necessity and sufficiency as it claims to have. (CHIHARA 1974) Lastly, it is usually assumed that the notion of ontological commitment is a semantic one as Quine himself has admitted that his criterion is a semantical formula. (QUINE 1953) But in his paper, James Oliver considers the question (his seventh) of the pragmatic basis of ontological commitment (OLIVER 1974)
ALTERNATIVES
In his paper, Chihara has provided other alternative criteria of ontological commitment and their formulations. He presented four criteria in all: Quine's, Chateaubriands's, Church's and Chomsky & Scheffler's. Chateaubriand's criterion which was originally presented in that writer's Ph.D. dissertation is extensively if not tortuously discussed by Chihara and involves the "notions of theorems of a theory" to "avoid talk about necessary and/or contingent truths." CHIHARA, 1974 p) On the other hand, Church's critierion is formulated as follows:
"The assertion of (Ex)(M) carries ontological commitment to entities x such that M," where the letter 'x' may be replaced by any variable, the italicized letter 'x' may be replaced by any name of a the same variable, the letter 'M' may be replaced by any open sentence containing only the above variable, and the italicized letter 'M' may be replaced by any name of this open sentence."
Noam Chomsky and Israel Scheffler state a criterion similar to Church's but in the form,
"A theory T makes a _____-assumption if and only if it yields a statement of the form "(Ex)(x is (a) _____)" (CHIHARA 1974)
According to Chihara's analsis, Quine's later writings aparently reflected a move towards a Chateaubriand-type of criterion. Of the four criteria, Chateaubriands's is deemed by Chihara as "the most adequate."
SIGNIFICANCE & APPLICATIONS
The notion of ontological commitment has important applications in philosophy particularly in the philosophy of language, philosophy of mathematics, philosophy of science, and in metaphysics. A specific application is the drawing of the distinction between nominalistic and realistic languages as expressions/manifestations of the metaphysical positions of nominalism and realism, where, in realistic languages, words of abstract or general nature" are "substituents for variables" unlike in the case of nominalistic languages. (CHIHARA 1974) Other important philosophical topics closely related with ontological commitment are:
-the nature and relation of intensionality & extensionality;
-the interpretation of the quantifier;
-the nature of theories and their interpretations;
-the nature of kinds;
-the question of existence of nearly possible entities; and
-the significance of Ockham's Principle (CONCISE ROUTLEDGE ENCYBLOPEDIA OF PHILO)
CONCLUSiON
THe foregoing brief considerations show that the logical system of first-order predicate calculus is closely intertwined with ontology via the semantical interpretation of the syntactical elements of the quantifier and its bound variables. Such is the case whether the intended universe of discourse involves individuals as numbers, words, or the entities of natural science, that is, whether the theory under study is mathematical, linguistic or scientific. Quine's criterion of ontological commitment may be applied by translating the given theory into the quantificational language of the predicate calculus. The notion of ontological commitment lays bare a fertile area of research of sufficient depth and breadth for aspiring researchers in philosophy especially in logic and metaphysics. Its usefulness and applicability in wide ranges
of philosophical topics provides enough warrant for a closer investigation of the deeper implications of the concept and the issues involved. Despite its origins from the highly- technical, forbidding realm of quantificational theory, the concept of ontological commitment, treated qualitatively, can serve as a unifying principle in tackling other less abstruse areas in philosophy. As with Tarski's Conception of Truth which evolved from scientific semantics and the calculus of classes, the concept of ontological commitment as propounded by Quine will continue to be a legitimate topic for scholarly philosophical papers, polemical or otherwise.
REFERENCES:
AUDI
CHIHARA
FLEW
MARTIN
OLIVER
PRESLEY
QUINE
QUINE
QUINE
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PAPER AS IT STANDS MAY BE CHARGED WITH PLAGIARISM...
STILL UNFINISHED ACTUALLY...
UPDATED Friday Feb 13, 2009
WARNING: COPY AT YOUR OWN RISK!
Tuesday, February 3, 2009
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