First of all, I need to explain the distinction between linguistic, and the most common conceptual memes. The later are purely semantic ones and owe their existence to culture and possibly historical contingencies. A good example may be the meme God. Linguistic memes carry no semantic weight in their own right. Instead they play the role of grammatical devices which serve as cues to the listener how the context where they appear should be interpreted. They resemble the function of case markers.
One common meme is the term "like". From my observation it serves (in most cases) the role of pre-empting a description.
Wednesday, December 7, 2011
Wednesday, November 23, 2011
The scientific method algorithm, computational complexity and TOE's.
I see science, or more precisely the method by which science proceeds (the scientific method) as an algorithm which asks progressively more complex questions, and subsequently receives answers of proportional complexity. Roughly, the science (theory) algorithm goes something like this:
Here we take the chain "(large)-(large with branches)-(large with leaves on some branches)-..." as an analogue for a chain of concepts of increasing complexity.
Science: Is the object General Sherman (largest tree in the world) larger (taller) than 10 metres?
Nature: Yes.
Science: Hoorray!
Science: Does General Sherman have branches (defined in one way or another)?
Nature: Yes.
Science: Hoorray!
Science: Do some of General Sherman's branches have feathers (defined in one way or another)?
Nature: No.
Science: Damn! Back to the drawing board.
Science: Do some of General Sherman's branches have leaves (defined in one way or another)?
Nature: Yes.
Science: Hoorray!
Science: ...?
Nature: ...
Science: ...!
:
:
:
:
:
Possibly ad infinitum.
A corollary of this model is, that it may be the case that our theories, as they evolve in discrete steps, one superseeding the previous one, will grow to arbitrary large complexity as they accommodate phenomena of gradually increasing complexity.
It is a theorem of Algorithmic Information Theory that the complexity of certain stings/objects/concepts put a lower bound on the complexity of the theory/program that generates them. Thus one cannot hope for a complete theory of everything (TOE) without some harboured assumptions about the complexity of the universe and the phenomena within it. It would certainly be presumptuous to harbour such assumptions since as far as I know the question whether the universe (multiverse) is finite or infinite, and hence it's complexity, remains an open question.
For it may be the case that our theories can only approach a limit which would be a TOE, but necessarily never reach it.
The above ruminations have been primarily inspired by the work of Gregory Chaitin.
Tuesday, November 22, 2011
Constructing a set disjoint from, and equinumerous with any other set.
A while ago I needed (a lemma in a larger proof) to construct for any arbitrary set A, another set that is disjoint from A and of the same cardinality as A (without the use of higher machinery of ordinals). The task is harder than it may seem. One may choose to attempt a proof by contradiction. One of my friends has in fact proceeded that way. I came up with the following construction which I’m quite fond of.
First, let me state a theorem (which is a direct consequence of Russell’s paradox) that I helped myself to.
Theorem. For each set S, the set
Is not a member of S.
For if we assume otherwise, an immediate contradiction ensues, for
Now for the construction. Let’s take some set A, and construct a set B that is disjoint from it, and equinumerous with it. Consider the set
It is clear that B has the same cardinality as A. (post-Clarification: just to be pedantic, it's not just the triple x, y, z that is in A, but rather I should have written "for x, y, z,... in A". I forgot the ellipsis.)
It is also clear that,
since,
Friday, November 18, 2011
Thoughts
I don’t think that it is too much of a presumption to claim that our conceptual repertoire developed thought human intellectual history is largely an outgrowth (and conditioned by) of the world we happen to occupy and the form we happen to posses.
We have had some success and seen a development of intellectual brevity which in effect made it possible to surpass our perceptual and rational limitations, leaving them behind as hindering properties of our human form (example: existence of the electromagnetic spectrum - our senses do not perceive X rays for instance, and peculiar existential nature of quantum entities - it seems that the distributivity theorem of propositional calculus fails at the quantum level ) . Nevertheless it is that very form that was the starting point, and persistently forces it’s limitations on our creativity and ingenuity.
Furthermore it should be noted that the universe we happen to occupy is a rather narrow experimental and severely conditioning ground – why, it just happens to be a world where sentient beings are possible. Empiria seems to suggest that other worlds are physically possible, and logic certainly doesn’t demand that possible worlds necessarily contain sentient beings.
One may find oneself wondering whether the universe (or the multiverse in its entire spatio- temporal and/or Hugh Everett’ian sense, or modal realism sense after David Lewis) is finite or infinite. If the latter than we should , with mature humility, accept epistemic defeat, in the “let’s ask nature what she is” game. If the former, then the problem doesn’t disappear but becomes more interesting when we consider questions of information, self-reference and computational complexity.
But notice that I have introduced a rather presumptuous classical dichotomy which in itself is an effect of intellectual conditioning to particular type of logic. Why not finite and infinite simultaneously (?), or neither finite nor infinite? And if simultaneously finite and infinite, then it appears that, again, I’m introducing crude notions concerning what we call time, yet have no conclusive answer to what it is or even whether it is a fundamentally coherent notion at all. What we refer to as time may just be a phenomenal manifestation of some more fundamental properties (Sean Carroll), peculiarities if you will, of the universe we happen to find ourselves in. So again one is reasoning from possibly perceptual limitations, making any inquiry fundamentally unsound.
But notice that I have introduced a rather presumptuous classical dichotomy which in itself is an effect of intellectual conditioning to particular type of logic. Why not finite and infinite simultaneously (?), or neither finite nor infinite? And if simultaneously finite and infinite, then it appears that, again, I’m introducing crude notions concerning what we call time, yet have no conclusive answer to what it is or even whether it is a fundamentally coherent notion at all. What we refer to as time may just be a phenomenal manifestation of some more fundamental properties (Sean Carroll), peculiarities if you will, of the universe we happen to find ourselves in. So again one is reasoning from possibly perceptual limitations, making any inquiry fundamentally unsound.
So we continue on what we supposedly do best – conjure up new concepts about what is, and what could be, with more lucid moments realizing that this is just a peculiar property of some peculiar (what we call, sentient and intelligent) entity occupying some peculiar universe – a property motivated by the desire and hope of finding an alibi for our existence, or maybe not even motivated at all, but it sure sounds more reassuring that such behavior is intended.
Wednesday, November 9, 2011
Super Intelligent Beings.
Imagine yourself, perceiving a plant, say a tree - its incredibly slow metabolic rate and predictable "behaviour" i.e growth. Also in some sense the plant is defenceless against your whim, whatever it may be - to cut it down, burn it, or eat (parts of) it. Better still imagine a stone - its practically unchanging crystal structure, and unique combination of minerals shaped by geological processes over billions of years. Again - the stone is a passive, and for all practical purposes (by definition!), a lifeless object. Now imagine, that a super-intelligent being would perceive YOU in a manner analogous to your perception of that tree, or rock.
In the time that you would take one breath, it would seem to such a being that eons have passed during which its leaps and strides of insight would dwarf the totality of our (human) current intellectual achievements.
Wednesday, June 15, 2011
A FALSE DICHOTOMY: Science vs Intelligent Design
I realised today something quite obvious, after reading a rather staunchly dismissive response to a book review of The Fallacy of Fine Tuning (Victor Stenger). The response was decisve in its dismissal but superficial if not outright vacuous in justifications for that attitude (I know the person hasn't read the book).
Intelligent and curious individuals with a desire to genuinely understand themselves and the world they live in will search with an opend mind. This philosophical inquiry will attract them to any information that brings them close to revealing yet another mystery, or a new conceptual approach which may make the search more efficient, or even revise and inform the previous approaches. Dubious and self contradicory views will naturally be either subjected to second rate or outhright rejected depending on the degree of internal inconsistencies or dissonances with common sense and direct observation. For example it is this sort of attitude that abandoned a once popular Flat Earth belief. This kind of apprach is a humbling attitude for it soberly aknowledges our epistemological limitations, but makes do with what it can, improving where it can, revising where possible.
The false dichotomy is between scientific thinking, which is highly correlated with the attitude described above, and just one theory/belief - namely Intelligent Design (clearly a subversive oxymoron) that deems itself special for all the wrong reasons (there's nothing genuine about it's pursuits for truth and understanding).
The point is that we don't choose between science and the latter, the distinction (and the correct dichotomy) is more along the lines of being genuinely curious about the world, and hence searching and discovering its wonderful mysteries or happy to conform to a simplistic set of views that quenches ones inner fears and desires.
Another way of putting this is considering Intelligent Design as one of a plethora of theories and philosophies. Succumbing to the false dichotomy creates an illusion of some special status for ID theory - a status it doesn't deserve.
Intelligent and curious individuals with a desire to genuinely understand themselves and the world they live in will search with an opend mind. This philosophical inquiry will attract them to any information that brings them close to revealing yet another mystery, or a new conceptual approach which may make the search more efficient, or even revise and inform the previous approaches. Dubious and self contradicory views will naturally be either subjected to second rate or outhright rejected depending on the degree of internal inconsistencies or dissonances with common sense and direct observation. For example it is this sort of attitude that abandoned a once popular Flat Earth belief. This kind of apprach is a humbling attitude for it soberly aknowledges our epistemological limitations, but makes do with what it can, improving where it can, revising where possible.
The false dichotomy is between scientific thinking, which is highly correlated with the attitude described above, and just one theory/belief - namely Intelligent Design (clearly a subversive oxymoron) that deems itself special for all the wrong reasons (there's nothing genuine about it's pursuits for truth and understanding).
The point is that we don't choose between science and the latter, the distinction (and the correct dichotomy) is more along the lines of being genuinely curious about the world, and hence searching and discovering its wonderful mysteries or happy to conform to a simplistic set of views that quenches ones inner fears and desires.
Another way of putting this is considering Intelligent Design as one of a plethora of theories and philosophies. Succumbing to the false dichotomy creates an illusion of some special status for ID theory - a status it doesn't deserve.
Saturday, June 4, 2011
Can't help but subject "Being" to logic
I certainly endorse an analytical approach to philosophy, but this is not the same as declaring oneself a rationalist. Recently however I've discovered that I may after all be falling within that category.
I have always endowed Being with a supreme status. That is, minds and their way of cognizing, perceiving and talking about Being and its manifestations have a somewhat secondary status. Gottfried Leibniz had introduced the principle of sufficient reason in aiding answering the question "Why is there something rather than nothing?". That principle is related to the anthropic principle which conditions the answer on the existence of intelligent life - us humans in particular . A related anthropic approach would be what I call the logical rather than the standard one with physical attributes criterion - there is something rather than nothing because we happen to occupy a possible world that actually exists - a necessary condition for the existence of beings that ask metaphysical questions.
This would be impossible in a possible world that happened to not be (physically non existent worlds are possible!). That is to say, out of the possible worlds there are those that are and those that are not. Those that happen to exist do not so necessarily i.e. it was possible for them not to be - a subtelty that Parmenides failed to see when he claimed "That which is, is and it is impossible for it not to be" - a modal fallacy identified by Aristotle.
Now that we have the sufficient logical layout, my problem can be clearly presented. It appears that although the question "Why is there something than nothing" has a partially satisfactory answer (in the negative sense, for it does not posit a positive cause to Being) in the sense that it points to the fact that we (physical beings) could not ask that question if nothing existed. Nevertheless, it follows (Alas! the point of my discovery that I'm a metaphysical rationalist) that in those non existent worlds it would be true to say that being is possible, since there is at least one possible world (namely the one we happen to inhabit - proof by inspection;) where being obtains.
The point is that it appears that ontological existence is a subset of modal (logical) existence. That being the case, plenty of worlds are not endowed with (physical) existence --- they don't acually exist, yet they posess modal existence and that is why we can talk about them - moreover some truths hold in those worlds despite them actually existing - "Being is possible" is one of such truths.
"Everyone knows that dragons don't exist. But while this simplistic formulation may satisfy the
layman, it does not suffice for the scientific mind. [...] Indeed, the banality of existence has been so
amply demonstrated, there is no need for us to discuss it any further here. The brilliant Cerebron, attacking the problem analytically, discovered three distinct kinds of dragon: the mythical, the chimerical, and the purely hypothetical. They were all, one might say, nonexistent, but each non-existed in an entirely different way." (Stanislaw Lem, The Cyberiad - Dragons of Probability)
I have always endowed Being with a supreme status. That is, minds and their way of cognizing, perceiving and talking about Being and its manifestations have a somewhat secondary status. Gottfried Leibniz had introduced the principle of sufficient reason in aiding answering the question "Why is there something rather than nothing?". That principle is related to the anthropic principle which conditions the answer on the existence of intelligent life - us humans in particular . A related anthropic approach would be what I call the logical rather than the standard one with physical attributes criterion - there is something rather than nothing because we happen to occupy a possible world that actually exists - a necessary condition for the existence of beings that ask metaphysical questions.
This would be impossible in a possible world that happened to not be (physically non existent worlds are possible!). That is to say, out of the possible worlds there are those that are and those that are not. Those that happen to exist do not so necessarily i.e. it was possible for them not to be - a subtelty that Parmenides failed to see when he claimed "That which is, is and it is impossible for it not to be" - a modal fallacy identified by Aristotle.
Now that we have the sufficient logical layout, my problem can be clearly presented. It appears that although the question "Why is there something than nothing" has a partially satisfactory answer (in the negative sense, for it does not posit a positive cause to Being) in the sense that it points to the fact that we (physical beings) could not ask that question if nothing existed. Nevertheless, it follows (Alas! the point of my discovery that I'm a metaphysical rationalist) that in those non existent worlds it would be true to say that being is possible, since there is at least one possible world (namely the one we happen to inhabit - proof by inspection;) where being obtains.
The point is that it appears that ontological existence is a subset of modal (logical) existence. That being the case, plenty of worlds are not endowed with (physical) existence --- they don't acually exist, yet they posess modal existence and that is why we can talk about them - moreover some truths hold in those worlds despite them actually existing - "Being is possible" is one of such truths.
"Everyone knows that dragons don't exist. But while this simplistic formulation may satisfy the
layman, it does not suffice for the scientific mind. [...] Indeed, the banality of existence has been so
amply demonstrated, there is no need for us to discuss it any further here. The brilliant Cerebron, attacking the problem analytically, discovered three distinct kinds of dragon: the mythical, the chimerical, and the purely hypothetical. They were all, one might say, nonexistent, but each non-existed in an entirely different way." (Stanislaw Lem, The Cyberiad - Dragons of Probability)
Sensless chatter analysis
Average teenager-mid twenties person wpm (words per minute - spoken) = 150.
Average teenager-mid twenties person wpm (words per minute - spoken) with the phrases (and the variations of) "so/and I/he/she/it/we/they am/is/are/was/were like..." = 45.
Average teenager-mid twenties person wpm (words per minute - spoken) with the phrases (and the variations of) "so/and I/he/she/it/we/they am/is/are/was/were like..." = 45.
Monday, May 23, 2011
Number Theory 4: The value of Euler's phi function for a product of Fibonacci sequence terms
There’s a corollary to the result fom the previous post. Each two consecutive terms of the Fibonacci sequence are relatively prime. Hence the following follows immediately, for the n product of Fibonacci terms.
Substituting into the formula, yields
Number Theory 3: A formula for calculating Euler's phi function
Below is a proof for calculating phi (Euler’s function) for any natural number. Phi(m) is defined to be the number of all natural numbers n less than m that are relatively prime to it i.e. each such n<m satisfies gcd(n,m)=1.
The p’s in the above formula denote primes. The proof will be relatively simple, as it will use the following fundamental results, where p denotes a prime and m and n are relatively prime.:
By the Fundamental Theorem of Arithmetic we express m in terms of its unique prime factorization,
Re writing the expression and, by associative law, multiplying the first two terms we have the following,
and substituting m’s prime factorization yields,
Further algebraic manipulation gives,
Now, by repeated use of associative law, factoring out each consecutive prime in m’s factorization and multiplying it by the term from the original formula where that prime appears,
leaves us with the product rearranged to the following form,
Finally we resort to (*), and (**) to get
Monday, May 2, 2011
The Way of a Saint: Beatification of John Paul II
Last Sunday Karol Wojtyla subsequently the previous head of the Catholic Church John Paul II, has been beatified by his successor, the present Pope Benedict XVI. Beatification is the necessary stage in the canonization (sainthood) process.
Becoming an orphan at the age of twenty, may have contributed to the decision to enter the path of priesthood thus allowing Karol Wojtyla actively embrace all of humanity as his extended family. This had been clearly apparent in his unprecedented reaching out to the peoples of all ethnic and faith backgrounds through over a hundred pastoral journeys he made to 129 countries – an activity surpassing in magnitude and scope any pontiff in history.
On a more personal level an act, exemplifying his compassion and the sincerity of his mission to embrace all humans as his family, can be seen in his attitude toward his would be assassin Mehmet Ali Ağca. After having recovered from the nearly lethal shooting he visited the prison cell of the man that attempted to kill him, and after having spent some time talking in private, his first official words to the world were ‘Pray for my brother, whom I have sincerely forgiven’.
While still as a Bishop, Karol Wojtyla participated in the Second Vatican Council (an official reform within the Catholic Church), where among his contributions were some of the most historic and influential products of the Council, among them the Decree on Religious Freedom (Lat. Dignitatis Humanae) – where it is declared and emphasized as the official stance of the Church that all persons have a right to religious liberty, a right with its foundation in the essential dignity of each human being.
Becoming the head of the Church certainly facilitated to implement the work he begun as a bishop. This again is marked by unprecedented acts of reaching out and solidarity with other faiths. It seems that in this respect John Paul II made history with every passing year - on the Christian level in his ecumenical work to unite on a common ground all the Christian denominations, and on a broader scale building bridges to other faiths like the pontifex maximus he was intended to be (a term informally adopted by the Catholic Church from ancient Rome, serving as an informal title for Popes – great builder of bridges). The notable acts of this interfaith solidarity were his historical visit and prayer in an Islamic mosque in Syria in 2001, and a similar act during his visit to Jerusalem whereby he was the first pope to visit and pray at the Western Wall – a sacred place of Judaism.
To my surprise however, Buddhism with its unique philosophy of salvation is considered by the Second Vatican Council to be in dissonance with the doctrine our builder of bridges served to implement. John Paul II inevitably adopted an attitude that faithfully reflects the doctrinal status quo of the Second Vatican Council. As a son and servant of the Catholic Church he professed the soteriology which his faith rendered into a conviction – through Christ is the only way to salvation.Hence the kind and declaredly tolerant, yet simultaneously patronizing attitude of the Second Council: ‘The Catholic Church rejects nothing that is true and holy in these [other] religions.’ (Nostra Aetate). This is the attitude which John Paul had, as the Council’s co-author and representative, consequently and inevitably embodied.
Below are relevant excerpts from his ‘Crossing the Threshold of Hope’ illustrating the core dissonance over the doctrinal contrast of salvation between Christianity and Buddhism – a dissonance declared existent by the authors of the Second Vatican Council.
All of the questions raised by the Second Vatican Council are reducible, finally, to this theme [soteriology]. Nevertheless, both the Buddhist tradition and the methods deriving from it have an almost exclusively negative soteriology […] in the conviction that it [the world] is only a source of suffering for man and that he therefore must break away from it […] The fullness of such a detachment is not union with God, but what is called Nirvana, a state of perfect indifference with regard to the world. (John Paul II Crossing the Threshold of Hope, 1994)
[in Christianity] It is in the world that man meets God. Therefore he does not need to attain such an absolute detachment […] The Church is guided by the faith that God the Creator wants to save all humankind in Jesus Christ, the only mediator between God and man, inasmuch as He is the Redeemer of all humankind. […] Buddhism is in large measure an ‘atheistic’ system. (John Paul II Crossing the Threshold of Hope, 1994)
The consistently respectful, tolerant yet ultimately patronizing attitude, of the Second Vatican Council is echoed in this homily addressed to the people of Thailand:Your Cultural heritage as Thai people is intimately linked to the indigenous Buddhist tradition, which provides a fertile terrain for the seed of God’s word, proclaimed by Jesus Christ to take root and grow. In the practice of Buddhism can be discovered a noble tendency to strive to separate oneself from ‘earthly wisdom’, in order to discover and achieve an interior purification and liberation. This aim is pursued through prayer and meditation, coupled with the practice of moral virtue. As the Second Vatican Council so clearly pointed out, the Church looks with sincere respect upon the religious wisdom contained in non-Christian traditions and rejects nothing that is holy and true in them. (John Paul II, Homily of the Mass for Peace in the World)
It appears from the compassionate, devoted to humanity and radiating with love yet somewhat conservative (i.e. faithful to the Council’s doctrine) attitude of John Paul II that the great builder of bridges is only as good as the guild whose engineering endeavors he is implementing.
In The Roman Triptych he writes ‘Non omnis moriar’ (not all of me will die). Indeed it is up to us to continue and improve upon the art of bridge building.
Becoming an orphan at the age of twenty, may have contributed to the decision to enter the path of priesthood thus allowing Karol Wojtyla actively embrace all of humanity as his extended family. This had been clearly apparent in his unprecedented reaching out to the peoples of all ethnic and faith backgrounds through over a hundred pastoral journeys he made to 129 countries – an activity surpassing in magnitude and scope any pontiff in history.
On a more personal level an act, exemplifying his compassion and the sincerity of his mission to embrace all humans as his family, can be seen in his attitude toward his would be assassin Mehmet Ali Ağca. After having recovered from the nearly lethal shooting he visited the prison cell of the man that attempted to kill him, and after having spent some time talking in private, his first official words to the world were ‘Pray for my brother, whom I have sincerely forgiven’.
While still as a Bishop, Karol Wojtyla participated in the Second Vatican Council (an official reform within the Catholic Church), where among his contributions were some of the most historic and influential products of the Council, among them the Decree on Religious Freedom (Lat. Dignitatis Humanae) – where it is declared and emphasized as the official stance of the Church that all persons have a right to religious liberty, a right with its foundation in the essential dignity of each human being.
Becoming the head of the Church certainly facilitated to implement the work he begun as a bishop. This again is marked by unprecedented acts of reaching out and solidarity with other faiths. It seems that in this respect John Paul II made history with every passing year - on the Christian level in his ecumenical work to unite on a common ground all the Christian denominations, and on a broader scale building bridges to other faiths like the pontifex maximus he was intended to be (a term informally adopted by the Catholic Church from ancient Rome, serving as an informal title for Popes – great builder of bridges). The notable acts of this interfaith solidarity were his historical visit and prayer in an Islamic mosque in Syria in 2001, and a similar act during his visit to Jerusalem whereby he was the first pope to visit and pray at the Western Wall – a sacred place of Judaism.
To my surprise however, Buddhism with its unique philosophy of salvation is considered by the Second Vatican Council to be in dissonance with the doctrine our builder of bridges served to implement. John Paul II inevitably adopted an attitude that faithfully reflects the doctrinal status quo of the Second Vatican Council. As a son and servant of the Catholic Church he professed the soteriology which his faith rendered into a conviction – through Christ is the only way to salvation.Hence the kind and declaredly tolerant, yet simultaneously patronizing attitude of the Second Council: ‘The Catholic Church rejects nothing that is true and holy in these [other] religions.’ (Nostra Aetate). This is the attitude which John Paul had, as the Council’s co-author and representative, consequently and inevitably embodied.
Below are relevant excerpts from his ‘Crossing the Threshold of Hope’ illustrating the core dissonance over the doctrinal contrast of salvation between Christianity and Buddhism – a dissonance declared existent by the authors of the Second Vatican Council.
All of the questions raised by the Second Vatican Council are reducible, finally, to this theme [soteriology]. Nevertheless, both the Buddhist tradition and the methods deriving from it have an almost exclusively negative soteriology […] in the conviction that it [the world] is only a source of suffering for man and that he therefore must break away from it […] The fullness of such a detachment is not union with God, but what is called Nirvana, a state of perfect indifference with regard to the world. (John Paul II Crossing the Threshold of Hope, 1994)
[in Christianity] It is in the world that man meets God. Therefore he does not need to attain such an absolute detachment […] The Church is guided by the faith that God the Creator wants to save all humankind in Jesus Christ, the only mediator between God and man, inasmuch as He is the Redeemer of all humankind. […] Buddhism is in large measure an ‘atheistic’ system. (John Paul II Crossing the Threshold of Hope, 1994)
The consistently respectful, tolerant yet ultimately patronizing attitude, of the Second Vatican Council is echoed in this homily addressed to the people of Thailand:Your Cultural heritage as Thai people is intimately linked to the indigenous Buddhist tradition, which provides a fertile terrain for the seed of God’s word, proclaimed by Jesus Christ to take root and grow. In the practice of Buddhism can be discovered a noble tendency to strive to separate oneself from ‘earthly wisdom’, in order to discover and achieve an interior purification and liberation. This aim is pursued through prayer and meditation, coupled with the practice of moral virtue. As the Second Vatican Council so clearly pointed out, the Church looks with sincere respect upon the religious wisdom contained in non-Christian traditions and rejects nothing that is holy and true in them. (John Paul II, Homily of the Mass for Peace in the World)
It appears from the compassionate, devoted to humanity and radiating with love yet somewhat conservative (i.e. faithful to the Council’s doctrine) attitude of John Paul II that the great builder of bridges is only as good as the guild whose engineering endeavors he is implementing.
In The Roman Triptych he writes ‘Non omnis moriar’ (not all of me will die). Indeed it is up to us to continue and improve upon the art of bridge building.
Tuesday, April 26, 2011
PROPOSITIONAL CALCULUS (Propositional Logic, Sentential Calculus, Sentential Logic) Emulator/Calculator
The Propositional Calculus Calculator (Propositional Logic Calculator) has been written in Matlab. So you'll need Matlab to run it. Just download the m.files from the links provided below and you're ready to use it. Alternatively copy the actual code which I pasted at the end of this post.
Using the truth tables, or truth trees (semantic trees) approach to evaluating formulae and arguments becomes a rather laborious task as the number of propositional variables (PV's) increases, and the number of interpretations grows exponentially i.e.
The most relevant applications it has are:
tautology.m
valid.m
consistent.m
Below are screenshots of the executed files upon entering relevant input. You'll need to read the syntax directions to get the idea how this is done. I'm posting them further below. You can download all the files from links provided at the end of this post.
The output here is self explenatory - 1 stands for affirmative.
Not a tautology, and we read the output as the counter-model (counter-example) being the following interpretation:
Again, for any contingency or in this case invalid inference a counter-model is provided (one is sufficient):
The tautology function, tells you whether some wff is a tautology or not,
plus gives a single counterexample (one is enough!)
The valid function, tells you whether an argument is valid or not,
plus gives a single counterexample
The consistent function determines whether the set of wff's is satisfiable by some interpretation
NOTE: for n pv's a model or counter-model of the form 0 0 1 0 ... 1
is to be interpreted as val(p1)=0, val(p2)=0, val(p3)=1, val(p4)=0, ..., val(pn)=1
NOTE: this program has not been optimized for efficiency, so for wff's containing in excess of
12 distinct propositional variables the execution time could take around 2 minutes, depending
on your CPU.
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=wff(x)
%x is the string array of the wff with pv's expressed as p1, p2,...
b=convert3(x);
a=char(b(1:length(b)-4));
npv=b(length(b)-3);
if b(length(b))~=0 || b(length(b)-1)~=0 || b(length(b)-2)~=0
y=[7,b(length(b)-2),b(length(b)-1),b(length(b))];%this condition obtains when a
%syntax error has been detected
else
%a will be the new wff string with pv's expressed as p(v,1),p(v,2),...
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%now for 2^2*npv rows we evaluate the wff with the values for v iterating
%over the number of rows. But we also need the truth table p.
p=truth_table(npv);
eval_array=zeros(1,2^npv);
for v=1:2^npv
eval_array(v)= eval(a);
end
y=[eval_array,b(length(b)-2),b(length(b)-1),b(length(b))];
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y = is_even(x)
if floor(x/2)==x/2
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=n_andf(x)
t=0;
for k=1:length(x)
if x(k)==0
t=t+1;
end
end
if t==0
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=max_row(h)
s=size(h);
v=[];
for k=1:s(1)
v=[v,row_pic(k,h)];
end
y=index_of(max(v),v);
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=notf(x)
if x==1
y=0;
else
y=1;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=npv_test(t)
%t is a row in the p matrix (in convert3.m) containing instances of pv's in the string
%here we're converting the matlab expressions of numerals into decimal
%form
r=[];
for i=1:length(t)
if numerical_string_entry(t(i))==1
r=[r t(i)];
end
end
sum=0;
for k=1:length(r)
v=10^(length(r)-k)*(r(k)-48);
sum=sum+v;
end
y=sum;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=numerical_string_entry(string_element_encoding)
c=0;
for i=48:48+9
if string_element_encoding==i
c=c+1;
end
end
if c==0
y=0;
else
y=1;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=row_pic(k,x)
r=[];
s=size(x);
for i=1:s(2)
if numerical_string_entry(x(k,i))==1
r=[r x(k,i)];%here we extract numerical entries from the character array
end
end
sum=0;%here each numerical row is transformed to a unique scalar quantity
%required to determine the largest numerical entry i.e. largest pv
for i=1:length(r)
sum=sum+(10^(length(r)-i))*(r(i));
end
y=sum;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=orf(x,z)
if x==1 || z==1
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=tautology(x)
w=wff(x);
if w(1)==7 && w(length(w)-2)==1
y='ERROR: the number of left-hand brackets must equal the number of right-hand brackets';
else if w(1)==7 && w(length(w))==1
y='ERROR: for any pn appearing in the wff, all pk such that 1=
else if w(1)==7 && w(length(w)-1)==1
y='ERROR: superflous brackets - double check the syntax!';
else if w(1)==7 && w(length(w)-1)==-1
y='ERROR: insufficient brackets - double check the syntax!';
else
npvs=log2(length(w)-2);
%we can easily extract the number of propositional variables from wff(x),
%since wff spits out a one dimensional array of the value of the wff for
%each interpretation.
t=truth_table(npvs);
counter_model=[];
if sum(w(1:length(w)-3))==length(w)-3%if wff is all 1's i.e. 1 on every interpretation
y=1;
else
y=0;
counter_model=t(min(index_of(0,w)),:);
%here we give just the nearest countermodel - it's sufficient!
%however the nearest_zero function can be altered to give all
%the counterexamples
counter_model
end
end
end
end
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y= truth_table(nPVs)
%this program generates binary strings required to construct a semantic
%table (table for Propositional Calculus of all interpretations for a given number of PV's)
%Basically here we generate thruth tables for a given number of PV's.
v=[];
for n=1:nPVs
for k=1:(2^nPVs)/(2^(n-1))
for i=(2^(n-1))*(k-1)+1:2^nPVs
if is_even(k)==0
v(i,n)=0;
else
v(i,n)=1;
end
end
end
end
y=v;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=valid(x,z)
conversion=['conf(n_andf([',x,']),',z,')'];
y=tautology(conversion);
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=all_pvs(x)
c=0;
for i=1:length(x)
if x(i)=='p'
c=c+1;
end
end
y=c;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=bconf(x,z)
if (x==1 && z==1) || (x==0 && z==0)
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=conf(x,z)
if x==0 || z==1
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=consistency(x)
w=wff(x);
if w(1)==7 && w(length(w)-2)==1
y='ERROR: the number of left-hand brackets must equal the number of right-hand brackets';
else if w(1)==7 && w(length(w))==1
y='ERROR: for any pn appearing in the wff, all pk such that 1=
else if w(1)==7 && w(length(w)-1)==1
y='ERROR: superflous brackets - double check the syntax!';
else if w(1)==7 && w(length(w)-1)==-1
y='ERROR: insufficient brackets - double check the syntax!';
else
npvs=log2(length(w)-2);
%we can easily extract the number of propositional variables from wff(x),
%since wff spits out a one dimensional array of the value of the wff for
%each interpretation.
t=truth_table(npvs);
Model=[];
if sum(w(1:length(w)-3))>0%if wff is all 1's i.e. 1 on every interpretation
y=1;
Model=t(min(index_of(1,w)),:)%Here is the model where the set is satisfied - we provide just one,
%but the nearest_one function can be altered to obtain all the models.
else
y=0;
end
end
end
end
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=consistent(x)
conversion=['n_andf([',x,'])'];
y=consistency(conversion);
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y =convert3(x)
%-------------------------------------------------
%syntax check if the number of brackets is equal
l=0;
r=0;
dd=0;
for i=1:length(x)
if x(i)==40
l=l+1;
end
if x(i)==41
r=r+1;
end
end
if r~=l
dd=1;%'ERROR: the number of left-hand brackets must equal the number of right-hand brackets'
end
%-------------------------------------------------
%the loop below extracts the pv numerals and for each pv inserts then in
%the p matrix where each pv numeral goes into a seperate row
b=0;
h=0;
for t=1:length(x)%this makes the code clean so any pn is a wff
if x(t)== 'p'
h=h+1;
end
end
f=0;
for t=1:length(x)%sufficient brackets for the number of functions
if x(t)== 'f'
f=f+1;
end
end
ee=0;
if 2*r<2 f="" font="">
ee=-1;%'ERROR: insufficient brackets - double check the syntax!'
else if 2*r>2*f
ee=1;%'ERROR: superflous brackets - double check the syntax!'
end
end
if h~=1 && x(length(x))~=41%just taking care of cases without last bracket
x=[x 41];
ee=-1;
end
%---------------------------------------------------------------
%here we create a matrix of x length and all_pvs height
siz=all_pvs(x);
p=zeros(siz,length(x));
for k=1:length(x)
for i=1:siz
p(i,k)='*';
end
end
%----------------------------------------------------------------
% if h==1
% y=[112 40 118 44 49 41 1 0 0 0];%here we just return the convertend string p(v,1) i.e. representing one pv
% else
for i=1:length(x)%this code extracts all numerial string entries from the wff and inserts them in the rows of p
if x(i)=='p'
b=b+1;
for k=1:length(x)
if numerical_string_entry(x(i+k))~=0
p(b,k)=x(i+k);
else
break
end
end
end
end
%-------------------------------------------------
%Next we determine the order of the pv's by finding the maximum length of
%numeral strings in the p matrix.
pdash=zeros(siz,length(p));
for i=1:siz
for k=1:length(x)
if numerical_string_entry(p(i,k))==1
pdash(i,k)=1;
end
end
end
pvorder=max(sum(pdash'));
numeralspace=sum(sum(pdash'));%the number of numerical entries in x
%-------------------------------------------------
%conversion proper
s=size(p);
nthpv=0;%n'th row in p corresponding to the n'th pv appearing in x
c=0;%is the number of accumulated entries through conversion i.e. 4 for each pv
lenxdash=length(x)+4*siz;
xdash=zeros(1,lenxdash);
for i=1:length(x)
if x(i)=='p'
nthpv=nthpv+1;
leng=sum(pdash(nthpv,1:s(2)));%number of numerical entries in pdash's i'th row
g=['p','(','v',',',p(nthpv,1:leng),')'];%replacement string considering each pv i.e. p row
xdash(i+c:i+c+length(g)-1)=g(1:length(g));%conversion
c=c+4;%accumulated entries through conversion
end
end
%---------------------------------------------------
%Here is the code that counts the number of pv's
c=[];
for i=1:s(1)
for j=1:s(1)
if npv_test(p(i,1:s(2)))==npv_test(p(j,1:s(2)))
c=[c 0];
else
c=[c 1];
end
end
end
if sum(c)==0%this implies tht all pv's are identical
npvs=npv_test(p(1,1:s(2)));%and here we just determine which one it is
else
max_row_p=max_row(p);%creating a variable so
npvs=npv_test(p(max_row_p(1),1:s(2)));%we pick one only instead of all the indexes where the maximum pv's appear
end
%---------------------------------------------------
%--------------------------------------------------
%Before we count the number of pv's we need to determine if the syntax is
%correct - this bit of code determines if there are any non sequential
%instances of pv's eg. p1,p2,p4 instead of p3
gg=0;
max_pv=npvs;%npv_test(p(max_row(p),1:s(2)));
for k=1:max_pv-1
t=0;
for i=1:s(1)
if npv_test(p(i,:))== max_pv-k
t=t+1;
end
end
if t==0
gg=1;%'ERROR: for any pn appearing in the wff, all pk such that 1
break
end
end
%--------------------------------------------------------------------------
empty_xdash=index_of(0,xdash);%this is ok
nonpvx=index_of_not_general('p',x);
xdash(empty_xdash)=x(nonpvx);
y=[xdash,npvs,dd,ee,gg];
end
%this is the PROPER EXECUTION LINE, just xdash will have to be
%transformed to char(xdash) in the program treating it as a wff string.
%eg. v=convert3('the actual wff string')
%x=char(v(1:length(v)-1); y=v(length(v)); where x is the wff and y is the
%number of pv's in that wff
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=index_of(x,z)
p=[];
for i=1:length(z)
if z(i)==x
p=[p i];
end
end
y=p;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=index_of_not_general(x,z)
p=[];
%for k=1:length(x)
for i=1:length(z)
if z(i)~=x && numerical_string_entry(z(i))==0
p=[p i];
end
end
%end
y=p;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=row_pic(k,x)
r=[];
s=size(x);
for i=1:s(2)
if numerical_string_entry(x(k,i))==1
r=[r x(k,i)];%here we extract numerical entries from the character array
end
end
sum=0;%here each numerical row is transformed to a unique scalar quantity
%required to determine the largest numerical entry i.e. largest pv
for i=1:length(r)
sum=sum+(10^(length(r)-i))*(r(i));
end
y=sum;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
Using the truth tables, or truth trees (semantic trees) approach to evaluating formulae and arguments becomes a rather laborious task as the number of propositional variables (PV's) increases, and the number of interpretations grows exponentially i.e.
Number of interpretations = 2Number of PV's
With this software you can evaluate a formula of any length and complexity in no time. It will take you longer to actually enter the formula, then the program to evaluate it.The most relevant applications it has are:
tautology.m
valid.m
consistent.m
Below are screenshots of the executed files upon entering relevant input. You'll need to read the syntax directions to get the idea how this is done. I'm posting them further below. You can download all the files from links provided at the end of this post.
v(p1) = 1, v(p2) = 0, v(p3) = 0
Wow! - Modus Ponens is valid - this thing works!
v(p1) = 0, v(p2) = 1
Here, since the set is consistent we're given the model (interpretation) that satisfies it:
v(p1) = 1, v(p2) = 1, v(p3) = 0, v(p4) = 0
Some comments on the input notation. The propositional binary functions we use are of the form:
Hence the negation, a unary function is just an operator on the set {0,1}:
So the notation in the program is naturally motivated. It is a functional notation.Hence complex formulae are just compositions of functions.
DOWNLOAD
Click on any of the links below to download The Propositional Calculus Calculator m.files. Make sure to download all 23 files!
Comments and feedback are welcome :)
INSTRUCTIONS
--------------------------------------------------------------------------------------
If you prefer the intuitive approach and don't want to read all the instructions
(which takes less than 5 minutes, and you should eventually!):
Just try entering for the wff ((p1 & p2) ==> ~p1), the following expression:
(I suggest a cut-and-paste approach):
tautology('conf(orf(p1,notf(p2)),notf(p1))')
and see what happens :)
Or try testing the inference {p1,(p1 ==> p2),p3} |= (p2 & p3), with the following:
valid('p1,conf(p1,p2),p3','andf(p2,p3)')
and see what happens :)
Or finally try testing the consistency of the set {p1,p2,(p1 v p3),~p3,(p3<==>p4)}
by entering:
consistent('p1,p2,orf(p1,p3),notf(p3),bconf(p3,p4)')
--------------------------------------------------------------------------------------
BUT, I DO ENCOURAGE YOU TO READ THE FOLLOWING:
--------------------------------------------------------------------------------------
SYNTAX
--------------------------------------------------------------------------------------
We have the standard functions:
notf(...) = NOT (a function whose domain/input is a single variable)
orf(...,...) = OR (all dyadic functions have a two variable domain/input)
andf(...,...) = AND
conf(...,...) = MATERIAL CONDITIONAL
bconf(...,...) = MATERIAL BICONDITIONAL
Before I will recursively define the language, note that in this program all
the wff's will have to be entered as strings i.e. 'whatever' is the input in
Matlab to convert the word whatever into a string array.
--------------------------------------------------------------------------------------
RECURSIVE DEFINITION OF THE LANGUAGE
--------------------------------------------------------------------------------------
B: any Propositional Variable is a wff i.e. p1, p2, p3, ...
HOWEVER! in any complex wff use the minimun number of distinct pv's
that is: do not use a pv with higher ordering then necessary
example: instead of (p3 v p27) ==> (p27 & p456) use (p1 v p2) ==> (p2 & p3)
R1: if A is a wff then notf(A) is a wff
R2: if A and B are wff's then orf(A,B) is a wff
R3: if A and B are wff's then andf(A,B) is a wff
R4: if A and B are wff's then conf(A,B) is a wff
R5: if A and B are wff's then bconf(A,B) is a wff
T: nothing else is a wff
--------------------------------------------------------------------------------------
META-FUNCTIONS
--------------------------------------------------------------------------------------
In order to get anything working in this emulator you'll have
to use one of the following functions:
tautology(...) - enter some some wff as a string
valid(...,...) - first string is a list of premises wff's, and the second string is the conclusion wff
consistent(...) - just a string listing some wff's
And remember - all ... entries are strings i.e. tautology('orf(p1,notf(p1))') - note the dashes.
(which takes less than 5 minutes, and you should eventually!):
Just try entering for the wff ((p1 & p2) ==> ~p1), the following expression:
(I suggest a cut-and-paste approach):
tautology('conf(orf(p1,notf(p2)),notf(p1))')
and see what happens :)
Or try testing the inference {p1,(p1 ==> p2),p3} |= (p2 & p3), with the following:
valid('p1,conf(p1,p2),p3','andf(p2,p3)')
and see what happens :)
Or finally try testing the consistency of the set {p1,p2,(p1 v p3),~p3,(p3<==>p4)}
by entering:
consistent('p1,p2,orf(p1,p3),notf(p3),bconf(p3,p4)')
--------------------------------------------------------------------------------------
BUT, I DO ENCOURAGE YOU TO READ THE FOLLOWING:
--------------------------------------------------------------------------------------
SYNTAX
--------------------------------------------------------------------------------------
We have the standard functions:
notf(...) = NOT (a function whose domain/input is a single variable)
orf(...,...) = OR (all dyadic functions have a two variable domain/input)
andf(...,...) = AND
conf(...,...) = MATERIAL CONDITIONAL
bconf(...,...) = MATERIAL BICONDITIONAL
Before I will recursively define the language, note that in this program all
the wff's will have to be entered as strings i.e. 'whatever' is the input in
Matlab to convert the word whatever into a string array.
--------------------------------------------------------------------------------------
RECURSIVE DEFINITION OF THE LANGUAGE
--------------------------------------------------------------------------------------
B: any Propositional Variable is a wff i.e. p1, p2, p3, ...
HOWEVER! in any complex wff use the minimun number of distinct pv's
that is: do not use a pv with higher ordering then necessary
example: instead of (p3 v p27) ==> (p27 & p456) use (p1 v p2) ==> (p2 & p3)
R1: if A is a wff then notf(A) is a wff
R2: if A and B are wff's then orf(A,B) is a wff
R3: if A and B are wff's then andf(A,B) is a wff
R4: if A and B are wff's then conf(A,B) is a wff
R5: if A and B are wff's then bconf(A,B) is a wff
T: nothing else is a wff
--------------------------------------------------------------------------------------
META-FUNCTIONS
--------------------------------------------------------------------------------------
In order to get anything working in this emulator you'll have
to use one of the following functions:
tautology(...) - enter some some wff as a string
valid(...,...) - first string is a list of premises wff's, and the second string is the conclusion wff
consistent(...) - just a string listing some wff's
And remember - all ... entries are strings i.e. tautology('orf(p1,notf(p1))') - note the dashes.
Remember that 'orf(p1,notf(p1))' is the input in Matlab to convert the expression orf(p1,notf(p1)) into a string array.
The tautology function, tells you whether some wff is a tautology or not,
plus gives a single counterexample (one is enough!)
The valid function, tells you whether an argument is valid or not,
plus gives a single counterexample
The consistent function determines whether the set of wff's is satisfiable by some interpretation
NOTE: for n pv's a model or counter-model of the form 0 0 1 0 ... 1
is to be interpreted as val(p1)=0, val(p2)=0, val(p3)=1, val(p4)=0, ..., val(pn)=1
NOTE: this program has not been optimized for efficiency, so for wff's containing in excess of
12 distinct propositional variables the execution time could take around 2 minutes, depending
on your CPU.
--------------------------------------------------------------------------------------
Have fun! This package certainly helps evaluating long wff's.
If you find any bugs, or have some suggestions, leave a comment below :)
--------------------------------------------------------------------------------------
Have fun! This package certainly helps evaluating long wff's.
If you find any bugs, or have some suggestions, leave a comment below :)
--------------------------------------------------------------------------------------
Mariusz
--------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------
Propositional Logic Calculator, Propositional Calculus Calculator, Propositional Logic Emulator, Propositional Calculus Emulator, Propositional Logic Simulator, Propositional Calculus Simulator,
Propositional Logic Application, Propositional Calculus Application, Propositional Logic Applet
Propositional Calculus Applet.
--------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------
COMPLETE CODE
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=wff(x)
%x is the string array of the wff with pv's expressed as p1, p2,...
b=convert3(x);
a=char(b(1:length(b)-4));
npv=b(length(b)-3);
if b(length(b))~=0 || b(length(b)-1)~=0 || b(length(b)-2)~=0
y=[7,b(length(b)-2),b(length(b)-1),b(length(b))];%this condition obtains when a
%syntax error has been detected
else
%a will be the new wff string with pv's expressed as p(v,1),p(v,2),...
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%now for 2^2*npv rows we evaluate the wff with the values for v iterating
%over the number of rows. But we also need the truth table p.
p=truth_table(npv);
eval_array=zeros(1,2^npv);
for v=1:2^npv
eval_array(v)= eval(a);
end
y=[eval_array,b(length(b)-2),b(length(b)-1),b(length(b))];
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y = is_even(x)
if floor(x/2)==x/2
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=n_andf(x)
t=0;
for k=1:length(x)
if x(k)==0
t=t+1;
end
end
if t==0
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=max_row(h)
s=size(h);
v=[];
for k=1:s(1)
v=[v,row_pic(k,h)];
end
y=index_of(max(v),v);
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=notf(x)
if x==1
y=0;
else
y=1;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=npv_test(t)
%t is a row in the p matrix (in convert3.m) containing instances of pv's in the string
%here we're converting the matlab expressions of numerals into decimal
%form
r=[];
for i=1:length(t)
if numerical_string_entry(t(i))==1
r=[r t(i)];
end
end
sum=0;
for k=1:length(r)
v=10^(length(r)-k)*(r(k)-48);
sum=sum+v;
end
y=sum;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=numerical_string_entry(string_element_encoding)
c=0;
for i=48:48+9
if string_element_encoding==i
c=c+1;
end
end
if c==0
y=0;
else
y=1;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=row_pic(k,x)
r=[];
s=size(x);
for i=1:s(2)
if numerical_string_entry(x(k,i))==1
r=[r x(k,i)];%here we extract numerical entries from the character array
end
end
sum=0;%here each numerical row is transformed to a unique scalar quantity
%required to determine the largest numerical entry i.e. largest pv
for i=1:length(r)
sum=sum+(10^(length(r)-i))*(r(i));
end
y=sum;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=orf(x,z)
if x==1 || z==1
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=tautology(x)
w=wff(x);
if w(1)==7 && w(length(w)-2)==1
y='ERROR: the number of left-hand brackets must equal the number of right-hand brackets';
else if w(1)==7 && w(length(w))==1
y='ERROR: for any pn appearing in the wff, all pk such that 1=
else if w(1)==7 && w(length(w)-1)==1
y='ERROR: superflous brackets - double check the syntax!';
else if w(1)==7 && w(length(w)-1)==-1
y='ERROR: insufficient brackets - double check the syntax!';
else
npvs=log2(length(w)-2);
%we can easily extract the number of propositional variables from wff(x),
%since wff spits out a one dimensional array of the value of the wff for
%each interpretation.
t=truth_table(npvs);
counter_model=[];
if sum(w(1:length(w)-3))==length(w)-3%if wff is all 1's i.e. 1 on every interpretation
y=1;
else
y=0;
counter_model=t(min(index_of(0,w)),:);
%here we give just the nearest countermodel - it's sufficient!
%however the nearest_zero function can be altered to give all
%the counterexamples
counter_model
end
end
end
end
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y= truth_table(nPVs)
%this program generates binary strings required to construct a semantic
%table (table for Propositional Calculus of all interpretations for a given number of PV's)
%Basically here we generate thruth tables for a given number of PV's.
v=[];
for n=1:nPVs
for k=1:(2^nPVs)/(2^(n-1))
for i=(2^(n-1))*(k-1)+1:2^nPVs
if is_even(k)==0
v(i,n)=0;
else
v(i,n)=1;
end
end
end
end
y=v;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=valid(x,z)
conversion=['conf(n_andf([',x,']),',z,')'];
y=tautology(conversion);
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=all_pvs(x)
c=0;
for i=1:length(x)
if x(i)=='p'
c=c+1;
end
end
y=c;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=bconf(x,z)
if (x==1 && z==1) || (x==0 && z==0)
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=conf(x,z)
if x==0 || z==1
y=1;
else
y=0;
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=consistency(x)
w=wff(x);
if w(1)==7 && w(length(w)-2)==1
y='ERROR: the number of left-hand brackets must equal the number of right-hand brackets';
else if w(1)==7 && w(length(w))==1
y='ERROR: for any pn appearing in the wff, all pk such that 1=
else if w(1)==7 && w(length(w)-1)==1
y='ERROR: superflous brackets - double check the syntax!';
else if w(1)==7 && w(length(w)-1)==-1
y='ERROR: insufficient brackets - double check the syntax!';
else
npvs=log2(length(w)-2);
%we can easily extract the number of propositional variables from wff(x),
%since wff spits out a one dimensional array of the value of the wff for
%each interpretation.
t=truth_table(npvs);
Model=[];
if sum(w(1:length(w)-3))>0%if wff is all 1's i.e. 1 on every interpretation
y=1;
Model=t(min(index_of(1,w)),:)%Here is the model where the set is satisfied - we provide just one,
%but the nearest_one function can be altered to obtain all the models.
else
y=0;
end
end
end
end
end
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%------------Created by Mariusz Popieluch 2011 prime4567@gmail.com---------
%--------------------------------------------------------------------------
%------------------http://www.mariuszessays.blogspot.com-------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=consistent(x)
conversion=['n_andf([',x,'])'];
y=consistency(conversion);
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y =convert3(x)
%-------------------------------------------------
%syntax check if the number of brackets is equal
l=0;
r=0;
dd=0;
for i=1:length(x)
if x(i)==40
l=l+1;
end
if x(i)==41
r=r+1;
end
end
if r~=l
dd=1;%'ERROR: the number of left-hand brackets must equal the number of right-hand brackets'
end
%-------------------------------------------------
%the loop below extracts the pv numerals and for each pv inserts then in
%the p matrix where each pv numeral goes into a seperate row
b=0;
h=0;
for t=1:length(x)%this makes the code clean so any pn is a wff
if x(t)== 'p'
h=h+1;
end
end
f=0;
for t=1:length(x)%sufficient brackets for the number of functions
if x(t)== 'f'
f=f+1;
end
end
ee=0;
if 2*r<2 f="" font="">
ee=-1;%'ERROR: insufficient brackets - double check the syntax!'
else if 2*r>2*f
ee=1;%'ERROR: superflous brackets - double check the syntax!'
end
end
if h~=1 && x(length(x))~=41%just taking care of cases without last bracket
x=[x 41];
ee=-1;
end
%---------------------------------------------------------------
%here we create a matrix of x length and all_pvs height
siz=all_pvs(x);
p=zeros(siz,length(x));
for k=1:length(x)
for i=1:siz
p(i,k)='*';
end
end
%----------------------------------------------------------------
% if h==1
% y=[112 40 118 44 49 41 1 0 0 0];%here we just return the convertend string p(v,1) i.e. representing one pv
% else
for i=1:length(x)%this code extracts all numerial string entries from the wff and inserts them in the rows of p
if x(i)=='p'
b=b+1;
for k=1:length(x)
if numerical_string_entry(x(i+k))~=0
p(b,k)=x(i+k);
else
break
end
end
end
end
%-------------------------------------------------
%Next we determine the order of the pv's by finding the maximum length of
%numeral strings in the p matrix.
pdash=zeros(siz,length(p));
for i=1:siz
for k=1:length(x)
if numerical_string_entry(p(i,k))==1
pdash(i,k)=1;
end
end
end
pvorder=max(sum(pdash'));
numeralspace=sum(sum(pdash'));%the number of numerical entries in x
%-------------------------------------------------
%conversion proper
s=size(p);
nthpv=0;%n'th row in p corresponding to the n'th pv appearing in x
c=0;%is the number of accumulated entries through conversion i.e. 4 for each pv
lenxdash=length(x)+4*siz;
xdash=zeros(1,lenxdash);
for i=1:length(x)
if x(i)=='p'
nthpv=nthpv+1;
leng=sum(pdash(nthpv,1:s(2)));%number of numerical entries in pdash's i'th row
g=['p','(','v',',',p(nthpv,1:leng),')'];%replacement string considering each pv i.e. p row
xdash(i+c:i+c+length(g)-1)=g(1:length(g));%conversion
c=c+4;%accumulated entries through conversion
end
end
%---------------------------------------------------
%Here is the code that counts the number of pv's
c=[];
for i=1:s(1)
for j=1:s(1)
if npv_test(p(i,1:s(2)))==npv_test(p(j,1:s(2)))
c=[c 0];
else
c=[c 1];
end
end
end
if sum(c)==0%this implies tht all pv's are identical
npvs=npv_test(p(1,1:s(2)));%and here we just determine which one it is
else
max_row_p=max_row(p);%creating a variable so
npvs=npv_test(p(max_row_p(1),1:s(2)));%we pick one only instead of all the indexes where the maximum pv's appear
end
%---------------------------------------------------
%--------------------------------------------------
%Before we count the number of pv's we need to determine if the syntax is
%correct - this bit of code determines if there are any non sequential
%instances of pv's eg. p1,p2,p4 instead of p3
gg=0;
max_pv=npvs;%npv_test(p(max_row(p),1:s(2)));
for k=1:max_pv-1
t=0;
for i=1:s(1)
if npv_test(p(i,:))== max_pv-k
t=t+1;
end
end
if t==0
gg=1;%'ERROR: for any pn appearing in the wff, all pk such that 1
break
end
end
%--------------------------------------------------------------------------
empty_xdash=index_of(0,xdash);%this is ok
nonpvx=index_of_not_general('p',x);
xdash(empty_xdash)=x(nonpvx);
y=[xdash,npvs,dd,ee,gg];
end
%this is the PROPER EXECUTION LINE, just xdash will have to be
%transformed to char(xdash) in the program treating it as a wff string.
%eg. v=convert3('the actual wff string')
%x=char(v(1:length(v)-1); y=v(length(v)); where x is the wff and y is the
%number of pv's in that wff
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=index_of(x,z)
p=[];
for i=1:length(z)
if z(i)==x
p=[p i];
end
end
y=p;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=index_of_not_general(x,z)
p=[];
%for k=1:length(x)
for i=1:length(z)
if z(i)~=x && numerical_string_entry(z(i))==0
p=[p i];
end
end
%end
y=p;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
function y=row_pic(k,x)
r=[];
s=size(x);
for i=1:s(2)
if numerical_string_entry(x(k,i))==1
r=[r x(k,i)];%here we extract numerical entries from the character array
end
end
sum=0;%here each numerical row is transformed to a unique scalar quantity
%required to determine the largest numerical entry i.e. largest pv
for i=1:length(r)
sum=sum+(10^(length(r)-i))*(r(i));
end
y=sum;
end
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
Subscribe to:
Posts (Atom)