It's also characteristic of this 3VL that it reduces to the two valued case when all the propositions involved are definitely true or false, and the middle value excluded. But a logic that has all the theorems of 2VL, and only those, would be equivalent to it. Not all the theorems of 2VL hold, just as not all the theorems of ordinary arithmetic hold for the integers, and not all those of real numbers apply to the complex numbers. Many theorems of 2VL hold in 3VL as well; typically those involving algebraic-type manipulations of expressions. However, some of them must be modified or restricted; typically those involving rules of inference.
As I've mentioned, direct proof via Modus ponens and transitive chains of inference doesn't work in standard Lukasiewicz logic; These rules need to be restricted to avoid dubious conditionals. Similarly, indirect proofs that rely on some form of "Reductio ad absurdum" also need to be restricted. It is not sufficient to prove P by assuming ~P and then deriving a contradiction Q and ~Q, because this isn't necessarily a contradiction in 3VL. Indirect proof is still possible, but it requires stronger contradictions of the forms "possible and impossible ("<>P & ~<>P) , or "Certainly and not necessarily []P & ~[]P", or even "Certain and impossible" (<>P & ~<>P).
There are also rules that express ideas that aren't available in 2VL. If Certainly P, then P; ([]P => P) and if P then possibly P (P => <>P) are both valid rules but their converses are not>
Wednesday, May 25, 2005
Tuesday, May 24, 2005
3VL - doubtful inference
The impression I have gathered is that Lukasiewicz 3-valued logic hasn't had a great deal of respect in the logical community. This is entirely understandable. As it has been presented so far, there are chronic difficulties with interpretation, and you can't do the same kinds of things with it that you can with classical logic. Most of the serious work that has been done with it has been using Lukasiewicz' "Polish notation", which is unfamiliar to most people who work with logic, and in Europe rather than the United States.
But the small step of defining a strict conditional for it makes an incredible difference. It's like giving it the power pill that turns lowly shoeshine boy into Underdog; like turning Bruce Banner into the Incredible Hulk; like turning a lightning bug into lightning; moonlight into sunlight, climbing over a mountain peak and seeing the Pacific Ocean on the other side. The difference in effectiveness is so huge, that it's amazing no one has seen it before. But if it has been seen, I haven't found it in the literature.
It was like a dazzling flash of hindsight. Revelation followed revelation so swiftly, and in such interconnected fashion, that I no longer recall their exact sequence of events. But I can describe some of them.
One of the early ones is that I realized the reason why Lukasiewicz logic hadn't been workable before.
I was already aware that, using the original Lukasiewicz conditional, Modus Ponens fails as a tautology. But it fails in only one case, Namely, when P is doubtful and Q is false. The truth table labels this as doubtful.
But of course!! The Lukasiewicz conditional allows the expression of doubtful conditionals, and if it were true without restriction, it would be possible to start with a doubtful premise and a doubtful conditional, and advance to a false conclusion. But by forbidding dubious conditionals and assuring that it is definitely the case that if P, then Q we repair the deficiency. The original truth table quite correctly labels a case where modus ponens can and should fail.
A few more examples, such as transitivity, yielded similar results, and the basis for a whole theory of doubtful inference falls out, naturally and easily.
Of course this must be so!! One of the purposes of logic, after all, is to assure that our rules of reasoning are correct and that we do not start from true premises and reason to false conclusions. And the strict conditional has just the kind of ordering properties, on three values, that the ordinary material conditional has for two values; when P =>Q is true, The conclusion Q is at least as true as the premises P.
But the small step of defining a strict conditional for it makes an incredible difference. It's like giving it the power pill that turns lowly shoeshine boy into Underdog; like turning Bruce Banner into the Incredible Hulk; like turning a lightning bug into lightning; moonlight into sunlight, climbing over a mountain peak and seeing the Pacific Ocean on the other side. The difference in effectiveness is so huge, that it's amazing no one has seen it before. But if it has been seen, I haven't found it in the literature.
It was like a dazzling flash of hindsight. Revelation followed revelation so swiftly, and in such interconnected fashion, that I no longer recall their exact sequence of events. But I can describe some of them.
One of the early ones is that I realized the reason why Lukasiewicz logic hadn't been workable before.
I was already aware that, using the original Lukasiewicz conditional, Modus Ponens fails as a tautology. But it fails in only one case, Namely, when P is doubtful and Q is false. The truth table labels this as doubtful.
But of course!! The Lukasiewicz conditional allows the expression of doubtful conditionals, and if it were true without restriction, it would be possible to start with a doubtful premise and a doubtful conditional, and advance to a false conclusion. But by forbidding dubious conditionals and assuring that it is definitely the case that if P, then Q we repair the deficiency. The original truth table quite correctly labels a case where modus ponens can and should fail.
A few more examples, such as transitivity, yielded similar results, and the basis for a whole theory of doubtful inference falls out, naturally and easily.
Of course this must be so!! One of the purposes of logic, after all, is to assure that our rules of reasoning are correct and that we do not start from true premises and reason to false conclusions. And the strict conditional has just the kind of ordering properties, on three values, that the ordinary material conditional has for two values; when P =>Q is true, The conclusion Q is at least as true as the premises P.
Monday, May 23, 2005
3VL - Success
I need to go back to the Principle of the Excluded middle. I say principle, because it's not a law, here: it's a contingent statement which applies to some propositions but not others. It comes in two forms: bivalence (p v ~p), and noncontradiction ~(P & ~P). In classical logic, these are equivalent. Both of these and their equivalence have been frequently challenged by various logicians or philosophers, but few people take these challenges seriously. In this logic, these two formulations are equivalent, but the necessity of the excluded middle may be either asserted or denied. The assertion, "It is necessarily (certainly) the case that either P or not P", or equivalently "It is not possible for both P and not P)" marks a dichotomously uncertain statement, one which must be either true or false: not neither and not both, although it may not be known which is actually the case. [](P v ~P) = ~<>(P & ~P) = !P. The denial, "it is not necessarily the case that either P or not P", or equivalently "It is possible for both P and not P", marks an equivocally uncertain statement, one with the middle truth value. ~[](P v ~P) = <>(P & ~P) = ?P.
When arguments are symbolized, one finds that that the middle may be consistently be either included or excluded, but the logic strictly enforces consistency once the choice is made. It is obviously inconsistent, and in fact results in a genuine contradiction, to allow the use of the middle truth value on the one hand, and then reassert the excluded middle in one of its forms, on the other. Yet the temptation to do so is both insidious and ubiquitous. More than one of the arguments that have been advanced against three valued logic employ just such an argument.
Perhaps more importantly, when I examined the various versions of 3VL, I found that several of them could be expressed in terms I had defined, which made this a more general system. And then, when I was looking at their connectives searching for such a definition, I noticed a certain definition of equivalence and said, "Hey, wait a minute! That's not an equivalence, that's only a biconditional!" Mathematically, an equivalence relation is reflexive, symmetric, and transitive, and these "logical equivalences" were none of those. It should also express the idea that two formulas should have the same truth value, and they didn't do that, either. Only one of them did (I believe it was Kleene's system, the one that had the conditional I had long ago discarded as inadequate). I had a use for that definition, and so I appropriated it.
At this point, I had a partially functional logic. I could establish commutativity, associativity, and the distributive laws for Conjunction (&) and disjunction (v); I had double negation, De Morgan's laws, and the interconversion of the modal functions, and the law of the contrapositive. I had my two types of uncertainty, and their behavior with respect to the other operators. Now I could add properties of equality (If P=Q, then ~P = ~Q), and properties of equality (If P and P=Q then Q; if P = Q and Q=R then P=R), which was an advance.
It also gave me an intepretation for the Lukasiewicz conditional: I could define it as (~P v Q v P = Q), which was curious, but didn't strike me as particularly useful or profound. And then, after I don't remember how long, I noticed that I didn't need a separate definition for equivalence. I could get it by applying necessity (or certainty) to the Lukasiewicz biconditional I was already using. P = Q = [](P <-> Q)
And then, on the basis that what was good for the biconditional was good for the conditional, I decided to define a strict Lukasiewicz conditional, P => Q as [](P -> Q), removing the uncertainty.
Duh. Of course. Obliviously. And the light came on, and suddenly I understood more than I had ever dreamed of, or anyone will believe.
When arguments are symbolized, one finds that that the middle may be consistently be either included or excluded, but the logic strictly enforces consistency once the choice is made. It is obviously inconsistent, and in fact results in a genuine contradiction, to allow the use of the middle truth value on the one hand, and then reassert the excluded middle in one of its forms, on the other. Yet the temptation to do so is both insidious and ubiquitous. More than one of the arguments that have been advanced against three valued logic employ just such an argument.
Perhaps more importantly, when I examined the various versions of 3VL, I found that several of them could be expressed in terms I had defined, which made this a more general system. And then, when I was looking at their connectives searching for such a definition, I noticed a certain definition of equivalence and said, "Hey, wait a minute! That's not an equivalence, that's only a biconditional!" Mathematically, an equivalence relation is reflexive, symmetric, and transitive, and these "logical equivalences" were none of those. It should also express the idea that two formulas should have the same truth value, and they didn't do that, either. Only one of them did (I believe it was Kleene's system, the one that had the conditional I had long ago discarded as inadequate). I had a use for that definition, and so I appropriated it.
At this point, I had a partially functional logic. I could establish commutativity, associativity, and the distributive laws for Conjunction (&) and disjunction (v); I had double negation, De Morgan's laws, and the interconversion of the modal functions, and the law of the contrapositive. I had my two types of uncertainty, and their behavior with respect to the other operators. Now I could add properties of equality (If P=Q, then ~P = ~Q), and properties of equality (If P and P=Q then Q; if P = Q and Q=R then P=R), which was an advance.
It also gave me an intepretation for the Lukasiewicz conditional: I could define it as (~P v Q v P = Q), which was curious, but didn't strike me as particularly useful or profound. And then, after I don't remember how long, I noticed that I didn't need a separate definition for equivalence. I could get it by applying necessity (or certainty) to the Lukasiewicz biconditional I was already using. P = Q = [](P <-> Q)
And then, on the basis that what was good for the biconditional was good for the conditional, I decided to define a strict Lukasiewicz conditional, P => Q as [](P -> Q), removing the uncertainty.
Duh. Of course. Obliviously. And the light came on, and suddenly I understood more than I had ever dreamed of, or anyone will believe.
Sunday, May 22, 2005
3VL types of uncertainty
I haven't given up on the other aspects of self-directed education (in case anyone is actually reading this), but in the last few weeks, I've found some logic-oriented blogs on the net. Starting with Logblog I'll start referring to those once I get done with my "confessions" here.
My attitude toward professional logicians is, If you can't join em, fight em. No one competent to understand what I am talking about has shown any interest. So, since I read that one of the components of a successful blog is to be controversial, and since I am clearly suffering delusions of enlightnment, I'm going to turn guerrilla and snipe at the Establishment.
At the point in my studies I had reached, I was not aware that I had almost independently reconstructed the 3-valued logic of Lukasiewicz, and I didn't understand the reasoning or philosophy behind the Lewis systems S1-S5 beyond what I could determine from the axioms.
I ventured onto Compuserve (The internet was just then beginning to grow), and asked there whether anyone had any comment. I was referred to Joe Celko, who was described as working on a 3-valued logic that dealt with missing values in Data bases, who referred me in turn to a discussion going on in the pages of Data Base Programming and Design. I read the articles with great interest, and found myself sympathizing with both sides on the debate. The 3-valued approach the proponents were using was similar to what I was doing, but I agreed that a sound theory was lacking. I didn't have the answers, either, but again, there were unanswered questions.
I went back to school to try to get my BS in Mathematics, and took a course in classical logic. The approach used in that course was natural deduction, and it basically covered propositional logic without going into predicate logic, but I took the opportunity to study that on my own. I took note of the fact that theorems of logic corressponded to truth-functionally true statements (that is, a statement that evaluated as true on every assignment of truth values), their negations were truth-functionally false, and others were contingent, and I wondered whether the middle value I was using could be used to describe these.
I transferred to ASU for a semester, and later lived next to the University for a year, and took the opportunity to examine the literature a little closer. I was dismayed to discover that my discoveries had indeed been anticipated, and I almost gave up. However, there were still unsettled questions. One of the comments I encountered was that "In spite of the promising combination of trivalence and modality, modal logic on this basis was never fully developed." I wanted to know why, and there was no further discussion, no references, no reasons why it didn't work. The other was the objection to interpretation. Lukasiewicz intended his truth value to represent the uncertainty of the future contingent, but an objector (no reference given) pointed to the "law of the excluded middle" and, apparently, there was no answer. When I worked on this, I decided that there were two different kinds of uncertainty involved. Using constants instead of variables or tables !P expresses the idea "True or false, but it's not certain which (!T=T, !U=F, !F=T) while ?P expresses the doubtfulness associated with the middle truth value (?T=F, ?U=T, ?U=T).
It's trivial to show that ~!P = ?P and ~?P = ~P, but !~P=P and ?~P=?P. This meant, to me, that "uncertainty" is an ambiguous concept, with two formally similar but contradictory interpretations. I didn't fully work out the details of how these were associated with the "and" and "or" at this point.
After this, I moved back to Utah, close enough to BYU that I could consult the literature there, and ventured onto the internet, this time at the newsgroup math.logic. One person noted that according to my tables, [](P v Q) = []P v []Q, and <>P & <>Q = <>(P & Q), which aren't accepted in traditional (e.g. Lewis-type) modal logic, while someone else referred me to Bolc & Borowic's work on multi-valued logic. I labored over these for some time, trying to figure how I could get [](P v Q) & ~[]([]P v []Q); and (<>P & <>Q) & ~[](<>(P & Q), but no matter how I transformed and tortured these statements, I got contradictions. Eventually, I decided that they were genuinely contradictions. To simplify the problem, supposing that P and Q are mutually exclusive, so that Q = ~P, and then applying the various transformation rules, these boil down to trying to assert the excluded middle on one side and deny it on the other. No wonder there's a contradiction!
My attitude toward professional logicians is, If you can't join em, fight em. No one competent to understand what I am talking about has shown any interest. So, since I read that one of the components of a successful blog is to be controversial, and since I am clearly suffering delusions of enlightnment, I'm going to turn guerrilla and snipe at the Establishment.
At the point in my studies I had reached, I was not aware that I had almost independently reconstructed the 3-valued logic of Lukasiewicz, and I didn't understand the reasoning or philosophy behind the Lewis systems S1-S5 beyond what I could determine from the axioms.
I ventured onto Compuserve (The internet was just then beginning to grow), and asked there whether anyone had any comment. I was referred to Joe Celko, who was described as working on a 3-valued logic that dealt with missing values in Data bases, who referred me in turn to a discussion going on in the pages of Data Base Programming and Design. I read the articles with great interest, and found myself sympathizing with both sides on the debate. The 3-valued approach the proponents were using was similar to what I was doing, but I agreed that a sound theory was lacking. I didn't have the answers, either, but again, there were unanswered questions.
I went back to school to try to get my BS in Mathematics, and took a course in classical logic. The approach used in that course was natural deduction, and it basically covered propositional logic without going into predicate logic, but I took the opportunity to study that on my own. I took note of the fact that theorems of logic corressponded to truth-functionally true statements (that is, a statement that evaluated as true on every assignment of truth values), their negations were truth-functionally false, and others were contingent, and I wondered whether the middle value I was using could be used to describe these.
I transferred to ASU for a semester, and later lived next to the University for a year, and took the opportunity to examine the literature a little closer. I was dismayed to discover that my discoveries had indeed been anticipated, and I almost gave up. However, there were still unsettled questions. One of the comments I encountered was that "In spite of the promising combination of trivalence and modality, modal logic on this basis was never fully developed." I wanted to know why, and there was no further discussion, no references, no reasons why it didn't work. The other was the objection to interpretation. Lukasiewicz intended his truth value to represent the uncertainty of the future contingent, but an objector (no reference given) pointed to the "law of the excluded middle" and, apparently, there was no answer. When I worked on this, I decided that there were two different kinds of uncertainty involved. Using constants instead of variables or tables !P expresses the idea "True or false, but it's not certain which (!T=T, !U=F, !F=T) while ?P expresses the doubtfulness associated with the middle truth value (?T=F, ?U=T, ?U=T).
It's trivial to show that ~!P = ?P and ~?P = ~P, but !~P=P and ?~P=?P. This meant, to me, that "uncertainty" is an ambiguous concept, with two formally similar but contradictory interpretations. I didn't fully work out the details of how these were associated with the "and" and "or" at this point.
After this, I moved back to Utah, close enough to BYU that I could consult the literature there, and ventured onto the internet, this time at the newsgroup math.logic. One person noted that according to my tables, [](P v Q) = []P v []Q, and <>P & <>Q = <>(P & Q), which aren't accepted in traditional (e.g. Lewis-type) modal logic, while someone else referred me to Bolc & Borowic's work on multi-valued logic. I labored over these for some time, trying to figure how I could get [](P v Q) & ~[]([]P v []Q); and (<>P & <>Q) & ~[](<>(P & Q), but no matter how I transformed and tortured these statements, I got contradictions. Eventually, I decided that they were genuinely contradictions. To simplify the problem, supposing that P and Q are mutually exclusive, so that Q = ~P, and then applying the various transformation rules, these boil down to trying to assert the excluded middle on one side and deny it on the other. No wonder there's a contradiction!
Saturday, May 21, 2005
3VL. Not quite.
I recall doing this sometime between 1984 and 1988, after I moved from Utah to Phoenix. When I was working out my version of 3-valued logic, the truth tables for "And" (&) and "or" (v) were satisfactory, but in order to express relationships between propositions, I needed a conditional and biconditional. My first attempt gave (using the constants) T -> T = T; T -> U = U; T -> F = F; U -> T = T; U -> U = U; U -> F = U; F -> T = T; F -> U = T; F -> F = T.
I then began exploring truth tables for the tautologies of classical logic, for instance de Morgan's law ~ (P & Q) <-> ~P v ~Q. I quickly found that there was a gaping hole in every truth table I could construct. For the most familiar logical laws, most entries turned out T, but when P and Q both had values of U, the proposition in question had values of U. This was hardly tolerable. It should be obvious that "if P then P" should be a tautology, but even for something this simple, I got U. At this point, I think I decided to see what else had been done on the subject. I went out to the ASU library to find either Rosser and Turquette's "Many valued logic" or Restall's "Three valued logic" (I'm not sure which), and found that this had been done. It's still not clear to me whether this was Kleene's "weak" system or his "strong" system, but in either case, it didn't work, for the very reason I had already discovered. I found that there was another alternative, Lukasiewicz 3-valued logic, which had the same truth tables I had already worked out, but differed in only one place: U -> U = T (instead of U. I didn't have much time to study it then, but I took this one idea to work with later.
Maybe a year or so later, I saw a friend working on truth tables for a different 3VL, and picked up my own studies again. This time, it occurred to me that I could use a function to distinguish "T or U" from "F", and called this "possible", and one to distinguish "T" from "U or F", and called this one "certain" or "necessary". In combination with negation, one of the first and easiest results were the formulas "Certainly not" = "not possible" and "Not necessarily" = "Possibly not".
When I found this, I considered it too simple and elegant to ignore. Also, using the Lukasiewicz conditional, and biconditional, I could prove a number of significant theorems of elementary propositional logic. However, one of the most important ones, Modus ponens, didn't work. I consulted the source most easily available to me, the Encyclopedia Britannica's article on "History and Kinds of logic", and learned a little bit more about the Lewis systems of modal logic. I particularly noted that there were several varieties, that these systems were not truth functional and could not be expressed with truth tables, but had to be constructed on an axiomatic basis. and that the decision problem (deciding whether a given proposition was or was not a theorem), was particularly difficult. I then tried evaluating a number of the axioms according to the three-valued tables I had developed, and found that some of them worked, and some of them didn't. At this point, I wondered "Why do some of these work, but not others?".
I then began exploring truth tables for the tautologies of classical logic, for instance de Morgan's law ~ (P & Q) <-> ~P v ~Q. I quickly found that there was a gaping hole in every truth table I could construct. For the most familiar logical laws, most entries turned out T, but when P and Q both had values of U, the proposition in question had values of U. This was hardly tolerable. It should be obvious that "if P then P" should be a tautology, but even for something this simple, I got U. At this point, I think I decided to see what else had been done on the subject. I went out to the ASU library to find either Rosser and Turquette's "Many valued logic" or Restall's "Three valued logic" (I'm not sure which), and found that this had been done. It's still not clear to me whether this was Kleene's "weak" system or his "strong" system, but in either case, it didn't work, for the very reason I had already discovered. I found that there was another alternative, Lukasiewicz 3-valued logic, which had the same truth tables I had already worked out, but differed in only one place: U -> U = T (instead of U. I didn't have much time to study it then, but I took this one idea to work with later.
Maybe a year or so later, I saw a friend working on truth tables for a different 3VL, and picked up my own studies again. This time, it occurred to me that I could use a function to distinguish "T or U" from "F", and called this "possible", and one to distinguish "T" from "U or F", and called this one "certain" or "necessary". In combination with negation, one of the first and easiest results were the formulas "Certainly not" = "not possible" and "Not necessarily" = "Possibly not".
When I found this, I considered it too simple and elegant to ignore. Also, using the Lukasiewicz conditional, and biconditional, I could prove a number of significant theorems of elementary propositional logic. However, one of the most important ones, Modus ponens, didn't work. I consulted the source most easily available to me, the Encyclopedia Britannica's article on "History and Kinds of logic", and learned a little bit more about the Lewis systems of modal logic. I particularly noted that there were several varieties, that these systems were not truth functional and could not be expressed with truth tables, but had to be constructed on an axiomatic basis. and that the decision problem (deciding whether a given proposition was or was not a theorem), was particularly difficult. I then tried evaluating a number of the axioms according to the three-valued tables I had developed, and found that some of them worked, and some of them didn't. At this point, I wondered "Why do some of these work, but not others?".
Friday, May 20, 2005
Three valued logic truth tables
I started with the intepretation of my third logical value as "True or false, but it's not certain which". For negation, I could extend the normal table for true or false, and reasoned, The negation of "true or false" would be "false or true": If I don't know whether or not a statement is true, I also don't know whether or not its negation is true. So, if P = U, then ~P = U.
The table for "or", I decided "if P is true, and Q is either true or false, the value of P or Q doesn't depend on the truth of Q", so if P = T and Q = U, P v Q = U, and likewise with P and Q interchanged. U or U should be U, U or F should be U.
Similarly with the table for "and".
Actually, as I found out later, this turns to be an unfortunate interpretation, but the truth tables still work. There is another interpretation that works better, though.
The table for "or", I decided "if P is true, and Q is either true or false, the value of P or Q doesn't depend on the truth of Q", so if P = T and Q = U, P v Q = U, and likewise with P and Q interchanged. U or U should be U, U or F should be U.
Similarly with the table for "and".
Actually, as I found out later, this turns to be an unfortunate interpretation, but the truth tables still work. There is another interpretation that works better, though.
Thursday, May 19, 2005
Three valued logic beginnings
I've recently begun following some logic blogs in an attempt to find an outlet for my work in three-valued logic. I've been told that my results are probably not publishable, but I want to discuss them. I mentioned earlier that I don't clearly recall when I became interested in logic, but I can describe some of how I developed this topic.
Between 1981 and 1984, I spent many hours in the mathematics section of the BYU library, and in the process kept my knowledge of logic from dissolving into rust, as well as picked up a smattering of predicate logic. For various reasons, related to the early-20th century "Crisis" in mathematical foundations, I encountered reason to suspect that classical two-valued logic was good for mathematics, but there were doubts that it was sufficient, especially when it dealt with infinite sets. Mathematicians became concerned about the difference between "True" and "provable". I read about various paradoxes in set theory, and did a little bit of playing with them. Russell's paradox attracted my special attention, as did the discussion that even going to a three-valued logic wouldn't necessarily resolve it. I also encountered an article which attempted to analyze Anselm's proof of the existence of God using modal logic, which attracted my attention to that subject.
As I recall now, it was about this time that I started tinkering with what I called a "logic of indecision", using three values. Among my early attempts were truth tables for negation, disjunction (using the OR), and conjuncion (AND), and I used symbols. T, U (or I), and F for the truth values.
Between 1981 and 1984, I spent many hours in the mathematics section of the BYU library, and in the process kept my knowledge of logic from dissolving into rust, as well as picked up a smattering of predicate logic. For various reasons, related to the early-20th century "Crisis" in mathematical foundations, I encountered reason to suspect that classical two-valued logic was good for mathematics, but there were doubts that it was sufficient, especially when it dealt with infinite sets. Mathematicians became concerned about the difference between "True" and "provable". I read about various paradoxes in set theory, and did a little bit of playing with them. Russell's paradox attracted my special attention, as did the discussion that even going to a three-valued logic wouldn't necessarily resolve it. I also encountered an article which attempted to analyze Anselm's proof of the existence of God using modal logic, which attracted my attention to that subject.
As I recall now, it was about this time that I started tinkering with what I called a "logic of indecision", using three values. Among my early attempts were truth tables for negation, disjunction (using the OR), and conjuncion (AND), and I used symbols. T, U (or I), and F for the truth values.
Electromagnetism
As part of my discussion on physics, it's useful to go into a little more detail on electromagnetism.
This includes three or four areas, at least from an elementary point of view. These are:
1) Electrostatics. This refers to the behavior of electricity and electric charge and related quantities, at rest.
2) Electric current. This refers to the behavior of electric charge in motion, and includes various kinds of electric circuits. Electrostatics and electric current are often combined under electricity.
3) Magnetism. This refers to magnets, both from magnetic fields and from electric currents.
4) Optics. This refers to light and electromagnetic radiation in general.
These subjects depend heavily on mechanics. Many of the concepts of electromagnetism are most easily introduced from classical mechanics. There is a certain analogy with gravitation, and relativity was developed as an attempt to reconcile certain findings in electromagnetic theory with mechanics. From an advanced point of view, these are closely connected. At small scales, electromagnetism is also closely tied to quantum mechanics as well. The role of thermodynamics is not greatly important except for thermal radiation. There are close connections with the structure of matter.
The other sciences of chemistry, astronomy, earth science, and biology mostly furnish examples of electromagnetism. Biographies of prominent and pioneering scientists, and various associations can be examined. There are numerous textbooks that include discussions of electromagnetism, most of them discussing mathematical relationships. The techniques of working with it vary according to the specific subject. Various instruments are required, since the quantities involved are mostly invisible. Practical application, the importance in various societies, and the history are also useful studies.
I don't have any particular recommendations, for study of this subject until some other time.
This includes three or four areas, at least from an elementary point of view. These are:
1) Electrostatics. This refers to the behavior of electricity and electric charge and related quantities, at rest.
2) Electric current. This refers to the behavior of electric charge in motion, and includes various kinds of electric circuits. Electrostatics and electric current are often combined under electricity.
3) Magnetism. This refers to magnets, both from magnetic fields and from electric currents.
4) Optics. This refers to light and electromagnetic radiation in general.
These subjects depend heavily on mechanics. Many of the concepts of electromagnetism are most easily introduced from classical mechanics. There is a certain analogy with gravitation, and relativity was developed as an attempt to reconcile certain findings in electromagnetic theory with mechanics. From an advanced point of view, these are closely connected. At small scales, electromagnetism is also closely tied to quantum mechanics as well. The role of thermodynamics is not greatly important except for thermal radiation. There are close connections with the structure of matter.
The other sciences of chemistry, astronomy, earth science, and biology mostly furnish examples of electromagnetism. Biographies of prominent and pioneering scientists, and various associations can be examined. There are numerous textbooks that include discussions of electromagnetism, most of them discussing mathematical relationships. The techniques of working with it vary according to the specific subject. Various instruments are required, since the quantities involved are mostly invisible. Practical application, the importance in various societies, and the history are also useful studies.
I don't have any particular recommendations, for study of this subject until some other time.
Thursday, May 12, 2005
Particle mechanics
I mentioned a while ago that I wanted to start discussing some of my thoughts on logic. I'm going to have to hold off on that for a little, while I discuss mechanics and the like.
One of the textbooks I've been looking at lately said something about mechanics being at the foundation of physics. To a large extent, I agree with this. Although by itself, particle mechanics doesn't seem to be a particularly large or useful area of knowledge, it's important to understand it in order to understand many other subjects.
I have divisions of particle mechanics:
1) Description. Many introductions tend to skip over this, but there are significant differences between particles and other extended bodies.
2) Kinematics. This subject includes the description of motion, without regard to its cause.
3) Force and Momentum. This includes Newton's laws of motion
4) Work and energy
5) Systems of particles.
One of the textbooks I've been looking at lately said something about mechanics being at the foundation of physics. To a large extent, I agree with this. Although by itself, particle mechanics doesn't seem to be a particularly large or useful area of knowledge, it's important to understand it in order to understand many other subjects.
I have divisions of particle mechanics:
1) Description. Many introductions tend to skip over this, but there are significant differences between particles and other extended bodies.
2) Kinematics. This subject includes the description of motion, without regard to its cause.
3) Force and Momentum. This includes Newton's laws of motion
4) Work and energy
5) Systems of particles.
Monday, May 09, 2005
Classical mechanics
I'm trying to tie this more directly into subjects I'm working on. I've done analysis of classical mechanics before, but my notes are buried and left behind in my last move, and it's been a while since I had convenient access to multiple sources. In a typical university library that uses the Library of Congress classification, classical mechanics can be found in several places. There are a few general works on the sciences that mention some basic principles. There are also more specialized texts on physics, and there are also more specialized texts on mechanics, each containing more general ideas. I was trying to reconcile a couple of these, and ran into problems in the order and style of presentation.
In the physics text, the approach started with the description of motion, (kinematics), and then went to Newton's laws, then to rigid bodies, and then other subjects. The mechanics text started with statics and equilibrium of forces, then went to the description of motion. The physical laws are the same with either approach, but I wanted to find one that would unify them. After much head-scratching and rearranging of topics, I decided upon a scheme that worked for me.
1) Particle mechanics deals with the laws of motion as they apply to particles with no (or negligible) parts, rotation or internal motion.
2) Rigid bodies deal with these same laws as applied to bodies that have shape and parts, and adds the topic of rotation.
3) Deformable body mechanics deals with bodies that can be deformed, and includes elastic bodies, fluids, and wave mechanics.
I noticed in one of the science texts a discussion of religion and science, and appreciated the comment that they deal with different approaches. Science deals with the "how" of nature, while religion deals with the "why" and with the building of communities. The text made mention of the fact that certain religious fundamentalists attempt to use scripture to pronounce on questions of science. The comment would have been balanced if it had mentioned that there are zealots of a scientific turn of mind and no appreciation of religion who go the other way and try to use the known laws of science to "disprove" religion.
In the physics text, the approach started with the description of motion, (kinematics), and then went to Newton's laws, then to rigid bodies, and then other subjects. The mechanics text started with statics and equilibrium of forces, then went to the description of motion. The physical laws are the same with either approach, but I wanted to find one that would unify them. After much head-scratching and rearranging of topics, I decided upon a scheme that worked for me.
1) Particle mechanics deals with the laws of motion as they apply to particles with no (or negligible) parts, rotation or internal motion.
2) Rigid bodies deal with these same laws as applied to bodies that have shape and parts, and adds the topic of rotation.
3) Deformable body mechanics deals with bodies that can be deformed, and includes elastic bodies, fluids, and wave mechanics.
I noticed in one of the science texts a discussion of religion and science, and appreciated the comment that they deal with different approaches. Science deals with the "how" of nature, while religion deals with the "why" and with the building of communities. The text made mention of the fact that certain religious fundamentalists attempt to use scripture to pronounce on questions of science. The comment would have been balanced if it had mentioned that there are zealots of a scientific turn of mind and no appreciation of religion who go the other way and try to use the known laws of science to "disprove" religion.
Saturday, May 07, 2005
Astronomy
Most of astronomy does not seem to be as practically useful as other areas of knowledge, but it still belongs. I've been drawn to it because of my interests in science fiction.
I have four major divisions of it:
1) Solar System astronomy. This includes studies of the sun, planets, asteroids, comets, and other things that exist and occur in our solar syste
2) Stellar astronomy. This includes studies of stars and star systems, the life cycles of stars, and the Milky Way Galaxy
3) Galactic astronomy. This includes studies of other galaxies and their composition, classification, and history.
4) Cosmology. This includes theoretical analysis and speculations on the structure, origin, and fate of the universe.
This depends heavily on the various areas of physics, to the point that astrophysics is an important combined field. Mechanics, including not only classical mechanics but gravitation, relativity, and quantum mechanics are all used in astronomy. Electromagnetism, thermodynamics, and the structure of matter are also used heavily.
Chemistry does not seem to be used as heavily, but chemical substances, changes, and systems are discussed in various areas. Areas of earth science are also useful, but there do not seem to be many applications of biology.
The application of other areas, such as psychology, and biography, the role of astronomical societies, language, literature, and philosophy, astronomy as a profession, instrumentation, the role of education and religion, national traditions, and the history of astronomy can be followed following the same patterns as for science in general and the other sciences mentioned.
This is an area in which we are collectively less informed than earlier generations. The majority of people today, particularly in the US, live in cities which are lit at night by electricity, which also lights up the sky to the extent that all but the brightest stars are invisible. Venus is regularly taken for a UFO. If I would suggest anything, it would be to go find a rural area on some clear night, and spend a few hours just looking. For those who are more ambitious and would like to find their way around the heavens, there are numerous guides available.
I have four major divisions of it:
1) Solar System astronomy. This includes studies of the sun, planets, asteroids, comets, and other things that exist and occur in our solar syste
2) Stellar astronomy. This includes studies of stars and star systems, the life cycles of stars, and the Milky Way Galaxy
3) Galactic astronomy. This includes studies of other galaxies and their composition, classification, and history.
4) Cosmology. This includes theoretical analysis and speculations on the structure, origin, and fate of the universe.
This depends heavily on the various areas of physics, to the point that astrophysics is an important combined field. Mechanics, including not only classical mechanics but gravitation, relativity, and quantum mechanics are all used in astronomy. Electromagnetism, thermodynamics, and the structure of matter are also used heavily.
Chemistry does not seem to be used as heavily, but chemical substances, changes, and systems are discussed in various areas. Areas of earth science are also useful, but there do not seem to be many applications of biology.
The application of other areas, such as psychology, and biography, the role of astronomical societies, language, literature, and philosophy, astronomy as a profession, instrumentation, the role of education and religion, national traditions, and the history of astronomy can be followed following the same patterns as for science in general and the other sciences mentioned.
This is an area in which we are collectively less informed than earlier generations. The majority of people today, particularly in the US, live in cities which are lit at night by electricity, which also lights up the sky to the extent that all but the brightest stars are invisible. Venus is regularly taken for a UFO. If I would suggest anything, it would be to go find a rural area on some clear night, and spend a few hours just looking. For those who are more ambitious and would like to find their way around the heavens, there are numerous guides available.
Friday, May 06, 2005
Mechanics
For some reason, I keep going back to mechanics as a starting point for my various studies. At this point, I am talking about mechanics as a division of physics.
There are several subjects that can be included in mechanics. Originally, this term had to do with machines, as in levers and screws, but with the addition and emphasis of other subjects, the meaning began to change. I recognize four principal areas.
1) Classical mechanics. This includes studies of particles, rigid bodies, and deformable bodies, including gases and fluids. This still is the most common and useful area of study. Before the 20th century, nearly all mechanics was classical mechanics.
2) Gravitation. This is usually included in classical mechanics, but it doesn't neatly fit within another organization of the subject, so I have separated it out.
3) Relativity. This deals with corrections which are needed to classical mechanics in the cases of very high speeds and strong gravitational fields. Since these are outside the realm of everyday experience and require some advanced mathematics to fully comprehend, I will not develop it in great deal.
4) Quantum mechanics. This deals with the corrections that are needed to classical mechanics in the case of atomic-sized and smaller particles. This is more applicable, especially in chemistry, but involves even more advanced mathematics, and I will set aside discussion of this subject as well.
There are some connections to electromagnetism and the structure of matter, and a few connections to thermodynamics, but for the most part, mechanics is considered more fundamental than these other subjects. Chemistry is useful when the particular properties of specific substances are important. Astronomy, earth science, and biology are useful for examples and illustrations. There are comparatively few people who specialize in mechanics as an area of theoretical study. It is somewhat difficult to find internet resources at an intermediate level of study, and this is most easily approached by examining general physics textbooks. It requires a fairly high level of mathematics, and is more closely associated with education than any of the other social institutions. The communities and peoples involved and the history of mechanics are also useful, though it can be difficult to find an introductory account of the history of mechanics.
My interest in the subject is more mathematical and theoretical than experimental, although what resources to suggest depends heavily on how much you already know.
There are several subjects that can be included in mechanics. Originally, this term had to do with machines, as in levers and screws, but with the addition and emphasis of other subjects, the meaning began to change. I recognize four principal areas.
1) Classical mechanics. This includes studies of particles, rigid bodies, and deformable bodies, including gases and fluids. This still is the most common and useful area of study. Before the 20th century, nearly all mechanics was classical mechanics.
2) Gravitation. This is usually included in classical mechanics, but it doesn't neatly fit within another organization of the subject, so I have separated it out.
3) Relativity. This deals with corrections which are needed to classical mechanics in the cases of very high speeds and strong gravitational fields. Since these are outside the realm of everyday experience and require some advanced mathematics to fully comprehend, I will not develop it in great deal.
4) Quantum mechanics. This deals with the corrections that are needed to classical mechanics in the case of atomic-sized and smaller particles. This is more applicable, especially in chemistry, but involves even more advanced mathematics, and I will set aside discussion of this subject as well.
There are some connections to electromagnetism and the structure of matter, and a few connections to thermodynamics, but for the most part, mechanics is considered more fundamental than these other subjects. Chemistry is useful when the particular properties of specific substances are important. Astronomy, earth science, and biology are useful for examples and illustrations. There are comparatively few people who specialize in mechanics as an area of theoretical study. It is somewhat difficult to find internet resources at an intermediate level of study, and this is most easily approached by examining general physics textbooks. It requires a fairly high level of mathematics, and is more closely associated with education than any of the other social institutions. The communities and peoples involved and the history of mechanics are also useful, though it can be difficult to find an introductory account of the history of mechanics.
My interest in the subject is more mathematical and theoretical than experimental, although what resources to suggest depends heavily on how much you already know.
Thursday, May 05, 2005
Chemistry
I have a chronic problem with selecting topics to study and develop. If I have a scheme, it becomes too rigid, but if I don't, I get lost among the possibilities. For now, I'm trying a scheme, but if it isn't satisfactory, I'll try something else.
I don't much like the traditional subdivisions of chemistry. It used to be that they were: General chemistry, analytic chemistry, inorganic chemistry, organic chemistry, and physical chemistry, and were studied in about that order. I've been trying a different approach:
1) Substances. The physical and chemical properties of elements, compounds, and mixtures.
2) Chemical change. Chemical equations and relationships of substances, thermodynamics and energy, rates and mechanisms, and types of change.
3) Chemical systems. One phase, two-phase, and multiphase systems. In this context, a Phase refers to one of the states states of matter: solid, liquid, or gas.
I'm not sure whether this approach really works yet or not; I'm still investigating it.
This depends heavily on physics. Mechanics, especially parts of classical mechanics is often useful. Electricity, magnetism, and optics are also important in chemistry. I've noticed some difference in the chemical approach to thermodynamics and the physical approach. Much of the structure of matter belongs as much to chemistry as to physics. There are various uses for astronomy, earth science, and biology in the study of chemistry. Discussions of the human body and psychology are somewhat useful, and I can sketch out how other areas are related to chemistry by identifying particular chemists. Like other areas of science, chemistry is largely a social endeavor. The chemical literature, mathematics, and measurement; chemistry as an occupation; chemical education; national approaches to chemistry and the history of chemistry are also subjects of interest.
I'm not mentioning any experiments or activities in chemistry: there are others more qualified to do so. What I can do is discuss some of the theoretical aspects of the subject.
I don't much like the traditional subdivisions of chemistry. It used to be that they were: General chemistry, analytic chemistry, inorganic chemistry, organic chemistry, and physical chemistry, and were studied in about that order. I've been trying a different approach:
1) Substances. The physical and chemical properties of elements, compounds, and mixtures.
2) Chemical change. Chemical equations and relationships of substances, thermodynamics and energy, rates and mechanisms, and types of change.
3) Chemical systems. One phase, two-phase, and multiphase systems. In this context, a Phase refers to one of the states states of matter: solid, liquid, or gas.
I'm not sure whether this approach really works yet or not; I'm still investigating it.
This depends heavily on physics. Mechanics, especially parts of classical mechanics is often useful. Electricity, magnetism, and optics are also important in chemistry. I've noticed some difference in the chemical approach to thermodynamics and the physical approach. Much of the structure of matter belongs as much to chemistry as to physics. There are various uses for astronomy, earth science, and biology in the study of chemistry. Discussions of the human body and psychology are somewhat useful, and I can sketch out how other areas are related to chemistry by identifying particular chemists. Like other areas of science, chemistry is largely a social endeavor. The chemical literature, mathematics, and measurement; chemistry as an occupation; chemical education; national approaches to chemistry and the history of chemistry are also subjects of interest.
I'm not mentioning any experiments or activities in chemistry: there are others more qualified to do so. What I can do is discuss some of the theoretical aspects of the subject.
Tuesday, May 03, 2005
Physics
Since I am personally more interested in science than in many other subjects, I'm going to shift to the other end of the subjects from history.
I consider physics to be the study of the laws or regularities of nature in general. As part of my studies in the past, I've devoted some hours to finding a uniform approach to it, and I have several categories:
1) Mechanics. This includes laws of motion, force, and energy. Subtopics include gravitation, relativistic mechanics, and quantum mechanics. The first two are most applicable to other people than scientists.
2) Electromagnetism. This includes electricity, magnetism, and optics, including light. These are less visible, but are regularly employed in our society.
3) Thermodynamics. This includes the study of temperature, heat and related quantities.
4) Structure of matter. This includes the study of subatomic particles, atomic and nuclear physics, molecular physics, and the forms of matter: solids, liquids, and gases, that we usually deal with.
These are fundamenal and more basic than the other sciences, although there is some overlap with chemistry. One of my own projects involves identifying particular prominent physicists, and societies of physicists.
I've recently taken a look at the classification schedules in the Library of Congress to reorganize the subjects of books according to my own preferences, and there is a great deal of popular literature that attempts to discuss very advanced concepts of physics without the use of mathematics. I find this situation very unsatisfactory: I like the mathematics, and there are some areas I would very much like to know about but still lack enough mathematical background. Physics depends heavily on measurement, as well as other areas of applied science. I have something of a distaste for philosophy of physics, although there may be a few useful ideas. One of my pet peeves is the way that physics has become such a specialized occupation and that it requires so much advanced training to even understand the ideas that are being discussed. Physics is generally considered difficult and serious, and there isn't very much that is recreational or fun, although a few creative educators have found some. I don't have any of the apparatus and equipment needed to do research in physics, although some of it is accessible.
I am interested in physics education and improving it, more than in commercial research, and I would prefer that the government not be the primary source of funding. Physics and religion deal with different subjects. To oversimplify, physics concentrates on what can be seen and observed by anyone, while religion deals with what is unseen, at least by most.
Physics is primarily a product of western civilization, but it is hard to get more specific without dealing more with its history. Although it has roots in ancient history, its recognized ancestry comes largely from Greek philosophers in the early classical period, and most of its development since the 16th century.
I consider physics to be the study of the laws or regularities of nature in general. As part of my studies in the past, I've devoted some hours to finding a uniform approach to it, and I have several categories:
1) Mechanics. This includes laws of motion, force, and energy. Subtopics include gravitation, relativistic mechanics, and quantum mechanics. The first two are most applicable to other people than scientists.
2) Electromagnetism. This includes electricity, magnetism, and optics, including light. These are less visible, but are regularly employed in our society.
3) Thermodynamics. This includes the study of temperature, heat and related quantities.
4) Structure of matter. This includes the study of subatomic particles, atomic and nuclear physics, molecular physics, and the forms of matter: solids, liquids, and gases, that we usually deal with.
These are fundamenal and more basic than the other sciences, although there is some overlap with chemistry. One of my own projects involves identifying particular prominent physicists, and societies of physicists.
I've recently taken a look at the classification schedules in the Library of Congress to reorganize the subjects of books according to my own preferences, and there is a great deal of popular literature that attempts to discuss very advanced concepts of physics without the use of mathematics. I find this situation very unsatisfactory: I like the mathematics, and there are some areas I would very much like to know about but still lack enough mathematical background. Physics depends heavily on measurement, as well as other areas of applied science. I have something of a distaste for philosophy of physics, although there may be a few useful ideas. One of my pet peeves is the way that physics has become such a specialized occupation and that it requires so much advanced training to even understand the ideas that are being discussed. Physics is generally considered difficult and serious, and there isn't very much that is recreational or fun, although a few creative educators have found some. I don't have any of the apparatus and equipment needed to do research in physics, although some of it is accessible.
I am interested in physics education and improving it, more than in commercial research, and I would prefer that the government not be the primary source of funding. Physics and religion deal with different subjects. To oversimplify, physics concentrates on what can be seen and observed by anyone, while religion deals with what is unseen, at least by most.
Physics is primarily a product of western civilization, but it is hard to get more specific without dealing more with its history. Although it has roots in ancient history, its recognized ancestry comes largely from Greek philosophers in the early classical period, and most of its development since the 16th century.
Monday, April 25, 2005
Concepts
As I mentioned in my last entry, I'm going to be including a history of some of my own discoveries in logic.
This gives a little more detail as a followup to March's post on Culture. The area of concepts is probably the single most practically important subdivision of knowledge: Nearly all people can converse, but literacy, or the ability to read and write, and its special subdivision numeracy, the ability to understand numbers and to calculate forms the foundation of all other learning.
I have divided this into several areas for convenience in study.
1) Language includes the study of language and linguistics, writing, and specific languages of the world.
2) Literature. I take this in a very broad sense, not only literary works of artistic merit, but nonfiction, scholarly works, and oral tradition.
3) Graphic arts. This includes drawing, painting, photography, and animation. Motion pictures could be grouped here, but I prefer to put them with others of the performing arts in the "Behavior" division of culture.
4) Mathematics. This is usually considered one of the sciences, and is often considered the language of science. However, it has roots in language and writing, so I group it with them.
5) Applied science. This includes bodies of knowledge such measurement and the calendar, accounting, electronics, and engineering. These are largely mathematical.
6) Philosophy. As usually recognized. I have already noted that this has some overlap with religion, but there is a something of a distinction.
Some of these subjects depend more heavily than others on the physical and natural sciences. These don't seem to connect directly with the human body, but psychology is useful, and biographies of linguists, authors, artists, mathematicians, philosophers, and other contributors are very useful. Understanding of their social connections and natural environment may also be useful. Other approaches include the creation of cultural works as an occupation, and the role of games and play. It is also useful to consider the physical books and other artifacts that are associated with cultural works. The role of families, education, economics, government, and religion can only be summarized here. Connections with social changes and movements, particular communities, and the peoples of the world are also important. There is little known with certainty about the prehistory of concepts, but it can be traced with increasing amounts of source material in antiquity, classical and medieval times, and modern history.
Most of these areas of knowledge are major areas in their own right, and I take it for granted that readers of this blog are literate and already have the foundations of this area. More specific suggestions will come with individual areas.
I don't clearly remember when I became interested in logic, but I do remember coming across the fundamentals of symbolic logic in my high-school geometry class. I also remember trying to reconcile algebraic and geometric styles of proof: they were quite different, and I didn't really appreciate how and why. This idea of attempting to reconcile different approaches has a lot to do with my later discoveries.
This gives a little more detail as a followup to March's post on Culture. The area of concepts is probably the single most practically important subdivision of knowledge: Nearly all people can converse, but literacy, or the ability to read and write, and its special subdivision numeracy, the ability to understand numbers and to calculate forms the foundation of all other learning.
I have divided this into several areas for convenience in study.
1) Language includes the study of language and linguistics, writing, and specific languages of the world.
2) Literature. I take this in a very broad sense, not only literary works of artistic merit, but nonfiction, scholarly works, and oral tradition.
3) Graphic arts. This includes drawing, painting, photography, and animation. Motion pictures could be grouped here, but I prefer to put them with others of the performing arts in the "Behavior" division of culture.
4) Mathematics. This is usually considered one of the sciences, and is often considered the language of science. However, it has roots in language and writing, so I group it with them.
5) Applied science. This includes bodies of knowledge such measurement and the calendar, accounting, electronics, and engineering. These are largely mathematical.
6) Philosophy. As usually recognized. I have already noted that this has some overlap with religion, but there is a something of a distinction.
Some of these subjects depend more heavily than others on the physical and natural sciences. These don't seem to connect directly with the human body, but psychology is useful, and biographies of linguists, authors, artists, mathematicians, philosophers, and other contributors are very useful. Understanding of their social connections and natural environment may also be useful. Other approaches include the creation of cultural works as an occupation, and the role of games and play. It is also useful to consider the physical books and other artifacts that are associated with cultural works. The role of families, education, economics, government, and religion can only be summarized here. Connections with social changes and movements, particular communities, and the peoples of the world are also important. There is little known with certainty about the prehistory of concepts, but it can be traced with increasing amounts of source material in antiquity, classical and medieval times, and modern history.
Most of these areas of knowledge are major areas in their own right, and I take it for granted that readers of this blog are literate and already have the foundations of this area. More specific suggestions will come with individual areas.
I don't clearly remember when I became interested in logic, but I do remember coming across the fundamentals of symbolic logic in my high-school geometry class. I also remember trying to reconcile algebraic and geometric styles of proof: they were quite different, and I didn't really appreciate how and why. This idea of attempting to reconcile different approaches has a lot to do with my later discoveries.
Saturday, April 23, 2005
Direction shift
It's been longer than I wanted since I created my last post here, but I've decided to shift directions a little. For one part, I'm going to start pursuing subjects in a little more depth. For another, as an example, I'm going to start describing the history of my development of one of my favorite topics; 3-valued propositional logic.
Saturday, April 16, 2005
Religion
Religion is systematically excluded from discussion in many of the American public schools. This means that the best way to learn about it, especially about religious traditions other than the one one was raised with, involves a process of self-education.
For my own purposes, I have divided this into four branches of study:
1) Particular religious traditions. These include the Abrahamic religions (Christianity, Islam, Judaism); Asiatic religions, Traditional or tribal religions, and Secularism, which includes atheism, agnosticism, secular humanism, and their relatives.
2) Religious organization
3) Religious practice and worship
4) Religious belief
Factual knowledge of nature and its laws tends to support some religious beliefs more than others, but not necessarily a secular point of view. Insight into the human body and psychology can also be applied. Biographies, particularly of religious figures and leaders, can also be infuential. Religious bodies and movements can be examined as social organizations, with demographics, patterns of relationship with nature, and geographic distribution. There are strong connections with areas of culture, including religious literature, and there is some overlap between religion and philosophy. Practices and behavior, and religious artifacts can also be considered. There are connections with families, education, economics, and government. The influence of religion on social structure and change, in different communities, and in different societies is also a useful topic. It is difficult to follow religion from prehistory, given the limitations of the sources, but in antiquity it becomes more possible to follow religion through groups such as the Egyptians and Babylonians. Several of the major religions of the world developed in the classical and medieval period, and can be followed through the modern period to today.
I don't have specific suggestions except to become familiar with your own religious beliefs, or if you claim none, what you do believe.
For my own purposes, I have divided this into four branches of study:
1) Particular religious traditions. These include the Abrahamic religions (Christianity, Islam, Judaism); Asiatic religions, Traditional or tribal religions, and Secularism, which includes atheism, agnosticism, secular humanism, and their relatives.
2) Religious organization
3) Religious practice and worship
4) Religious belief
Factual knowledge of nature and its laws tends to support some religious beliefs more than others, but not necessarily a secular point of view. Insight into the human body and psychology can also be applied. Biographies, particularly of religious figures and leaders, can also be infuential. Religious bodies and movements can be examined as social organizations, with demographics, patterns of relationship with nature, and geographic distribution. There are strong connections with areas of culture, including religious literature, and there is some overlap between religion and philosophy. Practices and behavior, and religious artifacts can also be considered. There are connections with families, education, economics, and government. The influence of religion on social structure and change, in different communities, and in different societies is also a useful topic. It is difficult to follow religion from prehistory, given the limitations of the sources, but in antiquity it becomes more possible to follow religion through groups such as the Egyptians and Babylonians. Several of the major religions of the world developed in the classical and medieval period, and can be followed through the modern period to today.
I don't have specific suggestions except to become familiar with your own religious beliefs, or if you claim none, what you do believe.
Tuesday, April 12, 2005
Social structure and change
One of the difficulties with a self-directed education program is that it's easy to begin with a burst of enthusiasm, but harder to keep it going when there aren't immediate rewards. Then again, there are many other such projects, such as weight loss, housecleaning, and practically anything else that takes longer or costs more than you had anticipated. The costs of a self-education project aren't so much monetary as in the form of time and other opportunities that may need to be sacrificed.
Social structure and change includes a significant part of sociology and anthropology. I have three principal divisions.
1) Social structure. This deals with the group structure of society. It may include, for instance, the distribution of people (urban or clustered, or rural and dispersed). There are social categories such as men, women, children, the elderly, and racial or ethnic groups, and there classes based on economic status. For societies with more than one community, the distribution of communities also belongs to this grouping.
2) Social types. These include categories based on the means of subsistence; for instance, hunting and gathering, horticultural, pastoral, seafaring, agricultural, and industrial societies.
3) Social change. These include things such as generational change, artistic styles, inventions, and the various mechanisms by which innovations are adopted, or fail to be adopted, by a society. I also include changes in society such as migration or settling, industrialization, modernization, and urbanization. Specific social movements, such as temperance, abolitionism, or world commmunism, to pick just a few examples, also belong here.
These are toward the opposite end of the scale of knowledge from the natural sciences, which are not as directly useful as they are in other areas. However, these depend heavily on leaders and prominent figures. These areas depend on and are sometimes hard to separate from social fundamentals such as social psychology, demography, and physical anthropology. Cultural elements such as idealogy and philosophy, customs, and objects are both products and instigators of social change, and the institutions of families, education, economics, government, and religion are all important. History is also useful in exploring the background and consequences of social structure and change.
It's difficult to make a specific assignment in this area, since there are so many possibilities and levels of involvement. Pick your own subject and try to consider it from a social point of view.
Social structure and change includes a significant part of sociology and anthropology. I have three principal divisions.
1) Social structure. This deals with the group structure of society. It may include, for instance, the distribution of people (urban or clustered, or rural and dispersed). There are social categories such as men, women, children, the elderly, and racial or ethnic groups, and there classes based on economic status. For societies with more than one community, the distribution of communities also belongs to this grouping.
2) Social types. These include categories based on the means of subsistence; for instance, hunting and gathering, horticultural, pastoral, seafaring, agricultural, and industrial societies.
3) Social change. These include things such as generational change, artistic styles, inventions, and the various mechanisms by which innovations are adopted, or fail to be adopted, by a society. I also include changes in society such as migration or settling, industrialization, modernization, and urbanization. Specific social movements, such as temperance, abolitionism, or world commmunism, to pick just a few examples, also belong here.
These are toward the opposite end of the scale of knowledge from the natural sciences, which are not as directly useful as they are in other areas. However, these depend heavily on leaders and prominent figures. These areas depend on and are sometimes hard to separate from social fundamentals such as social psychology, demography, and physical anthropology. Cultural elements such as idealogy and philosophy, customs, and objects are both products and instigators of social change, and the institutions of families, education, economics, government, and religion are all important. History is also useful in exploring the background and consequences of social structure and change.
It's difficult to make a specific assignment in this area, since there are so many possibilities and levels of involvement. Pick your own subject and try to consider it from a social point of view.
Thursday, April 07, 2005
Cities and communities.
A self-education program probably isn't going to compete with the NCAA finals or the lastest version of Law and Order or CSI, but I think the world would be a better place if more people took an interest in improving their minds.
There are far too many cities, towns, and communities in the world to make a comprehensive listing, and I have not yet found a source that gives them a significant treatment, although there may be one, or more, buried somewhere in the anthropological literature. The best approach I have been able to find is an indirect one.
Individuals who are connected to a city, by either having lived in one or visited it provide only a sample of what can be learned about it. A city's size or population is only a small step in learning about it, although quite a few facts can be guessed from its size and density of population. Large cities are more complex than small ones, and the complexity goes up faster than the population. The geography of a city site may tell a great deal about it. The buildings, streets, and other structures of a city are generally more enduring than particular individuals, and can be examined using methods of culture. The cultural institutions of a city are usually embodied in such buildings, and the size, prominence, and state of buildings are important indicators of the past or present importance. For example, there are churches, schools, government buildings, business and office buildings, and houses. These may or may not reflect ethnic groups and social classes, and may be easier to understand with history of the city or provide clues to the history of a city. Very few cities have existed throughout history: they tend to grow and decline, although in modern times, as the world's population has grown, more people have become concentrated in cities and there has overall been more growth than decline.
As an exercise, an excellent way to begin a self-education program is to begin with your own neighborhood and city. At some point, I may provide a sample analysis to suggest how the various subjects I have discussed so far can contribute.
There are far too many cities, towns, and communities in the world to make a comprehensive listing, and I have not yet found a source that gives them a significant treatment, although there may be one, or more, buried somewhere in the anthropological literature. The best approach I have been able to find is an indirect one.
Individuals who are connected to a city, by either having lived in one or visited it provide only a sample of what can be learned about it. A city's size or population is only a small step in learning about it, although quite a few facts can be guessed from its size and density of population. Large cities are more complex than small ones, and the complexity goes up faster than the population. The geography of a city site may tell a great deal about it. The buildings, streets, and other structures of a city are generally more enduring than particular individuals, and can be examined using methods of culture. The cultural institutions of a city are usually embodied in such buildings, and the size, prominence, and state of buildings are important indicators of the past or present importance. For example, there are churches, schools, government buildings, business and office buildings, and houses. These may or may not reflect ethnic groups and social classes, and may be easier to understand with history of the city or provide clues to the history of a city. Very few cities have existed throughout history: they tend to grow and decline, although in modern times, as the world's population has grown, more people have become concentrated in cities and there has overall been more growth than decline.
As an exercise, an excellent way to begin a self-education program is to begin with your own neighborhood and city. At some point, I may provide a sample analysis to suggest how the various subjects I have discussed so far can contribute.
Friday, April 01, 2005
Peoples of the world
I intend to go back to history and consider it in more detail, but for now, it's time to move on to study of peoples of the world. I may have mentioned before that this is connected to geography, sociology, and anthropology, but it's not necessary to be an expert in these to study various peoples. In order to study the history of the world, or even follow current events, it's necessary to become familiar with the various peoples of the world.
Peoples can also be called civilizations, nations, tribes, and ethnic groups. The exact definitions are rather fuzzy and I make no effort to be precise about them.
I have divided these into four major areas, largely on the basis of shared history, culture, and geography.
1) Western civilization and peoples. On the basis of shared history, religion, language, and other elements of culture, I include European, Modern North and South American, Russian, and Australia in this group.
2) Asiatic civilization and peoples. This is probably the largest division and is rather a catch-all category, and includes such diverse peoples as those of the Middle East, China, and India. I also include Australian aborigines, non-Russian peoples of central asia, and various others in this grouping.
3) African civilization and peoples. This includes sub-Saharan African peoples.
4) American Indian or Native American civilization and peoples. In modern times, these have been submerged by Western Civilization, but they still exist as distinct and groups within the larger communities.
Earth science and biology are the most useful of the natural sciences in study of peoples of the earth. A few particular areas of the human body and psychology are also useful, but most of these are directed toward particular individuals. However, each of the peoples of the earth has its own cultural heroes and famous or notorious personalities.
Studies of population and its growth and movement, relationships with the environment, race, and particular groups also form part of the foundation for studies of peoples. Although this area of corresponds roughly to human geography, people in widely different places are more similar in culture than some that are nearby.
The distinctive languages and philosophies, customs and arts, and differences in architecture, diet, and other elements of culture are part of the identification of peoples.
Likewise, important families, educational traditions, economic systems, and government are significant. Again, this corresponds only roughly to the nations of the earth, since some peoples have more than one government, and quite a few nations include more than one people.
Each people has its own distinctive structure, there are many different types, and history of the changes it has undergone. It is also composed of communities; usually more than one.
Many peoples that no longer exist can be identified from historical records. Some have been been absorbed or merged into others, or divided into more than one; while others have originated by division or combination of older peoples. These changes give world history much of its content and variety.
For an exercise, I would suggest choosing which of these peoples you most identify with, and describe as many features of it as possible.
Peoples can also be called civilizations, nations, tribes, and ethnic groups. The exact definitions are rather fuzzy and I make no effort to be precise about them.
I have divided these into four major areas, largely on the basis of shared history, culture, and geography.
1) Western civilization and peoples. On the basis of shared history, religion, language, and other elements of culture, I include European, Modern North and South American, Russian, and Australia in this group.
2) Asiatic civilization and peoples. This is probably the largest division and is rather a catch-all category, and includes such diverse peoples as those of the Middle East, China, and India. I also include Australian aborigines, non-Russian peoples of central asia, and various others in this grouping.
3) African civilization and peoples. This includes sub-Saharan African peoples.
4) American Indian or Native American civilization and peoples. In modern times, these have been submerged by Western Civilization, but they still exist as distinct and groups within the larger communities.
Earth science and biology are the most useful of the natural sciences in study of peoples of the earth. A few particular areas of the human body and psychology are also useful, but most of these are directed toward particular individuals. However, each of the peoples of the earth has its own cultural heroes and famous or notorious personalities.
Studies of population and its growth and movement, relationships with the environment, race, and particular groups also form part of the foundation for studies of peoples. Although this area of corresponds roughly to human geography, people in widely different places are more similar in culture than some that are nearby.
The distinctive languages and philosophies, customs and arts, and differences in architecture, diet, and other elements of culture are part of the identification of peoples.
Likewise, important families, educational traditions, economic systems, and government are significant. Again, this corresponds only roughly to the nations of the earth, since some peoples have more than one government, and quite a few nations include more than one people.
Each people has its own distinctive structure, there are many different types, and history of the changes it has undergone. It is also composed of communities; usually more than one.
Many peoples that no longer exist can be identified from historical records. Some have been been absorbed or merged into others, or divided into more than one; while others have originated by division or combination of older peoples. These changes give world history much of its content and variety.
For an exercise, I would suggest choosing which of these peoples you most identify with, and describe as many features of it as possible.
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