Section link: measurement of logic.
As a student, I read that the social sciences were not as advanced as the
natural sciences, largely because of the difficulty of applying measurement.
Simple measurements, such as classification and ordering of data, which are
the first two scales of measurement, characterise a science in its early
stages. More rational measurements, the so-called interval and ratio scales
were supposed to be rare in behavioral studies.
Sometime in the mid nineteen-seventies, I realised that the single transferable vote ( STV ) was the one electoral system that was built on the logical progression of all four measurement scales. ( There is a 1981 version of this theme -- in French -- on this site. Scientific Method of Elections is a later fuller treatment. )
Also in the mid nineteen-seventies, I was considering STV as a model for use in Constitutional Economics. This seemed right for economics as a science, because the structure of measurement is implicit in that voting method. The transfer of surplus and deficit votes made one think of incomes, in these terms.
In reviewer's comments, at the end of my web pages on Julian Barbour's The End Of Time, I mentioned Stevens' four measurement scales in the context of special and general relativity. This had occured to me from my early readings of popular science, perhaps as early as reading J W N Sullivan ( whose review is reproduced on site ).
Not until 1979, did I come across C S Lewis' The Four Loves. At first blush, it seemed too far-fetched to relate the four loves to the four scales of measurement. Now a justification for this measurement of love begins this site.
I wouldnt include that labor of love in the many failed attempts to be 'scientific' that have taken up so much of my time.
I didnt try to do more than show the sort of direction in which a
system of ideas might be based on the undefined concept of love. That is to say regarding the imbalances of the
emotions, with respect to lesser or greater love, up to its most broadly
balanced condition. I didnt want to dogmatise but encourage a new research
science of love.
I dont want to over-state the comparison, but Guy Claxton's book Hare Brain, Tortoise Mind shows how psychologists are testing the nature of the Unconscious mind, theorised by Sigmund Freud and Carl Jung.
Also I would caution against naively playing a numbers game, as seemingly I did by relating four loves to four measurement scales. For instance, there are three primary colors, red, green and blue. One might think these stand for three dimensions of color, whereby one could locate all the other colors as relative mixes. In fact, this idea has no basic significance. It is just a by-product of the way the human eye works. Other creatures have more primary colors.
Years later, I picked up the clue to a possible measurement of language. Languages may be classified according to the parts of speech they more or less possess. I cannot remember where I first read how the English language differs, for instance, from Latin. In English, the parts of speech may be recognised, for grammatical sense, by the order they appear in a sentence.
Latin depends on different endings to a word to signify the part of speech. Esperanto is based on this principle. I remember when well over a hundred, I think, nearer two hundred British MPs favored Esperanto as the world language. This modern man-made language -- man-made in that it was made by one man -- reckons to be an efficent reformed language.
But the grammatical sentence order of English involves an ordinal scale of
measurement. This is a more powerful scale of measurement than
classificaction. You dont have to remember a different word-ending for every
part of speech. There could be still less inflexion in English by just
relying on the word order in a sentence.
For example, all the personal pronouns need not have an object case as well as
a subject case, as the word 'you' does not.
Years later again, I may have something like an answer to how to take the measure of a language with the interval and ratio scales. In fact, I was thinking in terms of these measures for language reform, without realising they were measures.
Ive long noticed the tendency for language to lose its fluency, with
redundant usages, much as ships lose their stream-lines with barnacles.
American immigrants have rubbed off the rough edges of English, when it wasnt
their first language.
For example, 'We done good,' replaces 'We did well.' Americans tend not to
modify adjectives, when used as adverbs. American ( and British ) English
might use the adjective 'quick' as an adverb, instead of 'quickly'. To some
extent this happens in traditional English, speaking of whisky he drank
neat.
But there is also a contrary trend of redundant speech, largely American, that loves important-sounding catch phrases.
Americans and many British imitators dont plan, they have 'forward planning'. They dont report, they 'report back'. To plan is by definition forward looking. The prefix 're-' means 'back'. 'Check' is sufficient for 'check it out,' and 'listen' for 'listen up.'
Other examples, not necessarily American, include 'co-conspirators' for conspirators, 'sympathise with' for sympathise, 'co-operate with' for co-operate. Prefixes tend to lose the force of their qualificiation and so are liable to 'barnacle' words. Instance 'entrap' for trap, 'entice' for tice.
Ogden and Richards, the promotors of Basic English, found that most English verbs could be paraphrased by a core of 18 common verbs, without prefixes, followed by modifying words, mainly 20 space or time directives. They hoped the few hundred words of Basic English would be simple enough for everyone in the world to make themselves understood. Their attempt to make English more analytic contrasts with the German practise of running words together. Many web addresses follow a similar practise.
The Basic English approach has the advantage that words are not barnacled with prefixes, merely because that form has become customary. Wherever joined words are used, out of habit, one of the joined words may not be doing any meaningful work towards the message being conveyed.
Nearly all the Basic English verbs are irregular in the past tense. On my ESP alfabet page, I also suggested a past tense convention to make all English verbs regular in the past tense, as they are in the future tense.
Poets are concerned to pare off superfluous words. This works against the wear and tear to the language, as an instrument of communication, from those naturally more concerned with the communication than the instrument.
Redundant usage in language may be compared to redundant votes, those over what a candidate needed to win an election. The single transferable vote ( STV ) transfers these surplus votes to next prefered candidates. The third scale of measurement, the so-called interval scale, rations out the surplus vote between the proportions of all the elected candidate's voters for next preferences.
By analogy, joined words can have their surplus meaning transfered to a separate word. Or they could be eliminated if they have no further meaning to convey. A candidate is eliminated for want of further preferences.
Words in a sentence may not make uniform intervals, in that some words will
group together meaningfully. When the surplus meaning of a joined word is
rationed out to a separate word, the two words still form a meaningful unit
within the sentence, like 'co-operate' becoming 'work with,' 'excise' becoming 'cut out' or 'cut-out'.
As well as such spatial direction to verbs, auxiliary verbs, such as 'will' or
'would' or 'must' or 'can', direct the verb in time.
The fourth scale of measurement, the ratio scale operates in STV elections
to determine the proportion or quota of votes each candidate needs to win a
seat in a multi-member constituency.
Analgously, the sentence may be considered as a multi-member constituency of
words, to which meaning is rationed out. Each word is 'elected' as an equal
representative of a sentence's meaning.
Basicly, the parts of speech would refer to their ordered position in the sentence. Words would not have special status as nouns and adjectives, verbs and adverbs. Nor would words have special forms for each part of speech. The same word for all those four parts of speech would only be recognised as one or other part by its order of appearance in a sentence. It's not unusual in English for nouns to become used as verbs or adjectives or adverbs. For example, the noun, man, can become: to man, man-hole, man-made, etc.
Words have the plural form of adding an s to the singular form. But that is a distinction of observation, rather than grammar. This is complicated, in English, by also adding s to denote a possessive form of a noun. And it has to be 's or s' to denote singular or plural possessive, as in: The boy's sister ( or the boys' sister, if several boys have the same sister ).
The possessive form is short for: The boy has a sister ( or: The boys have a sister ). I remember a primary school teacher complaining to we children about over-using the verb 'to have.' So, the possessive form depends on changing the ending of words, as well as position in the sentence but it can be replaced by a sentence where the words are only order-dependent.
Another compromise, of the word-order sentence, is the use of connecting words beginning phrases or clauses, which are like twigs and branches on the stem of a sentence. A phrase is more than one word, without a verb, and a clause is several words, with a verb, that do not stand on their own as a sentence but can be added to one: "like twigs and branches" is a phrase; "which are on the stem of a sentence" is a clause.
A sentence need not have just one subject and one object or predicate ( what is said about the subject ). Tho, there is much to be said for simple sentences. These are the fashion, unlike the convoluted styles of nineteenth century novelists, such as Henry James.
Take an 'ecological' sentence, like the theme to the nursery rhyme, The House That Jack Built. An extremely high correlation was found between the number of spinsters living in an area and the abundance of its clover. This was because: Spinsters kept cats that ate rodents that reduced bees that pollinated clover that populated fields that surrounded houses that sheltered spinsters that...
The point of this circular sentence is that it breaks down the rigid division ( in a sentence as in life ) between subject and object. And in doing so, it replaces the need for clauses with their connecting words. The seven-times said word 'that' could be left out of the circular sentence about the spinster.
In conclusion, the grammar of language promises identification with measurement structure.
Watching 'black and white' films, as a child, I thought they were mainly grey. A fuzzy neutral color was more prevalent than sharply distinct black and white. This is true of reality in general, which does not consist of sharply defined categories. The forms of things tend to merge into each other. In taking account of the grey areas of experience, 'fuzzy logic' is more accurate than classical logic, not less.
Classical logic deals in extreme contrasts, which tends to make for polarised thinking and dogmatic opposition. Fuzzy logic allows for extremes approaching each other by degrees and encourages compromise, where applicable.
Author of Fuzzy Logic, Bart Kosko comments:
Fights break out when some person or some group or some government tries to round us off their way, tries to make us all A or all not-A...In this sense voting just asks for trouble.
This attitude was typified by premier Margaret Thatcher. When asked about some-one's politics, she retorted: He's not one of us. Her 'resolute approach' brought about her own down-fall, when it became nothing but doctrinaire rigidity against over-whelming opposition to her poll tax. Mrs Thatcher was only one of the most obvious cases of the orthodoxy of leaders.
Kosko's quoted criticism applies to the all-or-nothing X-vote. Preference voting admits of the saying he coins: everything is a matter of degree.
As a simple example, Kosko grades apples by the fraction they are red or green. ( Grading is on the ordinal scale and the fractions, as there used, are on the ratio scale of measurement. ) In this respect, he is applying more powerful measurement scales, in fuzzy logic, than classical logic, which just reasons in terms of the basic measurement scale of classification.
Critics have claimed probability theory would do the job just as well as fuzzy logic. Indeed, Kosko used it in the form of weighted decision-making ( which is on the interval scale of measurement ) to show how people make choices, when several more or less important considerations are involved.
Kosko has the intriguing idea that the part contains the whole: A sub-set
vanishing to nothing contains zero per cent of the whole, increasing in size
to the whole, contains all of the whole. In between, the sub-set contains part
of the whole, more or less.
This view is used to explain the paradoxes of self-reference in classical
logic, such as the paradox of the liar or Russell's paradox of classes. The
fuzzy view is that they are half-truths, due to the imprecision of
all-or-nothing logic.
Classical logic is based on a sharp distinction between statements as either true or false. There is no room for half-truths. Whereas Kosko was always asking his audiences to lift their hands up by just the amount they thought a given statement true.
We could make T or F stand, respectively, for when it is true or false that
certain candidates are a voter's choice. For a given number of candidates, k,
there are a certain number of logical possibilities of voters' choice. This
can be expressed in a truth table of all the logical possibilities, as shown
in texts of logic or finite mathematics.
After explaining how the binomial and multinomial theorems can give the
layouts of truth tables, it is indicated below how they are used to determine
correct reasoning.
The truth table shows all possible combinations of choices ( T ) and non-choices ( F ) between the different candidates. The table has a column for each candidate, with a row for each possibility, starting with all Ts and ending with all Fs, respectively meaning all are choices and none are choices of candidate by the voters.
For the two options, true and false choice, and three candidates, there are two, to the power of three, which equals eight, possible combinations of choice. Two, to the power of the number of candidates, expresses the sums of possibilities given by the binomial distribution. One could express the binomial theorem in terms of ( T + F )'k where the notation means the bracketed value is multiplied by itself k times, or, to the power of the number of candidates, k.
Expanding the binomial theorem in the normal way, in coefficients and powers of both T and F, gives an algebraic representation of the truth table of choices for a given number of candidates. The powers of T and F show how many choices or non-choices, respectively, there are in a given row of the truth table.
For example, row one of a truth table for three candidates, is symbolised by 1T³. This means one logical possibility that all three candidates are a true choice of the voters.
The coefficents of the binomial expansion give how many of the rows of logical possibilities have a given number of true or of false choices for the three candidates. Thus 3T²F means there are three rows in the truth table, with two true choices for any of the three candidates.
Thus, the classical logic options of true or false can be used in a binomial expansion of their logical possibilities. Fuzzy logic options admit of at least three logical options, crudely speaking, true, half-true and false. Consider the example on my web page about the diffusion equation. This could solve as a binomial distribution of constituencies according to the logicly possible combinations of rural or town districts.
To put it another way, some districts could be called truly rural. Whereas it would be false to say the town districts were rural. Fuzzy logic admits of other categories, more or less rural and town. There might be three categories, rural, suburban and town, or four categories, rural, suburban, town, and urban ( r, s, t, u, ) and so on.
Instead of the two categories in a binomial distribution, three categories imply a trinomial distribution. Multiple categories imply a multinomial distribution from expanding the multinomial theorem. If you like, this is a way of showing the vastly increased number of logical possibilities afforded by 'fuzzy logic'.
Instead of two added terms, the multinomial theorem has many added terms ( r, s, t, u, v, w...) to self-multiply their bracketed form, any number of times ( say n times ).
The expansion of ( r + s + t + +u + v + w )'n is found by adding all terms of the form:
r'a.s'b.t'c.u'd.v'e.w'f.{ n! / ( a! b! c! d! e! f! )}.
This form means r, to the power of a, times s, to the power of b, and so on, multiplied by the curly-bracketed term. The latter is the factorial number of n, or a given whole number multiplied by all its lesser whole numbers, which is divided by the factorials, of a, b, c, etc, multiplied by each other. The whole numbers a, b, c, etc, whatever they are, must add up to the number n.
The simplest example is the trinomial theorem: ( r + s + t )'n where n can be any whole number, say three. Hence:
r'a.s'b.t'c.{ 3! / ( a! b! c! ) }.
Then a, b and c are all the possible partitions of the number n. If n equals three, one partition is: a = 3, b = c = 0. Bearing in mind that numbers to the power of zero equal one and that factorials of zero equal one, this gives:
r³.{ 3! / 3! } = 1r³.
The coefficient, one, says that there is just one case of a cubic power of
r. The cube of r means that r combines only with itself, in this one case.
When b = 3, that is the one case when s combines only with itself. When c = 3,
then t combines only with itself.
In terms of our electoral example, these three cases mean one constituency
made up all of rural districts, another constituency made up all of suburban
districts and a third constituency all of town districts.
Another possible partition of n by a, b and c, is three split into two, one and zero. Note that either r, s or t can be two out of three districts in a constituency. That means three options or possibilities. But also when, say, there are two r-districts, either s can be the one other district with zero t-districts, or vice versa. Therefore the three options each have two options of their own, making, in all, three times two equals six options.
For the possibility of two r-districts, one s-district and zero t-districts, the formula is:
r².s'1.t'0.{ 3! /( 2! 1! 0! ) } = 3s.r²
Besides the chance that r can be two out of three districts in a constituency, the coefficient three in 3sr² says that there are three ways that r can be two out of the three districts. Three objects can pair in three different ways.
( This becomes apparent if one has to do the expansion without the formula. If the terms in the brackets are cubed, the exhaustive -- and exhausting -- calculation writes the bracketed terms down three times. The first term r in the first brackets is multiplied by r in the second set of brackets, then by s in the third brackets. First-brackets' r also multiplies by r in the third brackets and s in the second brackets. Thirdly, r, in the second brackets, multiplies by r, in the third brackets, and s in the first brackets. All this, just to get the term 3sr². )
This applies to the other five options, 3tr², 3rs², 3ts², 3rt², 3st². In all, there are three times six equals eighteen possibilities from the split of three into two and one.
There is one more split or partition, of n = 3, namely three ones, whereby a = b = c = 1.
The formula gives: r'.s'.t'.{ 3! /( 1! 1! 1! ) } = 3!r.s.t = 6rst.
This gives the six permutations of r, s and t. For instance, there are six possible ways that a constituency could have one each of three districts to be rural, suburban and town.
Add these six possibilities to the previous eighteen possibilities, as well as the three constituencies each all rural, all suburban or all town, and there are a total of 27 logical possibilities. This checks, taking one r plus one s plus one t, or three, to the power of three, equals 27.
Thus, the trinomial theorem, say, for three candidates gives a three-columned truth table with 27 rows of different possible ways of deciding whether to make the candidates true, half-true or false choices. In other words, there are 27 ways for the voters to make the candidates first, second or third preferences. The third preference does not help elect a candidate, because there is no fourth candidate to express a preference over. The third preference is, thus, no preference or a false choice.
For the case of only two candidates, who are simply true or false choices, there are four possibilities. Voters prefer both candidates or neither, or one or the other. But not all of these four cases constitute a true election. To 'elect' means to choose out. Some candidates must get excluded in favor of others. Therefore, if both candidates are chosen or neither are chosen, then no election has taken place. These two non-exclusive possibilities are false statements of an election.
True elections are the other two possibilities when one of the two candidates, but not the other, is chosen. These are two true statements of an election. This is an example of how truth tables decide false or true reasoning, when statements are linked by certain logical connectives.
The logic of an election uses a connective that goes by the text book jargon of 'exclusive disjunction'. There are a few other such connectives: conjunction ( and ), inclusive disjunction ( either...or, including both ), negation ( not ), conditional ( if...then ), biconditional ( if and only if...then ) found in texts on symbolic logic or finite mathematics.
Preference voting is of more than the X-vote's single preference. An order
of more or less true choice, in ranked choice or preference voting can be set
out in truth tables, of many candidates, expanded from the multinomial
theorem. As the number of candidates increases, the number of preference
permutations becomes astronomical.
Indeed, preferential elections generally
exercise maximum exclusiveness, with the condition that voters may express no
equal choices for two or more candidates, whether equally liked or equally
disliked.
For the case of three preferences for three candidates, maximum exclusiveness reduces the possibilities from 27 to 6. The latter are the six permutations of preference or ranked choice. An increase in candidates soon renders the number of straight permutations of preference more than astronomical enough.
There is the qualification that voters may not be obliged to state more preferences than they wish. With the honorable exception of Tasmania, Australian government compelled voters to rank a multitude of candidates or just X-vote for a party list that the patronising bosses have ranked for you.
Richard Lung
10 september 2003.
Sources:
Sidney Siegel, Nonparametric Statistics for the behavioral sciences.
Kemeny, Snell and Thompson, Introduction to finite mathematics.
Bart Kosko, Fuzzy Logic.