The STORY of LOGIC; AND THEN SOME !
The STORY of LOGIC; AND THEN SOME !
Zak Van Straaten
knowdule # 0
Logic: what’s in it for me ?
Logic: what’s in it for me ?
What’s in it for me ?
You will have a fun time going on a logic walkabout.
You will understand why others are not as logical as you are.
You will improve your logic skills.
After reading this book your thinking will be clearer and your reasoning skills will be honed.
You will be able to evaluate the logical thinking of your friends and competitors.
And if you are lucky your company's chairman may notice you and you will be selected to go on the fast track, accelerated management programme.
But even if that doesn’t happen any time soon, you will have started at the very beginning, which is of course a very good place to start. That’s the same place where logic super-heroes in training start.
Shall we start?
A friend of yours produces the following argument:
Nothing is better than a 100 years of bliss in Provence, and A baked New York cheese cake is better than nothing, so a baked New York cheese cake is better than a 100 years of bliss in Provence.
Lets assume that the dictate of reason is in your logic module which resides inside your brain.
What does the dictate of reason have to say about the argument ?
Does the conclusion follow logically from the premises?
Well, if by “nothing” you literally mean “the absence of things” then, yes. But who would assert the truth of, “the absence of things is better than a 100 years of bliss in Provence” ?
However, if "Nothing is better than a 100 years of bliss in Provence" means "a 100 years of bliss in Provence is better than everything" your logic module will not allow you to prefer a baked New York cheese cake.
If it did, your dictate of reason would come up with a counter-example and maybe a gentle reprimand. And the dictates of reason speaking from inside of some of your friends would also object by lodging numerous counter examples.
The following scenario is not a matter of logic; so your logic module will have nothing to say about it, except of course to say that it’s not a matter of logic.
Your wealthy French uncle has set up a foundation with the express purpose of guaranteeing you 100 years of the good life in the Provence village of your choice. All is paid for in advance, your accommodation, your wine cellar stocked with the best that France, Spain, Italy, South Africa, Australia, California, Chile can offer, and a brace of French chefs to see to your needs in the food department.
Your fiction module might say; Pretty neat idea, but alas very improbable.
_____________________________________________________________________________________________
knowdule # 2
Either it’s logic, or else it isn’t:
but what is it ?
knowdule # 2
Either it’s logic, or else it isn’t:
but what is it ?
It is a modern discipline. Modern computers or mobile phones or cars or airplanes or DVD players or PDA’s or iPods or Blackberrys or cruise missiles or space shuttles...couldn’t work without it. Logic is abstract reasoning that becomes very solidly silicon in logic circuits.
Logic, according to Richard Jeffrey (a Princeton logician and philosopher) is the science of deduction. It aims to provide systematic means for telling whether given conclusions do or do not follow from given premises
Logic is the purest part of Plato’s ideal world,-- abstract (like numbers & arithmetic), but also, for some super-heroes like Godel, more real than tables and chairs.
Logic is the spinal chord in software programs. Without it programs cannot be consistent and programs won’t work..
Logic, according to Wilfrid Hodges in his book LOGIC, can be defined as the study of consistent sets of beliefs.
Logic is also an ancient discipline. It goes back at least to 300 years before Julius Caesar crossed the Rubicon at the start of his war against Pompey and the Senate. A geometer called Hippocrates (not the same person as the celebrated doctor of Hippocratic oath fame) wrote four scrolls which later became the source for the first scrolls of Euclid’s Elements. The Hippocratic logical method was: make all your terms precise, definite and explicit so that all words and symbols are completely understood without any ambiguity. Then you make all your axioms or assumptions precise, definite and explicit. With all that in place you derive all the logical consequences of the system using only the accepted rules of inference applied to the axioms and any previously proved results or theorems. (see Leonard Mlodinow: Euclid’s window)
Logic is according to the three authors of Introducing Logic the backbone of Western civilization, holding together its systems of philosophy, science and law.
Logic, according to Graham Priest (a relevance logician and philosopher) in Logic: A Very Short Introduction, is the study of what counts as a good reason for what and why.
This logic of the philosophers, where the main interest is the study of logic applied to ordinary language, is better known as informal logic
Most logic books don’t bother to try to define logic. They just start doing it, and then let the reader work out for herself what sort of animal logic is or isn’t.
According to A.N. Prior, who was a Balliol logician, in his book Formal Logic, “the best way to discover what logic is about is simply by doing logic. I shall therefore dispense with a full introductory discussion of the nature of the subject, but will say just enough about it now to lead us rapidly to the particular topic within the subject with which our doing logic will begin.“ He goes on to say in a first stab at the task ;--- logic is about the systematic discrimination of good arguments from bad.
So far we have seen that logic seems to be many apparently different things. So we may ask: Yes, but, what is logic really ?
Of course, if you are patient (you pass the baked cheesecake test; ---you can resist eating all your cheesecake in one sitting; you are able to save some for later!) this book will tell you. In the meantime here is a kind of provisional answer.
It seems to depend on whom you ask! So we put the question to different characters.
Richard Feynman, one of the most accomplished physics gurus of the 20th century: “Scientific knowledge is a body of statements of varying degrees of certainty- some most unsure, some nearly sure, none absolutely sure.”
Nevertheless, science is reliable knowledge;--its based on the experimental method. Logic is the support for this reliable knowledge & reasonableness.
Joseph Mazur, a mathematician and author of EUCLID in the RAINFOREST; Discovering Universal Truth in Logic and Math: there are three kinds of logic:
1. the logic inside math: proof & disproof
2. the logic of infinity provided by Cantorian set theory: weird logic
3. the logic of science: plausible reasoning & rational criticism
an engineer: its circuits inside mobile phones, computers, cars, airplanes etc.
a computer scientist: it’s a chain of correct reasoning which runs through our programs and its also what makes our micro processors work.
a philosopher: its what makes good arguments good and lack of it makes bad arguments bad
a logic puzzle composer: it is what guarantees that there is only one correct answer in the puzzles I compose.
a maths guru: it’s the spinal chord in our proofs. Without it the proofs wouldn’t be proofs
a set theorist: its what we use to structure very large sets so that we can understand infinitely large numbers
Sherlock Holmes: Crime is common. Logic is rare. (in The Adventure of the Copper Beeches)
Logic, according to the well known HANDBOOK of Mathematical Logic edited by Jon Barwise is the study of proof theory, set theory, model theory & recursion theory.
Confusion reigns. So many definitions and they seem to be saying so any disparate things about it.
What kind of animal is logic? This you might say is very much like the six blind men of Indostan “to learning much inclined” describing an elephant from their own perspectives. Do you know the charming poem?
The Blind Men and the Elephant
by John Godfrey Saxe
American poet John Godfrey Saxe (1816-1887) based the following poem on a fable which was told in India many years ago.
It was six men of Indostan
To learning much inclined,
Who went to see the Elephant
(Though all of them were blind),
That each by observation
Might satisfy his mind
The First approached the Elephant,
And happening to fall
Against his broad and sturdy side,
At once began to bawl:
“God bless me! but the Elephant
Is very like a wall!”
The Second, feeling of the tusk,
Cried, “Ho! what have we here
So very round and smooth and sharp?
To me ’tis mighty clear
This wonder of an Elephant
Is very like a spear!”
The Third approached the animal,
And happening to take
The squirming trunk within his hands,
Thus boldly up and spake:
“I see,” quoth he, “the Elephant
Is very like a snake!”
The Fourth reached out an eager hand,
And felt about the knee.
“What most this wondrous beast is like
Is mighty plain,” quoth he;
“ ‘Tis clear enough the Elephant
Is very like a tree!”
The Fifth, who chanced to touch the ear,
Said: “E’en the blindest man
Can tell what this resembles most;
Deny the fact who can
This marvel of an Elephant
Is very like a fan!”
The Sixth no sooner had begun
About the beast to grope,
Than, seizing on the swinging tail
That fell within his scope,
“I see,” quoth he, “the Elephant
Is very like a rope!”
And so these men of Indostan
Disputed loud and long,
Each in his own opinion
Exceeding stiff and strong,
Though each was partly in the right,
And all were in the wrong!
Moral:
So oft in (theo) logic wars,
The disputants, I ween,
Rail on in utter ignorance
Of what each other mean,
And prate about an Elephant
Not one of them has seen!
Lets make a quick sketch of some of the subject matter of logic.
The engineers say that logic is mostly about embedding logic circuits in silicon or some other appropriate material.
The philosophers say it is about studying consistent sets of beliefs. It is the study of deduction and its aim to provide systematic means for telling whether given conclusions do or do not follow from given premises.
The mathematicians say it is the study of proof theory, set theory, model theory & recursion theory. Logic, according to Kleene in his book Mathematical Logic, is logic treated by mathematical methods. But mathematical logic has a double meaning, since it is also the study of the logic used in mathematics.
Logic, according to Bas van Fraasen in Formal Semantics and Logic, is both logic proper and meta-logic. We distinguish these subjects by their aims: the aim of logic proper is to develop methods for the logical appraisal of reasoning, and the aim of meta-logic is to develop methods for the appraisal of logical methods.
Logic, according to Jon Barwise, pays attention to the language used in mathematics. Maths is a science of abstract objects, e.g real numbers, functions, surfaces, or algebraic structures etc. Logic studies the ways abstract objects are defined, and to the laws of logic which govern us as we reason about these objects.
Logic according to many mathematicians is mostly all about logic chains: Logic chains are sequences of axioms, postulates, sentences and terms that are finely structured in their relation to each other. Some ideas in the sequence have to come before others and so some have to follow from others. Most of them have a precise place in the sequence.
Are the philosophers and mathematicians talking about the same subject matter? Apparently they are. Logic, according to Quine (a logician and one of the most distinguished philosophers of the 20thC) in Mathematical Logic, differs from the traditional formal logic so markedly in method, and so far surpasses it in power and subtlety, as to be generally and not unjustifiably regarded as a new science...The striking differences between the two must not be allowed to obscure the fact that they are both “logic” in the strictest sense of the word. They both have, vaguely speaking, the same subject matter”
Now that we have some idea of what its all about we can proceed.
___________________________________________________________________________
knowdule # 4
Does Logic make the world go round ?
Does Logic Matter ? Is Logic useful ?
knowdule # 4
Does Logic make the world go round ?
Does Logic Matter ? Is Logic useful ?
Does logic matter?
Logicians have usually argued that logic matters because it is useful. We use it to reason correctly and to avoid making mistakes in reasoning. We use logic to win arguments. We use it to cut out confusion. We use logic to nail the jelly to the tree, as professor Richard Bottom’s website so colourfully puts it. (www.csci.csub.edu/dick/maths/intro_logic.html) And although you cannot learn to be create you can learn logic.
Logic underpins a great deal of human activity. Sometimes the underpinning is literal. Our arguments (which use words like AND, OR, IF – THEN, NOT, THEREFORE) would literally fail without the use of these logical words. And our computers and digital devices like personal digital assistants and mobile phone handsets would also fail without logic circuits.
The Intel corporation makes chips that are the ‘brains’ of desktop and laptop computers. They have a clever marketing campaign to make us aware of these ‘brains’ inside digital devices. The sticker on my laptop says “intel inside” and ‘Pentium 4’. If we had to copy this strategy, not for Intel products, but for logic circuits we would have stickers made saying “logic circuits inside”. Where would we stick them so as to accurately reflect the prevalence of logic circuits?
Here are some places:
* desktop computers, laptops, personal digital assistants, Blackberrys, calculators & any software program
* car ignition systems, cars service monitoring systems, car cruise control systems, car digital display systems, etc.
* commercial planes
* industrial robots and A.I. robots
* rockets and space shuttles
* predator and aurora pilotless drone intelligence gathering military aircraft, smart weapons, cruise missiles
* chocolate and soft drink dispensing machines
* mobile telephone handsets and satellite phones
* quartz watches and clocks
* DVD iPod and CD players; personal portable music systems (MP3, CD, mini-disk or electronic tape)
* Game playstations and digital travel chess sets
* G.P.S. systems
* digital cameras
* smart dishwashers, smart washing machines, smart ovens, automatic bread makers, automatic rice makers, thermostats and household electricity switch systems etc.
Can you think of any others?
You would also have to stick one on your forehead!
To verify that there are logic circuits in your brain or nervous system you would to turn to the multidisciplinary field of cognitive science. In it you will find researchers who are neuroscientists, cognitive psychologists, neuro-anatomists, philosophers and artificial intelligence (A.I.) researchers to name only a few.
The present state of research knowledge in cognitive science and related subjects like A.I., neurobiology, brain science, and cognitive linguistics is not such that we know exactly what a logic circuit inside the human head looks like and how it works. But some of the cognitive scientists are convinced that whenever one uses language effectively that there must be ‘logic circuits’ at work or things which do the same job which logic circuits do. Whether these logic circuits in our heads function more or less like the logic circuits we have designed and put inside our electronic devices is an issue which cognitive science is addressing, and on which there will someday be a scientifically testable view.
We looked above at instances above where the logical underpinning is literal. Sometimes the underpinning is metaphorical. In talking about films, music, architecture and business people like to use ‘logic’. A music journalist in assessing Astor Piazzola’s tangled but logical counterpoints refers to his tangled but logical soul. (The Guardian Weekly, April 1-7, 2004)
Music lovers say things like, ‘I can’t always tell where the logic of Damien Rice’s music is leading, but I love it.’ Critics of business plans say, ‘Where is this going, I can’t follow your logic?’ when they mean ‘I don’t understand the order or rationale behind your thinking. Or maybe they want to be rude or dismissive, since if they can convince the audience that your business plans have no logic to them, that by itself would be good grounds for dismissing them out of hand. Sensitive critics wearing fashionable berets and wielding glasses of chic chardonnay listen to one of their colleagues talk of the logical flows of the sculpture they are looking at. And they all nod appreciatively. In the metaphorical sense the word ‘logic’ is used to mean the opposite of chaos or lack of order.
Logical ability matters if you wish to be an undergraduate in a leading research university like Cambridge. In 2002 9,288 candidates applied for 3,115 places. More than 7,144 had three or more A’s in A level school subjects. To get at the real intellectual performers 17 of the 24 colleges who admit undergraduates decided to set a new entrance test which the university described as;
“a combination of logic and problem solving, with an overlay of judgment and understanding.”
In other words, when the best scholars, with the best results, from the best schools apply and you wish to distinguish between them, you submit them to a logic test. The test is being applied to candidates who apply to read computer science, natural science, engineering and economics in the first instance, and it is likely to be applied to candidates in the humanities in the future. (The Times p.3 24/09/03)
Would you like to see one of the problems?
problem #1:
Vegetarian food can be healthier than a traditional diet. Research has shown that vegetarians are less likely to suffer from heart disease and obesity than meat eaters. Concern has been expressed that vegetarians do not get enough protein in their diet but it has been demonstrated that, by selecting foods carefully, vegetarians are able to amply meet their needs in this respect.
Which of the following best expresses the main conclusion of the above argument?
A. A vegetarian diet can be better for health than a traditional diet.
B. Adequate protein is available from a vegetarian diet.
C. A traditional diet is very high in protein.
D. A balanced diet is more important for health than any particular food,
E. Vegetarians are unlikely to suffer from heart disease and obesity.
________________________________________________________________________________________________________
knowdule # 6 Logic & Biology
How your biology MAY negatively affect the performance of your your logic module OR How the abstractness of the subject matter may negatively affect the performance of your logic module
How logical are human beings ? In some circumstances we are very logical, but in others our performance is rather poor.
Can you do this logic problem ? It consists of four cards placed on the table. You are told that each card has a letter on one side and a number on the other. The cards read:
Your task is to turn over only those cards which you need so as to be able to prove the following rule to be true: If a card has a D on one side, then it has a 3 on the other.
Take 3 minutes to think about it before you decide. Remember that you need to have a good logical reason for your choices.
Which card or cards did you choose to turn over ? Why ?
Now try this problem:
You are a bouncer in a Boston bar and you will lose your job unless you enforce the following law: If a person is drinking beer then he must be over 20 years old.
You are presented with 4 cards, and told that each card tells you what the person is drinking on one side and gives the person’s age on the other side. The cards read:
Your task is to turn over only those cards which you need so as to be able to prove that; If a person is drinking beer then s/he must be over 20 years old, is true.
Which card or cards did you choose to turn over ? Why ?
The answer to the first logical selection test is D and 7. When presented with this problem more than 75% of first year Stanford University students got it wrong. And we know that in order to get into Stanford you have to have excellent grades and be
academically very competitive.
When presented with the second problem 75% of them got it right. The cards you need to turn over are DRINKING BEER and
16 YEARS OLD.
So why do less than 25% get the first one right; and 75% correctly solve the second one?
According to Leda Cosmides, an evolutionary biologist who was at Stanford University, and her colleague John Tooby; and Gerd Gigerenzer and colleagues at Salzburg university humans find the second problem easy to solve because it involves checking on or enforcing a social contract. In other words it’s a case of cheater detection. When the logic involved has nothing to do with enforcing a social contract, as in the first puzzle, we humans find it difficult. (see Barkow, Cosmides & Tooby: The Adapted Mind; Evolutionary Psychology and the Generation of Culture. Oxford 1992, Oxford Univ. Press.)
The logic test was devised in 1969 by Peter Wason. And since then hundreds of Wason tests, with these cards and variations on these cards, have been administered to tens of thousands of subjects. More than 80,000 students have taken the tests in the USA and Germany.
Of course putting forward the hypothesis that humans are generally not good at purely abstract logic but very good at logic when it involves computing social advantage or disadvantage is not the same as proving the hypothesis true. But it is a very testable hypothesis, and despite various scientific attempts to try to refute it, the hypothesis has so far survived.
One of the tests involves the rule: “If you take a pension then you must have worked here ten years”. Some subjects are told that they are to imagine themselves to be employers and given the following cards:
As in the first two tests your task is again to turn over only those cards which you need so as to be able to prove the rule to be true.
The majority of “employer” subjects choose to turn over
Other subjects are told they are “employees”, and given the same rule and cards. The majority of “employee” subjects go on to choose
Suppose that the Leda Cosmides, John Tooby; and Gerd Gigerenzer hypothesis is true; the hypothesis that we find certain logic problems easy to solve because it involves checking or enforcing a social contract.
It doesn’t prove that we can’t do abstract logic (or maths) or that some of us cannot learn to reason logically about subjects other than social contracts. It only shows that for most humans its easier to reason logically about the social concerns that our evolutionary biology has wired us to reason about.
The hypothesis is also well embedded in a research programme which in a nutshell goes like this. The mind is not a unitary thing whose intelligence or development can be measured by a single IQ test. The typical mind is supposed to have “ a multitude of intelligences” (Howard Gardner: Frames of Mind, 1983) such as linguistic, mathematical, spatial, musical, inter-personal social relations etc. In Daniel Dennett’s vivid image the ‘mind’ is a kind of “Swiss-army knife” built up over millions of years to have special purpose gadgets to get things done quickly (Darwin’s Dangerous Idea p.490). So it is very good at doing certain things, such as threat detection because these skills have been selected for over millions of years of evolution. The Swiss army knife view, and the modular view of the ‘mind’ mesh very nicely with what is called the computational theory of ‘mind’, in which the ‘mind’ is just a name for all the software the brain runs in a particular human. The suggestion is that in the Swiss-army knife view, some of the software, like threat detection and being able to react quickly to any threat, has become hardwired, whereas other installed software such as the ability to write an intricate program to evaluate logical arguments in the JAVA or LISP language is not.
And when socio-biologists try to hunt down what some of these Swiss Army modules might be, they hunt for so called QWERTY phenomena. The QWERTY keyboard is a design dinosaur from the early days of the typewriter. Frequently used keys like ‘a’ and ‘i’ were separated by large distances because if you pressed one key and the key next to it in quick succession the keys used to stick. So the appropriate design solution at that time was to separate frequently used keys to speed up typing as much as possible. But its not relevant today, and these dinosaur keyboards now slow us down. Similarly the hunt is on for behaviours in us which might have been appropriate when the early humans separated out from the chimpanzee line about 7 million years ago. And Cosmides and Tooby have high hopes that our skill at logic when it comes to monitoring social agreements versus our faltering attempts when the subject matter is about other matters, not related to evolutionary advantage, might be a QWERTY phenomenon. We were selected to be good at the one and not the other. So maybe there is a hardwired Swiss Army knife module in our brains for logically monitoring social cheats, whereas our ability to ride the wild horses of hypothesis driven logic must be acquired through sweat and tears.
The Wason test together with the Cosmides, Tooby and Gigerenzer hypothesis tries to explain why we often get it wrong when we trust in human intuition.
Recent research has shown that when the Wason tests are modified to make the matter more concrete and to relate the problem given to the everyday experiences of the persons taking the test, the logical success rate improves dramatically. When the cards are:
Leeds
train
car
and the truth claim is: Every time I go to Manchester I travel by train.
then 10 from 16 subjects got the correct solution. They turned over the Manchester and car cards. In a control group which performed a task involving the same conditional logic, using abstract cards and a conditional rule which bore no relation to the experience of the subjects, only 2 from 16 got the right answer.
So degree of abstractness does negatively affect the performance of your logic module.
It turns out that all kinds of factors may affect your logical performance; factors such as, the instructions received about how to do the task / the number of response alternatives / linguistic variables in the sentences giving the instructions etc.
It is an instructive point for anyone writing a logic text or for any reader wishing to gain a better understanding of the subject of logic and its methods.
In Western civilization we have devised a whole range of logical tools that enhance and extend our human reasoning powers. Tools such as probability theory, statistics, decision theory, and game theory. These are often extremely helpful to us when we need to make a real world decision. However, sometimes even the experts pay no attention to these and other tools and try to reason to a conclusion just using their intuition or native wit. The results can be very humiliating and very sobering.
We shall try to incorporate this lesson and any other lessons to be learned from this into the way we present material to the reader, and hopefully thereby make this a very readable logic book for human intelligences.
_____________________________________________________________________________________________________________
more evenly numbered knowdules will be added in the coming weeks