Dear Organlearners,
John Gunkler <jgunkler@sprintmail.com> writes:
>When I refer to logical fallacies I am not referring to someone's
>opinion about an argument but to well-accepted rules that
>make what someone says invalid (or unworthy of belief.)
(snip)
>1. Fallacy of the Consequent (type 1 -- Denial of the antecedent):
>
>If p is true then q is true. p is not true. Therefore, q is not
true.
>
>["If I go to the mall I always buy a toy. I did not go to the mall.
>Therefore, I did not buy a toy." -- This is false, however, because
>I may have bought a toy at the corner drug store.]
>
>Note: The example above is often abbreviated as follows ...
>If p then q. Not p. Therefore not q. [This is taken to mean the
>same thing as the longer version above.]
Greetings John,
I have been trained as a natural scientist for 5 years up to MSc (physics)
level in subjects like mathematics, chemistry, physics, applied
mathematics, chemical technology, etc. During all those years I had never
been told once that logic itself could be an object of study. I did read
more widely than the average student, but even during those readings I
never stumbled on a book of logic. So when I began to look ten years later
(in the middle seventies) for something on logic, I was surprised to find
such a vast body of literature on the subject. I felt like being cheated
out of one of the treasures of humanity.
In my counseling of low performing students I discovered that many of them
did not have problems with scientific subjects (like chemistry or biology)
per se, nor with learning or psychological problems, but with with logic
itself. After twenty years I now feel strongly that "logic literacy", just
like "computer literacy", should be offered as a first year course for all
students at a tertiary education institution. I often even wonder if
"logical literacy" should not be one of the main topics in Systems
Thinking (one of the five discipline of LOs).
John, you listed seven typical fallacies in logical reasoning. There are
many more. How do we keep track of them? Should we memorise them? No. We
should rather learn how to think "logically" by studying logic as such.
Among other things, we should then try to discover logical fallacies
ourselves, or try to explain why a particular reasoning is fallicious when
it is pointed out to us. You, for example in the quote above, have pointed
out the "Fallacy of the Consequent". One way of explaining is to find an
example which illustrates the fallacy. You have followed this way. It is a
powerful way, but not without fallacies itself.
There are also two other ways. The one way is directly related to logics
itself. It is called Proof Theory (PT). The other way is indirectly
related and is called Model Theory.
PROOF THEORY
In Proof Theory (PT) we use the body of logic itself to show that the
opposite (contradictory) of the fallicious inference is true whereby,
in terms of the Law of the Excluded Middle, the questionable inference
is false. For the "Fallacy of the Consequent", i.e
If p is true then q is true. p is not true. Therefore, q is not
true.
we have to show that
If p is true then q is true. p is not true. Therefore, q is true.
But how do we show such the latter inference is true? This is where
"deduction" (or derivation) comes in. We begin with a small set of
*true statements* called "axioms" and a small set of *valid
inferences* called "inference rules". (The *true statements* are
being-like and the *valid inferences" are becoming-like.) We use these
"axioms" and "inference rules" to deduce (derive) some additional true
statements (truth beings) which are called "theorems" and additional
valid inferences (truth becomings) which are called "derived rules".
We deduce as much theorems and derived rules as that which will be
needed to conclude that
If p is true then q is true. p is not true. Therefore, q is true
is indeed a derived rule.
Unfortunately, deduction is not a purely mechanical thing as many people
have experienced it to be. Firstly, when a person has to MEMORISE the
deduction of a theorem or a derived rule, it is indeed a mechanical thing
to do. It is as worse as memorising a page from a telephone directory.
This is exactly where the fear or revulsion of many people for mathematics
begin. Secondly, as Goedle has proven, classical logic is incomplete. One
may actually try to prove a theorem or derived rule which fall outside the
scope of classical logic.
Deduction, despite the negative criticisms it has received of lately,
requires a high degree of creativity. Or to bring Lutzen Brouwer once more
into the picture, deduction requires intuition and the ability to make
constructions. Both intuition and constructions are activities at a lower
order than logic. In other words, they are necessary for logic whereas
logic is not necessary for them. Likewise creativity is necessary for
logic, but logic is not necessary for creativity. Even classical logic
tells us that should creativity had been neccessary for logic and logic
neccesary for creativity, (In other words, creativity is a neccessary and
sufficient condition for logic), then creativity and logic are equivalent
(isomorphic) to each other.
MODEL THEORY
Model theory is a 20th century development on classical logic. If my
memory serves me correctly, it was initiated by A Robinson following the
example set out by Goedel. One model is to consider propositions as truth
functions. A function has one or more inputs and only one output. An input
can have one one or more values. The same for an output. The inputs and
outputs of functions for classical logic have only two sharp values,
namely T (for "true") and F (for "false"). Hence classical logic is known
as a two-valued logic.
The creative person will immediately begin to think of the possibility of
three value, four value, .... logic and whether they will make any sense.
One ultimate extension of many valued logic is fuzzy logic where infinite
many values are considered between the two limits T and F. (Think about
the infinite number of points between the two numbers 0 (for F) and 1 (for
T) on a line. Thus a proposition in fuzzy logic is not like in classical
logic, i.e either T or F, but not both T and F as is required by Law of
the Excluded Middle (TEM). A fuzzy proposition is rather, for example, 25%
true (0.25 T) and thus 75% false (0.75 F). In other words, the TEM does
not operate in fuzzy logic, except "deep down below".
What do I mean by this "deep down below"? Well, if a fuzzy proposition
p = 0.25 T
then *****exactly*****
p = 0.75 F
or
NOTp = 0.75 T
In other words, once the unique value of p in terms of T is fixed (25% in
the example above), then its unique value in terms of F (75% in the
example above) is also fixed. No other value (eg. 33%, 87%) for NOTp is
allowed when p is 25% true. Consequently the LEM in terms of functions
means that a proposition has to have a UNIQUE output vluaes for a
particular (set of) of input values. This is a case where the essentiality
sureness ("identity-categoricity") begin to play its role.
Let me illustrate this uniqueness (sureness) in the output for
classical logical functions. The "monary" (one input) function NOT(p)
has the following "truth table". (A "truth table" take all the truth
possibilities into consideration.) See how, when p has the unique
value T, NOT(p) can have only one unique value, namely F. Only when p
has the unique value F can NOT(p) have the unique value T. In other
words, NOT(p) cannot have two output values (T and F) for one input
value for p.
p NOT(p)
T F
F T
In terms of functions the LEM now appears as the key feature of all
functions, namely unique value for the output. Any action which
produces more than one value for the ouput in the same case, is called
a relation. This can be summarised by
p REL(p)
T T, F
F T, F
There are three other "monary" truth functions. They are
p SURE(p)
T T
F F
p TRUE(p)
T T
F F
p FALSE(p)
T F
F F
Some of these four monary truth functions may sound far fetched, but there
are much sense in them. Some people are "negativists" uphelding
"negativism". It means that whatever proposition p someby else uses, they
deal with the negation of it. For example, if somebody else say p is true
or false, they will say it is respectivley false or true. Some people are
positivists uphelding "positivism". It means that whatever proposition p
someby else uses, they deal with the proposition as it stands. For
example, if somebody else say p is true or false, they will say it is
respectively true or false also. Some people are "naivitists" uphelding
"naivitivism". It means that whatever proposition p someby else uses, they
deal with the it as a piece of truth. For example, if somebody else say p
is true or false, they will say it is true. Some people are "fallibalists"
uphelding "fallibalism". It means that whatever proposition p someby else
uses, they deal with it as a fallacy. For example, if somebody else say p
is true or false, they will say it is false.
Let us consider the binary truth functions. There are sixteen of them
for classical logic. The two most common ones to us are AND and OR.
The truth table for AND is:
p q AND(p, q)
1 T T T
2 T F F
3 F T F
4 F F F
This function models a composite proposition like
"the leaf is green AND the snow is black"
with p="the leaf is green" and q="the snow is black".
This example applies to case 3 so that the composite proposition is
false.
The truth table for OR is:
p q OR(p, q)
1 T T T
2 T F T
3 F T T
4 F F F
The AND is "narrow" in its identification of true while the OR is much
"broader" in its identification of true. The EQUIV (equivalent)
function is intermediate as its truth tabel shows:
p q EQUIV(p, q)
1 T T T
2 T F F
3 F T F
4 F F T
The (in)famous function IMPLY us also broad in its scope like the OR.
The truth table for IMPLY is:
p q IMPLY(p, q)
1 T T T
2 T F F
3 F T T
4 F F T
Let us go deeper into the IMPLY function. It uses the words "if.....,
then......"
Case (1) is a sentence like "If Joe has fever, then Joe is ill." The
two simple propositions are "Joe has a fever" and "Joe is ill". In
this case the implication is true.
Case (2) is a sentence like "If Joe has fever, then Joe is not ill."
The two simple propositions are "Joe has a fever" and "Joe is not
ill". In this case the implication is false.
The IMPLY is infamous because of cases (3) and (4). Many people want
these latter two cases to be false also. But in that case they
actually need the function AND above and not IMPLY. A vew others are
not so strict and want to weed out only case (3) or case (4). When
they weed out case (3), they are actually in need of the function
EQUIV. When they weed out case (4) they are in need of the function
p q LAST(p, q)
1 T T T
2 T F F
3 F T T
4 F F F
In other words, LAST projects the composition on its last part q.
Can we really have implications for case 3 and 4? Yes. Here is an
example for case (3). "If the sun revolves around the earth, then the
sun is attracted by the earth". The simple sentence "the sun revolves
around the earth" is true in the sense of Aristotle, but false in the
sense of Copernicus. The simple sentence "the sun is attracted by the
earth" is true in the sense of the law of gravitation because the
earth and the sun exert equal forces on each other. People before
Copernicus, Galileo and Newton argued in terms of case 1 (which is
actually case 3). They did not accept Copernicus, Galileo and Newton's
arguments, fearing that they were arguing in terms of case 3 which the
people considered to be false. Were people better informed about case
3, namely that it is a true implication, they would have accepted
Copernicus, Galileo and Newton's arguments far easier.
Today the whole situation has reversed! People accept the sentence
"the earth revolves around the sun" like Copernicus, Galileo and
Newton's. Thus they infer "If the earth revolves around the sun, then
the earth is attracted by the sun". They think about bodies revolving
along other bodies like the moon around the earth, sattelites around
the moon or Mars, etc. Hence they make use of case 1. But they make
fun out of people who still say "If the sun revolves around the earth,
then the sun is attracted by the earth". But this is case (3) of the
implication. It is not merely somebody's "democratic right of freedom
of speach" which allows case (3), but the very spirit of logic!
What about case (4). Here is an example, uttered sarcastically: "If
Joe says that he is clever, then I am the uncle of the king's dog".
The two simple sentences are "Joe says that he is clever" and "I am
the uncle of the king's dog". Sarcasm, although often hurting, is a
powerful way of uncovering truth.
MODEL EXAMPLE
Before I stop my rambling, let me show what the fallacy of
Denial of the Antecedent amounts to:
with an example
"If I go to the mall, then I buy a bread. I did not go to the mall.
Therefore, I did not buy a bread."
Consider the following extended truth table
p q IMPLY(p, q) NOT(p) NOT(q)
1 T T T F F
2 T F F F T
3 F T T T F
4 F F T T T
The two simple sentences are p = "I go to the mall"
and q = "I buy a bread". The denial of the antecedent is
"I did not go to the mall" with respect to "I go to the mall"
as the antecedent. The Denial of the Antecedent points to cases (3)
and (4) for the column NOT(p). It is BOTH the two cases where NOT(p)
is true (T). The consequent is "I did not buy a toy", ie. NOT(q). But
in case (3) NOT(q) is false (F) whereas in case (4) NOT(q) is true
(T). In other words, we have two possible outcomes for NOT(q). It
means that the Law of the Excluded Middle is not valid any more. To
pinpoint it in terms of functions, the inference sheme
IMPLY(p, q) // Assume ###(p) = T, thus conclude ###(q)
has lost is functionality (unique value for output) where ###(p) refer
to a monary function.
The following two cases are of this inference scheme are valid
IMPLY(p, q) // Assume (p) = T, thus conclude (q)
This inference is known as Modus Ponens which is case (1) of the
extended truth table. Look at the first three columns p, q and IMP(p,
q) to pinpoint this case.
IMPLY(p, q) // Assume NOT(q) = T, thus conclude NOT(p)
This inference is known as Modus Tollens which is case (4) of the
extended truth table. Look at the last three columns IMP(p, q), NOT(p)
and NOT(q) to pinpoint this case.
But the following case of this inference scheme is invalid. It is
known as Affirming the Consequent. Its scheme is
IMPLY(p, q) // Assume (q) = T, thus conclude (p)
Look at cases (1) and (3) in the extended truth table at column two
for q. But in column (1) for P we have two possibilities for p.
Functionality is lost!
FINAL NOTE
I hope that by now some of you may have become more interested in
logic on the one hand and the functionality of function on the other
hand. It almost seems as if logic depends on the functionality (unique
output value) of functions. Is that so? On the other hand, the Gibbs
expression
/_\F < W
is an (order) relation. What comes first, the function or the
relation -- the chicken or the egg -- logic oor free energy --
mechanics or dynamics of creativity?
Best wishes
--At de Lange <amdelange@gold.up.ac.za> Snailmail: A M de Lange Gold Fields Computer Centre Faculty of Science - University of Pretoria Pretoria 0001 - Rep of South Africa
Learning-org -- Hosted by Rick Karash <rkarash@karash.com> Public Dialog on Learning Organizations -- <http://www.learning-org.com>