John Gunkler" <email@example.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.) In general an
>argument may be logically fallacious in three ways:
>In its MATERIAL content -- through a misstatement of the facts.
>In its WORDING -- through incorrect use of terms.
>In its FORM or structure -- through the use of an improper
>process of inference.
Thank you very much for explaining in more detail what you meant. When
you warned against fallicious thinking, I had the uneasy feeling that
you were using the shot gun with the sawn off barrel, spraying hail
all over the place, hoping to hit someone's thoughts. Now I know that
you had something much more clear in mind.
Just two points on the name of the topic. If we are going to stay on
the topic of logic, we should have changed the topic to "Inconsistent
logic". Science itself is more than merely logic. Science has also an
empirical character to it. In other words, how to obtain true
statements (facts) without the use of logic and how to test logical
conclusions (which ought to be true) without the use of logic. But
there are two other most important characterestics of science which
are seldom contemplated. Science has a creative character and a humane
character to it. Hopefully we will shift our learning spirits some day
on these two characters.
The other point is that I am not happy with the qualification "Junk",
even if it is placed in quotation marks. Whether it is "junk" science
or "junk" logic, is not important. The word "junk" stinks with
preconceived judgement. If there was not such a thing as a "paradigm
shift" (See T Kuhn), then something like "junk" science would not have
had any special meaning to science. The same for logic.
Alas, "paradigm shifts" do exist. Here is an example from science.
When Max Planck concluded mathematically that the energy of
electromagnetic waves has to be quantified in terms of its frequency,
based on empirical data, his conclusion was judged as "junk" science.
It took an Einstein (yes, Einstein himself) to perceive the gold in
this piece of "junk" when he used it to explain photo-electricity.
After that, Bohr began to use Planck's conclusion to explain the
emmision of light form a hydrogen atom. Then Schroedinger used it to
set up his wave equation through which quantum mechanics emerged.
Here is another example from logic. Early this century Lutzen Brouwer
felt that logic was flawed in its most basic axiom, namely the Law of
the Excluded Middle (LEM). The LEM may be formulated as follows with p
symbolising any proposition:
not (p and (not p))
It means that its not true that a proposition p and its denial (not p)
are both true. Either p is true so that
(not p) is false or p is false so that (not p) is true. He based his
feeling not on logic, but on his intuition and constructions which he
What happened to Brouwer? He was laughed out of every organisation
(journals, societies, meetings) because of his "junk" logic. Why? The
LEM had been used in every system of logic (Frege, Hilbert, Russel,
...) as an axiom. The various logical systems differed because of the
other axioms or inference rules which they incorporated. So he had
only one thing to do -- to work all alone and create his system of
intuitionistic/constructivist logic. But some genius himself began to
take notice of Brouwer, namely Kurt Goedel. Eventually, with brilliant
insight, he used the classical logic which logicians all over the
world were so proud of to prove a theorem which stunned those very
logicians. It is now known as the famous Incompleteness Theorem. It
states that there are theorems in classical logic which classical
logic will not be able to prove!
In the mean while one of Brouwer's students Arend Heyting created a
logical system to generate the intuitionistic theorems which Brouwer
so painstakedly had developed. Furthermore, it was supplemented by an
algebra which modelled that system just as Boolean algebra modelled
classical logic. Great was the surprise when mathematicains began to
realise that Boolean algebra and Heyting algebra were complementary
duals to each other. The "junk" logic of Brouwer finally became a
What sort of logic was laying behind this complementary dual of
Boolean and Heyting algebras? The answer stayed out in the cold for
many decades. Then it came from a direction which nobody even
expected. Soon after WWII, S Eilenberg and S Maclane (1945) produced a
63 page paper with the title "General theory of natural equivalences".
Another two decades had to flow by before FW Lawvere realised that in
this paper a new foundation for mathematics was hidden. So in 1966 he
proposed category theory as a foundation for mathematics. Category
theory works with two kinds of entitities, namely sets ("beings") and
functors ("becomings"). A revolution took place. Many mathematicans
did not even call it "junk" mathematics, but outright "abstract
nonsense". In 1974 MP Fourman took this revolution into logic with his
PhD thesis "Connections between category theory and logic". In 1975 he
took the bold step with his paper "The logic of Topoi". Toposlogic
(not topology which is something different) emerged -- the logic
behind Boolean and Heyting algebras.
Let us now see how Brouwer defended himself:
* Let those who come agter me wonder why I built these
* mental constructions and how they can be interpreted
* in some philosophy; I am content to build them in the
* conviction that in some way they will contribute to the
* clarification of human thought.
What on earth has Toposlogic to do with learning individuals and
In 1975, like Brouwer in the beginning of this century, I became
convinced that classical logic do not serve the logical needs of
students who have to learn chemistry. Classical logic was not
consistent with the logical thinking required in chemistry. What was
the culprit? Take any textbook on chemistry. A number of chapters are
needed to delineate the theory of acids and bases. The following is an
NaOH + HCl = NaCl + H2O
Then, at some other place in the book a number of chapters are needed
to delineate the theory of oxiders and reducers as well as its
relation to electrochemistry. The following are oxidation-reduction
2Na + Cl2 = 2NaCl
2H2 + O2 = 2H2O
Thus students get the idea that a chemical reaction can be either an
acid-base reaction or a redox reaction, but not both. The (in)famous
LEM in operation. Nowhere in those textbooks will you find an
statement which either affirm the LEM with respect to chemical
reactions or refute it. Classical logic with its LEM is simply assumed
to be applicable. However, as soon as we have to do with COMPLEX
reactions, LEM fails. A typical example is the following (unbalanced)
FeSO4 + KMnO4 + H2SO4 =
Fe2(SO4)3 + MnSO4 + K2SO4 + H2O
In this equation on the lefthand side the oxidiser is Fe in FeSO4 and
the reducer is Mn in KMnO4. The student is told that it is a redox
reaction. So the student infer from classical logic with its LEM that
it is not an acid-base reaction also. Neither does the textbooks
clarify the issue. Why? Because LEM is involved.
This COMPLEX reaction is indeed also an acid-base reaction. The acid
is the H in H2SO4 and the base is the O in KMnO4. Thus my long
learning journey through the literature on logic began, trying to
understand why the LEM failed! I asked my mathematical colleagues to
help me, but that failed. I began to learn classical logic and all its
variants. Then one day I stumbled on the collected works of Brouwer.
My hair began to raise -- here was a person who felt the same as me.
So I began to follow the creative course of time in intuitionistic
logic. Then I came into contact with category theory. What appeared to
be "abstract nonsense" to others, was a new viewpoint on the world of
mathematics, one which made sense to me. Eventually I had to learn
toposlogic. By then I was not surprised anymore -- toposlogic and
chemical logic were very much the same thing!
Finally, what point did I wish to make with the long contribution
above? The Law of the Excluded Middle (LEM) holds for most simple
systems. I am even willing to make the conjecture that LEM is the
defining logical property of simple systems. But as soon as we move
from simple systems to COMPLEX systems, the LEM becomes complexified
itself. We cannot think about LEM in a simple manner anymore. I am not
willing to make the conjecture that whenever LEM fails, we have to do
with a complex system. I have always been able to find LEM acting on
some lower leavels of the system
>Here are some examples (there are many others) of logical
>fallacies. In the examples, the letters p, q, r (etc.) stand for
>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
>7. The Converse Fallacy of Accident: Arguing improperly from
>a special case to a general rule.
John, again I want to thank you for bringing such fallacies under the
attention of all of us. I hope it will encourage fellow learners to
begin learning the art of logic. If they had not previous experience
with formal logic, begin with a textbook like "Symbolic Logic" by IM
>The point is, when people present ideas fallaciously we
>should be skeptical about their conclusions. That's not to
>say that their conclusions are necessarily false; it is only
>to say that we have no more reason for believing the
>conclusions to be true after we read the statement than we
>did before we read it.
Thank you John.
I want to extend this warning. When people present their ideas, we
should not only be sceptical about the logical reasoning in such
ideas, but also about the logical SYSTEM used in presenting such
ideas. I have indicated above how important it is to know that there
is not only one kind of logical system, namely the classical system.
The seven fallacies which you have discussed, are very common in
classical logic. But classical logic rests on The Law of the Excluded
Middle. Likewise the seven fallacies which you have pointed above rest
on the LEM. Pull LEM from underneath them and they are not classical
The one thing which I do not try to do, is to become sceptical of
As for my own logic, it is definitely not classical any more. It is
also not intuitionistic nor toposlogical anymore. I have not made up
my mind what it "is" because I still learn more and more about it. For
example, as time goes by I become more sensitive to the role of
transdisciplinray thinking in logic. What I do know, is that it has to
be a logic which can deal with the "deep complexity" of reality --
Creation and Creator. Classical logic cannot do it as Goedel pointed
out long ago.
Please note that I am not advocating the expulsion of logic from our
thinking as many thinkers are doing today -- a retirn to irrational
thinking. The fact that even logical systems can fail is no proof,
neither logical nor empirical, that the idea of logical thinking is a
At de Lange <firstname.lastname@example.org> Snailmail: A M de Lange Gold Fields Computer Centre Faculty of Science - University of Pretoria Pretoria 0001 - Rep of South Africa
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