- Hide quoted text — Show quoted text -
Christian Fermueller (chr…@csdec1.tuwien.ac.at) wrote:
} Can anyone answer the following question and/or give
} appropriate references?
}
} Is there a Hilbert-style calculus of (classical) propositional
} logic, where each axiom schema and derivation rule
} contains at most two different propositional variables? If
} not, how do you prove that such a system cannot exist? (I
} do not care which connectives the system is based on, as
} long as we have functional completeness.)
}
} To avoid some misunderstandings: By Hilbert-style
} calculus, I mean a finite number of axiom schemata (like:
} A -> (B -> A)) and rules (like modus ponens or:
} |- A => |- A v B). Usual systems contain axiom
} (schemata) like: (A -> B) -> [(B -> C) -> (A -> B)], which
} contains _three_ propositional variables. The question is,
} whether this is necessarily so, and why (not)?
In Hilbert and Ackermann’s _Principles of Mathematical
Logic_ 1950 p. 27, four axioms of classical propositional
logic are given. Only the last axiom uses more than two
variables.
1. (A > B) > [(C > A) > (C > B)]
The previous formula is equivalent to the following:
2. [(C > A) & (A > B)] > (C > B)
Formula 2 represents the transitivity of implication.
Transitivity, a necessary property of classical logic, requires
at least three variables for its representation.
Neil Nelson
n_nel…@ix.netcom.com(Neil Nelson) wrote:
>[...]
>1. (A > B) > [(C > A) > (C > B)]
>The previous formula is equivalent to the following:
>2. [(C > A) & (A > B)] > (C > B)
>Formula 2 represents the transitivity of implication.
>Transitivity, a necessary property of classical logic, requires
>at least three variables for its representation.
Augustus de Morgan is known only in connetion with his "2 sentences".
Actually he was one of the greatest logicians besides Aristotle. His
logic, "Formal Logic: Or, The Calculus of Inference" etc. London 1847
doubled not only the than known log. propositions but also the
syllogisms.
To the problem above the following might be of interest:
Chapt.1, First Notions of Logic (p. 1-25) de Morgan shows that there
must be sthg. wrong with classical logic trying to do math and using
inequations as propositions:
"Every one should be aware that there is much false form of inference
arising out of badness of style, which is just as injurious to the
habits of the untrained reader as if the errors were mistakes of logic
in the mind of the writer.
’X is less than Y; Y is less than Z: therefore X is less than
Z…’"
________
Or + X < + Y
+ Y < + Z
so + X < + Z
("+" meaning "the whole" or "all")
________
"…This at first sight, appears to be a syllogism; but, on reducing
it to the usual form, we find it to be…not a syllogism, since there
is no middle term." p.20
Euclid tought us: Two quantities equal each other, if they equal a
third quantity. This third quantity is the so-called middle term in
logic. It must be one and the same in both cases. Neither logic nor
mathematics would be possible without this.
But what do we have here if we make two equations out of the first two
inenquations?
+X = k * (+Y)
+Y = l * (+Z) ,
Factors k and l "make" +X and +Y equal the left sides. So they are <1,
cause +Y>+X and +Z>+Y. Two quantities +X and (l * (+Z)) do not equal a
third, but each one equals another, namely (k * (+Y)) and +Y, so they
cannot equal each other. Unpleasant but true. But any "inner voice"
rebells against this truth and tells us that it is not a truth. And if
that voice is right there is no "badness of style", but a bad logic.
We just have to devide the first equation by k to "get" the common
third, namely +Y:
+X : k = +Y
+Y = l * (+Z)
+X = k * l * (+Z)
Why does nobody do that in propositional logic of all day life? It is
so easy. It only seems so, cause mathematical abstraction seems to be
as self-evident as all day life abstraction. But you will get a
surprise if you take for +X, +Y, +Z quantities of the physical world:
let
+X = all humans
+Y = all mammals
+Z = all animals
"k times l times l all animals" or "all humans devided by k" seems
problematic. And if you devide all humans by k, you get the quantity
of all mammals…
So multiplication/division in logic seems not to be the solution.
BUT: The above equations are true and not false. So there _must_ be
some kind of factors in logic like in math which tell us the same.
"k times l times all animals" must be able to be represented by a
logical proposition. When I say proposition I mean a sentence which
everybody can understand and not that mishmash which is often to be
seen in modern logic books. Math as the "art of dealing with abstract
quantities" and logic as the "art of dealing with real _and_ abstract
quantities" _must_not_differ_. It seems, as long as we do not call all
humans: "k times l times all animals" we must keep on searching.
Lothar Seidel
Am Foersterpfad 10
60529 Frankfurt, Germany
tel.:+49-69-6663185
100664.3…@compuserve.com
In article <4aueeu$…@dub-news-svc-4.compuserve.com>,
- Hide quoted text — Show quoted text -
Lothar Seidel <100664.3…@compuserve.com> wrote:
>n_nel…@ix.netcom.com(Neil Nelson) wrote:
>>[...]
>>1. (A > B) > [(C > A) > (C > B)]
>>The previous formula is equivalent to the following:
>>2. [(C > A) & (A > B)] > (C > B)
>>Formula 2 represents the transitivity of implication.
>>Transitivity, a necessary property of classical logic, requires
>>at least three variables for its representation.
>Augustus de Morgan is known only in connetion with his "2 sentences".
>Actually he was one of the greatest logicians besides Aristotle. His
>logic, "Formal Logic: Or, The Calculus of Inference" etc. London 1847
>doubled not only the than known log. propositions but also the
>syllogisms.
>To the problem above the following might be of interest:
>Chapt.1, First Notions of Logic (p. 1-25) de Morgan shows that there
>must be sthg. wrong with classical logic trying to do math and using
>inequations as propositions:
>"Every one should be aware that there is much false form of inference
>arising out of badness of style, which is just as injurious to the
>habits of the untrained reader as if the errors were mistakes of logic
>in the mind of the writer.
> ‘X is less than Y; Y is less than Z: therefore X is less than
>Z…’"
>________
>Or + X < + Y
> + Y < + Z
>so + X < + Z
>("+" meaning "the whole" or "all")
>________
>"…This at first sight, appears to be a syllogism; but, on reducing
>it to the usual form, we find it to be…not a syllogism, since there
>is no middle term." p.20
>Euclid tought us: Two quantities equal each other, if they equal a
>third quantity. This third quantity is the so-called middle term in
>logic. It must be one and the same in both cases. Neither logic nor
>mathematics would be possible without this.
[ . . . ]
>seen in modern logic books. Math as the "art of dealing with abstract
>quantities" and logic as the "art of dealing with real _and_ abstract
>quantities" _must_not_differ_. It seems, as long as we do not call all
>humans: "k times l times all animals" we must keep on searching.
Whatever the vices of mathematical logic, one hopes they are
not elementary!
The gist of De Morgan’s complaint seems to be: R(a,b) and R(b.c) do
not necessarily imply R(b.c); the missing "middle" is precisely the
statement of transitivity– axiom or theorem. Surely logic does not miss
this; and everybody knows that ‘divides’ or ‘subset’ are transitive,
while E (‘membership’) is not.
Typically, the Propositional Calculus is not enough to clarify these
matters. The Predicate Calculus does better. Consider aEb, which is in a
sense contingent, and compare with Ax ( xEy => xEz ); you cannot be fooled
about transitivity then.
More generally, De Morgan bemoans fabrication of relatedness out of
unrelatedness — out of thin air. That, roughly, is Euclid’s point too;
the need for a "common measure". I should think the message has been
received. Early example: when the edifice of Greek geometry, with its
integral multiples and subdivisions, was shaken by the discovery of irrati-
onals, Eudoxus found a resolution, which fairly interpreted is the notion
of Dedekind cut. Intuitions about ‘length’ and ‘continuity’ are fine, but
logic is logic, and it was respected.
.
Ilias
ikas…@alumni.caltech.edu (Ilias Kastanas) wrote:
>In article <4aueeu$…@dub-news-svc-4.compuserve.com>,
>Lothar Seidel <100664.3…@compuserve.com> wrote:
>>n_nel…@ix.netcom.com(Neil Nelson) wrote:
>>>[...]
> Whatever the vices of mathematical logic, one hopes they are not elementary!
>[...]
> Typically, the Propositional Calculus is not enough to clarify these
> matters. The Predicate Calculus does better.
>[...]
Ilias,
First of al I must excuse: I took the ">" for the greater-than sign
rather than for implication. So my article does not belong to the
question at all.
If you are engaged in a certain thing you sometimes get blind. My job
is researching on elemantary logic as Aristotle did, trying to find
the things he did not find.
Do you know that the reason for the predicate calculus was the
propositional logic? Sure you do: All researching for the complete
propositional logic and a real formalim of propositional logic seemed
to be in vain. The elementary thing: the proposition A=B was too
complicated to be formalized. So locic gave up to find it. I think
that was wrong.
I wonder if you can give me an answer to a very elementary
propositional logic problem from the standpoint of Euclid
(part<whole).
Let A, B, C be three quantities.
If A is a part of B,
and a part of B is the whole C,
Which relation between A and C is then always true? Is there any or
not?
Please try to say it in words.
Lothar
Lothar Seidel
Am Foersterpfad 10
60529 Frankfurt, Germany
tel.:+49-69-6663185
100664.3…@compuserve.com
n_nel…@ix.netcom.com(Neil Nelson) wrote:
>1. (A > B) > [(C > A) > (C > B)]
>The previous formula is equivalent to the following:
>2. [(C > A) & (A > B)] > (C > B)
>Formula 2 represents the transitivity of implication.
>Transitivity, a necessary property of classical logic, requires
>at least three variables for its representation.
>Neil Nelson
Sorry,
my follow up with de Morgan was a fault, an answer not belonging to
the question. I took ">" as greater than rather than as "implication".
I should have seen it cause the >[ and the ]> don’t give sense as a
greater than. First read than write, Mr. Seidel!
Lothar Seidel
Am Foersterpfad 10
60529 Frankfurt, Germany
tel.:+49-69-6663185
100664.3…@compuserve.com
In article <4avp37$…@gap.cco.caltech.edu>,
ikas…@alumni.caltech.edu (Ilias Kastanas) writes:
>The gist of De Morgan’s complaint seems to be: R(a,b) and R(b.c) do
>not necessarily imply R(b.c); the missing "middle" is precisely the
>statement of transitivity– axiom or theorem. Surely logic does not
>miss this; and everybody knows that ‘divides’ or ‘subset’ are
>transitive, while E (‘membership’) is not.
>Typically, the Propositional Calculus is not enough to clarify these
>matters. The Predicate Calculus does better. Consider aEb, which
>is in a sense contingent, and compare with Ax ( xEy =xEz ); you
>cannot be fooled about transitivity then.
It’s surely slightly unfair to criticise De Morgan for failing to take
account of logical developments since his time: what you are saying is
perfectly obvious today, but De Morgan lacked not only the predicate
calculus but a generally accepted formulation of the propositional
calculus and even "transitivity" as a defined concept. All of these, in
fact, were developed on foundations which De Morgan had no small part in
constructing, with "complaints" like the above playing a significant role
in clarifying vital issues. Of course, when we stand on the shoulders of
giants, we can see further than the giants; but that is no reason for
denying the giants’ perceptiveness.
Peter Wilkinson
pwilkin…@cix.compulink.co.uk
In article <DJyL3o….@cix.compulink.co.uk>,
- Hide quoted text — Show quoted text -
Peter Wilkinson <pwilkin…@cix.compulink.co.uk> wrote:
>In article <4avp37$…@gap.cco.caltech.edu>,
> ikas…@alumni.caltech.edu (Ilias Kastanas) writes:
>>The gist of De Morgan’s complaint seems to be: R(a,b) and R(b.c) do
>>not necessarily imply R(b.c); the missing "middle" is precisely the
>>statement of transitivity– axiom or theorem. Surely logic does not
>>miss this; and everybody knows that ‘divides’ or ‘subset’ are
>>transitive, while E (‘membership’) is not.
>>Typically, the Propositional Calculus is not enough to clarify these
>>matters. The Predicate Calculus does better. Consider aEb, which
>>is in a sense contingent, and compare with Ax ( xEy =xEz ); you
>>cannot be fooled about transitivity then.
>It’s surely slightly unfair to criticise De Morgan for failing to take
>account of logical developments since his time: what you are saying is
>perfectly obvious today, but De Morgan lacked not only the predicate
>calculus but a generally accepted formulation of the propositional
>calculus and even "transitivity" as a defined concept. All of these, in
>fact, were developed on foundations which De Morgan had no small part in
>constructing, with "complaints" like the above playing a significant role
>in clarifying vital issues. Of course, when we stand on the shoulders of
>giants, we can see further than the giants; but that is no reason for
>denying the giants’ perceptiveness.
>Peter Wilkinson
>pwilkin…@cix.compulink.co.uk
I agree; in fact, I was praising DeMorgan, not criticizing him. I
gave an opinion as to what his "complaint" may have been, and credited him
with discerning some matters related to transitivity, logical Calculi, and
other things yet to come. Sorry if I was not entirely clear.
Ilias