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From: Bruno Marchal <marchal.domain.name.hidden>

Date: Fri, 16 Aug 2002 15:23:43 +0200

At 10:29 -0700 13/08/2002, Wei Dai wrote:

*>Does it mean anything that S4 and intuitionistic propositional
*

*>calculus (= 0-order intutionistic logic, right?) ...
*

Right.

*>.... have the same kind of models, or is
*

*>it just a coincidence? I guess Tim is saying that it does mean something,
*

*>but I don't understand what.
*

I almost missed this fair question. It is not a coincidence.

I choose S4 for not frightening Tim with irreflexive or non

transitive accessibility relation :-).

Kripke knew a 1933 result(*) by Godel according to which, (with IL

standing for Intuitionist logic):

Theorem: S4 proves T(A) if and only if IL proves A

where T is a function from the propositional language to the modal

propositional logic given by

T(A) = A if A is a propositional letter = A belongs to {p, q, r ...}

T(-A) = []-T(A)

T(A->B) = [](T(A) -> T(B)

Kripke first discovered his semantics for general normal modal logic.

By Godel 1933 this provides him (and us!) the "S4" possible world

semantics of IL.

Note that Beth developed a more awkward but similar semantics before.

The "world" of the S4 models (= of the IL) models are sometimes

interpreted as *state of knowledge*. This fit well with the fact

that the S4 logic in my thesis describes a pure first person knower.

(But I get S4Grz, its antisymmetrical extension, good for subjective

irreversible time).

The following remarks may help.

S4, which is a *classical* extension of CL

(Classical Logic), is capable of simulating IL. This is not an argument

that CL is better, for Godel (again) found that IL can simulate CL:

basically IL proves (- - A) when CL proves A. It's a key of IL that

IL does not prove (- - A) -> A. (but IL proves A -> (- - A))

Like CL admits an algebraic semantics in term of Boolean Algebra, ...

parenthesis:

(where the propositions A, B, C ... are interpreted by subsets of

a set W, the "and" by intersection, the "or" by union, the "-" by the

complementary, the constant f by the empty set and the constant t by

the whole set W. You can even (in our modal context interpret the element

of W as worlds verifying the formula, for example a tautology being

true in all world you see a tautology, like the constant t, is

interpreted by W). For example [A union -A] = W, i.e the exclude

middle principle is universally valid, in CL.

end of parenthesis

... IL admits a topological interpretation, where the propositions

are interpreted by open sets in a topological space. The "and" by

intersection, the "or" by union, the not by ... the interior of

the complementary. For exemple take as topological space the real

line. Interpret A by the open set (-infinity 0) then -A is (O infinity)

you see [A union -A] does not give the whole space, and this

shows the exclude middle is not universally valid, in IL.

Quantum Logic (QL) like IL, is a syntactically weaker logic than CL.

(And thus IL and QL are semantically richer). Algebraic semantics

for QL is given by the lattice of subspace of vector spaces (or Hilbert

spaces).

A good shape to remember is the following drawing (Arrows from bottom

to up = "syntactically extend":

S4 B

\ /

\ /

CL

/ \

/ \

IL QL

S4 gives a classical representation of IL, and B gives (thanks to

Goldblatt result) classical representation of QL.

(*) Godel was so famous that he did'nt need no more to prove his

affirmation/conjecture. It is a two pages paper without proof.

The Godel's "result" will be proved by McKinsey and Tarski in 1948.

Bruno

Received on Fri Aug 16 2002 - 06:32:44 PDT

Date: Fri, 16 Aug 2002 15:23:43 +0200

At 10:29 -0700 13/08/2002, Wei Dai wrote:

Right.

I almost missed this fair question. It is not a coincidence.

I choose S4 for not frightening Tim with irreflexive or non

transitive accessibility relation :-).

Kripke knew a 1933 result(*) by Godel according to which, (with IL

standing for Intuitionist logic):

Theorem: S4 proves T(A) if and only if IL proves A

where T is a function from the propositional language to the modal

propositional logic given by

T(A) = A if A is a propositional letter = A belongs to {p, q, r ...}

T(-A) = []-T(A)

T(A->B) = [](T(A) -> T(B)

Kripke first discovered his semantics for general normal modal logic.

By Godel 1933 this provides him (and us!) the "S4" possible world

semantics of IL.

Note that Beth developed a more awkward but similar semantics before.

The "world" of the S4 models (= of the IL) models are sometimes

interpreted as *state of knowledge*. This fit well with the fact

that the S4 logic in my thesis describes a pure first person knower.

(But I get S4Grz, its antisymmetrical extension, good for subjective

irreversible time).

The following remarks may help.

S4, which is a *classical* extension of CL

(Classical Logic), is capable of simulating IL. This is not an argument

that CL is better, for Godel (again) found that IL can simulate CL:

basically IL proves (- - A) when CL proves A. It's a key of IL that

IL does not prove (- - A) -> A. (but IL proves A -> (- - A))

Like CL admits an algebraic semantics in term of Boolean Algebra, ...

parenthesis:

(where the propositions A, B, C ... are interpreted by subsets of

a set W, the "and" by intersection, the "or" by union, the "-" by the

complementary, the constant f by the empty set and the constant t by

the whole set W. You can even (in our modal context interpret the element

of W as worlds verifying the formula, for example a tautology being

true in all world you see a tautology, like the constant t, is

interpreted by W). For example [A union -A] = W, i.e the exclude

middle principle is universally valid, in CL.

end of parenthesis

... IL admits a topological interpretation, where the propositions

are interpreted by open sets in a topological space. The "and" by

intersection, the "or" by union, the not by ... the interior of

the complementary. For exemple take as topological space the real

line. Interpret A by the open set (-infinity 0) then -A is (O infinity)

you see [A union -A] does not give the whole space, and this

shows the exclude middle is not universally valid, in IL.

Quantum Logic (QL) like IL, is a syntactically weaker logic than CL.

(And thus IL and QL are semantically richer). Algebraic semantics

for QL is given by the lattice of subspace of vector spaces (or Hilbert

spaces).

A good shape to remember is the following drawing (Arrows from bottom

to up = "syntactically extend":

S4 B

\ /

\ /

CL

/ \

/ \

IL QL

S4 gives a classical representation of IL, and B gives (thanks to

Goldblatt result) classical representation of QL.

(*) Godel was so famous that he did'nt need no more to prove his

affirmation/conjecture. It is a two pages paper without proof.

The Godel's "result" will be proved by McKinsey and Tarski in 1948.

Bruno

Received on Fri Aug 16 2002 - 06:32:44 PDT

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