On 09 Nov 2012, at 13:19, Roger Clough wrote:
No doubt Bruno has already figured out the relationship
between the necessary and the contingent, perhaps
in his levels of sigma, but at any rate, some logician
has done this, where below
the necessary (Platonia?): [] or it is necessary that.. and
the possible (world ?) <> or it is possible that
At any rate, there are a number of other forms of existence
given by modal logic as indicated next which provide a
sort of theology of existence:
http://plato.stanford.edu/entries/logic-modal/
"What is Modal Logic? (a theology of existence)
Narrowly construed, modal logic studies reasoning that involves the
use of the expressions ‘necessarily’ and ‘possibly’.
However, the term ‘modal logic’ is used more broadly to cover a
family of logics with similar rules and a variety of different
symbols.
A list describing the best known of these logics follows.
Logic Symbols Expressions Symbolized
Modal Logic □ It is necessary that ..
◊ It is possible that …
Deontic Logic O It is obligatory that …
P It is permitted that …
F It is forbidden that …
Temporal Logic G It will always be the case that …
F It will be the case that …
H It has always been the case that …
P It was the case that …
Doxastic Logic Bx x believes that …
2. Modal Logics
The most familiar logics in the modal family are constructed from a
weak logic called K (after Saul Kripke). Under the narrow reading,
modal logic concerns necessity and possibility. A variety of
different systems may be developed for such logics using K as a
foundation. The symbols of K include ‘~’ for ‘not’, ‘→’
for ‘if…then’, and ‘□’ for the modal operator ‘it is
necessary that’. (The connectives ‘&’, ‘∨’, and ‘↔’
may be defined from ‘~’ and ‘→’ as is done in propositional
logic.) K results from adding the following to the principles of
propositional logic.
Necessitation Rule: If A is a theorem of K, then so is □A.
Distribution Axiom: □(A→B) → (□A→□B).
etc. etc. etc.
Yes, and G is K (above, same axiom, same Rule) + the formula []([]p-
>p)->[]p. (Löb's formula)
G captures what any sound platonist machine having enough beliefs in
arithmetic will be able to prove about herself when described at some
correct 3p-level. The 3-I.
The main axiom for the machine 1-I are []p -> p, and the more
sophisticated []([](p->[]p)->p)->p. (Grzegorczyk's formula). Again,
same rules.
For the notion of (intelligible, sensible) matter, the main formula
will be p->[]<>p, with <>p put for ~[]~p. But without the
necessitation rule, but still the axiom []p -> p.
K is the common part of all modal logics known as "normal modal
logic", and they main characteristic is that they have a semantic in
term of many-worlds (in a very general sense), with worlds being
accessible or not between each others.
Modal logic is a part of mathematical logic. Many different modal
logics exist, and have their corresponding applications. Modal logic
has been invented by Aristotle, to reason in metaphysics and theology,
and mathematicians get serious about it after Kripke found a very
handy mathematical semantics, capable of distinguishing easily many
theories. To be sure, before and after Kripke, other semantics exists,
notably in term of relational algebra, topological spaces, etc. It is
a large field. G is a normal modal logic, but G* is already not.
Bruno
http://iridia.ulb.ac.be/~marchal/
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