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.

<SNIP>


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/



--
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to 
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at 
http://groups.google.com/group/everything-list?hl=en.

Reply via email to