inaccesible worlds
TRANSCRIPT
Enigma, carefully concealed from outsiders
…though logical necessary sentence should be true in any logical world but (alas!)
in Kripke type semantic nothing is talking about whether such worlds belong to the alternatives to an actual world
Individuals domain
N. Cocchiarella:
supply Kripke semantics with the condition which requires that in alternatives to actual world all models be incorporated (in usual meaning of the first-order logic) with the same individuals domain like ones in actual world
Cocchiarella’s condition leads to that individuals which are quantified in alternative worlds should be completely defined in actual world that in turn implies their obligatory presence (or, at least, complete determination) in actual world.
domains of alternatives to an actual world should be apparently somehow limited for otherwise inevitably set-theoretic paradoxes arise induced by consideration of arbitrary big powers of individual alternatives to an actual world
Besides,
Accessibility
Splitting modal operators
ΩA means “A is logically necessary” and □A means “A in a sense is logically necessary”
In semantics for relative alethic modal logic with those operators one have to consider not the one but the whole family of binary (accessibility) relations on a set of possible worlds
A is logically possible in given world iff one discovers A in at least one possible world accessing by means of all spectrum of accessibility relations;A in a sense is logically possible in given world iff one discovers A in at least one possible world accessible from this world by means of at least one accessibility relation.
If enrich now our language with the two
more operators of possibility and ◊ ∇
(where ∇A means “A is logically
possible” and ◊A means “A in a sense is
logically possible”) then one has to
accept too more postulates:
I. Humberstone:
A is logically impossible in given world iff one discovers A in all possible world inaccessible by means of any accessibility relations
systems of inaccessibility logic which is obtained by
introducing into language a new
connective ▲ (▲A means “A is logically
inaccessible”)
Transcendent inaccessibility
If we consider an operator of logical necessity Ω as the operator of “transcendental” necessity, then an operator ▲ would be called the operator of ‘transcendent” inaccessibility or simply transcendency operator since it supposes that neither under any conditions, nor in any sense we will be able to access a possible world in which some needed formula will be true.
Such terminology is especially transparent for epistemic logic when the case in question will be the epistemic inaccessibility of alternatives to given world
J.Hintikka
■A means “A in a sense is logically inaccessible” featuring as
Transcendency in a sense:
A in a sense is logically inaccessible in given world iff one discovers A in all possible world inaccessible from this world by means of at least one accessibility relation
Duals ▼ and ♦ of the operators ▲ and ■
A is possibly transcendent for given world iff one discovers A in at least one possible world inaccessible from this world by means of all spectrum of accessibility relations
A is in a sense possibly transcendent for given world iff one discovers A in at least one possible world inaccessible from this world by means of at least one accessibility relation
▼A means “A is serial transcendent” and ♦A means “A in a sense is serial transcendent” (by analogy with the seriality property of relations where for any element always at least one would be found connected with it by given relation)
Besides, the pairwise duality holds:: ■A is equivalent to negation of a serial
transcendence of non-A,
▼A coincides with negation of in a sense transcendence of non-A while serial transcendence of A one can understand as negation of the transcendence of non-A.
Forbidding redoubling transcendence in a sense If one will adhere the comprehension of ■A as “А
in a sense is transcendent” then since there does not fixed meaning in which A is logically inaccessible (it is just maintained that this sense exists) then from informal considerations it follows that we have to forbid (as it was remarked in case of the operator □ of logical necessity in a sense) redoubling transcendence in a sense.
Such redoubling is based on the transitivity of accessibility relations. But if some world is accessible in a sense and then the following one is accessed from that, again in a sense, then this in no way means that the first sense is the same as the second one. And if so, then our third world is accessible just by virtue of some composition of senses, but we have not any language resources for yielding such sense constructions,
`
An inaccessibility by means of some inaccessibility relation does not mean the lack of accessibility with the help of another one which forbids, in essence, a transcendence of given world in respect to another one. In this case it rather have to talk on (logical) transcendentality since an accessibility of one or another possible world is determined now by the specific accessibility relation i.e. by some logical abilities which do define properties of that relation revealing through an accessibility for him some possible worlds and an inaccessibility of others. Especially that the transcendence requires (by its meaning) a principal inaccessibility by means of no one relation from families of accessibility relations belonging to given ontology.
COMMENTARY
Operator of transcendentality
Trascendentality of A means that А is transcendent in a sense and is logically possible in a sense
⇓A is read “A is transcendent”
Serial trascendentality of A means that А is serial transcendent or is logically necessary
Transcendentality considered and Hintikka’s transcendentality
If regards that logical inaccessibility means the transcendence of respective possible worlds
and bimodality in respect to alethic modal logics means accounting the specificity of logically possible worlds
then the treatment of the notion of logical
transcendentality really means some attitudes of our conceptual system
(a theory of logical transcendence as inaccessibility of alternative worlds, understanding transcendentality as the result of simultaneous inaccessibility and accessibility of worlds in different senses and etc.)
Ontology of negativeness
we have to take into account the relationship of those, put forward the hypothesis of the satisfiability of some assertions in them, to discuss their logicality or non-logicality etc
in the process of analysis of such modal systems we can obviate some hidden rocks on which according to Hintikka’s analysis quantified alethic modal logic can be crashed
not only possible worlds are taking into account but also possible worlds inaccessible
for one another
During the whole study of the system of relative alethic modal logics the point was just the propositional alethic modal logics and problems arising does not required an essential revision of the notion of logically possible worlds on the base of analysis of individual domains of possible worlds and respective ontological commitments of alethic modalities. But such problems inevitably appear during the transfer to quantified transcendental modal systems.
The end
Thank you for your attention