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Leibniz's programme for mathematization of human knowledge was formulated in the following words: "Actually, when controversies arise, the necessity of disputation between two philosophers would not be bigger than that between two computists. It would be enough for them to take the quills in their hands, to sit down at their abaci, and to say (as if inviting each other in a friendly manner): Let's calculate! (Calculemus!)"
Unfortunately, Leibniz's attempts
to realize this idea were unsatisfactory. In a series of papers I succeeded
in translating different fragments of logic into arithmetic of integers.
To know them push the button:
You will see here the cover of my textbook on logic for 15-year-old
pupils of the Bulgarian secondary schools (they are in the 9th class). My paper
dedicated to the 70th anniversary of Prof. Dimiter Vakarelov is also included
You will find here: different axioms for intuitionistic necessity and possibility; adequate algebraic, topological, Montague and Kripke semantics for them; decidability of many logical systems; intuitionistic modal logics without the finite model property; intuitionistic temporal logics; axiomatization of the intuitionistic double negation as an intuitionistic necessity together with its adequate Kripke semantics.
All results shown here are known
since 1978. It is amusing to see publications of the late 90s considering
some particular intuitionistic modalities, direct analogues of the classical
ones. It is more amusing to meet intuitionistic possibility introduced
on the base of the intuitionistic necessity using the classical definition.
In fact, both basic modalities are rather different in intuitionistic logic.
Something more, the non-trivial operation is namely possibility being connected
with the intuitionistic disjunction. To obtain answers to your questions
push the button:
According to Arrow's celebrated Impossibility Theorem it is impossible to formulate a choice function satisfying en bloc certain natural conditions, if the voters must choose between 3 or more alternatives. One of the natural conditions is the non-existence of a dictator, i. e., a person able to impose all decisions. When the alternatives are only 2, Arrow showed that majority voting satisfies all "good" conditions and avoids the existence of a dictator.
However, the majority choice
between two alternatives is vulnerable in another sense: the set
of the decisions consequently taken by the voters is inconsistent only
if a fixed group (a clique, or an oligarchy) exists, able
to dictate all decisions. To see the details push the button