Aristotle's Thesis is a negated conditional, which consists of one propositional variable
with a negation either in the antecedent (version 1) or in the consequent (version 2).
In this paper we focus on the intuitionistic propositional logic with one propositional variable.
A valuation from the set V ar of propositional variables to a (B, [less than or equal to]) is a function h : V ar [right arrow] B.
n); xj being a literal (a propositional variable
or its negation).
We can see it as a proposition and we can associate it a propositional variable that represents it.
h]-tuple, that can be considered as a propositional variable p.
r], then associating to each valuation a propositional variable, we obtain nr propositional variables, and each one can take a true or false truth-value, depending on the satisfaction of the atomic formula.
More generally, in this intensional setting a single propositional variable about which we know nothing else but that it is assumed to be true logically implies nothing but itself.
Confronted with this potential multitude of variables, we have no right to single out ~q': since the ~ p' of the consequent is true under the constraint of the antecedent, k is in true disjunction not only with any propositional variable but with every propositional variable, and so in effective disjunction with none in particular.
These expressions may be propositions in natural language in which the phrase ~logically implies' or some equivalent appears, but they may also be formulas in which propositional variables replace propositions and in which the existence of the relation of logical implication is explicitly asserted by virtue of a symbol designed to do just that.
These formulas extend propositional formulas by allowing both universal and existential quantifiers over propositional variables
, and are useful for modeling problems in artificial intelligence and computer science.
Its preliminary assumptions include the following: (i) Quantification over propositions and propositional variables