I. EPISTEMIC EXPRESSIONS
Those who deal with propositions we shall call propositioners. Assume that a is the name of the pro positioner (a term denoting the propositioner), and x is a proposition. Every proposition represents a special perceptible (visible, audible, etc.) object and possesses a meaning and a truth value. We classify as epistemic propositions only those propositions that pertain to the relation of the propositioner a to the proposition x in these three aspects. These are the following propositions. We shall say that a propositioner a is familiar with a proposition x as a perceived object if she can either pronounce or write x, has heard or seen x, remembers it, etc. The propositions recording this knowledge will be schematically written as 'a has x' (or 'a possesses x') and called the propositions of possession. The possession of a proposition represents a certain capability or a certain state of the propositioner. A propositioner can possess a proposition without knowing its meaning and truth value. We shall say that a propositioner a is familiar with the meaning of a proposition x (understands the meaning of x) if she knows the value of all terms occurring in x, and the properties of all logical operators occurring in x. Obviously, there may be various degrees and forms of understanding the meaning of a proposition depending on the level and the nature of education, the speech skills, and so on, which is of no importance here. For us it would be sufficient to assume that if a propositioner is familiar (in some form and to some degree acceptable among a given circle of people) with the value of terms and the properties of logical operators occurring in x, then she is familiar with the meaning of x (understands x) from the point of view of this circle of people. In order to understand the meaning of x, one has to possess this proposition in one or another way. The propositions recording the fact that a knows the meaning of the proposition x will be schematically written as 'a understands x' and called the propositions of understanding. In everyday speech the expression 'a does not understand x' is ambiguous. In one sense it means that a possesses a proposition x but does not know the value of certain terms or properties of some logical operators occurring in x. In another sense it means that it would be wrong to say that a understands x. And such negation can result not only from what has been said above but also from the fact that a does not possess x at all. Thus, it is necessary to distinguish between two kinds of negation here. If a propositioner a believes (thinks, states, etc.) that a proposition x is true (or if the situation is indeed such as stated in x), we shall say that a recognizes (or accepts) x. The propositions in which this fact is recorded will be called propositions of acceptance (recognition) and schematically written as 'a accepts (or recognizes) x\ With the propositions of acceptance, one does not imply that the accepted proposition as true. A propositioner can recognize a proposition as true if in reality it is not. The acceptance of a proposition x by a propositioner a represents a certain action of a, in particular, such as: a says to other persons or to herself that she recognizes x (considers it as true); in response to the question whether a accepts x or not she answers 'yes', nods her head, and so on. It is in the propositions of acceptance that this kind of the propositioner's actions are recorded. These actions are diverse. We shall agree that a accepts x if and only if she realizes or can realize under certain conditions (for instance, on request, if compelled, voluntarily) a certain action which can be classified as that of the acceptance of propositions. The negation of the statement 'a accepts x9 is also ambiguous. It is used actively, i.e., as 'a accepts not-x' and passively, i.e., as 'a does not perform any action which represents the acceptance of x\ We shall use it in the second sense. Actions of acceptance of propositions are diverse. Thus, situations are possible (and they do occur in reality) when the propositioner accepts x in some way (e.g., she says to herself or to close acquaintances that she regards x as true) and refrains from doing so in another, in particular, she says before a certain group of persons that she does not accept x. According to our convention with respect to the passive meaning of the negation of the acceptance of a proposition, one should not see any contradiction in this fact. If the propositioner states that she does not accept x, in terms of our agreement this means only that she does not accept x in one way (through an official declaration of acceptance), which does not make it impossible that she accepts x in a different way. A propositioner will be in the state of logical contradiction only if she accepts x and at the same time accepts not-x. In order to accept x as a proposition, the propositioner should understand it. Thus, here again there may be two possibilities for the negation of the statement 'a accepts x\ namely, the following ones: (1) a understands x but performs no actions to accept it; (2) one cannot say that a accepts x for the reason which not only coincides with that given in (1), but also of not understanding x. The propositions of acceptance are related to those of rejection and indifference. The former will be written as 'a rejects x'. They represent abbreviations or replacements of propositions of the form 'a does not accept 'not-*'. The latter we shall write as 6a is indifferent with respect to x\ They are abbreviations for such propositions as 'a does not accept x and does not accept not-x' or 'a does not accept and does not reject x\ Finally, epistemic propositions include propositions to the effect that a propositioner a knows that the proposition x is true (a is informed of x; a knows that x). Such propositions will be called the informative propositions or the propositions of knowledge; they will be written schematically as 'a knows that x\ In the case of informative propositions it is assumed that x is true, and a accepts it in some way (and, therefore, possesses the proposition x and understands it). A special case of such statements are propositions of the form 'a guesses that x\ 'a has heard that x\ etc. In general, there are certain states of propositioners which are recorded in the informative propositions. For the informative propositions it is also essential to distinguish between two kinds of negation. Without this, paradoxical consequences result. Thus, it is intuitively obvious that when they say 'a knows that x' the truth of x is implied. Quite similarly, one assumes that x is true when one says 'a does not know that x\ This logical association between statements can be expressed through the conditionality operator as follows: (1) if a knows that x, then x; (2) if a does not know that x, then x. By applying the rule of contra-position, we obtain from the first statement: (3) if not-x, then a does not know that x. And from assertions (2) and (3), according to the transitivity rule, follows that: (4) if not-x, then x. An assertion (5) if x, then x holds. From (4) and (5) we obtain: (6) if x or not-x, then x. And from (6), by virtue of the fact 'x or not-x' is logically true, it results that any proposition x is true. But regardless of such paradoxes the following is obvious. The negation of the statement 6a knows that x' may mean that x is true but a does not accept it, and that the statement 'a knows that x' is wrong.