Logic Of Certainty Research Articles

In the logic of certainty, the material conditional corrective is stronger than the indicative conditional connective

It is almost universally assumed that the indicative conditional connective is stronger than the material conditional connective. In the logic of certainty, however, the deduction theorem for the material conditional connective fails, and consequently the material conditional connective is stronger than the indicative conditional connective. One implication of this is that the import–export rule and modus ponens for the indicative conditional connective can both hold, without the indicative conditional connective collapsing into material conditional connective.

Geometric interpretation of subjective probability: random numbers and objective conditions of coherence

In the domain of the logic of certainty we study the objective notions of the subjective probability with the clear aim of identifying their fundamental characteristics before the assignment, by the individual, of the probabilistic evaluation: probability is an additional and subjective notion that one applies within the range of possibility, thus giving rise to those gradations, more or less probable, that are meaningless in the logic of certainty. When we study the criteria for evaluations under conditions of uncertainty and their corresponding conditions of coherence we show an inevitable dichotomy between the subjective or psychological aspect of probability and the objective or logical or geometrical one. The affine properties are the basis of essential concepts of probability theory and only they make sense, being independent of the choice of a coordinate system; however, the importance of the metric properties appears in order to represent random numbers and analytical conditions of coherence.

On the Dichotomy between the Psychological or Empirical Aspect of Subjective Probability and the Logical or Geometrical One

In the domain of the logic of certainty we examine the objective notions of the subjective probability with the clear aim of identifying their fundamental characteristics before the assignment, by the individual, of the probabilistic evaluation. Probability is an additional and subjective notion that one applies within the range of possibility, thus giving rise to those gradations, more or less probable, that are meaningless in the logic of certainty. Each criterion for evaluations under conditions of uncertainty is a device or instrument for obtaining a measurement; it furnishes an operational definition of probability or prevision P and together with the corresponding conditions of coherence can be taken as a foundation for the entire theory of probability. When we examine these criteria and their corresponding conditions of coherence we show the inevitable dichotomy between the subjective or psychological or empirical aspect of probability and the objective or logical or geometrical one.

The Logic of Subjective Probability

There is a sense in which our logics of truth and of certainty should coincide. They should satisfy what I call the logical correspondence principle. By a logic of truth I mean a system of logic such as the propositional calculus of Whitehead and Russell (PC) the theorems of which are truth tautologies, (i.e. propositional formulae all instances of which are propositions that are true in all possible worlds). By a logic of certainty I mean a probability system in which the range of possible probability values is restricted to i and o. Now I say that a probability system A and a system of propositional logic B satisfy the logical correspondence principle if (a) P(a) = i is a theorem of A iff a is a theorem of B, and (b) P(o) = I,