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THE THEORY OF ASSERTORIC CONSEQUENCES IN ALBERT OF SAXONY* CHAPTER III QUANTIFICA TION In the preceding chapter we examined Albert's theory of consequences among unanalyzed propositions, and we shall be concerned in the three following chapters with Albert's exposition of consequences among assertoric analyzed propositions. The present chapter will deal with the analysis of propositions in regard to their quality, quantity, opposition, and ampliation. The fourth chapter will study consequences from one analyzed assertoric proposition to another. These analyzed assertoric propositions are called 'quantified propositions'. The fifth chapter will study consequences from two quantified propositions to a third. These three chapters might be called the theory of the categorical assertoric syllogism ; the first two chapters being the theory of quantity of propositions, and the third may be regarded properly as the analyzation of the syllogistic. In our times some modern logicians have remarked that there are several errors in the so-called traditional theory of the syllogism; for instance, the contention of the traditional position that a particular proposition follows from a universal. The following examples are given to prove that such a contention leads to errors, and hence, the theory is inconsistent in some aspects. 'AU my dimes are shiny, therefore some of my dimes are shiny', 'AU centaurs are Greek, therefore some centaurs are Greek' ; If there are no centaurs, and I have no dime, then, the universal propositions are true, but the particulars are false; and, so, the inferences are invalid. If round squares are impossible, then, the following universal is true: 'AU round squares are round,' but the following particular is false: 'Some round squares are round;' consequently, to infer the last proposition from the first is invalid. Sometimes the universal proposition will be true whether there are, or there are not objects such as the propositions indicates ; but if there are not, then the partieSee Francican Studies 18 (1958) pp. 290—354. 13 14A.GONZÁLEZ ular wiU be false; and, therefore, the inference from the former to the latter is invalid, as the following examples show : 'All golden mountains are mountains, therefore some golden mountains are mountains/ 'AU Pegasi are winged horses, therefore some Pegasus is a winged horse,' 'AU brakeless trains are dangerous, therefore some brakeless train is dangerous,' 'AU dragons breathe flame, therefore some dragon breathes flame.' The preceding and similar instances prove that if V is the subject of a universal proposition, and 'x' is an empty term or null class; then, the universal having 'x' as subject is true, but the particular with 'x' as subject is false. Hence, the inference from universal to particular is invalid. Furthermore, all the laws of the logical square, except those of contradiction, are invalid; the same holds for the laws of conversion per accidens, and for all syUogisms with two universal premises and a particular conclusion.1 Lukasiewicz has proven in an historical and systematic analysis that Aristotle's theory of the categorical syUogism, is free from the errors that have been imputed to it regarding the existential import of propositions. The Aristotehan theory is exceedingly exact, although it has a narrow frame. The categorical syllogism is a deductive system of true propositions or theses concerning the constants: A (every), E (none), I (some), O (some-not). Almost aU theses of the Aristotelian doctrine are implications, or 'if . . . then' propositions. The variables of the system are term-variables, the values of which are exclusively positive, universal , non-empty terms. Thus, singular, negative, and empty terms are not introduced into the theory; which is the case also with the quantifiers . Another great and fundamental shortcoming of Aristotle's system is the supposition that all problems can be expressed by the four types of syUogistic premises: viz., universal affirmative, particular affirmative , universal negative, and particular negative; and that the syUogism is the only instrument of proof. The logic of propositions is absent from the system, although some of its laws are used intuitively. The important invention of the propositional calculus was due to the Stoics.2 After an historical and textual analysis of Aristotle's doctrine, Lukasiewicz builds up a system of the Aristotelian non-modal categorical syUogism according to the requirements of...