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31 Hunie on Intuitive and Demonstrative Inference This paper is divided into four sections. The first section deals with Hume's attempt to resolve a dilemma concerning the objects of intuitive and demonstrative inference. In the second section I try to show that his resolution of the dilemma is hard to reconcile with his phenomenalist doctrine of the origin of ideas. In the third section I examine tne meaning of "intuition" in Hume. Finally, in the fourth section I compare his view of mathematical inference with his view of causal inference, a kind of inference to which he tends to assimilate the former. What are the objects of intuitive and demonstrative inference in Hume supposed to be? In the Treatise they are co-extensive with four kinds of philosophical relations resemblance , contrariety, degrees in quality and proportions in quantity or numbers. The distinguishing characteristic of the group, according to Hume, is that they all depend entirely on the ideas which we compare together. Tne example he gives is that of a triangle 2 whose three angles are equal to two right ones. And this presumably is at the same time an example of a relation of proportion in quantity or numbers. Needless to say, what cries out for explanation here, apart from the notion of comparison with which we shall deal later, is the phrase "depend entirely upon the ideas." Hume himself does not provide the explanation. Consequently, a certain amount of speculation becomes inevitable. One fairly plausible although, as we shall see, not entirely satisfactory interpretation of the phrase in question would associate it with a thesis concerning the reducibility of relations between exemplars of properties to relations between the properties themselves. Thus it may 32 be argued that the equality holding between the three angles of a triangle and two right angles is reducible to an equality holding between certain of their properties. The properties would be the 180° that each set of angles exemplifies. Indirect evidence for this interpretation is provided by Hume's account of the relations of contiguity and distance, species presumably of relations of time and place, which do not, according to him, depend upon the ideas wnich are compared together. Indeed, these relations may be changed by "an alteration of their place, without any change on the objects themselves or on their 3 ideas." "The place" in its turn "depends on a hundred 4 different accidents which cannot be foreseen by the mind." Leaving aside Hume's failure or refusal to draw a clear distinction here between ideas and objects which probably has to do with his belated realization that only the latter have places and are contiguous to or distant from one another, one notes his insistence that an object's place is not an essential property of the object. For his use of the word "accident" in the above-quoted passage has very little to do with its ordinary use and everything to do with its logical one where it is contrasted with "essential." Furthermore, the sentence immediately preceding the one containing that passage gives some indication of what Hume takes its logical use to be : A determinate property is an accidental one if loss of that property does not affect the object's identity or, as he would put it, the object itself. Nor is the accidental nature of the property anything but reinforced by its dependence logical or otherwise upon other accidental properties which like all such properties require an appeal to observation and induction in order to determine whether they are exemplified by that object. The loss of an essential property, on the other hand, would affect the object's identity because such properties are contained in the very definition of the object. And it suffices to understand the definition - Hume's foreseeing by the 33 7 mind comes in here - in order to determine that the property is exemplified by the object. How does this account provide indirect evidence for our interpretation of Hume? The answer is relatively straightforward. According to that interpretation, relations which depend entirely on the ideas which we compare together are tnose where relations between exemplars of properties are reducible to relations between...