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chapter 3: On Mediate Judgment or Discourse - Gershom Carmichael, Natural Rights on the Threshold of the Scottish Enlightenment: The Writings of Gershom Carmichael 
Natural Rights on the Threshold of the Scottish Enlightenment: The Writings of Gershom Carmichael, ed. James Moore and Michael Silverthorne (Indianapolis: Liberty Fund, 2002).
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On Mediate Judgment or Discourse
On mediate judgment or discourse.
Whenever the mutual relationship of any two ideas is not adequately understood by the direct comparison of them, some other intermediate idea has to be brought in (as we have pointed out above), so that one may infer the relation of these ideas to each other from the perceived relation of both of the ideas in question with this mediate idea (or with several ideas properly related to each other). The act by which we give a verdict on the relation of ideas with each other in turn, resting on judgments previously given about the relation of both with the third, is called a mediate judgment or discourse, as we said above.
It makes no difference to the consequence, whether both of the extreme ideas are directly compared with the same intermediate idea, or whether only one of the extreme ideas is directly compared with the first mediate idea, and this with the second, and so on, until we end at the other extreme idea, in accordance with rules to be established below. However, because the order of nature and of teaching requires us to begin with the simpler ideas, we assume for the time being merely that the two extreme ideas are compared with one and the same intermediate idea in the same number of propositions; for the moment we ignore the question whether the relation of both of the extreme ideas with that mediate idea is known directly or by the assistance of one or several further mediate ideas.
Of the affirmative and negative syllogism and of their principles and parts.
A statement by which a mediated judgment as such is expressed, together with the two other judgments on which it immediately rests, is called a syllogism; in other words, it is a statement by which one proposition is explicitly inferred from two others, by which its two terms are compared with the same middle term. This is a verbal syllogism; for the actual progress of the mind through two judgments, by which the same number of ideas are separately compared with the same middle idea, to a third judgment by which the relation of the same two ideas with each other is determined, is called a mental syllogism. For reasons analogous to those I noted above under the name of proposition, most of what will be said below about the syllogism, should be particularly and primarily understood of the mental syllogism, and secondarily of the verbal syllogism.
A syllogism may be affirmative or negative. The affirmative syllogism is defined as that by which one concludes that the objects of two ideas are the same as each other, because it had previously affirmed the identity of both of them with the same object of a third idea. It rests on the axiom that things that are the same as some third thing are the same as each other; for example:
A negative syllogism is that by which one concludes that the objects of two ideas are not the same as each other; because it had affirmed the identity of one of them, and denied the identity of the other, with the same object of some third idea. It rests on the axiom that things of which one is the same as some third thing to which the other is not the same, are not the same as each other; for example:
In both cases we are considering only a syllogism which proceeds correctly (i.e., only a valid syllogism).
The conclusion from what has been said is that in every syllogism three ideas are compared. They are two ideas whose congruence or incongruence is inferred and which are usually called the extremes (in the example given above of the affirmative syllogism, these are the human mind and immortal), and a third idea, with which both of the extreme ideas are individually compared, so that either both are seen to be congruent with it, or one is and the other is not. This third idea is commonly called the middle; an instance, in the same example, is thinking thing. These three ideas, together with the words in which they are expressed, are commonly called the three terms of the syllogism.
Likewise it is also agreed that three propositions are contained in every syllogism. By one proposition the congruence or incongruence of the two extremes is inferred, and it is called the conclusion; as in the example given above, the human mind is immortal (of which the predicate, immortal, is called the greater extreme, and the subject, the human mind, is called the lesser extreme). In addition there are two other propositions, by which the two extremes are individually compared with the middle, and these are called premises. Of these, that which compares the major extreme with the middle is called the major proposition, as in the aforesaid example, every thinking thing is immortal; that which compares the minor extreme with the middle, is called the minor proposition, or assumption, as the human mind is a thinking thing.
On the general rules of syllogisms.
From the definitions given of both syllogisms (affirmative and negative), and from the axioms on which they are constructed, it is quite clear that in order for two extremes to be united with each other in a conclusion, it is necessarily required that both of them, in the premises, be united with the middle; and, in order for the extremes to be separated from each other in the conclusion, it is no less necessarily required that one of them be united with the middle in the premises, the other separated from the same middle. It follows that, if both extremes are separated, in the premises, from the middle, neither the congruence of the extremes with each other, nor their incongruence, will be able to be inferred. Hence naturally flow the rules which are commonly but rather confusingly (since they are mixed with others) taught as the rules on the quality of syllogistic propositions:
[Further specifications of these rules of the syllogism follow (sections 3–9), which follow the rules given in The Art of Thinking about the extension of terms, which as Carmichael notes, are not found in “the scholastic logics.”1 However the rules on “the quantity and quality of syllogisms,” which are found in the scholastic logics, and which include the “moods” and “figures” of the syllogism, may be deduced from the rules of the extension of terms. This Carmichael proceeds to do. In accordance with the plan of an elementary textbook, there is no discussion of sophisms and paralogisms. The chapter ends with the following general account of the nature of argument.]
On the syllogism considered with reference to its content.
Every perfect argument (such as all, in intention, seem to be) which is constructed according to the rules laid down above, if it rests on true premises, yields a true conclusion, and if it rests on certain premises, yields a certain conclusion; this is called demonstration. The certain intuition of the connection between the extreme ideas which is inferred in the conclusion is normally called knowledge. This intuition may be achieved in the very act of drawing the conclusion (an act which gets its own self-evidence and certitude directly from a distincter perception of the truth of the premises), or in a subsequent act which relies on a more general recall of an earlier demonstration.
But if even one of the premises on which the argument rests is uncertain, then the conclusion deduced is also doubtful and uncertain (so far as it flows from these premises). Such a mental verdict is called an opinion. It may be reached either in the very act of drawing the conclusion, by a particular inspection of the premises, or accepted from our confidence in a previous chain of reasoning.
Finally if any of the premises is false, it carries no weight, supposing it is false, toward establishing the truth of the conclusion. However it sometimes happens that a conclusion deduced from false premises is true, just as a conclusion drawn from uncertain premises is sometimes certain on other grounds.
But a syllogism which does not conform to the rules given above is said to be a paralogism, although this name is normally applied particularly to syllogisms in which the fault in form is obvious. Other paralogisms, in which the fault of form is concealed by some convoluted fallacy of ambiguous words or phrases, are usually called by the more particular name of sophisms.
[1.] E.g., Franco Burgersdyck, Institutiones Logicae. Carmichael’s colleague John Law reported to the faculty in August 1712 that “in the logicall years I began always the logicks by teaching those of Burgersdick.” (Glasgow University Archives 43227).