Chapter XII: On Questions, and, in passing, on the way in which Sciences are extendedEdit
- Corresponding to the special principles of a science are special questions which must not be transferred from one genus to another, so that no discussion of a science with persons ignorant of it can lead to valid results. Two kinds of opposites to a science exist:—questions or demonstrations entirely outside its range and those which involve a breach of some of its laws.
If a syllogistic question be the same as one of the members of an alternative, and if there be premises in each science from which the syllogism belonging specially to each science may be deduced, there must be some scientific question from which the special syllogism corresponding to each science is derived.
It is plain then that not every question can be a geometrical or a medical question, and similarly with all other special sciences, but only those questions can be geometrical proceeding from which some of the matters connected with geometry are proved, or something proved on the same principles as geometry; e.g. optical theorems. The same is the case with other sciences. Now with regard to these questions, in the case of geometry they must be explained in accordance with the principles and conclusions of geometry, but no account need be given of the principles themselves by the geometrician as such, and this applies to other sciences also.
One should not then ask every possible question of a person acquainted with a particular science, nor need he answer every question asked of him, but only a question concerning the definite subject of the particular science. If one enter into a discussion with a geometrician as such, it is clear that the proof he gives will be a sound one if drawn according to these principles, otherwise unsound. It is also clear that in such circumstances one cannot confute a geometrician except accidentally, so that we must not discuss geometry before persons ignorant of that science, for any unsound arguments put forward will remain unnoticed. The same is the case with other sciences.
Since then there are geometrical questions, it may be asked whether there are also ungeometrical, and what kind of ignorance in connection with each science causes certain questions to bear the same relation to that science as ungeometrical bear to geometrical questions. Further is a syllogism resting on ignorance a syllogism formed from premises which contradict the science it belongs to, or rather a fallacy which nevertheless does belong to the science in question, e.g. geometry? Or, again, is a question belonging to another pursuit, such as a musical question, ungeometrical as regards geometry? Again, is the supposition that parallel lines can meet in one sense geometrical and from another point of view ungeometrical? ‘Ungeometrical’ is in fact an ambiguous expression, as is ‘unrhythmical.’ One thing may be ungeometrical or unrhythmical from not possessing the quality in question at all, another from having it defectively. So too the form of ignorance resulting from bad or defective principles is contrary to Science. In mathematical sciences the fallacy is more easily perceived than in other sciences, because in them the middle term is always expressed twice, something being predicated distributively of the middle term, and the latter in turn predicated distributively of another subject. The predicate is not however used distributively. In mathematics one may, as it were, see by an immediate act of thought the relations of the middle term, while in words they remain unnoticed. E. g. as regards the question, ‘Is every circle a figure?’ If one describe a circle on paper it clearly is so. If the conclusion be drawn ‘then the epic cycle is a figure,’ this is clearly untrue.
No objection should be raised to a science on the ground that its premises are inductive, for just as nothing can be a premise which does not apply to several instances (otherwise it would not be universally predicable, and Syllogism is drawn from universals), so an objection must have a universal application. Premises and the objections to them correspond to one another, and any objection one urges against a premise should be capable of serving either as a demonstrative or as a dialectical premise.
The laws of the syllogism are violated when the common attribute of both major and minor terms is treated as their predicate. An instance is the syllogism of Caeneus that ‘fire increases in geometrical proportion’; ‘for,’ as he says, ‘fire increases rapidly and so does geometrical proportion.’ No syllogism can, however, be formed thus. The truth is: if the proportion which increases most quickly in respect to quantity be the geometrical, and if fire be that which increases most quickly in respect to motion . . . .
Thus it is sometimes impossible to draw a conclusion from two premises of this kind, at other times it is possible, though the possibility may not be observed. If it were impossible to draw any true conclusion from false premises, it would be easy to bring the syllogism to a conclusion, for it would necessarily be convertible. For instance let A exist by hypothesis, and when A exists let something else (B for instance) exist also, which one knows in this instance does exist. By conversion then it may be shewn from B that A exists. Conversion is more frequent in pure mathematics because these admit of no accidental qualities (and in this differ from dialectical arguments) but only of definitions.
Mathematical science is advanced not by the use of a number of middle terms, but by the subsumption of one term under another (as A under B, B under C, C under D, and so to infinity). The process may also take two directions, A being predicable both of C and E. Suppose A represents any number definite or indefinite.
B any odd number of definite magnitude.
C any odd number whatsoever.
(Then A will be seen to be predicable of C).
Again:— Let D be an even definite number.
E any even number whatsoever.
Then A is predicable of E.