85. Demonstration is a syllogism which produces science, or it is a reasoning which, by the aid of premises evidently true, gives a certain and evident conclusion. -- The sophistical syllogism is a source of error; the probable syllogism gives only verisimilitude; the demonstrative syllogism alone produces science, that is, certain and evident knowledge of a truth.
86. Demonstration is necessarily preceded by that species of doubt called methodical, and which is defined as Doubt which is supposed to attend a thesis before it is demonstrated. -- A truth to be demonstrated is first proposed in the form of a question, and the intellect is supposed to be in suspense between its affirmation and its negation; that is, it is supposed to doubt. This doubt, called methodical, bears only on the truth or truths to be demonstrated, and not on the indemonstrable principles. Unlike the systematic doubt of sceptics, or real doubt, methodical doubt is not actual, permanent, or universal; unlike the Cartesian doubt, it not only admits the veracity of consciousness, but also that of all the cognitive faculties, and does not touch self-evident truths.
Methodical doubt may bear on one of these four questions: 1. Does the thing exist? 2. What is its essence? 3. What are its accidents? 4. Why does it exist? The first question presupposes at least the nominal definition of the thing; the question regarding its essence supposes that of its existence already answered; the third question presupposes at least the notion of attribute; and the question of the wherefore of a thing can find its answer only in the principles or reasons of the thing; hence it is this last question that properly comes under the head of scionce.
87. All demonstration presupposes three notions: 1. that of the subject; 2. that of the predicate; 3. that of the middle term. -- All demonstration has for its end to show that a certain predicate agrees or disagrees with a certain subject by comparing both with a third term; hence it is clear that, prior to all reasoning, we must have the notion of these three terms.
88. The middle term of demonstration must fulfil three conditions: 1. It must contain the reason of the thing; 2. It must be known as the reason; 3. This reason must be eertain. -- Demonstration produces scientific knowledge by means of a middle term; but to know a thing scientifically, we must know the reason of it, know that it is the reason of it, and know it with certainty; hence the middle term must comply with these three conditions of science.
89. Demonstration is divided into a priori and a posteriori; direct and indirect or ad absurdum. -- A priori demonstration is that which descends from cause to effect, as when "from the existence of Providence we infer the order of the universe;" a posteriori demonstration ascends from effects to their cause, as when "from the order of the universe we infer the existence of Providence."
Direct demonstration proves not only that a thing is, but, moreover, why it is; as, "The soul is immortal, because it is a spirit." Mathematics abounds in examples. Indirect or apogogic demonstration simply shows that we must admit the thing on account of the absurdities which would flow from its denial; as, "If the soul is not immortal, there can be no mortal order." This kind of demonstration serves to prepare the way for science and to defend it, but it does not constitute science.
To indirect demonstration may be referred the argument called ex datis, so designated because from the concessions of an adversary we draw conclusions which are evidently against him; as, "You grant that the world could not make itself; then God must have created it." The demonstration called circular or regressive is at the same time a priori and a posteriori; a posteriori, since it ascends from effect to cause; a priori, since from the cause better known, it returns to the effect for a better knowledge of it; as, "The order we behold in the world proves the existence of Providence; and as there is a Providence, we are certain that even events unknown to us are ordained by it."
A demonstration is pure when the premises are analytical; as, "An infinitely perfect being is necessary being; but a necessary being is eternal; therefore an infinitely perfect being is eternal." It is empirical when the premises are experimental; as, "Water seeks its level; but this stream is water; therefore this stream seeks its level." It is mixed when one premise is analytical and the other synthetical; as, "There can be no effect without a cause; but this building is an effect; therefore it must have a cause."