Within the framework of classical Aristotelian logic, a paralogism is defined as an invalid syllogism. These paralogisms of deduction are “arguments of traditional syllogistic form which break one or another of a well-known set of rules” (Hamblin 1970, p. 44).
1. Syllogism rules
Traditional logic has established the following rules, which make it possible to eliminate invalid syllogisms. The following syllogisms respect all the rules of the syllogism; they are valid.
A syllogism contains three terms.
From two negative premises, nothing can be concluded
no M is P
no S is M
If a premise is negative, the conclusion must be negative
no M is P the major premise is negative.
some S are M,
so some S are not P the conclusion is negative
In a valid syllogism, the medium term must be distributed at least once
no M is P M is distributed (universal).
all S are M,
so no S is P the conclusion is valid.
If a premise is particular, the conclusion is particular
no M is P
some S are M the minor premise is particular.
So, some S are not P the conclusion is particular.
A paralogism is a syllogism that does not respect one or several preceding rules. Of the 256 modes of the syllogism, 19 modes are valid; therefore, a syllogism can be fallacious in 237 different ways. The question of whether it “seems” conclusive or not is irrelevant. The term paralogism designates nothing other than a mistaken calculation.
The following are key forms of syllogistic paralogisms. The first form corresponds to the paralogism of homonymy, the others to an inadequate distribution of qualities and quantities.
(1) Paralogism of four terms.
(2) Paralogism concluding from two negative premises.
(3) Paralogism drawing a positive conclusion from a negative premise.
(4) Paralogism of the undistributed middle term.
(5) Paralogism of universal conclusion from a particular major.
(6) Paralogism of universal conclusion from a particular minor.
— The following paralogism consists of four terms:
Metals are simple bodies.
Bronze is a metal.
* Therefore bronze is a simple body.
Bronze is not a simple body but an alloy. In the minor premise, bronze is said to be a metal because it looks like an authentic metal such as iron, it can be melted and molded, etc. In the major premise, metal is used with its strict meaning. Metal is homonymous, and the syllogism actually has four terms; S. Ambiguity.
— The following paralogism concludes from two negative premises:
Some B are not C some rich are not arrogant
No A is B no poet is rich.
* Therefore No A is C * no poet is arrogant.
— The following paralogism concludes universally from a particular major:
all A are B all men are mortal
no C is A no dog is man
* Therefore No C is B * no dog is mortal.
In the major premise, “all men are mortal”, the major term, mortal, is not distributed: this premise says nothing of all mortals, but only of certain mortals, namely, that “they are men”. Yet the conclusion “No dog is mortal” claims something of all mortals: “no mortal is a dog”. So the major term is distributed in the conclusion and not in the major premise. The conclusion thus affirms more than the premise, which is impossible.
3. Evaluation using the rules of the syllogism
Syllogisms are traditionally evaluated on the basis of a system of rules (§1), in a step-by-step process:
— Check the number of terms, and propositions.
— Identify the middle term, the major term, and the minor term.
— Determine the quantity and quality of the premises and conclusion.
— Identify the distribution of terms.
— Check the organization of the distribution of terms: check that the middle term is distributed at least once. If the major term or the minor term is distributed in the conclusion, make sure that they are also distributed in the premises; etc.
This laborious method is based on the notion, at the very least unintuitive, of the quantity of the predicates. It shifts the analyst’s attention from the understanding of the structure and articulation of the syllogism, from what the syllogism asserts, to the fragmented application of a system of rules. It may develop the ability to apply an algorithm, but it is far from an everyday critical thinking process.
4. Evaluation with Venn diagrams
The use of Venn diagram provides a more intuitive and clear base for syllogism assessment. Three intersecting circles represent the three sets which correspond to the three terms. The assertion made by each premise is carried to the corresponding circle. If a premise asserts that a set (made up of a circle or a portion of a circle) contains no elements, that circle or the portion of a circle is blacked out (striped). If a premise asserts that a set (id.) contains one or more elements, a cross is placed in the circle or portion of a circle. A portion of a circle is therefore either black, has a cross, or remains white. When white, nothing can be said about it.
The data of the premises having thus been plotted on the diagram, the result can be compared with what the conclusion asserts, the diagram shows whether the syllogism is or is not valid.
Consider the syllogism:
Some rich people are not arrogant
No poet is rich
* No poet is arrogant
The three intersecting circles represent the rich (R), the poets (P) and the arrogant (A), respectively.
— “Some rich are not arrogant”: consider the circle of the rich and that of the arrogant; put a cross outside of their intersection: there are some people within this zone.
— “No poet is rich”: consider the circle of the poets and that of the rich, and blacken their intersection: there is nobody within this zone.
— Finally, look at the circle of poets and that of the arrogant people. The conclusion affirms that the intersection of the circle of poets with that of the arrogant is black; but we see that this is not the case; it is partly white. This syllogism is a paralogism.
Consider the syllogism:
No M is P
All S is M
Therefore No S is P
The three intersecting circles represent the M set, the S set and the P set.
— “No M is P”: the intersection of the circles M and P is black (empty).
— “Every S is M”: the non-intersecting zone of the circles S and P is black (empty).
— Looking at the S circle the P circle, we can see that the intersection is black (empty); this is precisely what the conclusion claims, “No S is P”. This syllogism is valid.
5. Paralogism of quantifier permutation
By generalization, the word paralogism can refer to any error made in applying the rules of formal logic. For example, the paralogism of quantification is an error committed when the existential and the universal quantifier are permuted:
All human beings have a father; so they have the same father
For every human H, there is a human F, such that F is the father of H
* Therefore There is a human being F such that for every human being H, F is the father of H.
The following passage may contain such a paralogism, albeit complicated by a fallacious verbiage that is to say an eloquent amplification, S. Verbiage:
And all the geniuses of science, including Copernicus, Kepler, Galileo, Descartes, Leibnitz, Buler, Clarke, Cauchy, speak like [Newton]. They all lived in true adoration of the harmony of the worlds and of the all-powerful hand that threw them into space and sustained them.
And this conviction is not based on impulses, like poets. Figures, theorems of geometry give it its necessary basis. And their reasoning is so simple that children would follow it. They establish, first, that matter is essentially inert. It follows that, if a material element is in motion, it is because another has constrained it; for every movement of matter is necessarily a communicated movement. They thus claim that since there is an immense movement in the sky, which carries away in the infinite deserts billions of suns of a weight which crushes the imagination, it is because there is an all-powerful motor. They establish, secondly, that this movement of the heavens presupposes the solving of the problems of calculation, which have required thirty years of study, etc.
Ém. Bougaud, [Christianity and the Present Times], 1883.