Claudio Gutierrez

Translated from Spanish by Inés Gutiérrez

96. Formal Fallacies

In the previous chapters we have studied inference patterns, i.e. the steps and processes of valid reasoning. We have done it rigorously, establishing rules that allow us certain transformations. In order to complete the picture we must now say that in logic, contrary to what regulates the field of individual rights, what is not allowed is prohibited. We may only perform those transformations expressly authorized by the rules, none other. Any move contravening the rules –as in games such as chess, checkers, or football– is reputed invalid and is branded a formal fallacy. At the beginning of this text we analyzed informal fallacies, which depend exclusively on non-structural features of language. Now is time to study this other type of fallacy, which depends on the existence of rules, even if it also obtains some persuasive power from non-logical aspects of language. The number of formal fallacies is indefinite, since rules may be broken in many different ways. In this chapter we will mention some of the most frequent and dangerous, so our readers may guard themselves against them.

97. Assertion of the Antecedent

One of the inference tactics studied, that of separation, allows us to reason based on a conditional proposition and another proposition identical to the antecedent of the conditional, so as to get the independent assertion of the consequent. In this manner:

If this remedy is effective, I recover soon It is the case that this remedy is effective therefore I recover soon

This is a perfectly correct argument, as we have already seen and may prove again by opening up the conclusion, introducing the second premise, and applying the rule of promotion over the consequent. But there is an “argument” very similar to this one that is not valid but we feel favorably disposed to accept it as such, simply because of its superficial similarity with the former. To wit:

If this remedy is effective, I recover soon I recover soon therefore This remedy is effective

The first premise is identical to the previous one and we have the same reasons to accept it as before; the second premise turned out to be true, since I got well. We should proceed to separate the antecedent, but there is no possible way to do so according to our rules. If I recovered, it may be perfectly possible that I did so for any other reason having nothing to do whatsoever with the remedy. However, the psychological force of this fallacy’s similarity with the previous valid argument may confuse us and lead us to assert the consequent with a presumptuous logical force.

98. Negation of the Non-Asserted

Another valid argument form is this:

Either society or delinquents are guilty It is so that society is not therefore Delinquents are guilty

We may prove it valid through the separation tactic. It would be applicable if we were certain that society is so well organized that it does not promote nor instigate crime in any form. But frequently enough we tend to “reason” the following way:

Either society or delinquents are guilty It is so that delinquents are therefore Society is not

which is invalid, for the same considerations that invalidate the previous fallacy.

99. Illegitimate Generalization

Another common formal fallacy consists in observing a few cases where two attributes coincide in the same individuals, and concluding that they always occur together, in all individuals. People with racial, political, or religious prejudices use this fallacy as argument in their attempt, bound to fail before analytical minds but often successful with the unprepared in strengthening their own faulty positions. Example:

Some Birlanders (1) exploit their costumers therefore All Birlanders exploit their costumers

All attempts to prove this conclusion based on this premise are doomed to failure: one may eliminate the universal quantifier from the conclusion and open up the premise in a double board, but these will not produce on the strong board. The strategy of indirect proof will not work either, as the reader may easily verify by himself.

100. Illegitimate Instantiation

Not only a mismanaged universal quantification can lead to fallacies; existential quantification may also do so, if one is careless. Thus, the following “argument form” is often accepted in spite of its lack of theoretical base on the logical rules:

Some lay organizations are secret societies Some Christian communities are lay organizations therefore Some Christian communities are secret societies

This scheme is not valid because, although you may open one of the disjunctions –let us say the second premise– the restriction in the rule of disjunction prevents you from opening a second quantification while a first one has not yet been closed ⁽²⁾. As traditional logicians used to say, “from two particular premises nothing follows”. The strength of this formal fallacy would be null without the passionate collaboration of prejudices and dogmatisms of diverse nature.

101. Absence of Link

Another case of formal fallacy occurs from the absence of link within syllogisms. For instance, when both premises are negative:

No friend returns books Some neighbors are not friends therefore Some neighbors return books

As you can see, the common term which should serve as link between the two premises is denied in both its occurrences; hence they cannot serve as counter twins for the rule of separation. Traditional logician dictum; “from two negative premises nothing follows”.

Many other cases of absence of link were treated by classical logic under different names and rulings, but for us they are all reduced to the very same anomaly. Two more examples:

All are No is therefore No is

All are All are therefore All are

102. Persuasion and Fallacies

It is interesting to ask ourselves whence do fallacies get their persuasive power; that is, what could the psychological trap be that makes many of the described fallacies seem more “evidently valid” than the following syllogism, perfectly correct:

No mortal is perfect Some perfect beings are men therefore Some men are not mortal

The trap may be different according to two cases: quantification arguments and propositional ones. Let us see.

Quantification formal fallacies may be explained by the passionate proclivities of humankind, who craves for easy condemnation of adversaries and of people simply different from our own group. Without the concurrence of these passions, fallacies of this kind would be all but incapable of exerting any persuasive power.

The other case is offered by fallacies such as assertion of the antecedent, negation of the non-asserted or syllogistic absence of link. Interestingly enough, here the “arguments” have not one but two formal discrepancies drawing them away from the corresponding valid structure: not only the two occurrences of the connecting term, appearing in the premises, are not counter twins as they should; there is also de opposite situation between the two occurrences of the other term: they are counter twin and should be identical. It does seem that one fault covers up the other, what apparently results in our tendency to take the scheme for valid. A confirmation of this diagnostic is the fact that, if we take the first syllogism and correct any of its two errors, it no longer seems as convincing as when both mistakes occurred jointly:

No friend returns books Some neighbors are friends therefore Some neighbors return books

No friend returns books Some neighbors are not friends therefore Some neighbors do not return books

These two semi-fallacies (each one has half the mistakes of the original) no longer cheat anyone; only by reading them we realize there is no argument there, since the conclusion “strays away” from the premises. However, they are “less invalid”, so to speak, than the aforementioned. The reader may do the same experiment with the other fallacies and verify that, by partially correcting them, they become less and not more convincing.

103. Inconsistent Premises

The validity of an argument is not enough to assert that the conclusion we have reached is true. As we explained in the introduction, one thing is that the process through which we pass from the premises to the conclusion be correct, and another, very different, that there be truth at the finishing point; for this to be the case there must also be truth at the starting point. The valid syllogism we consigned in the previous section, for instance, has a false conclusion, even though it is perfectly correct. This is so because one of the premises, the second, is clearly false.

We must then be especially careful of not accepting or proposing premises of whose truth we are not sufficiently sure. To protect ourselves, we must research the case in archives, in libraries, wherever necessary, so as to make sure that the propositions from which we depart are all true. Sometimes, nevertheless, there is no need to make any research: we know that the premise is true “by definition,” just as we explained in chapter VI. We must now add that, often enough, no research is needed to decide that a premise is false: this happens if the premises are inconsistent with each other, that is, one denies what the other asserts.

In chapter XII we saw that there may be contradiction between premises and an assumed conclusion (demonstrating thus the contradictory proposition of the conclusion through reductio ad absurdum). We now add that the reductio strategy is sometimes applicable to the premises themselves, without any need to pay attention to the conclusion we are intent in resisting. If you are able to prove that your opponent contradicts himself i>in the very premises, you would have achieved an enormous dialectic feat. It would be like destroying his weapons before he even had a chance to fire the first shot. So, it is very convenient to do take advantage of this whenever possible. But it is also indispensable, because if you let your opponent work with inconsistent premises, you are giving him the chance to prove anything, and to do so validly. Let us see an example to understand why:

Either there is social justice or short-term development, but not both Either there is social justice or long-term development, but not both Either there is short-term development or long-term development, but not both therefore There is no development, short-term or long-term

The conclusion is obviously false, assuming the premises as true, since it is the mirror image of the first part of the third pair of premises; however, it can be proved validly starting from the premises. Let us see; we begin by opening up the suspected premise:

We introduce some other premises appropriately
in each one of the twin yards we apply
separation twice, followed by conjunction
We now consolidate the twin yards by compaction, promoting to the main yard; open the conjunction and, with the help of the remaining premise and the separation tactic, produce and , assembling finally –by conjunction– the searched-for conclusion.

Notice that just after the compaction we had produced two mirror twins in the main yard, a fragrant contradiction! This is a clear proof that our set of premises is inconsistent. And now that we have at our disposal an unconditional contradiction, we know that we could have begun by applying the rule of creation to request any arbitrary conclusion (and its mirror twin), for instance: “we all must commit suicide immediately.” Let us do it.

We proceed as in the previous demonstration, but within the secondary upper yard.
Otherwise we follow same path as before:
we promote through compaction

Finally, by introducing the premise in the upper twin yard, a contradiction is generated, allowing us to promote to the main board the ridiculous conclusion of the universal suicide.

To fend ourselves against these awkward kind of valid arguments we should directly inspect the premises which, we suspect, are mutually inconsistent. One method to do it is to work with them independently from any conclusion, trying simply to derive any absurdity. For the above case, the steps could be:

We introduce premises
apply, as before, separation and conjunction
promote, and the searched-for contradiction gets generated.

Incompatibility of the premises thus proved, our opponent will have been disarmed and will be suffering the worst opprobrium dreaded by any genuine logician: to have spoken in an incongruous way.

Nota 1: Natives of Birland.

Nota 2: Nevertheless, see qualification in section 67.