Claudio Gutierrez

Translated from Spanish by Inés Gutiérrez

88. Rule of Creation

The rules we have explained heretofore, and which we have applied within the direct strategy, are rules that allow us to operate with the material existent on the board; they allow us to open it, close it, reproduce or introduce it in innermost yards, promote it to outermost yards. With such rules we are able to resolve a great number of problems from elementary logic. However, we cannot resolve them all. Certain demonstrations which respond to indirect strategy require a rule that allows the creation of new material, that is, generating formulae different from and in addition of those initially given us as premises. Would it be possible to count with a rule of such a special nature? Of course, we could never authorize the creation of new formulae in boards of strong truth; that would be equivalent to accepting premises without any foundation whatsoever. But weak boards offer themselves as a promising possibility. Let us remind ourselves of the meaning of a double board: it asserts that at least one of its twin yards contains formulae as true as those in the yard where it exists. If now we consider the fact that of two opposing propositions (mirror twins), at least one has to be true ( tertium non datur⁽¹⁾, as medieval logicians used to say), we realize that it will be perfectly legitimate to introduce double boards, each containing only one of two mutually mirror-image formulae. So, it is possible to have a rule which allows us to create new material; but such creation must occur in secondary boards especially introduced to that effect, and the two formulae created must be the specular reflection of each other, each one being initially alone in each of the two linked yards. Here is the rule:

We may create within any space a double board, whose yards contain each only one formula, provided that both formulae are vertically symmetrical with each other (mutually mirror images along vertical axis).

This rule applies equally well to propositions and quasipropositions. It does not apply to the expression “,” which by itself is neither one nor the other, and whose mirror image is itself. We establish this as the only restriction to the current rule.

89. Logical Truths

The rule of creation has a very interesting feature: it allows us to demonstrate certain propositions without the need to use any preexistent material at all; that is, it offers us the possibility of arriving to certain conclusions without the help of premises. For instance, it renders possible the solution of a problem with no premises, such as the following:


As we have the freedom to create any formula and its contradiction in s double board, we proceed to create the two formulae that integrate the conclusion, which happened to be mirror twins of each other:

Closing the disjunction, we obtain a formula identical to the model formula.
Logically, we have created it from nothing, that is to say, without starting from any preexistent material. We have encountered a necessary truth, independent of any particular experience.


This kind of truths, commonly known as “logical truths”, do not require of any empirical research in order to be substantiated; it is enough to understand what they say to realize that we cannot but to assert them as true. For instance, the conclusion of the previous example conveys the message that either two particular conjoined propositions are both true, or that is not the case (i.e., at least one of them is false), statement which we immediately perceived as being necessarily true. So, the rule of creation means, in substance, that we may always have as additional premises in our arguments any of an infinite number of propositions, namely, logical truths. Their argumentative strength is as great as that of any good definition, inasmuch as they not only constitute true premises but also premises which are by necessity immune to the attacks of any opponent.

90. “Laws of Thought”

Within logical truths, infinite in number, there are three very famous ones that deserve special attention. The ancients considered them the laws of thought, the very foundation of logic. Today we find them in a more modest place, as some among a large number of propositions that may be proved without the need of premises, by means of the rule of creation. Their illustrative importance is, nevertheless, very great, and that is why we are dwelling on them with some care. They are three, namely: the principle of identity, the principle of contradiction, and the principle of excluded middle.

The principle of identity asserts that “what is, is” or, in a more logically exact form: “what is true, is true”; better yet, “if something is true then it is true”. The principle of contradiction asserts that “what is, cannot not be at the same time and in the same sense” or, more strictly speaking: “two self contradictory propositions cannot both be true”; better yet, “it is not the case that a proposition is true and also not true”. Finally, the principle of excluded middle asserts that “between being and not being there is no middle term” (tertium non datur), that is: “two contradictory propositions cannot both be false”; better yet, “a proposition is either not true or true”. Curiously, in our notation, the three “laws of thought” are simultaneously represented by means of the same formula, namely: “,” which is straightforwardly demonstrated without the need of premises by creating mirror twins in a double board, like this:

We create the two formulae that integrate the conclusion
close the disjunction

and get the desired conclusion. Easy enough!

As a matter of fact, the principle of identity as normally quoted has a conditional form: “if something is true then it is true”; that is, “if then .” As we already know, in order to represent the conditional in our formal vocabulary we must deny the antecedent and place it in a disjunction with the consequent: “.” The principle of contradiction, on its part, is quoted as the negation of a conjunction: “it is not the case that a proposition is true and not true”, or “it is not the case that and not .” We represent the conjunction “ and not ” as “.” Upon negation, it will look as the mirror image: “.” Finally, the principle of excluded middle says: “a proposition is either not true or true”; that is, “either or ,” represented as “.” In short: we find that, in our graphic system, the three “laws of thought” are reduced to one and the same law, a profound discovery.

There is an important relationship between each one of these laws and one of our rules of inference. Thus, the principle of identity is related with the rule of introduction, because what is strong truth is always strong truth (our rule of introduction allows us to repeat any formula considered true). The principle of contradiction, on its part, relates to the rule of promotion: two contradictory formulae in the same yard assure us that the content of the board is vitiated of falsity. The principle of excluded middle, finally, asserts that at least one of two contradictory propositions is true, allowing the creation of contradictories within a disjunction, which is the gist of our rule of creation.⁽²⁾

91. Indirect Strategy

The rule of creation, as the other rules, may be used tactically; that is, to directly transform propositions into other propositions, as need arises. Thus, for instance, it may be used to transform “” into “” by means of creating “” and its mirror twin, and introducing the “” in one of the yards of the double board (the one where there is no “”); this would be a new case of weakening. Nevertheless, the main use of the rule of creation is not tactic but strategic: it allows us a completely new style of deduction, which we call indirect strategy. In contrast to direct strategy, where the rules are applied to the material at hand, the rule of creation opens the novel path of actually creating a part or the whole of what we want to prove. In this manner, we are spared a lot of work. Furthermore, some conclusions can only be attained through this strategy. It constitutes, then, an irreplaceable resource of the logical arsenal.

Indirect strategy comes in two flavors, depending on the way we apply the rule of creation to our problem: it may be that we create exactly a replica of one of the parts of a disjunctive conclusion (and, of course, its mirror twin in the other yard), and in that case we would be building a hypothetical proof. Or it may be that we create the whole of the conclusion searched after (and its counter twin), in which case we would be employing a proof by reductio ad absurdum. We will now explain each one of these styles of indirect strategy.

92. Hypothetical Proof

Hypothetical proof consists in assuming something and seeing the consequences that follow. It is important when the requested conclusion is of a hypothetical character, that is, when it includes weak truths (a disjunction or a conditional). If, for example, we have to prove the conclusion “if then, if then ,” with the help of the premise “if and then ,” it may be convenient to assume that “” is true, in order to see what derives from it. For practical purposes, we can take “to assume” as equivalent to “to create” through the mirror twin technique. Let us represent the example:

The premise by itself does not allow us, through direct transformation, to achieve the conclusion, but we may request and its mirror image
, introduce the premise where the counter twin occurs
, open the disjunction and introduce
to finally apply separation of and close both disjunctions.

Observe that this problem is a case of association of disjunction, law which we could not prove in chapter XI because we still did not have at our disposal the rule of creation.

93. Reductio ad Absurdum

Often, in the middle of a discussion, we find ourselves unable to refute a proposition which, nevertheless, we deem false. Whenever this happens, we can think that there are not enough premises, and consequently simply request one, by means of the rule of creation. But which one? It does not require too much reckoning to find out: the most practical thing is to ask for the conclusion itself that we wish to form, paying, of course, the price of also creating, in the twin yard, its negation which would naturally coincide with the thesis of our opponent. So, what we are really doing is assuming that what he asserts is true, but with the mischievous intention of showing how easily a contradiction (mirror twin in same yard) will be generated by that thesis. Hence the name of this proof, reductio ad absurdum, since we are reducing our opponent’s thesis to a contradiction or absurdity. And, of course, if a contradiction occurs in one of the yards in a double board everything contained in the other yard may be promoted to the immediately surrounding board (rule of promotion).

Let us examine an example. The premises are: “if then ” and “if then ”; we want to prove the conclusion “if then and .” The example is artificially simple, so our assumed opponent must be a little limited, since he does not believe that our conclusion is true. We proceed to create in a double board the theses of our opponent and ours, which are mirror twins:

opponent's thesis  our thesis

We open the conjunction and introduce the premises in the same yard

separates and from premises and, through conjunction, the desired contradiction –which will allow us to promote our thesis– is formed.

It is important to note that reductio is such a powerful strategy that everything that can be proven through another strategy can also be proven, usually more easily, through this method. Therefore, anytime we do not see our way clearly, it is worthwhile to increase our premises by requesting the conclusion and its mirror image. If the desired conclusion is unqualifiedly demonstrable, this will surely derive a contradiction in the yard where its negation occurs.

94. The Logic of Irony

In section 44 we discussed an ironical text taken from the Commonwealth Magazine; premises commonly used to attack revolutionary movements of our time were used in it, as foundation to “demonstrate” that Herod’s actions, which led to the slaughter of the innocents, were perfectly justified. As we all agree that the slaughter of the innocents was a barbaric crime, the fact that the demonstration hold can mean only one thing: one of the premises used in the proof has to be false. So, the logical structure of an ironical text clearly corresponds to the strategy of reductio.

95. Proof and Contraposition

In this chapter we have been speaking about about “proofs,” either “hypothetical” or “by reductio ad absurdum.” But we have been presenting proofs since the previous chapter. A proof is a graphical expression of an argument. When using ordinary language, each step will probably be expressed as a line; in our case, where we use boards and other graphical elements, we might say that each step is expressed by a box, something like a movie film composed of frames, which correspond to instants in a movement sequence. A proof is then the movement of logical thought passing from one situation to another, in accordance with certain rules (in the case of movies there are also rules: they are the rules of kinematics and optics, the branches of physics that have most to do with cinematography). Logical proofs may be long and complicated, but they may also be short and simple. The tactics studied, for instance, are of the latter kind

In our system of logical representation, as you surely have noticed, the metaphor of “mirror reflection” plays a crucial role. It serves us, above all, to express the idea of negation; but it also plays an important function in the rules of promotion and creation. Now that we have outlined the concept of “proof” we may put the same metaphor to an even more powerful use. It consists in the following: if we have a proof with only one premise, we may reflect in the mirror not only a single formula but the whole proof. Surprisingly enough, we will obtain another valid proof! ⁽³⁾ This proof, called proof by contraposition, is a great additional resource in the logical arsenal: if we confront a difficult problem, we may “reflect it in the mirror” to see if the resulting problem is easier to solve, since it is certain that if a proof is valid, its contraposition is also valid.⁽⁴⁾

Let us use weakening as material to test this strategy:

Ignore the conclusion enclosure.

reflect the whole proof, producing another proof
, namely the division tactic:

Another example with compaction:



by reflecting the whole proof
, you obtain another proof, the repetition tactic:

We say about proof pairs like these that both specular symmetrical proofs relate to each other by contraposition. In the case of proofs with two or more premises, we can also contrapose, provided we first unite all the premises through conjunction, transforming them into a single proposition. From then on, in fact, we have only one premise, and we may apply –to it and its conclusion– the same strategy. We recommend the reader to try out the procedure with some of the proofs with several premises that we have shown heretofore.