Claudio Gutierrez

Translated from Spanish by Inés Gutiérrez

71. Applying the Board Representation as a Method of deduction

At the end of chapter VIII, we announced that when tackling the topic of deduction we would thoroughly illustrate the application of the board representation to it. This method offers us a wonderful quasi-mechanical technique that takes us by the hand along the reasoning process which constitutes what we call deductive method. We call deduction the logical process that allows us to go from the known truth of certain propositions to the truth (unknown at the beginning) of other propositions. Such process is extremely important, as our readers will be ready to concede, in all intelligent activities of ordinary life. But it is still more important in the creation and systematization of scientific doctrines, as we will examine with some detail in the last chapters of this work. Notwithstanding this importance, deductive basic operations are so simple that they can be represented graphically, and can even be treated as elemental moves of a board game.

Reasoning or deduction, in the same way as chess, military science, politics, or business, is founded on two easily discerned logical pillars: strategy, which is the long range planning for the solution of problems, and tactics, which comprises all the basic procedures capable of taking us, step by step, to the final goals anticipated by our strategic decisions. We will approach tactics first and proceed later to integrate them into alternative strategic perspectives. Both these stages will make ample use of graphic representation. Without further ado, let us begin the presentation of our basic elements of deduction, our deductive tactics. They consist basically in direct transformations of a logical structure into another logical structure, in accordance with a few very simple rules.

72. Tactics of Division and Composition

Let us begin with the rule of conjunction, assuming as already known the board graphic-representation technique previously discussed. The rule says:

A conjunctive formula can be opened, leaving its two parts free within the yard (board or part of double board) where the formula stands. Likewise, any two standing formulae can be connected under a conjunctive brace to produce a conjunction within the current yard.

Thus, the formula “” can transform itself into the following:

And, of course, also into the following:

where each one of the atomic formulae stands as independent. This application of the rule of conjunction will be known as the division tactic..

Inversely, the previous square can give rise to any of the conjunctive combinations of the four elements it contains. For instance the following:

This application of the rule of conjunction will be called the composition tactic..

73. Tactics of Commutation and Association

The same conjunction rule that allows us to use the previous tactics also permits the use of commutation or association. The first one consists of going from “” to “”, through the combined application of division and composition; the second, in going from “” to “”, also by means of applying division (twice) and composition (twice).

There is also a commutation tactic for disjunctive formulae; in order to apply it we need the rule of disjunction, which we formulate in the following way:

A disjunctive formula can be opened in a double board, created for that very purpose within the yard where that formula originally stands, by placing in each new twin yard each of the two constituents of the disjunction (the disjuncts); the formula thus opened can be closed again in the original yard by taking a complete independent formula from each of the yards and enclosing them under a disjunctive brace, to be placed in the yard that contains the double board.

Thus, the formula “”, opened once, would be represented as:

and, opened thrice, also as:

The first open boards from the inside out, can then generate –by closing up– alternate commuted formulae like “” or “.”

Disjunction, as conjunction, is associative, so that from the original formula analyzed one may also form “.” However, we cannot prove this case yet, based on the rules presented thus far. So, we will leave the proof of the associativity of the disjunction for later on. It is not a simple tactic since it makes essential use of strategic thinking. Hence, we will not count it as one of our basic tactics or elemental procedures.

74. Tactics of Repetition and Weakening

Let us recall the difference between the single board (strong truth) and a double board (weak truth). The single board means that all the propositions included in it are all surely true; the double board, on the other hand, says only that at least one of the twin yards contains only true propositions (or, in the case of existential quantification, that the quasi-proposition contained in one of them is true about some individual, not necessarily about all). Let us also remember that within twin yards there can be other boards still weaker than the yard that includes them. A double board inside a yard of another double board means that if that yard which includes it includes only true propositions, at least one of the two yards of the weaker double board will also be a yard that includes only true propositions. Now then, what is strong truth is also weak truth (as lawyers say: he who can do more can also do less), which means that we may voluntarily reproduce inside any weak board something that is already in the stronger yard that contains it. For example, we can go from

to

We will call this rule, rule of introduction. It says:

Any formula standing in a board can be introduced again in the same board, even within weaker yards comprised in it.

This rule does not allow the introduction of a formula in areas external to the particular yard where the formula originally stands; nor into the yard of an existential quantification flag. For example, it would be illegitimate to go from this configuration

to this one

Based on the rule of introduction we can now formulate two more tactics of inference. The first one is the repetition tactic: From “” one can go to “” (by means of introduction and conjunction). The second is the weakening tactic: From “” one can go to “”. For that we use the three rules, like this:

1. premise:       2. After applying the conjunction rule:

3. After twice applying the introduction rule:     4. After applying the disjunction rule:

All is good Something is beautiful therefore: Something is good and beautiful

1

2

3

75. Tactics of Separation and Compaction

When we formulated the disjunction rule, we saw that it was possible to move a formula from a secondary board to the immediate stronger yard by means of placing it within the disjunctive sign along with another formula from the complementary twin yard. This is legitimate because two twin yards mean that the total content of one of them is all true. Now we will give another rule, which we will call rule of promotion, allowing us to move a formula from a secondary yard to the immediately external one, in two very special cases, without the need to include it in a disjunction. The first one is when it so happens that two identical free-standing formulae exist in the two twin yards, one in each; consequently, that formula will have to be unconditionally true, since double boards tell us that the total content of at least one of them is true. Thus, from

we can go to

The second case occurs when, for particular circumstances, in one of the twin yards two formulae occur, one being the mirror image of the other. That, as we know, means that one is the negation or contradictory of the other). In such case, we also know that one for sure is false; therefore, the yard where both contradictory formulae are cannot be where the formulae are all true according to the meaning of the disjunctive board; the “true yard” will rather be the other one, which –we would have just discovered– contains only true propositions. Accordingly, we can proceed to move all its content to the immediately external board. Thus, from

we may legitimately infer

We eliminate the mirror twins as a penalty for their joint presence in the same yard since, logically, to contradict oneself constitutes a capital sin. As a result, we promote all the content of the other yard, since we now know that all is true.

We can now formulate our rule of promotion:

We can promote to the immediately outer board any independent formula of a yard which also occurs in its twin yard; or all the contents of a yard whose twin yard contains a contradiction, i.e. two formulae that are mirror images of each other.

The second case covered by the rule constitutes the separation tactic. As an illustration, let us use the famous problem from classical logic, modus ponendo ponens:

1. premises:       2. rule of disjunction:

3. rule of introduction:     4. rule of promotion:

Observe that the rule of promotion is the complement of the rule of introduction: the latter allows one to move formulae from a strong board to a weaker included board; the former, on the other hand, allows one to move formulae from weak board to a stronger encompassing board. Notice an important difference, however: the rule of introduction allows you to write the formula in any board internal of the yard where the original formula is; in contrast, the rule of promotion allows you to move it only to the immediately following encompassing board.

76. Tactics and Rules

Before we summarize the tactics stated thus far, let us clarify the difference between the concepts of “rule” and “tactic”. Rule is a general logical principle that tells you which transformations are legitimate; all that is not legitimate is considered illegitimate, that is it cannot be done. Tactic, on the other hand, is the application of one or more rules to a simple problem that appears often in the course of our arguments; it implies the application of one or more rules, one or several times.

We could very well learn only the rules, and not the tactics, and directly resolve with them our logical problems. But this would make our intellectual work harder. It is better to identify the rules most frequent applications, give them an easy name, and learn them by rote. It is a good thing to understand the rules well, to know why they are valid. They are conceptual, they help us to understand, but it is not indispensable to memorize them.

These are the tactics explained thus far:

1. Division:: Going from the conjunction of two propositions to the assertion of only one of them. Example: from “there is a tax and a financial crisis” to “there is a tax crisis”.
   
   

2. Composition: Going from two independent propositions to the conjunction of them. Example: from “there is a tax crisis” and “there is a financial crisis” to “there is a tax and a financial crisis”.
   
   

3. Commutation: Going from a conjunctive or disjunctive proposition in a certain order of constituents to the same molecular proposition but with a different order of constituents. Example: from “there is a tax and financial crisis” to “there is a financial and tax crisis”.
   
   

4. Association: Going from a conjunctive proposition which has a conjunction as one of its constituents, to the same structure but with the internal conjunction changed so that it includes the other constituent of the larger conjunction as one of its own constituents, the displaced one being promoted as the other constituent of the outer conjunction. Example: from “there is a tax crisis, unemployment and economic recession” to “there is unemployment, tax crisis and economic recession”.
   
   

5. Repetition: Going from a proposition to the conjunction of that proposition and another instance of it. Example: from “we must save” to “we must save and we must save”.
   
   

6. Weakening: Going from a conjunctive proposition to a disjunction composed of the same two elements. Example: from “there is a tax and financial crisis” to “there is a tax or financial crisis”.
   
   

7. Compaction: Going from a proposition disjunctive with itself to simply that proposition. Example: from “we must save or we must save” to “we must save”.
   
   

8. Separation: Going from a disjunction and the negation of one of its elements to the simple affirmation of the other element. Example: from “we do not save or resolve the crisis” along with “we save” producing “we resolve the crisis”.

77. Remarks on Quantification

The rules we have formulated and the tactics in which we have summarized their use are perfectly applicable to quasipropositions. Thus, from “” and “” I can go to “”; from “” and “” I can go to “”, etc. On the other hand, the “” or flag that appears in universal and existential quantifications raises some problems and compels us to establish certain restrictions to the rules; these are due to the quantifications character which, as we know, are very special “conjunctions” and “disjunctions”. Let us formulate those restrictions explicitly: Since “”, the first element of the universal quantification, does not mean anything, it must disappear when the quantification is opened. In practice, this means that the rule of conjunction allows us to open universal quantifications but does not allow us to close them. On the other hand, you may open or close a disjunction where an “” appears, but we cannot simultaneously open two existential quantifications, so as to avoid the fallacy of “illegitimate instantiation.” ⁽¹⁾. Finally, the “” can neither be introduced nor promoted, since these operations do not adjust to the particular sense of this symbol when it appears independently, namely, to serve only as a cautionary flag that warns you that what is asserted refers not to every but solely to at least one individual. However, this is no hindrance for the “” to be promoted or introduced as a part of another formula, namely “” or “.”