Claudio Gutierrez

Translated from Spanish by Inés Gutiérrez

78. Strategy and Tactic

Anyone who has ever been interested in military sciences, or at least in the games of chess or football, has some notion about the concepts of “tactic” and “strategy”. However, before using them, let us try to explicate each one in comparison to the other.

There is an important difference between tactic and strategy: Tactic is neither interested in any particular goal, nor in following a model or seeking a particular conclusion. It simply tells us what can be done starting from certain materials. Strategy, on the other hand, is eminently finalist: it is interested in a pursued end, in a model to imitate. Once we have clearly identified our goal, we go on to consider the available materials and the tactics or instruments to deal with them. A path towards the expected end is thus sketched. If we have enough imagination and the problem is solvable, we will always be able to reach our objectives. However, we should always take into consideration the possibility that a path exists which is ready-made for us to benefit from: we should look into the types of strategy minted for us by the tradition of logic. Although we might resolve the problems just by using rules or applying tactics, it should be easier to do so within the framework of one of well-known deductive strategies. This will not spare us from the use of imagination, but it will require much less effort than working always from scratch.

Even if we have still not formulated all the inference rules needed for our deductions, we have already a large enough armory of tactics to begin their application in relation to interesting problems. So, we are ready to begin our practical study of strategy. We will deal first with direct strategy, leaving the indirect type for later, when our repertoire of rules is complete.

79. Logic as a Game

The deduction process may well be considered a board game. The player is asked to resolve a problem, applying the “rules of the game”, and if he does, then he wins. This involves patience, and is for only one player, as the card game solitary. Nothing prevents, however, its being turned into a competitive one, two or more players trying to resolve the same problem, the one who resolves it first –or in less moves– being declared the winner. The problem constituting each match will have the following data: A few formulae, which we will call premises, our starting point. We place them on a board –which we will qualify as “main”– of maximally strong truth (not subject to any condition). And the model formula, goal of the game, which is our task to construct. We might place it also on the main board, but within a special enclosure to recall that it cannot be used as “construction material”, just look-at, as the plan of a building is placed somewhere at the construction site to be consulted periodically by the master builder. We call that formula, the conclusion. The game will consist in the transformation, step by step, of the available material into a formula identical to the model occurring in the enclosure. At the end, there may occur more formulae on the main board than that of the conclusion; the conclusion, though, must forcefully be present for one being declared the winner.

80. Simple Syllogism

The first strategic type we shall explain has a distinguished history that goes back to Greek Antiquity; it is called syllogism. Its constituents are two conditional premises, reducible as such to disjunctions, with an element in common; that common element serves as a bridge or link to integrate another two elements in a new proposition, the conclusion. Our first example will be a simple syllogism, so called because no quantifications occur in it. It is the following:

If we respect the law, there will be progress If there is progress, there will be welfare therefore If we respect the law, there will be welfare

Strategic reckoning makes us realize that the conclusion is a disjunction and, consequently, it is reasonable to decide that our last step in its formulation must be to close a disjunction; and the first, to open a disjunction. As we have two disjunctions as premises, and each one has a variable equal to a variable in the conclusion, it will amount to the same thing to open any of them. For instance, the first one:

One of the yards of the double board, the upper one, already contains what it must at the time of closing the disjunction; we must then focus on the other yard, where “” occurs but we would like to have “”. At this time, we should begin putting the other premise to good use, by including it in the frame containing “” so that they could interact together:

Until now, the only thing we have done is applying rules directly, following elemental intuitions about what is useful to do. But now, since we have a clear case of modus ponendo ponens in the lower yard, we proceed to apply the separation tactic in conformity with what was explained in sección 75. As a result, we obtain “” where we want it, along with the “”. We close a disjunction by taking a proposition from each yard of the double board, forming in the main board the conclusion we were looking for: “”.

81. Quantified Syllogism

Syllogisms can have quantified premises and conclusion; however, the strategy for their resolution is very similar to that of simple syllogism, especially if the quantification is universal in all its propositions.

Every country that progresses is enterprising No country where the citizens expect everything from the Government is ever enterprising therefore No country where the citizens expect everything from the Government ever progresses

First, we eliminate the conjunctions of both premises, making all the “” disappear, in conformity with what was said in section 77. Next, we apply the strategy of simple syllogism, beginning by opening any one of the premises. After applying the separation tactic, we close the disjunction to the result “.” The difference between this result and the conclusion is only that in the latter a universal quantifier occurs. The premise, on the other hand, is a quasiproposition asserted as strong truth, since it occurs in the main board. Consequently, it is sensible to add a universal quantifier, with which we win the game, the premise having been assimilated to the target proposition. But for legitimately doing this, we would need to be supplied another rule. For simplicity sake, tough, we prefer to establish that the rule of conjunction applies also within the virtual yard of the conclusion, which is entirely reasonable, since proving a conjunctive proposition is the same as proving each of its conjoints separately. ⁽¹⁾

82. Existential Syllogism

Another type of quantified syllogism is the one where existential generalizations occur. For instance:

Nobody who promises what he cannot fulfill is sincere Most politicians promise what they cannot fulfill therefore Most politicians are not sincere

As a preliminary step, we eliminate the universal quantifier from the first premise. We immediately recognize the other premise and the conclusion as existential propositions. The last step of our strategy must then be to close an existential quantification, and our next step, to open it. So, we open the existential quantification, with the following result:

Right after that, we eliminate the conjunction brace in the secondary yard. Introducing the other premise in that yard, we become ready to apply the separation tactic, given the coexistence of “” and “” within the same frame:

We now have in the same frame a “” and a “”, which we can easily join by means of the composition tactic.

In closing the existential quantification, we win the match.

There are many other syllogisms, formed by different combinations of categorical propositions. Their proofs are all very similar to those above.

It is important to take into consideration that some of the syllogisms taken as valid in traditional logic are not so considered by modern logic. The reason is that traditional logic labors under the hypothesis of existential content mentioned in section 69. For instance, the following traditional syllogism:

No is ; all are ; therefore: some is not
Try to prove it and you will see that it is not possible. On the other hand, by adding as additional premise the hypothesis of existential content “there is at least a ,” the argument becomes valid, as it is easily demonstrated by our method.

83. Weak Dilemma

The second type of direct strategy that concerns us, the dilemma, has also an illustrious history dating back to ancient times. The dilemma is considered as one of the most powerful persuasive recourses in existence, due to the clearness and elegance of this argument. It is characterized by the fact that it has three premises –a disjunctive one and two conditionals–; and that the components of the disjunction are, at the same time, antecedents of the conditionals. The weak dilemma culminates in a disjunction; the strong dilemma, in a simple affirmation. Let us first examine the weak dilemma:

If income tax increases, investment is discouraged If sale tax increases, consumption is discouraged Either income tax or sale taxes increases therefore Either investment is discouraged or consumption is discouraged

Since the conclusion is disjunctive, the last step of the deduction must be to close a disjunction; consequently, the first will be to open one. Given the fact that we have three disjunctions as premises, we might begin the game in three different ways; all leading to the same result. We take the path we deem most natural, namely, to open the disjunction where the two antecedents occur; immediately afterward, we include the other two premises in the twin yards, so that each conditional occurs as an independent formula in the yard where its antecedent also occurs. That simple.

We will then have two opportunities, one in each yard, for applying the separation tactic, getting free “” above and free “” below.

Taking these two formulae for closing a disjunction

we win the game.

84. Strong Dilemma

The strong dilemma differs from the weak one by the fact that the consequents of the two conditional premises are identical; this allows a unified conclusion. Example:

If income tax increases, there will be discontent If sales tax increases, there will be discontent Either income tax increases or sales tax increases therefore There will be discontent

We begin by applying the same strategy of the weak dilemma

but we will obtain the same formula in every yard
what allows us to apply the rule of promotion in its second variety
finishing as the match winners.

85. Polisyllogism

These two types of strategy, syllogism and dilemma, can be used several times in the resolution of the same problem; either separately or in combination, according to your needs. That is to say, there can be a multiple strategy of a polisyllogistic, polidilemmatic, or syllogistic-dilemmatic nature. Let us see some examples. Above all, a polisyllogism, whose formal strategy is very similar to that of the dilemma, as you will easily recognize:

All regressive taxes decrease the income of the masses All decrease in the income of the masses diminishes social demand All decrease in social demand provokes a recession in the economy therefore All regressive taxes provokes a recession in the economy

The universal quantifiers of all formulae eliminated, the chain reasoning becomes clear:

We open the central disjunction
introduce de remaining premises
use modus ponens
close the disjunction

                        and win.

Notice the structural similarity between the polisyllogism and the weak dilemma studied above.

86. Mixed Strategy

All these different strategies may be used in combination. The following is an example of the result of combining the syllogism with the dilemma:

Every increase in income tax decreases the revenue of entrepreneurs Every decrease in the revenue of entrepreneurs contracts social demand Every increase in sales tax contracts social demand Either there is an increase in sales tax or there is an increase in income tax therefore There is a contraction in social demand

The first two premises integrate a syllogism with the conclusion “”. This syllogism resolved, our problem transforms into a strong dilemma, complicated by two existential quantifications:

We open the disjunction
introduce premises
open upper existential and introduce formula

do modus ponens

close existential quantification
apply same strategy in lower yard
promote twins as one formula

and win.

87. Tacit Premises

It is not always the case that ordinary language provides us with all the propositions integrating an argument. Some are often omitted; because the person speaking or writing assumes that the hearer or reader will provide them him or herself, due to the fact that the propositions are clearly true. For instance, the polisyllogism analyzed previously can be formulated with tacit (meaning “silent”) premises, the following way:

All regressive taxes decrease the revenue of the masses therefore All regressive taxes provokes a depressive effect in the economy

Likewise, the syllogysm-dilemma that served to illustrate the mixed strategy could be formulated enthymematically (that is, with tacit premises) in this manner:

Every increase in income tax decreases entrepreneurs revenue Either there is an increase in sales tax or there is an increase in income tax therefore There will be a contraction in social demand

In all these cases what needs to be done before anything else is to state explicitly all what is assumed in a tacit manner. Only when we are reasonably sure that we have all the propositions “on the table” can we begin analyzing the argument structure and checking its soundness.