Claudio Gutierrez
Translated from Spanish by Inés Gutiérrez
Claudio Gutierrez
We have studied deductive arguments that allow us to ascertain the truth of a conclusion based on the truth of its premises. We may call those arguments strong, since they prove a proposition with all the force of conviction already possessed by the premises. But there are also other arguments, important in discussion and research, which do not prove so much; they do not prove that a conclusion is true, neither do they assume that the premises are: the only thing asserted by them is that a set of propositions are consistent between themselves, that there is no reason to establish a priori (from the start) that at least one of them is false. Because of their being less pretentious, not searching for truth but just for simple consistency, we call these arguments, weak deductive arguments.
In the previous chapter we saw the importance of demonstrating that certain premises are inconsistent; this may simply destroy our opponent’s argument before he even has a chance to launch them against us. This led us to say that the argument to prove the consistency of premises is of paramount importance: it allows us to defend our argument against the accusation that our premises are capable of prove anything, true or false. So, in every controversy there may be an introductory stage where the discussion does not refer to whether the premises have grounds or the argument scheme is valid, but rather to the question of the premises being consistent or not by themselves. In order to prove that they are not, the techniques explained in the previous chapter should be used; we will currently explore the techniques used to prove that they are.
Let us work with an example of apparently inconsistent premises:
| Many unbalanced conditions are due to repression | Some  are  |
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| Self-control is a repression | All  are
|
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| No unbalanced condition is worth sustaining | No is
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| Self-control is worth sustaining | All is
|
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First of all we notice that none of our premises is the mirror image of any other. We eliminate the universal quantifiers and open up the existential one and its conjunction:
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We next introduce in the weak yard the premises bound for chain separation
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which is obviously no raw material for producing any contradiction, since there are no mirror twins in sight.
We may verify the mutal consistency of any group of premises through a similar procedure.
The most common application of the consistency proof occurs in the preparatory stage of a discussion, in cases when our opponent has expressed distrust towards the compatibility of our premises. If the premises are apparently consistent, such as the following:
there is no problem. The question only arises when the problem is sufficiently complex so as to become uncertain whether the premises have the logical possibility of all being true.
This procedure has also application in criminology and in those situations where it is necessary to give excuses (alibis) of some sort; if the excuse is apparently consistent, one may proceed to check it for its factual support. But it may happen that what you declare as an alibi becomes one more reason for incrimination, if not everything you is mutually consistent. The same may happen in the so called “confrontation” of suspects, or in the separate cross-examination of different witnesses. In all theses cases, the strategy is to obtain a larger number of depositions, which may be considered as premises, to produce a combination of statements with maximal possibility of being inconsistent, in the case that some one is lying. It is then up to the counsel for the defense, or the accused himself, to demonstrate that, in spite of appearances, their depositions are actually consistent; they may do so through the aforementioned proof procedure.
Another important usage of the consistency proof deals with the legitimate use of the appeal to authority. If a person deserves your trust, it is enough that what he states be consistent for you to accept it as true. We practice this principle every day and at any time; in our conversations with relatives, friends or partners, we assume that what is being said is true; we only pause to doubt if we notice a contradiction, when “something strange” in what we hear awakens our suspicions. Something similar may happen even in religion, where some of the dogmas are sometimes presented as mysteries, that is to say, truths that may not be proved in a logical sense. Nevertheless, and this is recognized by any theology deserving its name, it is always necessary to demonstrate that what is proposed as object of religious belief is perfectly reasonable, i.e. does not involve logical contradictions.
108. Consistency of Hypothesis
The consistency proof is also used profusely in scientific research. Once a phenomenon has been studied empirically, i.e. by means of experimental methodology –for instance, in a laboratory–, scientists must then formulate hypotheses explaining the phenomena. The number of possible hypotheses is in theory infinite; in practice, however, it reduces considerably, seeing that only credible hypotheses are considered, that is, those most likely to be true. But even some of the credible hypotheses may turn out later to deserve rejection due to their inconsistency, in themselves or in regard to other well reputed hypotheses or theories. This facilitates the scientist’s work, since she or he will not need to verify the truth value of every hypothesis presented, often through long, complicated, and expensive empirical methods; they will be able to get rid of many by means of a simple logical analysis, which neither require field trips or surveys, nor expenditures in laboratory materials.
It is up to the person proposing a hypothesis the minimal obligation of demonstrating that his/her idea is consistent, not only pertaining to compatibility of the different propositions that compose it, but also in relation with propositions concerning the phenomenon in question accepted by the relevant sciences. What we do not know, and the hypothesis intends to tell us, must at least be consistent with what we already know through other laws or principles from the pertinent field. Compliance with this consistency requirement may also be carried out by means of the procedure shown above.
109. Possibility and Instantiation
Consistency lies, as we have seen, in the logical possibility of premises being conjointly true; to say that two propositions are mutually consistent is the same as saying that the facts described by those propositions may both occur in reality, none of them implicating that the other should not occur. In this sense we say that it is logically possible that there be life in Mars, since the fact of being alive and the fact of being in Mars do not contradict each other in principle, none being the negation of the other or implying the denial of the other.
This logical possibility must be distinguished from real possibility: many things are logically possible, but they are not given in practice; Thus, it is very unlikely, although not contradictory, that there is life in Mars. In contrast, we can always say that what is in fact possible is also logically possible: if traveling to the moon is already possible, it is of course also logically possible. If two events occur in reality, it is not possible that the propositions which state those facts contradict each other. Where there is real possibility there is always logical possibility, but the reverse is not true: there is not always real possibility where there exists logical possibility.
This provides us with a very effective method, although not always applicable, for demonstrating logical consistency or possibility: if we are able to show a real case where the propositions examined are all evidently true, the propositions cannot be inconsistent. This kind of argument is called proof by instantiation; something is stated as not possible and we, instead of getting into an argument, show a real and palpable example of what is said to be impossible. If it is said that being vertebrate and having cold blood are two incompatible attributes, we do not lose time arguing; we immediately present fish as an instance. If it is said that a robust unarmed democracy is not viable, we do not argue; we just point to the case of the Republic of Costa Rica.
The inverse logical technique is also quite useful. It is called counterexample, and profusely used in science to attack universal propositions (like hypotheses and scientific theories) assumed to be true. Universal propositions, as you would recall, state that two attributions occur always together: if the antecedent occurs, then the consequent must also occur. If we now present a case where two attributions do not occur together, where one of them occurs but the other does not, we will have undermined the trust deposited in such universal quantification.
Counterexample is especially strong when used against an assumed definition. Definitions, as we studied before, are universal propositions whose truth does not need any demonstration: they are true because of the constitution itself of the language we use. If someone tells you that vertebrates have hot blood “by definition”, or that “by definition” a democracy has to defend itself with a strong army, to answer with an example to the contrary proves in a smashing way that our interlocutor is using language incorrectly. By doing this you deprive him of a very powerful premise, as every definition is, and may be destroying his whole argumentation.
Reasoning by instances performs a very important role in science development. Many complicated and abstract theories, let us say on the field of physics or economics, become intuitively understandable by including simplified examples which scientists call models of the respective theories. Thus, gas kinetic theory models heat as what occurs to minuscule and perfectly spherical balls of equal mass and volume, whose dimension is negligible compared to the distance between them. It was been briskly discussed in the first half of the XX century whether light should be modeled as a particle or as a wave. In the field of economics, it is common to model business cycles by means of simple machines which tend, or not, to a point of equilibrium.
We might say that all those scientific models of important theories have enormous pedagogic value, for they help us to understand the content in question, in spite of its exceedingly abstract nature. But models have also a deep epistemic value, since researchers never feel completely satisfied with a purely abstract or formal presentation of a theory unless they are able to forge an intuitive model of it, (1) which –they seem to feel– is the only warranty of theories or hypotheses being consistent.
Note 1: Nevertheless, there are models in the field of mathematics itself, although these mathematicas should be considered simpler than those of abstract theory explained herein. In this work, we have used this technique to explain the nature of universal and existential quantifications, when we presented a model of its meaning relative to a very simple universe of a few individuals: in this case, universal quantification transforms themselves into a conjunction of singular propositions, and existential quantification in a disjunction of the same. See
section 62.
