Claudio Gutierrez

Translated from Spanish by Inés Gutiérrez

43. Determining the Function of a Text

The first task of the analysis of structure is directly linked to the analysis of sense: it is a matter of determining whether or not a linguistic text or passage performs a descriptive function. If it does, then it has structure (from the point of view of general logic). Otherwise, it does not. How do we know if the text is descriptive? Let us recall what we studied in the introduction about the functions of language. The text is descriptive if it gives us information about the real world, or at least about some possible world. The text is descriptive if I can assert what it says as true or false, unconditionally or within certain assumptions (such as the suspension of judgment in literary fantasy). The text is descriptive if there is an event, real or imaginary, to which the text might correspond. If I say: “there is a crisis in the Mediterranean”, the text is clearly descriptive because it matches a real event in our 1968 world. It is also descriptive if I say (in fiction) that Pedro Páramo was the owner of all the land in the region, because it responds to an imaginary event created by a novelist. And if I say (in science fiction) that during a long interplanetary journey cosmic rays can make people diabolically intelligent, that is descriptive too, because it corresponds to a general event within the respective fantastic framework. But if, on the other hand, I say “Well, what an idea!” or “Let’s sing together” or “Thank you very much!”, it is clear that there is no description here, nothing true or false, no events –neither real nor fictional– to possibly match what I am saying.

44. Identifying Propositions

That step of the analysis concluded, and supposing the text has turned out to be descriptive, our next task will be to determine if we have many independent thoughts before us or, on the contrary, only one, long or short, but integrated into a single unit. We will call these units, as it was explained in the second chapter, propositions. Propositions can be usually recognized by their punctuation: they are separated from each other by periods, semicolons, commas, colons, or other such signs. But punctuation marks are only one element of judgment for the structure analysis, in which we cannot trust absolutely.

Sometimes we must make certain minor changes in the text, not altering its sense but clarifying the structure. For example, we must eliminate the interrogative form from so-called rhetoric questions. A question is rhetorical when we expect it to be answered in a certain way; it is, in fact, equivalent to an affirmation or negation. Thus, it is becoming to transform this kind of questions into indicative propositions, by which we greatly facilitate text analysis. The following ironical text, full of rhetoric questions, can be rephrased as shown immediately below:

What choice did Herod really have? If he had allowed the Christ to live, wouldn’t he have failed in his duty? Wouldn’t it have been possible, or rather very probable, that afterwards any average Joe could have felt sufficiently inspired to proclaim himself the Messiah? Isn’t it likely that, if Jerusalem had fallen under the hands of the new king, other Middle East nations would also have fallen, and later the whole world? What responsible king would wish to have that on his conscience? ⁽¹⁾

Changing the questions for declarative phrases, the text would look like this:

Herod had no choice. If he had allowed the Christ to live, he would have failed in his duty. It would have been possible, or rather very probable, that afterwards any average Joe could have felt sufficiently inspired to proclaim himself the Messiah. It is likely that if Jerusalem had fallen under the hands of the new king, other Middle East nations would also have fallen, and later the whole world. No responsible king would wish to have that on his conscience.

Another minimal change we could make is omitting the expressions which simply indicate that what we are saying is true, or probable, or likely, or possible, on the condition that those indications of intensity of belief have no real relevance within their context. Indeed, they have none in the passage we are examining. Simplified accordingly, it would look as the following:

Herod had no choice. If he had allowed the Christ to live he would have failed in his duty. Afterwards any average Joe could have felt sufficiently inspired to proclaim himself the Messiah. If Jerusalem falls under the hands of the new king, other Middle East nations would also fall, and later the whole world. No responsible king would wish to have that on his conscience.

We have here five independent propositions. Some are short –such as the first one– others larger –e.g. the one before the last one–. But each is a logical whole and can be thought of independently from the others. We are ready now for our next step.

45. Molecular and Atomic Propositions

Once the propositions have been identified, we must pay attention to each in particular. Not from the point of view of content, this does not interest us in a logical analysis, but from the point of view of their structure. In the case of our example, we are not interested in the historic or ethical resonances of the text, but only in the form of propositions, for example, in the difference in length between the first and the one before the last one. That proposition is longer for the simple reason that it contains more information than the others. In fact, it can be divided in (at least) two propositions. It would be enough to omit the particle “and” and replace it with a period for us to have two propositions, namely:

If Jerusalem falls under the hands of the new king, other Middle East nations would also fall. Later, the whole world.

The second phrase, of course, does not explicitly formulate a proposition; something already said in the first is understood. We can supplement that to facilitate the analysis. Both propositions will now read:

If Jerusalem fell under the hands of the new king, other Middle East nations would also fall. If other Middle East nations fell, the whole world could also fall.

So far, so good. But, are not these two propositions still too large? Indeed, they are. And for a reason similar to the one given before: each one of them can be divided into sub-propositions. In the previous case, the internal propositions were connected by the particle “and”. Now we are faced with another particle, or system of particles, namely, “if… then”. Let us look again at the example, changing the grammatical mode for the sake of clarity:

If (Jerusalem falls...) then (other Middle East nations fall). If (other Middle East nations fall) then (the whole world falls).

Suppressing the connecting particles, we have now three independent propositions. These are:

Jerusalem falls.
Other Middle East nations fall (twice).
The whole world falls.

Each one of these propositions makes sense by itself; all could be asserted independently. However, they are not stated thus in our example, but as part of a larger proposition. By applying a simple analogy borrowed from physical chemistry, we will call these minimal-sized propositions, which cannot be divided into other propositions, atomic propositions. On the other hand, we will call propositions composed of others propositions, inwardly linked by connecting particles, molecular propositions. Atomic propositions are those not composed out of any other proposition; molecular, those composed out of other propositions that can, in turn, be either atomic or molecular. In our example, the net to last proposition in the text is molecular and consists of two propositions, also molecular, linked by the particle “and”; each of these being itself composed of two atomic propositions, linked by connecting particles of the “if… then” type.

46. Logical Connectives

As we have seen, an analysis of structure begins by determining the independent propositions present in the text. The next step is to determine if these propositions are molecular or atomic. The next step, only applicable to molecular propositions, will be to determine what kind of connecting particle (connective, for short) contributes to their formation. We have already found two types of these connectives, namely, “and” and “if… then”. There are many others, for instance “or”.

Let us examine the following passage:

Authority does not lose its ethical base whenever it makes a mistake or allows malice to influence its decisions. Every society is made up of human beings and must tolerate the weakness of its components.

In the first proposition, which is the one that interests us, the author is denying (that is, asserting as false) that authority loses its ethical base whenever it makes a mistake or allows malice to influence its decisions. Thus, he is denying the following proposition:

Authority loses its ethical base whenever it makes a mistake or allows malice to influence its decisions.

This is, of course, a molecular proposition. What connectives hold its parts together? To answer this question we should first answer the following one: Which atomic propositions integrate the molecular proposition? The answer is not difficult; with a little observation we realize that the minimal parts that could be stated independently are:

Authority loses its ethical base.
Authority makes a mistake.
Authority allows malice to influence its decisions.⁽²⁾

How do we bind these atoms to form the molecular proposition? Of course, by means of connectives which, in this case, are: “or” and “whenever”. If we think of it, we notice that “whenever” has the same meaning as the connective “if… then”; the sole relevant difference is that the positions of the atoms are reversed. The molecular proposition can then be read as follows:

If [(authority makes a mistake) or (authority allows malice to influence its decisions)] then (authority loses its ethical base).

Within this large molecule, there is a smaller molecule, i.e.:

Either (authority makes a mistake) or (authority allows malice to influence its decisions).

Here the connective is “or”, which is now thereby being introduced to you.

Let us now recall that the large molecular proposition of our example was denied, not asserted, by the author of the text. This provides us with another important logical connective which we must take into account when analyzing structures: the particle “no” or, if you wish, “it is not the case that." Including it in our formalization,⁽³⁾ the final version of the proposition would be as following:

It is not the case that [if (either authority makes a mistake or authority allows malice to influence its decisions) then authority loses its ethical base].

47. Conjunction and Disjunction

Let us now analyze the meaning of each of the connectives we have uncovered. The connective “and”, which we will call conjunction, asserts that the two propositions it links are both true. In other words: a conjunctive molecular proposition, formed by propositions united by the connective "conjunction", is true if and only if each one of the connected propositions is true.

The connective “or”, which we will call disjunction, asserts that at least one of the propositions it links is true (it might be that all are true, but not that all are false). In other words: a disjunctive molecular proposition, formed by propositions united by the connective "disjunction", is true if and only if at least one of the connected propositions is true.. Back to our example, it would be enough that authority made a mistake in order to be true that either it made a mistake or it allowed malice to influence its decisions. But it could happen that authority made both things, and the molecular proposition would still be true. On the other hand, if it does not do any of the two things, the proposition would be false, but only in this unique case.

The disjunction we are studying is the inclusive disjunction (at least one of the two or more sub-propositions is true, but all of them might be true). There exists another kind of disjunction in ordinary language, called exclusive disjunction. It says that one of the two sub-propositions is true and the other false. But in practice, the inclusive disjunction is sufficient to express both ideas. In the following example you will see both usages of the disjunction combined in a unique molecular proposition:

If authority becomes unjust or inefficient it must yield its place voluntarily or involuntarily.

This proposition is formalized like this:

If (authority becomes unjust or authority becomes inefficient) then (authority must yield its place voluntarily or authority must yield its place involuntarily).

It is clear that the second disjunction establishes that its two parts cannot both be true, in addition to saying that at least one is true. Even interpreting the disjunction as inclusive, the idea of exclusion remains, for the words “voluntarily” and “involuntarily” demand that at most one of the alternatives be true. This information is implicit in the words used. But in general we can achieve the same effect by adding to the inclusive disjunction the injunction: “but not both”. This additional proposition is easily formalized using the logical connectives "and" and "not".

48. The Conditional

The next connective that we should examine is “if… then”. The proposition formed with its help is called a conditional. Its first part is the antecedent (the component statement following the word “if”); its second part, the consequent (the component statement following the word “then”). The connective tells us that the antecedent cannot be true and the consequent false; i.e. if the antecedent is true then its consequent has to be true; it is enough that the antecedent occurs to make the consequent occur too. But this does not mean that both must occur. The consequent can very well occur without the antecedent occurring, and the conditional would still be true. For example, “if Jerusalem falls then other Middle East nations fall” can be true in spite of the fall of other Middle East nations without the fall of Jerusalem (they could have fallen for other reasons).

Sometimes we are interested in the conditional acting in both directions, i.e. demanding that if the antecedent occurs the consequent must occur, but also that if the consequent occurs the antecedent must occur. We have another connective for this in ordinary language: “if and only if… then." We could very well have used it in the previous example regarding authority, since what was said there could better have been said this way: “If and only if authority becomes unjust or inefficient then…” This proposition is equivalent to two propositions of similar form, united by a conjunction, both being conditionals of a unique direction. Namely: “If authority becomes unjust then…” and “only if authority becomes unjust then…”

It is easy to show that the conditional “only if… then” is the reverse of the conditional “if… then”; it is equivalent to the latter with the antecedent and consequent positions interchanged: “only if authority becomes unjust or inefficient then it must yield its place” is equivalent to “if authority must yield its place then it is the case that it is unjust or inefficient.” For simplicity sake, we will use only one type of conditional: “if… then.” We can always represent the other idea indirectly, by using more words, as shown above.

49. The Negation

The last particle to occupy us is “not” ("no", "it is not the case", etc.). A proposition containing it is called a negation. Negations are molecular or atomic, according to what is denied being molecular or atomic. For more clarity in the analysis, it is convenient to put the negation at the beginning of the proposition. For instance, transforming “authority does not lose its ethical base” into “it is not the case that authority loses its ethical base.”

The meaning of this logical particle is very simple: the negation is true if what is denied is false; false otherwise. .

50. Other Compound Propositions

It is important to observe that not all compound propositions occurring in ordinary language can be analyzed in the way indicated here. Particularly, there are compound propositions the truth of which does not depend on the truth of the atomic propositions that compose them. This is especially true about propositions expressing subjective attitudes, for example, propositions that use the verbs “to know”, “to believe”, “to suspect”, “to doubt”, “to expect”, “to allow”, “to obligate”, etc. The proposition “John believes that he, John, is the world’s most intelligent person” is a compound proposition, in the sense that part of it is also a proposition, namely, “John is the world’s most intelligent person”. However, the truth of the larger proposition does not depend absolutely on the truth of the atomic proposition noted. John believing something is a fact that can happen or not in reality, and John being intelligent is another fact, which may or may not be true; both facts are totally independent from each other. On the other hand, if I say “John is conceited and is intelligent”, this molecular proposition is and can only be true if “John is conceited” and “John is intelligent” are both true propositions. This kind of compound proposition, whose truth wholly depends on the truth of its conforming propositions, receives the technical name of extensional proposition, maybe because its truth value “extends” (in a functional way) from the atomic propositions to the molecule; please notice that it is only to this kind of propositions that we can apply the structural analysis techniques referred to in this book. The other kind, which includes subjective attitudes, receives the name of intensional proposition. From the point of view of general logic it cannot be analyzed by parts and must be considered as an atomic proposition. Its analytical treatment is exceedingly difficult and forms the subject of different new branches of logic, still experimental, highly complex and of problematic validity.