Claudio Gutierrez

Translated from Spanish by Inés Gutiérrez

51. Structure and Graphic Representation

In the previous chapter, we explained how descriptive language can be analyzed from the point of view of its logical structure; that it is possible to distinguish diverse units of independent thought, called propositions; and that those units can be atomic or molecular, the latter formed out of the atomic ones with the help of connectives. We presented three of these, namely, conjunction, disjunction, and conditional; in addition, the special logical operation called negation and which can be applied to any proposition, be it atomic or molecular. We will now make a more rigorous presentation of the same subjects, but using different methods of logical-structure graphic representation. All of them contribute to depict what we have already said is fundamental in the structure of molecular propositions; namely, that their truth or falsehood depends on, and exclusively on, the truth or falsehood of their component atomic propositions.

52. Conjunction and Disjunction Tables

Let us agree on representing two different atomic propositions with the letters “” and “”. Each one of these letters will represent any proposition, which can be true or false and refer to any subject. For example, “authority exists in society”, “insects lay eggs”, or “we can never be absolutely sure about anything”. Inasmuch as we are doing logic, we are not interested in the content of propositions; we are only interested in their structure. In this case, the only thing that interests us is that the propositions represented by “” and “” are atomic propositions, that is, they are not composed of other propositions.

Once we have these two variables representing any propositions, we can also represent the conjunctive and disjunctive molecules, as follows:

conjunction :
disjunction :

As you can see, the symbol “”, a brace that encloses the atomic propositions from above, will represent the conjunction connective; while the symbol “”, a brace that embraces them from below, will represent the disjunction connective.

If we now ask ourselves how best to check whether a molecular proposition is true or false, the answer is –of course– that we have to make use of the truth value ("true" or "false") of the constituent atomic propositions. We can rigorously do that with the help of the following table, which tells us directly the truth value of the molecular proposition according to whether the constituent propositions are themselves either true or false, as follows:

true	true	true	true
true	false	false	true
false	true	false	true
false	false	false	false

53. Negation Table

Turning now to negation, we can easily see that it is also possible in this case to build a similar truth table. Let us represent the negation of a proposition with the mirror image, along the longitudinal axis, of the proposition to be denied. If what is denied is “”, then we will represent its negation as “”; so, if we have doubts about whether a particular formula is the negation of another, we should only check whether the former is the mirror image of the latter.

The respective truth table is very simple; if we are dealing with an atomic negation, we have:

true	false
false	true

But, surprisingly enough, we discover that the same holds for molecular formulae. For instance:

true	false
false	true

The reading of the formula offers no difficulty. But, what about formula ? If the readers have not already discovered it by themselves, the reading is this: “either it is not the case that or either it is not the case that "; which –surprisingly again– is another way of saying “it is not the case that and ". So, the representation of a molecular proposition has at least two readings! We will discover later that this fact has a tremendous potential for the simplification of logical operations.

If we try molecular negation over a disjunction, instead of over a conjunction, we will have:

true	false
false	true

The reading of the formula is the following: "it is not the case that and it is not the case that ; which of course is equivalent to “it is not the case that either or ".

Molecular negation tables make conspicuous the relations of meaning that exist between conjunction and disjunction: to deny a conjunction is equivalent to affirm disjunctively the negations of each of the propositions that constitute the conjunction. To deny a disjunction is equivalent to affirming conjunctively the negations of each of the propositions that constitutes the disjunction. We invite the student to assign propositions from ordinary language to the "" and "" variables and check by himself that these relations effectively hold.

54. Conditional Table

Before making a table for the conditional we must remember exactly what it is that we convey with “if then ". What we assert is that it is not the case that the antecedent, “”, is true and the consequent, “", be false. That is, we deny the molecular proposition “”. And as negation is represented by reflecting in a mirror whatever we want to deny, the conditional is equivalent to “”. There are two more ways to read this formula. The first is, as we already said, “it is not the case that and not ". The second is “either not or ”. To verify that this last reading is correct, let us imagine it de-formalized back to what a tired mother says to her little kids: “Don’t make noise or you'll go to bed”. Evidently, this is equivalent to saying: “If you make noise you'll go to bed.”

We are now ready to represent the conditional molecule:

So, to represent the conditional we must deny the antecedent and place it, along with the (unnegated) consequent, inside a disjunctive brace. Now, if the antecedent is a molecule, we should –of course– apply the general recipe for negation: the antecedent is reflected in our virtual mirror as a precondition to its disjunctively being connected with the consequent. Let us represent the following conditional as an example: “if authority becomes inefficient or unjust then it must yield its place”. Let us take “authority becomes inefficient” as “”; “authority becomes unjust” as “”; and “authority must yield its place” as “”. The complete proposition is, as expected, a conditional with molecular antecedent. A provisional translation would be: “If then “”. Applying the rule for representing the conditional, it would turned into “”, since the antecedent’s negation is ““ ”. Observe that in the conditional formula the disjunctive brace encompasses the conjunctive one (negation of the antecedent), for this is a constituent parts of the larger formula.

As you can see, we have made conspicuous the relations that exist between the conditional and the other connectives –disjunction and conjunction–, through the application of the negation operation.

We are now ready to build the table of the conditional:

true	true	true
true	false	false
false	true	true
false	false	true

If the conditional contains a molecular antecedent or consequent, the table values do not change, in as much as we are considering the three formulae as un-analyzed wholes:

true	true	true
true	false	false
false	true	true
false	false	true

55. Conjunction Board

We have seen that the upper brace “” serves to represent graphically the conjunction or conjunctive molecule. There is another way to represent the same thing. Let us call “board” a leveled and closed surface, for example a sheet of paper or a blackboard. Since the components a conjunction are both true, we can consider a conjunction as a strong affirmation of those connected propositions. Hence, there is no problem with convening that any board that confines those propositions by themselves (without braces) within its limits would be a graphical representation of their conjunctive assertion as good as the upper brace. The board by itself becomes, so to speak, a great conjunction of all the propositions contained in it.

If our analysis has led us to discover that the propositions “”, “”, and “” are true, we may represent the set of them all on our board like this:

Reciprocally, if there are several propositions in our board, we can join any number of them, by pairs, with a conjunction brace, within or outside the board, forming any of their possible combinations; in the example, “”, “”, “”, “”, etc. In short, for any proposition it would be the same to be within a board or comprised by an upper brace. Our logical interpretation of them will be the same: they are true.

56. Disjunction Board

Just as there is a board for the conjunction, which is simple enclosure, we can create a double board to represent the disjunction. Conjunction asserts that their connected propositions are all true; disjunction, on the other hand, that at least one of them is true, although we still not know which one of the two. Then, we can represent graphically the disjunction with a double board inside the main board, the previously described conjunction board. The two propositions connected by the disjunction will appear on the double board each by itself (with no use of a disjunctive brace) within one of the two “rooms” or “yards” of the double board. This will be an “open” representation of disjunction, as opposed to the “closed” one that makes use of a brace. The disjunctive proposition “” found closed in the conjunctive board that contains it

can then be opened to give the following alternative representation:

This representation is equivalent to the previous one, and it affirms that one of the twin yards –we do not know which, for the moment– could allow eventually all its formulae to move to the surrounding outer board, since it would have been demonstrated that it comprises only true propositions; that demonstration having consisted in the fact that the proposition in the other yard do not deserve our trust. Uncertainty about which one of the yards is “the good one” makes us brand the double enclosure as a weak board, in opposition to the one presented before, the strong board, which asserts that all its propositions are definitely true (of course, from the perspective of an observer who considers that board as the main board).

The usefulness of this open representation is the following: in contrast to the closed one, it allows us to manipulate the formulae contained in the twin yards ⁽¹⁾, in the hope of producing something obviously false in one of them, which will tell us that the formulae on the other yard are the ones which are true. This will allow us to unify its propositions with those in the immediately surrounding board.

Open representations of conjunction and disjunction can be combined. Thus, in the case of the proposition the open representation would be

.

An open disjunction as the previous one can again be closed in the main board –even several different times–; for doing that, it would be enough to take one complete proposition from each of the twin yards, and connect them by means of the disjunction brace:

To avoid confusion over the truth of propositions that appear within disjunctions, it is required that, once a disjunction is opened, no other be open within the same double board. Once it is closed and the set of twin yards discarded, then –and only then– one can proceed to open another disjunction on a new set of twin yards. If we have the formula

                                      or else this other way:

    but never ever this way:

57. Translating Ordinary Language

In ordinary language we do not always find the studied connectives in their pure form, such as “and”, “or”, “if… then”. Those ideas are sometimes expressed with words like “but”, “though”, “unless”, and many others. All these words add some psychological shade to the logical or structural relationship of the propositions. From the point of view of logical analysis, though, we must always consider them equivalent to the simple connectives. Thus, “John does not love you, but I do” is the same, from the logic structure point of view, as “John does not love you and I love you”; in this version, the comforting shade that was present in the original has been lost, but that emotional tint is of no interest to logic.

Immediately below we give some equivalents to orientate the reader in the formalization of texts from ordinary language:

" only if ", " if ", " provided that ", " in case that " = "if then "

There are many more other equivalents which the student will grasp with practice.

When formalizing a text, it is convenient to proceed top-down, i.e. from the outside to the inside; we find the outermost connective, and then, step by step, formalize the other ones. Consider the following example.

Our starting point is a text of ordinary language:

If food production does not increase or the population does not cease to grow at such a vertiginous pace, there will be hunger and turmoil in our country.

If [food production does not increase or the population does not cease to grow at such a vertiginous pace] then [there will be hunger and turmoil in our country].

[there will be hunger in our country] and [there will be turmoil in our country]

It is not the case that [either [food production increases] or [the population ceases to grow in such a vertiginous pace]]

If it is not the case that [ or ] then [ and ]

If it is not the case that then

If then .

.

58. Applying Graphic Representation

The method of truth tables allows us to decide the value, true or false, of a molecular proposition, knowing the values of the propositions that it comprises. This is valid even when the connectives apply not only to atomic propositions but also to molecular propositions. In this latter case, the truth tables are applied first to decide the value of the innermost molecular formulae; using the resulting values, the values of the encompassing connectives are decided, and successively so outbound. In the example, if it is given that “” and “” are true propositions, but “” and “” are false, by applying the conjunction and disjunction tables we will find out that “” is true and that “” is false; notwithstanding, the total proposition turns out to be true, being a disjunction with a true component. We do not have to apply the conditional table since, as we indicated before, we represent the conditional as a disjunction with its first element denied.

The application of the board representation method occurs in the solution of reasoning problems and will be extensively illustrated in Chapter XI.

59. Well-formed Formulae

Thus far, we have left somehow undefined the number of propositions which can be directly embraced by a conjunction or disjunction, although we have limited that number in the examples to only two. In fact, everything said heretofore about both connectives is compatible with the existence of braces encompassing more than two propositions at a time, provided we are furnished with boards with a commensurate number of yards each disjunction,. However, in order to avoid unnecessary complication and mistakes in relation to the subject matter of the following chapters, it is convenient that we restrict conjunctive or disjunctive formulae to two propositions per brace only. Formulae with more than two elements can always be expressed with the help of multiple conjunctions and disjunctions, for example in the following ways: for conjunction, or ; and for disjunction, or .

For the sake of rigor, we can give a set of rules of formation which will tell us strictly which formulae are well formed in our logical language. One of the possible sets which would do the job is the following:

1. “”, “”, “”, “”, and in general any horizontally symmetric variable, is a well-formed formula.
2. “” embracing exactly two well-formed formulae is also a well formed formula.
3. 3. The result of reflecting vertically in a mirror a well formed formula is also a well-formed formula.

We leave to the reader the task of creating another set of rules, different from this but very similar to it, which can also fulfill the required function.