The Marlo diagram can represent and solve, at least in theory, logic problems with an unlimited number of terms. In López Aznar (2014), (2018b), it was confirmed that the diagram far exceeded the requirements which, according to Moktefi & Shin (2013), are expected from any diagrammatic system. Diagrams are created when trying to add the ability of expressing implicit possibilities to the Venn diagram. This idea was developed in 2013 when studying Johnson Laird's theory of mental models, in González Labra (1998). Diagrams prioritise the concept of “being associated to” instead of the concept of belonging and inclusion. This has allowed the recombining of their DNA with the connective network models, described in Crespo (2006). In López Aznar (2017), (2018a), the Marlo diagram is translated into tree diagrams that show the logical connectives as the simple elimination of different possibilities. This idea is already present in the classic truth tables, but logical trees make it possible to simulate how truth and falsity propagate through the networks of a cognitive system. The diagram has been in development since 2013, in high school classes, with the help of students and other teachers who are experts in mathematics.
Thought is surrounded by a halo. —Its essence, logic, presents an order, in fact the a priori
order of the world: that is, the order of possibilities, which must be common to both world
and thought.
The Marlo diagram uses propositional models to communicate the associations that a variable maintains with others in a mental space S. As we already know, in truth tables the number of elements of S is determined with the formula 2 . In this work, for reasons of simplicity, we will use, like the tables of truth, dichotomous systems in which A means to be and ¬A means not to be. For example, if “b” means being beautiful, then “¬b” means not beautiful.
To construct the models of the simplest propositions, we must place the subject in the middle of what we will call the propositional model. For its part, the predicate must also be placed within the model, but on one of the sides or regions thereof (see figure 1). All the variables within the same region of a model are necessarily associated and form a unique and distinct type of object in which the contradiction is not possible. A very important characteristic of our diagram is that we will also write the predicate on the external margin of the model whenever we can conceive that it is possible to have that predicate not associated with the subject of the proposition. For example, if we declare that all numbers that finish in zero are divisible by two, we are explicitly communicating the association between ending in zero (subject) and being divisible by two "t" (predicate). That is why in the first model of figure 1 we can see that "z" and "t" form a type of object "zt", although now we see that relation from the perspective of the set of "z". However, it is implicit in our proposition that the possibility of being divisible by two without ending in zero is left open, and that is why we write a “t” apart from the "z" model. The letter “t?” stands for the possibility “t¬z”. We have put a question mark in “t?” because, if we eliminate prior knowledge, we cannot know based only on the proposition, if in fact that type of numbers exists or does not exist. Variables outside any model A should be read as associated with ¬A.
Suppose we now add the premise all the numbers ending in zero are divisible by five “f”. We represent the model of this proposition also in figure 1, with the same rules as before, and we realize that the two models have the common denominator “z”. When different models share the same subject, we can synthesize them into one according to the law of identity. It is in this way that we obtain the synthesis -or model 3- of figure 1. We can see that the "z" model associates "t" and "f" necessarily within it in the same region, providing the object type "ztf”. We can already affirm that, at least in theory, there are such objects which end in zero and are divisible by two and by five in the space of S. However, as we have also compiled the implicit possibilities in the synthesis, we see that, apart from the numbers that end in zero, there may be numbers divisible by two and others divisible by five, of which we know nothing. On these marginal terms, we can only state with certainty that these are possibilities that must be taken into account when we draw our conclusions. To draw solid conclusions from model 3, we must make the correct questions in an orderly manner about all the associations shown in it. First question: How many types of Z are associated with F and T? All or part? Answer: we see in the image that all Z, the only one that appears, is F and T. Second question: Are all types of T necessarily associated with F or only a part of them? Answer: we see that only a part of the types of T is necessarily associated with F, so we can conclude that at least part of T is F. We cannot say that all T is F because there is a possibility, outside Z, of T not being associated to F. Although we cannot say, based solely on the premises, that there is a part of T which is not F. Third question: Can we say that all F is T? or can we only affirm with certainty that some numbers divisible by five are divisible by two? Answer: we observe model 3 and we see that only a part of F is necessarily associated with T; the other part of F is outside Z, neither united nor separate from T.
When drawing conclusions, the model functions as an external memory device that reminds us of all the possibilities that we must consider, which helps to avoid the most common errors when reasoning. This turns the diagram into a powerful didactic tool of logic. In fact, the simplest version of the diagram was tried as a didactic tool with seventy high school students of sixteen years of age. The students received both theoretical (seven hours) and practical instruction with a booklet of exercises (3 hours). The test consisted of solving six syllogisms using the diagram: three by synthesis (universal, particular and probable) and three by exclusion (universal, particular and null). 88.3% of the students passed the test. And 36.6% of them without making a single mistake. See López Aznar (2014). 1.2
Every simple proposition may be reduced to the expression of a relation between two ideas:
the two ideas are represented by the two terms (subject and predicate) […] These terms may
be either both of them individual, or one of them individual and the other collective, or both
of them collective.
The second decision we must make to represent sentences in the diagram is whether or not we will write the predicate P outside the model of the subject, that is, outside the circle. But we must forget our previous knowledge! When a child first hears that all vertebrates have a trunk, he can still assume that there are other beings with a trunk but that they are not vertebrates. But if the child hears the propositions only vertebrates are mammals or hears the proposition donkey is the same as burro, then he has to consider the predicates burro and mammal universally. And the child should think that the terms are universal because the propositions forbid him to think of mammals that are not vertebrates or burros that are not donkeys. The main idea is that when the proposition does not allow the possibility of conceiving the predicate outside the subject, the predicate is universal. And when the predicates are universal, they are not represented separately from the subject.
I acknowledge that there is a certain amount of difficulty in getting used to
considering whether the predicate of a proposition is taken universally or not, because there are
no explicit quantifiers for the predicate of propositions. However, I have verified with
my students that it is almost always possible to reach a complete agreement in this
sense. And besides, this way of proceeding has several advantages: Hamilton lamented
that classical logic had forgotten the extensional dimension of the predicate and he
claimed that syllogistic logic could be improved if not only subjects were quantified,
but also the predicates in premises, see
During the research processes our models maintain spaces of uncertainty, but all the possibilities do not have the same weight. For instance, suppose you meet my family for the first time and ask all my brothers if they are hunters. Everyone answers yes. You can now affirm with certainty that all my brothers hunt “ ℎ”, and you can also establish with certainty that at least part of those who hunt in my family are my brothers “ℎ ”. Nevertheless, can you say there are hunters in my family other than my brothers? Is there “h¬b”? It is possible, but only at the level of conjecture. And if I told you now that I'm going hunting with a member of my family, what option would you choose if your life was at stake? That hunter of my family, is he or is he not my brother? Is the so-called fallacy of the affirmation of the consequent always irrational?
Figures 5, 8, 9, 10 and 11 can facilitate understanding of the diagrams because they contain lots of examples of propositions that combine particular and universal subjects and predicates. 1.3
We have observ’d (sic), that whatever objects are different are distinguishable, and that
whatever objects are distinguishable are separable by the thought and imagination. And we
may here add, that these propositions are equally true in the inverse, and that whatever
objects are separable are also distinguishable, and that whatever objects are distinguishable
are also different.
thetically combinable with the variables of other regions, provided that
contradictory pairs are not established. The main idea is that each region of a propositional
model contains a brick or element with which it is possible to create different types of
objects by combining them. This idea has repeatedly appeared in the history of logic.
See Ramón (1990). For example, “sdic¬a” is a probable conjunction of types of
patients, however, others are possible (but not probable). In models that contain many
terms, it would be impossible to explicitly express all the possibilities at stake.
However, we can establish as a rule that all variables not explicitly confirmed in the
models, as well as all their possible combinations, are acceptable at the level of
conjecture. That is, variables that do not appear explicitly in the diagram, or those that appear
only with a question mark, should be considered as acceptable possibilities at the level
of conjecture. In our example, eighteen of the thirty-two combinations we can make
with the dichotomous criteria a, s, i, d, c are still possible. And this is because we only
discard the fourteen combinations that contain "ca", "ac", "i¬s" or "¬si". But remember
that those fourteen possibilities that have not yet been discarded do not have the same
evidence in their favour. In
We should note that this information is the same as that included in the truth table of ⊻
when we affirm that 0110. Therefore, we see that the whole extension of P is associated
with ¬Q, being impossible to find ¬Q outside of P in the Marlo diagram. Written in
formal language: p ⊻ q = p↔ ¬q = ¬ . On the other hand, the tree diagram of the
inclusive disjunction p ∨ q tells us that starting from P it is both possible to reach Q and
¬Q. Thus, it would be a bad option to select that route to communicate information
because it is a path that does not discard anything. However, we see that starting from
¬P we necessarily arrive at Q, so: ¬ . Likewise, when reading the inclusive tree
diagram from the perspective of criterion q, we find that the relevant information is on
the line of ¬Q. Starting from ¬Q we are sure to obtain P, so: ¬ . However, starting
from Q, it is possible to reach P following the upper route and it is possible to reach ¬P
through the lower route. So this does not inform us of anything. The same information
is offered by the Marlo diagram, the tree diagrams and the truth tables of p ∨ q, where
by affirming 1110 we are communicating that we eliminate the possibility ¬P¬Q. The
zero in the truth tables equals a removed path in the diagrams. All the logical connectors
can be defined as conditionals,
Belief models do not only inform about what things are or are not. Cognitive systems
also communicate information about the subjective certainty of their beliefs. When
eliminating possibilities, actual facts prevail over theories and theories over
suppositions. In this way, the belief models expressed by the Marlo diagram have the capacity
to adapt to the continuous evolution of the environment, following the spirit of the
Pragmatist Philosophy of Life in Ortega y Gasset, see
%abc (4) ● Evidence-based hypothesis: When solving complex problems such as the one with schizophrenia, we notice that it is possible to combine an affirmed variable with another possible one. Those combinations are conjunctions which are more possible than those based only on conjectures. This is, ab? starting from a and b? is more likely than ab? starting from a? and b?. ● On-going facts: they determine the presences and absences (5) which must be taken into account now, and are capable of changing emotions, behaviour, perception or thought in a possible world defined as 0 in the universe of discourse. These are expressed in capital letters:
A¬B = presence a¬b
̅̅̅¬̅̅̅ = absence a¬b
(5)
A VERB, is that which, besides something else, signifies time [..] But I say that it signifies
time, besides something else, as for instance, “health” is a noun, but “is well” is a verb; for
it signifies besides being well, that such is the case now.
[…] first, if two terms agree with one and the same third, they agree with each other:
secondly, if one term agrees and another disagrees with one and the same third, these two
disagree with each other
thesis, the basic laws of identity, uncertainty and distinction must be applied. They are the basis of the logic inferences expressed in figure 9. The principle of uncertainty states that what is uncertain in premises must remain uncertain in conclusions. The principle of distinction forces to provisionally separate the variables when there is not sufficient reason to associate them as a unit in the same region of the model (see probable synthesis in figure 9). Consequently, those variables will fill different sides in the model. However, as long as it is not explicitly expressed that those variables represent incompatible objects, their combination is probable. For example, if we have been informed that a primate species hunts small monkeys in Borneo, and that a primate species cleans fruit in Borneo; therefore, it is likely that the species which hunts and cleans fruit is the same. It is a probable belief, but it is not necessary.
Inferences by exclusion. If two variables are respectively related to incompatible variables (between themselves), the mutual relation is impossible. Therefore, if 1 is B and 1 is ¬B, then, it is impossible that 1 could be associated to 1. Nevertheless, in this case, the relations between other types of A and other types of C remain uncertain. When confronting two models defined with any excluding variable, three types of inferences are obtained. See figure 9. We have highlighted in grey colour the incompatible regions of the models.
Universal: No part of A is C and no part of C is A. We reach this conclusion when there are no parts of A or C that could be combined without incurring in the contradiction B¬B. For example: Minerals are electrical conductors. Carbon is not an electrical conductor.
Partial: One part of C is not A, although every A may be related to the other supposed part of C. We reach this conclusion when there is a part of C that could be combined with A. This is the most difficult type of logical inference for my students. We can see an example in Figure 11. 3. Null: in this case we cannot say anything with certainty because there are no impossible combinations. We cannot affirm that two variables are excluded when a part of both remains uncertain. We can only say that a certain part of A cannot be associated with a certain part of C, or vice versa.
Exclusion is a difficult process and can be avoided by transforming the models in order to operate by synthesis. However, it is important to understand its principles to get a more complete picture of what it means to reason. In Synthesis, the logical inference is based on understanding the necessary relationships. In Exclusion, the logical inference is based on the understanding of impossible associations, when two variables are placed on different paths. Sometimes the relations between two variables are not symmetrical and it is possible, as we have seen in Figure 11, that although every A are C, a part of C is not A. In such cases, it is very difficult for students to consider possible and impossible relationships in an orderly manner. Teaching logical reasoning is simply teaching to look in the correct order and show that a shift of focus is always necessary to review all conclusions. In Figure 11 we get that it is only true that, based on the premises, some primates cannot be hominids, although every hominid could be a primate. It is evident that primates that are not bipedal cannot be bipedal hominids. However, considering only the premises and forgetting our previous knowledge, it would be possible, although not necessary, to associate every "h" with the indeterminate "p" part. Figure 12 shows a problem in which the models have been transformed to be able to operate by synthesis. 4
In theory, the Marlo diagram makes it possible to work with an unlimited number of
terms, although in practice it must fit the available space. Furthermore, the
understanding of the differences between the visual processes necessary to solve inferences by
synthesis and by exclusion, allows us to explain, for example, why the EI syllogism of
the fourth figure, which is solved in the diagram by exclusion, is more difficult than the
syllogism EI of the first figure, which is solved by synthesis, although both have a
particular negative conclusion, García Madrugá (1982). The main idea underlying the
operation of the diagram is to specify which possibilities are eliminated and which are
maintained, an idea that we already found in the explanation of the Carroll diagrams
that appears in
It is also worth remembering that the Marlo diagram is a powerful tool for the didactics of logic. I have been able to see for myself repeatedly that the students fully understand their principles at the beginning of the high school period. But understanding a language is one thing and knowing how to produce messages with it is another. In any case, the diagram facilitates, in my classroom, the coordination and integration of language and eye movements while troubleshooting a problem. And a student, who looks at where he points and points out what he names, is a student who makes himself understood and because of that convinces me that he understands the problem himself.
Explaining the logic with the Marlo diagram has allowed me to obtain another point of view about what reasoning means. The tree diagrams show us all the mental space simultaneously, while the Marlo diagrams are like the pieces of a jigsaw which fit together.
[...] we must, I say, observe two sorts of propositions that we are capable of making: -First,
mental, wherein the ideas in our understandings are without the use of words put together,
or separated, by the mind perceiving or judging of their agreement or disagreement.
Secondly, Verbal propositions, which are words, the signs of our ideas, put together or separated
in affirmative or negative sentences.