PROMETEICA - Revista de Filosofia y Ciencias. 2026, v. 33
Artículos
https://doi.org/10.34024/prometeica.2026.33.21454
LA TEORÍA DE LOS MODELOS MENTALES DENTRO DE UNA LÓGICA NO AXIOMÁTICA
A TEORIA DOS MODELOS MENTAIS DENTRO DE UMA LÓGICA NÃO AXIOMÁTICA
Miguel López-Astorga
(Universidad de Talca, Chile)
Recibido: 14/11/2025
Aprobado: 29/04/2026
ABSTRACT
This paper tries to capture the theory of mental models from Non-Axiomatic Logic. The theory of mental models describes people’s natural inferential processes and has strong empirical support. Non-Axiomatic Logic allows building a computer system based on itself. So, if the theory of mental models is right, and Non-Axiomatic Logic can capture the theory, it is possible to build a computer system based on the latter logic coming to conclusions like those of the human natural inferential ability. I attempt to show that with several components of Non-Axiomatic Logic, this logic can represent the basic inferential processes the theory of mental models proposes.
Keywords: computer system. human reasoning. inference. non-axiomatic logic. theory of mental models.
RESUMEN
Este trabajo intenta abordar la teoría de los modelos mentales desde Non-Axiomatic Logic (Lógica No Axiomática). La teoría de los modelos mentales describe procesos inferenciales naturales de las personas y tiene un fuerte apoyo empírico. Non-Axiomatic Logic permite construir un sistema computacional basado en ella. Por tanto, si la teoría de los modelos mentales es correcta y Non-Axiomatic Logic puede representar la teoría, es posible elaborar un sistema computacional basado en esta última lógica que llegue a conclusiones como las de la capacidad inferencial natural humana. Intento mostrar que, con varios elementos de Non-Axiomatic Logic, esta lógica puede recoger procesos inferenciales básicos que la teoría de los modelos mentales propone.
Palabras clave: sistema computacional. razonamiento humano. inferencia. non-axiomatic logic. teoría de los modelos mentales.
RESUMO
Este trabalho procura abordar a teoria dos modelos mentais a partir da Non-Axiomatic Logic (Lógica Não Axiomática). A teoria dos modelos mentais descreve os processos inferenciais naturais das pessoas e tem um forte suporte empírico. A Non-Axiomatic Logic permite construir um sistema computacional baseado nela. Portanto, se a teoria dos modelos mentais está correta e a Non-Axiomatic Logic pode representar a teoria, é possível desenvolver um sistema computacional baseado nesta última lógica que chegue a conclusões semelhantes à capacidade inferencial natural humana. Procuro demonstrar que, com vários elementos da Non-Axiomatic Logic, esta lógica pode representar processos inferenciais básicos propostos pela teoria dos modelos mentais.
Palavras-chave: sistema computacional. raciocínio humano. inferência. non-axiomatic logic. teoria dos modelos mentais.
We can ask a question: will we be able to build a computer program thinking like us? The answer is complex. We could build that program from a psychological theory describing the way human beings think. Do we have such a theory? Perhaps we do. At least, we have a psychological theory with strong empirical evidence: the theory of mental models (e.g., Johnson-Laird, Byrne, & Khemlani, 2024).
In this paper, I will assume that the theory of mental models (from now on, TMM) is correct, and its descriptions of the manner people make inferences are suitable. My aim is to show that we can create a computer program coming to the same conclusions as TMM. I will argue that if TMM is right, it is possible to build a computer system thinking as human beings.
Specifically, I will attempt to show that the general principles and theses of TMM can be captured from Non-Axiomatic Logic (Wang, 2013). Non-Axiomatic Logic, or NAL, is the logic underlying a computer system such as Non-Axiomatic Reasoning System (e.g., Wang, 2006), or NARS. If NAL can accept and reject the same conclusions as TMM, NARS can do that, too. But if a computer system such as NARS can come to the same conclusions as those TMM predicts, it is possible to argue that NARS can work as the human mind.
The first section of this paper will explain the general principles and theses of TMM. After that, I will describe the components of NAL necessary to translate those principles and theses into ‘Narsese’, that is, the language to communicate with NARS (e.g., Wang, 2013). Finally, I will try to show that NAL can come to the same conclusions as TMM.
TMM, conjunctions of possibilities, and necessity
If a concept is important within TMM (at least, within its latest version), it is that of ‘conjunction of possibilities’ (e.g., Khemlani, Hinterecker, & Johnson-Laird, 2017). The theory assigns conjunctions of possibilities (i.e., sets of possibilities linked by copula ‘and’) to classical logic connectives such as the conditional (i.e., copula ‘if…then’), biconditional (i.e., copula ‘if and only if’), disjunction (i.e., copula ‘or’, and conjunction (i.e., copula ‘and’) (see also, e.g., Khemlani, Byrne, & Johnson-Laird, 2018). Resorting to the language of modal logic just as a metalanguage, we can express the conjunction of possibilities of each connective as follows.
1 Acknowledgments: Project “Dialéctica virtuosa de la educación. De la fundamentación filosófica y antropológica a implementaciones concretas.” Project code: 15/I139-SAL160/23. UNMdP.
-A conditional such as ‘if I have milk, then I have lactose’: (Milk Lactose) (¬Milk Lactose)
(¬Milk ¬Lactose) (see also, e.g., Quelhas, Rasga, & Johnson-Laird, 2017).
Where ‘’ indicates that what follows it between brackets is possible, ‘’ stands for copula ‘and’, and ‘¬’ expresses that what is to its right is negated, i.e., represents negation.
The proponents of TMM usually claim that the theory is not a standard modal logic (see also, e.g., Espino, Byrne, & Johnson-Laird, 2020). I am respecting that here. As said, the language of modal logic, especially symbol ‘’, is being used only as a metalanguage (for a discussion about whether TMM is a modal logic, see also, e.g., Johnson-Laird & Ragni, 2025). The same applies to the cases of other connectives in this section.
If a conditional is mentally processed as a biconditional (i.e., as a proposition with copula ‘if and only if’), then the second possibility in the example above (that in which Milk is negated and Lactose is affirmed) is removed (see also, e.g., Johnson-Laird & Byrne, 2002). On the other hand, only the first conjunct (that in which both Milk and Lactose are affirmed) is intuitively obvious. The other two conjuncts need deliberative effort to be identified (see also, e.g., Barres & Johnson-Laird, 2003). Besides, TMM claims that the conjuncts do not always match the cases in which the conditional holds in classical logic. Semantics and pragmatics can change, or modulate, them (e.g., Johnson-Laird & Byrne, 2002). If the conditional is ‘if I go to China, then I go to Beijing’, the conjunction of possibilities is: (China Beijing) (China ¬Beijing) (¬China ¬Beijing). The second conjunct has a different structure now. The clause that is negated is Beijing, that is, the second one. The reason is that I cannot travel to Beijing without going to China.
-A disjunction such as ‘I have milk or I have juice’: (Milk Juice) (Milk ¬Juice) (¬Milk
Juice) (see also, e.g., Johnson-Laird, Khemlani, & Goodwin, 2015).
If a disjunction is mentally processed as exclusive (i.e., as a disjunction in which the two disjuncts cannot be true at the same time), the first conjunct (that in which both Milk and Juice are the case) is eliminated (see also, e.g., Quelhas, Rasga, & Johnson-Laird, 2019). There is also a difference in disjunction between the conjunctions of possibilities grasped by intuition and the conjunctions of possibilities apprehended by deliberative effort. The previous conjunction only is considered when people make deliberative effort. If not, the conjunction of possibilities is, in the case when individuals understand the disjunction as inclusive (i.e., as a disjunction in which the two disjuncts can be true at the same time), as follows:
(Milk Juice) (Milk) (Juice) (see also, e.g., Espino & Byrne, 2013). Semantics and pragmatics can also modulate the possibilities of a disjunction (see also, e.g., Orenes & Johnson-Laird, 2012). For example, given the disjunction ‘I go to China, or I go to Beijing’, its conjunction of possibilities would be: (China Beijing) (China ¬Beijing). The third conjunct is missing because Beijing is the capital of China.
-A conjunction such as ‘I have milk and I have lactose’: (Milk Lactose) (see also, e.g., Khemlani, Orenes, & Johnson-Laird, 2014).
Conjunctions in natural language generally refer to facts. This is because, according to TMM, when a sentence is linked to just one possibility, it is a fact (e.g., Khemlani et al., 2017). Modulation is also possible in the case of conjunction, too (see also, e.g., Johnson-Laird, 2010). For instance, the possibilities corresponding to ‘you have milk and you feel bad’ can be equivalent to those of a conditional such as ‘if you have milk, then you will feel bad’. The conjunction of possibilities would be:
(Milk Bad) (¬Milk Bad) (¬Milk ¬Bad).
There are many interesting details within the framework of TMM. However, what is essential to achieve the goal of the present paper is the way it derives conclusions. On the one hand, a piece of information can remove possibilities and to lead to a conclusion. Let us see some examples.
Let us suppose that (1) holds.
If I have milk, then I have lactose
Following TMM, if (1) is the case, and the necessary deliberative effort is made, (2) is the case as well.
(Milk Lactose) (¬Milk Lactose) (¬Milk ¬Lactose)
The information that I drank milk eliminates the second and third conjuncts in (2). I do not have milk in them. Only one possibility of the conjunction remains: the first one. In it, not only do I have milk, but I also have lactose. For this reason, given (2) and the information that I have milk, I can conclude that I have lactose.
This is an application of Modus Ponendo Ponens, that, is the logical schema ‘if p then q, p; therefore, q’ (see also, e.g., Byrne & Johnson-Laird, 2009). The theory explains the application of, for example, Modus Tollendo Tollens, that is, the logical schema ‘if p then q, not-q; therefore, not-p, too. If (2) and the information that I did not have lactose are correct, the possibilities in which I have lactose in (2), that is, the two first ones, are removed. In the third one, I do not have milk. Therefore, I can infer the latter fact, that is, that I do not have milk either (e.g., Byrne & Johnson-Laird, 2009).
From these explanations about Modus Ponendo Ponens and Modus Tollendo Tollens, one might think that there are not any differences between TMM and classical logic. But this is not so. Beyond the fact that deliberative effort is required to use Modus Tollendo Tollens (it is necessary to identify the third conjunct in (2), i.e., the possibility in which the two clauses are negated), there are cases in which modulation prevents Modus Tollendo Tollens from being applied (e.g., Byrne & Johnson-Laird, 2009). If the conditional is (3),
If I go to China, then I go to Beijing The conjunction of possibilities will be (4).
(China Beijing) (China ¬Beijing) (¬China ¬Beijing)
In this situation, Modus Tollendo Tollens does not allow inferring any conclusions. The information that I was not in Beijing only enables to eliminate the first possibility in (4). I was not in Beijing in two possibilities: the second and third conjuncts. In one of them, I was in China. In the other one, I was not in that country. Hence, the fact that I was not in China cannot be inferred. On the contrary, the fact that I was not in Beijing could be derived from the fact that I was not in China. The latter fact would remove the two first conjuncts in (4). In the remaining possibility, I was not in Beijing. This is against classical logic, which does not allow making that.
Similar examples can be given in the case of disjunction. If (5) is true,
I have milk or I have juice
Conjunction (6) is also true (if the deliberative effort needed is made):
(Milk Juice) (Milk ¬Juice) (¬Milk Juice)
This allows applying rules valid in classical logic such as Modus Tollendo Ponens (i.e., the rule enabling to derive a disjunct from its disjunction and the negation of the other disjunct, e.g., the rule with this structure: ‘p or q, not-p; therefore, q’) (see also, e.g., Khemlani & Johnson-Laird, 2009). If it is known that (5) is true, and I state that I do not drink milk, only the third conjunct in (6) is acceptable. In that conjunct, I drink juice.
TMM moves away from basic requirements of classical logic with its account of disjunction, too. Modulation can block the use of rules such as Modus Tollendo Ponens as well (see also, e.g., Orenes & Johnson-Laird, 2012). If the disjunction is (7),
I go to China, or I go to Beijing The possibilities will be two:
(China Beijing) (China ¬Beijing)
We cannot deduce that I was in Beijing from the fact that I was not in China. In one conjunct in (8), I was in Beijing. In the other one, I was not in that city.
Lastly, we can think of an example related to conjunction. Conjunctions usually indicate facts, but there are cases in which this is different. For example, the case of (9).
You have milk and you feel bad Its conjunction of possibilities is (10).
(Milk Bad) (¬Milk Bad) (¬Milk ¬Bad)
As pointed out above, the possibilities in (10) are those of the conditional. So, we can apply a rule like Modus Ponendo Ponens. I can derive that I will feel bad from the fact that I drank milk. Likewise, we can use a rule akin to Modus Tollendo Tollens. I can conclude that I did not have milk from the fact that I do not feel bad.
Another important thesis of TMM is that it rejects the validity concept as understood in classical logic. According to this concept, a conclusion must be true if its premises are so. Instead, it proposes an alternative concept: necessity (e.g., Johnson-Laird et al., 2024). This concept is essential because it allows explaining interesting phenomena such as that of the deduction in (11) (see also, e.g., Orenes & Johnson-Laird, 2012).
I have milk
Therefore, I have milk or I have juice
Inference (11) is correct in classical logic. If the premise (‘I have milk’) is true, given that the conclusion is a disjunction (‘I have milk or I have juice’) in which the premise is one of the disjuncts, the conclusion must be true, too. However, people tend to think that it is false (Orenes & Johnson-Laird, 2012). According to TMM, the reason is that the conclusion is not necessary. It refers to a conjunction of possibilities such as (6), that is, (Milk Juice) (Milk ¬Juice) (¬Milk Juice). That makes it a non-necessary conclusion: it implies more possibilities than the premise (e.g., Johnson-Laird et al., 2024). For example, the third conjunct in (6) provides that it is possible that I do not have milk and that I have juice. That possibility is not compatible with the premise.
These are the main components of TMM (at least, as far as my aims in this paper are concerned). I will try to show that NAL has the resources to create a level or module of NARS working following those components. Given its empirical support, we can think that TMM describes the manner our mind makes inferences. If this is the case, the result of my account will be a level or module of NARS built a posteriori, and being able to come to the same conclusions as human beings.
Inheritance, extensional union, and extensional difference
A key copula in Narsese is ‘’, that is, the ‘inheritance copula’ (e.g., Wang, 2023). It establishes relations between extensions and intensions in a sense different from the habitual one in logic (see also, e.g., Wang, 2013). Let us suppose that the inheritance copula links the term Turtle to the term Chelonian. This means that Turtle is a term included in the set of the extension of Chelonian (Turtle is an example of Chelonian), and that Chelonian is a term belonging to the set of the intension of Turtle (Chelonian is a property of Turtle) (e.g., Wang, 2013; Definition 2.8). This can be expressed as follows:
Turtle Chelonian
If TurtleE represents the extension of Turtle, and ChelonianE stands for the extension of Chelonian, (12) implies that
TurtleE ChelonianE (Wang, 2013; Theorem 2.4).
NAL resorts to set theory language as a metalanguage (e.g., Wang, 2013). Here, ‘’ indicates that the left set is equal or a subset of the right set.
On the other hand, if TurtleI indicates the intension of Turtle, and ChelonianI refers to the intension of
Chelonian, (12) also provides that
ChelonianI TurtleI (Wang, 2013; Theorem 2.4).
NAL is a non-axiomatic logic because it can never be sure about any statement. Evidence changes (e.g., Wang, 2011). For this reason, truth values in NAL needs two numbers: it is necessary both a ‘frequency value’ (‘f’) and a confidence value (‘c’). If ‘w+’ represents the set of ‘positive evidence’, and ‘w’ is the set of ‘total evidence’, then “f = w+/w” (Wang, 2013, p. 35; Table 3.1). If ‘k’ is a constant being able to have any value, the system being less confident as k is higher (see also, e.g., Wang, 2023), then “c = w/(w + k)” (Wang, 2013, p. 35; Table 3.1). A statement such as (12) must include the values for f and c in the same way as (13) (e.g., Wang, 2013).
Turtle Chelonian f, c
If the system has seen 500 turtles that are chelonians, it will consider (14).
Turtle Chelonian 1, 0.998
Where f = 500/500 = 1, and c = 500/501 = 0.998 (for k = 1).
NAL also admits different kinds of compound terms. For my aims in the present paper, we only need those built from ‘extensional unions’ and ‘extensional differences’. Using First-Order Predicate Calculus as a metalanguage, which is also frequent in NAL (e.g., Wang, 2013), the extensional union, or ‘intensional intersection’, Turtle ∪ Salamander means that
x {[(Turtle ∪ Salamander) x] [(Turtle x) (Salamander x)]} (see, e.g., Wang, 2013; Definition 7.7).
Where ‘’ is the universal quantifier, which indicates that what is in the formula applies to any x in the universe considered, and ‘’ is the symbol for the biconditional.
Extensional union Turtle ∪ Salamander refers to the set of all the turtles and all the salamanders, and to the properties shared by Turtle and Salamander (in their intensions). So, (Turtle ∪ Salamander)E includes both the extension of Turtle and the extension of Salamander (Wang, 2013; Theorem 7.4). Both
and (16) hold.
Turtle (Turtle ∪ Salamander) 1, 0.9
Salamander (Turtle ∪ Salamander) 1, 0.9
In (15) and (16), f = 1 and c = 0.9 because those are the numbers NAL attributes whenever f and c are not explicitly established (Wang, 2013).
Extensional difference Turtle – Red indicates that
x {[x (Turtle – Red)] [(x Turtle) ¬(x Red)]} (see, e.g., Wang, 2013; Definition
7.8).
Extensional difference Turtle – Red represents the set of turtles that are not red. Therefore, only (17) holds. (18) cannot be accepted (Wang, 2013; Theorem 7.10).
(Turtle – Red) Turtle 1, 0.9
(Turtle – Red) Red 1, 0.9
Finally, it is only necessary to add that evaluative questions such as (19) are possible in NAL.
Turtle Chelonian?
If Turtle Chelonian is in the system, the answer will be positive, and the values for f and c will be shown.
For several reasons that I will not consider here (to address them would take us away from my goal in the present paper), a statement can be in the system with different values. In that case, given an evaluative question on that statement, the system would indicate the highest c and the value f for that highest c (e.g., Wang, 2013). This is an important characteristic of NAL. However, as will be seen below, it is not relevant here.
NAL is much more complex than my description in this section. I have presented just a limited number of components of the system, and none of its inference rules. But that number suffices to capture the way TMM works.
TMM from NAL
What I will show below is not the first or the only possible computer program working as TMM. There are already programs trying to computationally implement TMM (e.g., Johnson-Laird, Byrne, & Khemlani, 2023; Khemlani & Johnson-Laird, 2022; Ragni, Kola, & Johnson-Laird, 2018). As far as I know, the present paper proposes the first computational treatment of TMM from NAL, and hence from NARS.
To build a level or module of NAL being able to derive conclusions like those of TMM, the first task to do is to explain how the conjunctions of possibilities can be translated into Narsese. I will try to do that in this section. I will consider each of the examples above. They are illustrative enough to describe the manner TMM understands the human inferential activity. To capture each of them can mean to capture the essential basis of the latter theory.
The first example was that of (1), that is, ‘if I have milk, then I have lactose’, and (2), that is, (Milk Lactose) (¬Milk Lactose) (¬Milk ¬Lactose). (2) can be translated as set of inheritance statements (20).
{(Milk ∪ Lactose) Possible 1, 0.9, (Lactose – Milk) Possible 1, 0.9, (Milk –
Lactose) Impossible 1, 0.9}
The first element of the set, that is, (Milk ∪ Lactose) Possible 1, 0.9, stands for the possibility in which I have milk and lactose. The second one, that is, (Lactose – Milk) Possible 1, 0.9, represents the case in which I have lactose without having milk. The third one, that is, (Milk – Lactose) Impossible 1, 0.9, refers to the circumstance in which I have milk without having lactose, which is deemed as impossible.
One might think that one possibility is missing, and that one of the statements in (20) does not need to be there. The possibility that seems to be missing is the last one in (2), that is, (21).
(¬Milk ¬Lactose)
Which indicates the situation in which I do not have milk or lactose. The statement that appears to be unnecessary is the last one in (20), that is, (22).
(Milk – Lactose) Impossible 1, 0.9
This is because to derive conclusions within the framework of TMM, as shown in the literature about the latter theory, what is possible is enough. What is impossible is not required. However, to get NAL to make inferences as TMM, (22) is useful and (21) is not necessary. (22) expresses that having milk without lactose is impossible. Therefore, as provided by (21), in a circumstance in which lactose is not had, it is impossible to have milk. We know that we can remove lactose from milk, but NAL only considers the accumulated experience to date (see, e.g., Wang, 2011). If it must think of just (1), the system cannot know that milk without lactose is possible in the real world. In this way, if we assume that (22) is the statement corresponding to (21) in Narsese, we can come to the same conclusions indicated above following TMM. We only need one more assumption: PossibleE ∩ ImpossibleE = ø (where ‘∩’ and ‘ø’ work exactly as in set theory; ‘∩’ is the symbol for intersection, i.e., the symbol referring to the elements that are both in the left set and the right set; ‘ø’ expresses empty set), that is, nothing can be both in the extension of Possible and in the extension of Impossible.
Let us start with the case of Modus Ponendo Ponens. If the first premise is (1), the second premise must be ‘I have milk’. Let ‘NALTMM’ be the level or module in NAL working as TMM. In NALTMM, the first premise should be processed as the statements in (20), that is, (22), (23), and (24).
(Milk ∪ Lactose) Possible 1, 0.9
(Lactose – Milk) Possible 1, 0.9 The second premise should be translated as (25).
Milk Possible 1, 0.9
From now on, I will ignore the values for f and c. The reason is that in all the statements in Narsese below the values will be the same: f = 1 and c = 0.9. I will remove them to simplify.
If (25) is true, NALTMM must ignore (22), as it can derive (26) from (22).
Milk Impossible
An objection can be presented here. Contradictions are admitted in NAL. The system can use contradictory statements and come to different conclusions following different paths (e.g., Wang, 2013). Against this, I can say that, while it is correct, we can also implement different levels or modules within NAL that can be at different tiers. Those levels or modules can be mutually independent (e.g., Wang, 2013).
Given that, as I pointed out above (and it can be deduced from Wang, 2013; Theorem 7.4), (Milk ∪ Lactose)I = MilkI ∩ LactoseI (where ‘∩’ works exactly as in set theory), (25) can be derived from (23), too. But what is relevant is that (27) can be inferred from both (23) and (24).
Lactose Possible
And this can be understood as an application of Modus Ponendo Ponens.
The following example was that of the derivation of the conclusion in Modus Tollendo Tollens. If (1) is the first premise in an inference of Modus Tollendo Tollens, then the second premise should be ‘I do not have lactose’, that is, (28).
Lactose Impossible
In this case, (23) cannot be considered. It allows concluding (27). The same can be said about (24): it also enables to infer (27). Only (22) remains. What (22) allows deriving is (26).
One might think that this does not totally capture the proposal of TMM. According to the latter theory, to apply Modus Tollendo Tollens, it is necessary to make the deliberative effort required to identify (21), and my account seems to ignore that. Nevertheless, this problem is easy to solve. Two different levels or modules could be built, for example, NALTMM1 and NALTMM2. One of them would work with all the possibilities deliberative effort can note. The other one only would consider the possibilities that can be recovered without effort. In the latter level or module, Modus Tollendo Tollens could not be used. Similar arguments apply to any other example in this paper implying deliberative effort. In fact, this distinction between two levels or modules can help in any circumstance in which effort and lack of effort are related to different expected results within the framework of TMM. In the cases in which the predictions of the theory are different when there is effort and when there is no effort, the building of two diverse levels or modules such as NALTMM1 and NALTMM2 is always a feasible solution.
My next example was that of conditional (3), that is, ‘if I go to China, then I go to Beijing’, whose possibilities needed to be modulated. Those possibilities could be expressed in Narsese as (29), (30), and (31).
(China ∪ Beijing) Possible
(China – Beijing) Possible
(Beijing – China) Impossible
In an inference of Modus Tollendo Tollens, the second premise would be ‘I do not go to Beijing’, which would be in Narsese:
Beijing Impossible
Given that (29) enables to infer (33),
Beijing Possible
It must be removed. On the other hand, (31) just allows deducing (32) again. Only (30) can be considered, but it is not useful either. (30) leads to (34).
China Possible
Which is the opposite of the conclusion in Modus Tollendo Tollens.
Regarding disjunction (5), that is, ‘I have milk or I have juice’, NALTMM could attribute statements (35), (36), and (37) to it (if it is understood as inclusive).
(Milk ∪ Juice) Possible
(Milk – Juice) Possible
(Juice – Milk) Possible
The second premise in Modus Tollendo Ponens would be ‘I do not have milk’, that is, (38).
Milk Impossible
Statement (38) is incompatible with (35), as the latter enables to come to (25), that is, to Milk Possible. The same can be said about (36). (37) can be accepted. It leads to (39).
Juice Possible
That is, to the conclusion in Modus Tollendo Ponens.
In the case of (7), that is, ‘I go to China, or I go to Beijing’, the disjunction requires modulation of its possibilities. In NALTMM it would be linked to statements (40), (41), and (42).
(China ∪ Beijing) Possible
(China – Beijing) Possible
(Beijing – China) Impossible
Statements (40), (41), and (42) are equivalent to, respectively, (29), (30), and (31). If the second premise is now ‘I do not go to China’, it would be expressed as (43).
China Impossible
Thus, (40) should be ignored. It allows coming to (34), that is, to China Possible. The level or module could not accept (41) either. NALTMM can deduce (34) from it, too. Only (42) is coherent with (43). But
implies (32), that is, Beijing Impossible.
Apparent conjunction (9), that is, ‘you have milk and you feel bad’, can refer to statements (44), (45), and (46).
(Milk ∪ Bad) Possible
(Bad – Milk) Possible
(Milk – Bad) Impossible
These statements are related to conjunction (9) because of modulation. As the possibilities of (9) in TMM, statements (44) to (46) correspond to a conditional in natural language (‘if you have milk, then you will feel bad’), not to a conjunction. Therefore, the application of a rule like Modus Ponendo Ponens is viable.
Let us suppose that it is stated that ‘you have milk’, that is, (25), that is, Milk Possible. (46) needs to be removed, as it enables to derive (26), that is, Milk Impossible. The statements that keep holding are (44) and (45). (47) can be concluded from both.
Bad Possible
The latter statement would be the conclusion in Modus Ponendo Ponens.
Lastly, regarding inference (11), that is, ‘I have milk. Therefore, I have milk or I have juice’, the premise just offers the information in (25), that is, Milk Possible. However, the conclusion refers to statements
(35) to (37), that is, respectively, (Milk ∪ Juice) Possible, (Milk – Juice) Possible, and (Juice –
Milk) Possible. From both (35) and (36), NALTMM can derive that Milk belongs to the extension of
Possible. Nonetheless, an evaluative question applied to the content in (37), that is, (Juice – Milk)
Possible?, would give a negative answer and prevent (11) from being accepted.
Conclusions
Hence, it seems that we can build a level or module of NAL such as NALTMM. That level or module captures the essential inferential processes of TMM by means of a few components of the logic. Beyond the basic inheritance copula, only two of its types of compound terms are necessary: extensional union and extensional difference. The idea of TMM related to the substitution of logical validity for necessity can be considered resorting to evaluative questions. No inference rule of NAL is required.
NAL has much more resources. It can deal with much more details of the conjunctions of possibilities. The literature of TMM cited in the present paper reveals many of those details, which often refer to relations between the clauses. NAL can address them. I have not described above how NAL can do that because the account presented suffices to show the general way NAL can consider the processes proposed by TMM. Maybe this is a point to develop in a future work.
This means that TMM not only can be emulated by means of a non-axiomatic logic such as NAL. It can also be implemented using a computer system such as NARS. This is consistent with the theory. As said, there are already computer programs working as TMM. On the other hand, NAL has the potential to work like, for example, Prolog, a programming language based on classical logic (e.g., Wang, 2013).
If TMM is correct (and, as also indicated, its strong empirical support can lead us to think that), the result of my account can be relevant. If the theory describes the manner our mind acts, a level or module such as NALTMM can come to the same conclusions as us. I am not claiming that the human mind works following requirements such as those of the inheritance copula, or the concepts of extensional union and extensional difference. What I state is just that, because of NALTMM, NARS can infer conclusions like those predicted by TMM.
Some challenges need to be addressed, and further research is required. The main point is natural language processing (for general lines to process natural language from NARS, see, e.g., Wang, 2013). We need algorithms to translate expressions in natural language into statements such as those used by NALTMM. To work in practice as TMM, NALTMM must be able to store and relate information in a particular way: in the way TMM proposes. Modulation is especially important regarding this. Not all human beings have the same information and knowledge about the same topics. Perhaps, different levels or modules are also necessary to consider the differences in information and knowledge individuals have. This is the path to follow in next studies.
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