2. Development of LoT
The target theory is minimally developmental. Being limited in infancy, it avoids to account for changes in LoT or specifies how rules, principles, and constraints of LoT change in concern to LoT properties. We outline some general principles of a developmental theory of LoT (DLoT) as complementary to the target theory. DLoT claims that a probabilistic language-of-thought (PLoT), defended in the target article, provides early foundations of LoT, but it does not accommodate later development. DLoT argues that rules emerge from PLoT with development, upgrading pragmatic reasoning in early childhood into deductive and analogical reasoning later. Mental awareness is a critical factor in this development (Demetriou, Makris, Kazi, Spanoudis, & Shayer, Reference Demetriou, Makris, Kazi, Spanoudis and Shayer2018). In psychometric terms, DLoT is a systematic expansion of a core relational integration capacity into representations and rules prescribing optimal inference-based integration to handle novel encounters capitalizing on experience (Demetriou, Golino, Spanoudis, Maris, & Greiff, Reference Demetriou, Golino, Spanoudis, Maris and Greiff2021).
This is obvious in mastering the four basic schemes of syllogistic reasoning: Modus ponens (MP), modus tollens (MT), and the fallacies, affirming the consequent (AC) and denying the antecedent (DA). MP, a logical primitive, emerges as a Bayesian product from pragmatic contexts, at 5–6 years of age (Oaksford & Chater, Reference Oaksford and Chater2020). Pragmatic MP is an induction binding two representations (“A occurs” and “B occurs”) into an inductive rule (i.e., “When A occurs, B also occurs”). Transition to rule-based thinking at 6–7 years lifts the alignments of representations of preschool age into a rule-based representational imperative (A and B, A, therefore B), involving a sense of logical necessity. MT is grasped at 8–9 years, when inferential rules emerge as explicit constraints on how representations may be combined in syllogistic chains. By the end of childhood, representational imperatives are fluent enough to be read both ways (A and B, not B, therefore not A) (Christoforides, Spanoudis, & Demetriou, Reference Christoforides, Spanoudis and Demetriou2016).
The integration of MP and MT into a fluent inferential ensemble transforms inductive imperatives into deductive necessities constrained by rules explicitly metarepresented in principles specifying how inferential spaces are interrelated. Rules specify that different representational spaces may have different inferential constraints (e.g., birds fly, mammals walk, fish swim, etc.) yielding different inductive implications about individual elements in each space (e.g., blackbirds fly, elephants walk, sharks swim, etc.). Moving across representational spaces is possible when relations are abstracted unifying observable differences, such that flying, walking, and swimming is movement in space. Thus, initial premises define the constraints of the mental space in which inference occurs (e.g., birds fly) and premises following specify an application subdomain of this space where property transfer is necessary (e.g., accepting that dogs are birds necessarily implies that dogs fly). Therefore, actual properties (e.g., dogs are not birds) are overwritten by logical constraints connecting mental spaces. Checking consistency of representations in reference to these constraints enables understanding logical fallacies: Accepting “If A then B” does not allow drawing any conclusion about A if only knowing that B occurred (AC) or about B if only knowing that A did not occur (DA), because B may be caused by causes other than A. That is, the space of the argument is embedded in a context of possibilities, defined by principles integrating inferential rules.
Developing awareness is critical in the formalization of inferential and truth-evaluation rules crystallizing into DLoT. Transition from Bayesian reasoning to rule-based inference at 6–7 years depends on awareness of representations interlinked into predicate–argument structures transforming Bayesian possibilities into emerging logical necessities. Awareness of inferential processes at 8–9 years allows us to represent rules underlying inference and differentiate between them, according to logical operators, engendering biconditionality. Awareness of the relations between rules at 11–13 years allows inducing principles defining relations between rules, involving an awareness of inferential promiscuity enabling conception of an infinite number of alternative premises (Kazi, Kazali, Makris, Spanoudis, & Demetriou, Reference Kazi, Kazali, Makris, Spanoudis and Demetriou2019). Hence, PLoT develops in parallel with syllogistic reasoning, with the second formalizing Bayesian principles into logical schemes. Children learn complex concepts by running probabilistic inductions over representations of the world. These inductions and their associated inferential processes are represented with increasing accuracy with development. These representations crystallize inferential imperatives into schemes of syllogistic reasoning. Awareness is a fundamental mechanism in this crystallization raising system 1 into system 2 reasoning. In actual life they often interchange in use akin to a system 1.5 where both Bayesian and logical rules are used as it currently fits.
3. Multiple LoTs
The target article alludes (sect. 6) that there may be more than one LoT. Multiple LoTs are preferable over one LoT. They are established with development, such as a mathematical, a causal, a spatial, and a social LoT, reflecting ability to use multiple symbolic systems obeying different syntaxes. Objects and relations in different domains, such as causal, quantitative, spatial, and social relations, generate different types of symbols and different rules for their transformation. These rules reflect specificities in search, encoding, and evaluation of relations in each domain akin to the differentiation of a common LoT into largely autonomous LoTs. These express the rules and constraints for representing and processing relations specific to each domain (Demetriou et al., Reference Demetriou, Spanoudis, Christou, Greiff, Makris, Vainikainen and Gonida2023). Dehaene, Roumi, Lakretz, Planton, and Sablé-Meyer (Reference Dehaene, Roumi, Lakretz, Planton and Sablé-Meyer2022) concur, proposing multiple LoTs, “akin to computer languages, which encode and compress structures in various domains (mathematics, music, shape…)” (p. 1). In development, these languages diverge in the fashion that Indo-European languages emerged from a common protolanguage, often becoming mutually unintelligible, such as an LoT of mathematics, music, chemistry, and so on. Translating them into each other is possible drawing on a common protoLoT, but it requires special learning. Assuming different LoTs accounts for intra- and interindividual differences in cognitive development and learning difficulties, such as dyslexia or dyscalculia. Therefore, specifying a developing general LoT and local LoTs helps integrate different disciplines, such as cognitive, developmental, and brain science and enables mapping human development on artificial intelligence (Demetriou et al., Reference Demetriou, Golino, Spanoudis, Maris and Greiff2021).
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1. Introduction
This commentary focuses on section 5 of the target article, dealing with language-of-thought (LoT) in children. We agree with the target article that assuming LoT is useful for further development of cognitive and developmental theories. Specifying constructs in LoT offers a system for exploring relations between representations and their development. The six properties of LoT proposed in the target article are useful, enabling to specify mental units carrying information, rules for binding and transforming units, and model the development of mind. Examples of the developmental implementation of properties are given below. However, the target article is weak in two themes: Development and multiplicity of LoT.
2. Development of LoT
The target theory is minimally developmental. Being limited in infancy, it avoids to account for changes in LoT or specifies how rules, principles, and constraints of LoT change in concern to LoT properties. We outline some general principles of a developmental theory of LoT (DLoT) as complementary to the target theory. DLoT claims that a probabilistic language-of-thought (PLoT), defended in the target article, provides early foundations of LoT, but it does not accommodate later development. DLoT argues that rules emerge from PLoT with development, upgrading pragmatic reasoning in early childhood into deductive and analogical reasoning later. Mental awareness is a critical factor in this development (Demetriou, Makris, Kazi, Spanoudis, & Shayer, Reference Demetriou, Makris, Kazi, Spanoudis and Shayer2018). In psychometric terms, DLoT is a systematic expansion of a core relational integration capacity into representations and rules prescribing optimal inference-based integration to handle novel encounters capitalizing on experience (Demetriou, Golino, Spanoudis, Maris, & Greiff, Reference Demetriou, Golino, Spanoudis, Maris and Greiff2021).
This is obvious in mastering the four basic schemes of syllogistic reasoning: Modus ponens (MP), modus tollens (MT), and the fallacies, affirming the consequent (AC) and denying the antecedent (DA). MP, a logical primitive, emerges as a Bayesian product from pragmatic contexts, at 5–6 years of age (Oaksford & Chater, Reference Oaksford and Chater2020). Pragmatic MP is an induction binding two representations (“A occurs” and “B occurs”) into an inductive rule (i.e., “When A occurs, B also occurs”). Transition to rule-based thinking at 6–7 years lifts the alignments of representations of preschool age into a rule-based representational imperative (A and B, A, therefore B), involving a sense of logical necessity. MT is grasped at 8–9 years, when inferential rules emerge as explicit constraints on how representations may be combined in syllogistic chains. By the end of childhood, representational imperatives are fluent enough to be read both ways (A and B, not B, therefore not A) (Christoforides, Spanoudis, & Demetriou, Reference Christoforides, Spanoudis and Demetriou2016).
The integration of MP and MT into a fluent inferential ensemble transforms inductive imperatives into deductive necessities constrained by rules explicitly metarepresented in principles specifying how inferential spaces are interrelated. Rules specify that different representational spaces may have different inferential constraints (e.g., birds fly, mammals walk, fish swim, etc.) yielding different inductive implications about individual elements in each space (e.g., blackbirds fly, elephants walk, sharks swim, etc.). Moving across representational spaces is possible when relations are abstracted unifying observable differences, such that flying, walking, and swimming is movement in space. Thus, initial premises define the constraints of the mental space in which inference occurs (e.g., birds fly) and premises following specify an application subdomain of this space where property transfer is necessary (e.g., accepting that dogs are birds necessarily implies that dogs fly). Therefore, actual properties (e.g., dogs are not birds) are overwritten by logical constraints connecting mental spaces. Checking consistency of representations in reference to these constraints enables understanding logical fallacies: Accepting “If A then B” does not allow drawing any conclusion about A if only knowing that B occurred (AC) or about B if only knowing that A did not occur (DA), because B may be caused by causes other than A. That is, the space of the argument is embedded in a context of possibilities, defined by principles integrating inferential rules.
Developing awareness is critical in the formalization of inferential and truth-evaluation rules crystallizing into DLoT. Transition from Bayesian reasoning to rule-based inference at 6–7 years depends on awareness of representations interlinked into predicate–argument structures transforming Bayesian possibilities into emerging logical necessities. Awareness of inferential processes at 8–9 years allows us to represent rules underlying inference and differentiate between them, according to logical operators, engendering biconditionality. Awareness of the relations between rules at 11–13 years allows inducing principles defining relations between rules, involving an awareness of inferential promiscuity enabling conception of an infinite number of alternative premises (Kazi, Kazali, Makris, Spanoudis, & Demetriou, Reference Kazi, Kazali, Makris, Spanoudis and Demetriou2019). Hence, PLoT develops in parallel with syllogistic reasoning, with the second formalizing Bayesian principles into logical schemes. Children learn complex concepts by running probabilistic inductions over representations of the world. These inductions and their associated inferential processes are represented with increasing accuracy with development. These representations crystallize inferential imperatives into schemes of syllogistic reasoning. Awareness is a fundamental mechanism in this crystallization raising system 1 into system 2 reasoning. In actual life they often interchange in use akin to a system 1.5 where both Bayesian and logical rules are used as it currently fits.
3. Multiple LoTs
The target article alludes (sect. 6) that there may be more than one LoT. Multiple LoTs are preferable over one LoT. They are established with development, such as a mathematical, a causal, a spatial, and a social LoT, reflecting ability to use multiple symbolic systems obeying different syntaxes. Objects and relations in different domains, such as causal, quantitative, spatial, and social relations, generate different types of symbols and different rules for their transformation. These rules reflect specificities in search, encoding, and evaluation of relations in each domain akin to the differentiation of a common LoT into largely autonomous LoTs. These express the rules and constraints for representing and processing relations specific to each domain (Demetriou et al., Reference Demetriou, Spanoudis, Christou, Greiff, Makris, Vainikainen and Gonida2023). Dehaene, Roumi, Lakretz, Planton, and Sablé-Meyer (Reference Dehaene, Roumi, Lakretz, Planton and Sablé-Meyer2022) concur, proposing multiple LoTs, “akin to computer languages, which encode and compress structures in various domains (mathematics, music, shape…)” (p. 1). In development, these languages diverge in the fashion that Indo-European languages emerged from a common protolanguage, often becoming mutually unintelligible, such as an LoT of mathematics, music, chemistry, and so on. Translating them into each other is possible drawing on a common protoLoT, but it requires special learning. Assuming different LoTs accounts for intra- and interindividual differences in cognitive development and learning difficulties, such as dyslexia or dyscalculia. Therefore, specifying a developing general LoT and local LoTs helps integrate different disciplines, such as cognitive, developmental, and brain science and enables mapping human development on artificial intelligence (Demetriou et al., Reference Demetriou, Golino, Spanoudis, Maris and Greiff2021).
Acknowledgment
Thanks are extended to George Spanoudis for his comments on an earlier version of this article.
Competing interest
None.