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This is a model of the syntax of sentential and quantified logic made of LEGO.
Sentential Logic
For sentential logic, each of the smallest yellow block-like assemblies represents an atomic sentence, and each red, light and dark grey, green, and blue T-shaped assembly represents a sentential logical connective ¬ (not), & (and), ∨ (or), ⊃ (if … then), and ≡ (if and only if), respectively. Each connective can be coupled with up to two of atomic sentences and other connectives via technic pins and pinholes, forming complex sentences of sentential logic. 'A & ¬ B', for example, would be depicted as follows.
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The basic elements of the model (the assemblies representing atomic sentences and connectives) are easiest to combine only in ways that the basic elements of the syntax of sentential logic can be combined to form sentences. Thus exploring this model can aid in one’s understanding of the syntax of propositional logic.
There are some differences between the syntax of sentential logic and the model that are worth noting. First, an advantageous way the models differs from the syntax concerns the fact that the design of the connectives obviates the need for parentheses. A sentence can be built by combining one or more atomic sentences with a connective. Then a more complex sentence can be built by combining the previously built sentence, perhaps with another previously built sentence or atomic sentence with another connective, resulting in a more complex sentence. The main connective of the resulting sentence will the the "uppermost" connective--the one combined with last. Below, for example, is '(A ∨ ¬ B) ⊃ ¬ C'.
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A limitation of the model's construction is that the connectives aren't stretchy. As a result, for some sentences, such as '(A & (B ⊃ C)) ∨ ((A ⊃ B) & C)', extenders must be used on the main connective in the model, so that there is enough physical space to fit low-level connectives between medium-level connectives.
Quantified Logic
When the yellow atomic sentence elements are replaced with the yellow block-like property and relation assemblies, and the small tan individuals and the black and white quantifiers and their variables (the last consisting of white labels connected to black ropes), the model can be used to represent sentences of quantified logic. Each quantifier will be fitted with an appropriate number of appropriate variable "ropes" (with variables matching the variable of the quantifier). (I should add that the idea of representing variables as ropes from quantifiers to argument positions in properties and relations can be traced back to a very similar idea communicated to me by Cody Gilmore at UC Davis of how he imagined variables in his theory of propositons, slot theory. For more on slot theory, see, e.g., Gilmore's paper "Slots in Universals".)
The number of variables fitted to a quantifier will depend on the number of variables one needs to bind in the formula that falls within the scope of the quantifier. In '∀x x = a', for example, only one variable must be fitted to the ∀ quantifier, since there is only one variable that needs to be bound in 'x = a'.
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In '∀x x = x', however, two variables must be fitted, since there are two variables that need to be bound in 'x = x'.
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If the length of a formula bound by a quantifier is large, ropes can be added to extend the physical range of a variable, as in the following model of '∃x (Fx & ∀y (Fy ⊃ x = y)'.
The higher complexity of the model of quantified logic means there is more freedom of movement in it. In some cases, this freedom is reflected in the freedom one has in concatenating the vocabulary of quantified logic. For example, individuals could be inappropriately fitted to quantifiers.
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In other cases, however, the physical design of the model introduces more freedom of movement than is present in the concatenability of the vocabulary of quantified logic. For example, properties and relations can be inappropriately fitted to the bottom of connectives.
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Moreover, the elements of the model must be internally modified sometimes, e.g., when fitting a quantifier with a certain number of variables.
Factors like these may lower the pedagogical value of the LEGO model of quantified logic relative to that of the model of sentential logic. Students having trouble with the complexity of the syntax of quantified logic may have just as much trouble with the complexity of the LEGO model, including all the rules one must follow to avoid combining the elements in both ways possible with and ways not possible with the vocabulary of quantied logic. Still, some may find it more intuitive to learn rules when they concern allowable combinations of physical objects rather than of abstract objects. Moreover, the model can be explored in concert with the rules of syntax of quantified logic, thus providing opportunities for multi-modal exploration and reinforcement. So plenty of pedagoical value remains.
Instructions for the model, as well as ideas for using connective extenders, fitting various numbers of variables to quantifiers, and a parts list, can be found at the link below.
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