This is a follow-up to my MathOverflow question: Is the consequence relation of a finite set of boolean connectives finitely generated?. Suppose we are working in a logical system of three values $\{0,1,2\}$. Let us denote that set by $3$. Let us also consider a countably infinite set of propositional variables $PROP$. We can define, for any subset $S$ of all finitary operations on $3$, a consequence relation based on that set $S$ and $PROP$, using $1$ as the designated value. However, we can also consider adjoining that set $S$ to $3$, forming an algebraic structure $(3;S)$. Is it the case that whenever the algebraic structure has a finite basis of identities, then the associated consequence relation is also finitely based, and vice versa? Or, are there cases where the consequence relation is finitely based, but the algebraic structure is not? Also, are there cases where the algebraic structure is finitely based, but the consequence relation is not? I asked this on math stack exchange, but it didn't receive an answer or even a comment.
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$\begingroup$ Can you explain what it means, exactly, for an algebraic structure to have a finite basis of identities? Do you mean that it can be finitely presented? That is, there is a finite list of generators and identities so that if I form all words and quotient by those identities I get the given structure? $\endgroup$– Joel David HamkinsCommented Jun 29, 2021 at 8:35
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1$\begingroup$ And could you also say precisely what it means for the consequence relation to be finitely based? $\endgroup$– Joel David HamkinsCommented Jun 29, 2021 at 8:41
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$\begingroup$ @JoelDavidHamkins I mean, an algebraic structure is finitely based, when the universally quantified equations that are valid in that structure can be generated from a finite set. And a consequence relation is finitely based when there are a finite number of rules that, when the rules are closed under substitution, give the whole consequence relation. $\endgroup$– user107952Commented Jun 29, 2021 at 16:33
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