Definition of a protoalgebraic deductive system

Question

I'm trying to understand the definition of a protoalgebraic logic given in Blok and Pigozzi.

I'm specifically interested in two things:

1. Exactly which family of Leibniz operators the authors are referring to with $\Omega T$.
2. Other equivalent characterizations of protoalgebraizability.

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A deductive system $S$ in Blok and Pigozzi is a tuple $(L, \vdash_S)$ is a tuple consisting of a language $L$ and a consequence relation $\vdash_S$, with $\vdash_S$ additionally constrained to be finitary.

Here is the definition of a protoalgebraic logic from page 12 of Algebraizable Logics. (The [and only if] is mine; I like being explicit about using iff in definitions.)

A deductive system $S$ is called protoalgebraic if [and only if], for every $S$-theory $T$, every pair of $\Omega T$-equivalent formulas are interderivable relative to $T$, i.e. $$ \langle \varphi, \psi \rangle \in \Omega T \Rightarrow T, \varphi \vdash_S \psi \;\; \text{and} \;\; T, \psi \vdash_S \varphi $$

This definition sort of makes intuitive sense. $\vdash_S$ induces a family of interderivability relations on $S$ which I'll call $[\simeq]_{ST}$ (for each theory $T$).

$$ [\simeq]_{ST}(\varphi, \psi) \;\;\textit{if and only if}\;\; T, \varphi \vdash_S \psi \;\; \text{and} \;\; T, \psi \vdash_S \varphi $$

And thus

$$ \Omega T \;\subset\; [\simeq]_{ST} $$

So the interderivability relation is a superrelation of the Leibniz operator $\Omega T$ for all $T$.

This also makes sense, but I'm having trouble figuring out exactly what relation $\Omega T$ refers to.

Definition 1.4 on page 10 gets me part of the way there.

Definition 1.4 Let $A$ be any algebra. For each $F \subseteq A$, let $\Omega_{A}F$ be the binary relation on $A$ defined by:

$$ \Omega_{A}F = \{ \langle a, b \rangle : P(a) \Leftrightarrow P(b) \;\; \text{for every $P$ definable over $\langle A, F \rangle$} \} $$

And this makes sense. $A$ is an algebraic structure, $F$ is essentially a collection of parameters. So the notion of definability makes sense here.

Blok and Pigozzi's definition of a theory is a little different than the one I'm used to. For them, a theory is deductively closed and not merely a set of sentences (page 6). I don't think we lose anything by considering arbitrary collections of sentences, since $\vdash_S$ is constrained to be a finitary consequence relation.

I am having trouble interpreting what exactly $\Omega T$ is supposed to be though. It's clearly really $\Omega_{A}T$ where $A$ is the algebra where the connectives (or, equivalently, function symbols) in $T$ are receiving their interpretations. But the role of $T$ itself is confusing. $T$ is not a subset of the carrier of $A$ like $F$ is in the earlier examples. I'm not sure how to use $T$ and $A$ to fashion a notion of definability and thus $\Omega_{A}T$.

I'm interested in how to get the specific Leibniz operator, but I'm also interested in alternative definitions of protoalgebraizability that don't involve the Leibniz operator since that might help me with the intuition behind this definition.

Protoalgebraic logics appear in some form in answers on this site here, here, and here, but those answers do not directly address what a protoalgebraic logic is. They instead explain why books that mention protoalgebraic logics are good resources for learning about the algebraic approach to understanding nonclassical logics (at least I think that's a fair summary of their intent).

Answer 1

Leibniz operator definition

Algebraizable Logics has a definition of this operator on page 11 that nails down what definable means in this context. I've paraphrased for brevity and apologize for my bad typesetting.

$$ \Omega_{A}F = \bigg\{\langle a, b \rangle : \varphi^A(a, \vec{c}) \in F \Leftrightarrow \varphi^A(b, \vec{c}) \in F \\ \text{for all $\varphi(p, \vec{q}) \in \text{Fm}_L$ and all $\vec{c} \in A$ } \bigg\} $$

So, this is a really interesting definition. This is like ordinary first-order definability in a first-order structure whose function symbols are the connectives of the logic and whose sole relation symbol is a chosenness predicate that is equvalent to membership in $F$. Also note that we always get every element of $A$ (wff's in this case) as parameter for the purposes of assessing definability.

With this in mind, we can examine the contrapositive of the definition of protoalgebraicity.

$$ T, \varphi \nvdash_S \psi \;\;\text{or}\;\; T, \psi \nvdash_S \varphi \;\;\text{implies}\;\; \langle \varphi, \psi \rangle \notin \Omega{T} $$

In this setting, we fix a set of formulas $T$. We then consider two wffs $\varphi$ and $\psi$.

Suppose they are inequivalent given $T$ as hypotheses.

This means we can construct a definable function (with arbitrary parameters from $A$), that sends $\varphi$ inside $T$ and $\psi$ outside $T$ or vice versa.

It also now seems clear why BP insisted on a theory $T$ being deductively closed on page 6.

An alternative characterization of protoalgebraicity

We can come up with a set of well-formed formulas in (at most) two free variables $\Delta(x, y)$ with properties inspired by the natural language phrase $\text{If $x$, then $y$}$.

This amounts to an explanation of the introductory paragraph of The simplest protoalgebraic logic by Josep Maria Font.

Note that the third source is Algebraic semantics for universal Horn logic without equality. This is written by Blok and Pigozzi, but is not Algebraizable Logics.

According to Blok and Pigozzi (3), a logic is protoalgebraic when there is a set $\Delta(x, y)$ of formulas in at most two variables satisfying the following two very elementary properties.

$$ \vdash_L \Delta(x,x) \tag{$\mathrm{R}_\Delta$} $$ $$ x, \Delta(x, y) \vdash_L y \tag{$\mathrm{MP}_\Delta$} $$

Here's my attempt to analyze what this means.

I don't have evidence that the intended reading of $\vdash_L \Delta(x, y)$ is like $\vdash_L \bigwedge \Delta(x, y)$ and not $\vdash_L \bigvee \Delta(x, y)$. However, the conjunctive reading makes sense if we think of the set of wff's $\Delta(x, y)$ as generalizing the role of $a \to b$ in modus ponens.

I'm going to invent two new terms, a diagonal tautology and a diagonal set.

Let $\varphi$ be a well-formed formula. $\varphi$ is a tautology if and only if $\vdash_L \varphi(\vec{x})$ holds for all compatible variable assignments $\vec{x}$.

Let $\varphi$ be a diagonal tautology if and only if:

• $\varphi$ has at most two free variables.
• If $\varphi$ has zero or one free variables, then $\varphi$ is a tautology.
• If $\varphi$ has exactly two free variables, then $\varphi(x, x)$ is a tautology.

Suppose $X$ is a set of well-formed formulas.

$X$ is a diagonal set if and only if:

• $\mathrm{FV}(X)$ has cardinality at most two.
• Every element $\varphi$ of $X$ is a diagonal tautology.

So, $\Delta(x, y)$ witnesses the protoalgebraicity of $\vdash_L$ if $\Delta(x, y)$ is a diagonal set and

$$ x, \Delta(x, y) \vdash_L y \;\; \text{holds for all $\{x, y\}$-valuations} $$

So, this makes sense. $\Delta(x, y)$ is like $x \to y$ in modus ponens, but it can be cobbled together out of random connectives and doesn't necessarily have to be $\{x \to y\}$ (or even finite).