Are there any natural examples of generalized consequence relations that are not tame (i.e. don't "distribute" over each element of the conlusion)?

Question

Are there any natural examples of generalized consequence relations that are not tame (in a sense made precise below)?

I'm calling a generalized consequence relation a relation on $\mathrm{Set}[\mathrm{Wff}] \times \mathrm{Set}[\mathrm{Wff}]$ rather than $\mathrm{Set}[\mathrm{Wff}] \times \mathrm{Wff}$.

Tameness of $\models$ means that the following property holds of $\models$:

$$ \Gamma \models \Delta \;\;\textit{if and only if}\;\; \;\; \text{for all $\varphi$ in $\Delta$, $\Gamma \models \varphi$} $$

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For context, I'm interested in ways of characterizing what a logic is abstractly. I know of two, the consequence relation (axiomatization given below) and the deductive closure à la Tarski (straightforwardly equivalent to the consequence relation characterization).

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In Algebraizable Logics, Blok and Pigozzi list out some rules that a consequence relation must follow. I'll write $\vdash$ where they write $\vdash_S$. I'll also number the properties (101, 102 ...) instead of (1, 2, ...) so it is easy to compare corresponding properties in different lists.

101. $\varphi \in \Gamma$, then $\Gamma \vdash \varphi$.
102. $\Gamma \vdash \varphi$ and $\Gamma \subset \Delta$, then $\Delta \vdash \varphi$.
103. $\Gamma \vdash \varphi$ and (for all $\psi$ in $\Gamma$, $\Delta \vdash \psi)$, then $\Delta \vdash \varphi$.
104. $\Gamma \vdash \varphi$, then there exists a finite $\Gamma_0$ of $\Gamma$ such that $\Gamma_0 \vdash \varphi$.
105. $\Gamma \vdash \varphi$, then $\sigma(\Gamma) \vdash \sigma(\varphi)$.

If we imagine a binary relation where both sides are sets, namely $\models$, then we can simplify a lot of these rules and some of them familiar names from the study of binary relations more generally.

200. identity: $\Gamma \models \Gamma$
201. some kind of monotonicity type property: $\Gamma \models \Phi$ and $\Phi' \subset \Phi$, then $\Gamma \models \Phi'$.
202. another kind of monotonicity: $\Gamma \models \Phi$ and $\Gamma \subset \Delta$, then $\Delta \models \Phi$ .
203. transitivity: $ \Gamma \models \Phi $ and $\Delta \models \Gamma$, then $\Delta \models \Phi$
204. finitariness: $\Gamma \models \Phi$ and $\Phi$ is finite, then there exists a finite $\Gamma_0$ such that $\Gamma_0 \models \Phi$.
205. subtitution closure: $\Gamma \models \Phi$, then $\sigma(\Gamma) \models \sigma(\Phi)$

This paraphrase is kind of nice. It lets us describe what a logic is in an interesting way.

A logic is a set $\Lambda$ (really $\mathrm{Set}[\mathrm{Wff}]$) that is acted on by a monoid $\Sigma$ (the subtitutions, which form a monoid under composition).

$\Lambda$ is additionally equipped with two preorders $\subset$ and $\models$.

Both preorders respect $\Sigma$, so we can just insist on the following extra rule. Note the direction of the rule. $\models$ is quite different from the deductive closure.

301. $\Gamma \subset \Delta$ implies $\Delta \models \Gamma$.

And this rule for finitariness below, if we choose to insist on finitariness.

304. finitariness: $\Gamma \models \Phi$ and $\Phi$ is finite, then there exists a finite $\Gamma_0$ such that $\Gamma_0 \models \Phi$.

However, without the tameness property given in the first paragraph (reproduced below), not every $\models$ corresponds to a $\vdash$.

$$ \Gamma \models \Delta \;\;\textit{if and only if}\;\; \;\; \text{for all $\varphi$ in $\Delta$, $\Gamma \models \varphi$} $$

I can kind of sort of see a possibility for using a non-tame generalized consequence relation to represent a defeasible reasoning system, but I'm curious if there are any standard examples of such a system.

Answer 1

It seems like you intend the intuitive meaning of $\Gamma\models \Delta$ to be "If all sentences in $\Gamma$ are true, then all sentences in $\Delta$ are true." This is fine by me (and it agrees with the notation used in model theory, where we might write $T_1\models T_2$ with $T_1$ and $T_2$ theories or $\varphi(x)\models p(x)$ with $\varphi(x)$ a formula and $p(x)$ a complete type). But I want to point out that it conflicts with the usual notation for sequents, where $\Gamma\vdash \Delta$ means "If all the sentences in $\Gamma$ are true, then some sentence in $\Delta$ is true". See the Wikipedia page I linked to for discussion of the motivation for this notational convention. The conflict might make looking for references a bit tricky.

A natural kind of example where your tameness condition can fail is the world of substructural logics. "Substructural" here refers to a logic that lacks some of the usual "structural rules". One interpretation of substructural logics is that they take into account the "resources" (hypotheses) used to prove things. A famous example of a substructural logic, mentioned by Noah in the comments, is linear logic.

For example, in a substructural logic, $\Gamma\vdash \varphi$ might mean "$\varphi$ is provable using each assumption in $\Gamma$ exactly once."

Under this interpretation, if we have $\Delta = \{\varphi_1,\dots,\varphi_n\}$ with $n>1$ and $\Gamma\vdash \varphi_i$ for all $1\leq i \leq n$, instead of $\Gamma\vdash \Delta$, we would naturally get $$\underbrace{\Gamma,\Gamma,\dots,\Gamma}_{n \text{ times}}\vdash \Delta.$$

To make sense of this failure of tameness, we would want the things on the left and right of the turnstile to be lists or multisets instead of sets.

Your monotonicity properties 201 and 202 correspond to the structural rule called "weakening", while the failure of tameness described here corresponds to the absence of the structural rule called "contraction". In a substructural logic with weakening but without contraction, you could get a non-tame generalized consequence relation by interpreting $\Gamma\models \Delta$ as "we can prove all of the sentences in $\Delta$, and in all these proofs taken together, we use each hypothesis in $\Gamma$ at most once".

Of course, this is just one way tameness can fail. There might be other natural examples using sets instead of lists or multisets.