What is the point of algebraic logic?

Question

The question is worded a bit unfortunately. Honestly, I find algebraic logic to be one of the most interesting subjects I've ever encountered. But I find myself unable to articulate why it's interesting, other than for its own sake. Is there any reason, practically speaking, for taking this approach to logic?

Answer 1

One advantage of algebraic logic is that the distinctions and relations between the meta levels become very clear. However, as long as algebraic logic stays on the level of propositional logic, and doesn't try to capture predicate logic (or at least equational reasoning from universal algebra or syllogistic reasoning), this advantage risks to become trivial by not reaching situations which would benefit from this sort of clarification. Now cylindric algebra and polyadic algebras seem to capture predicate logic, but they only capture classical predicate logic, while Heyting algebras only capture intuitionistic propositional logic.

---

So let me instead try to explain why algebraic logic is useful for me:

Using algebraic logic allows to leave the strictly logical context during the study of logical systems. This allows to continue investigations even if some things don't fit together. It also allows to study duality, even so dual logics in general fail to be logical systems in any reasonable sense. And you can really investigate many aspects of logical systems in an algebraic way, independent of whether they need investigation or not.

Let's try to illustrate this with an example which feels natural from an algebraic point of view, but dubious from a logical perspective. Start with intuitionistic logic, or rather a Heyting algebra, and define the tenary operation $t(x,y,z):=((z\to y)\to x) \land ((x\to y)\to z)$. Notice that $t(x,x,x)=x$ and $t(x,y,1)= (y\to x)$, i.e. $t$ is idempotent and allows to define implication. So Heyting algebras can be specified by purely idempotent operations, which is important in universal algebra in the context of Mal'cev conditions. Actually $t$ is a Mal'cev operation, i.e. $t(x,x,z)=z$ and $t(x,z,z)=x$. Now look at the dual Heyting algebra, where implication turns into minus (or non-implication, if you really prefer). Then $t'(x,y,z):=(z-(y-x)) \lor (x-(y-z))$ and classically (i.e. in a Boolean algebra) we have $t=t'$, but effectively the dual Heyting algebra is no longer able to talk about implication in any meaningful way. (I suspect the same is true for equivalence, but I haven't checked it yet.) But isn't the ability to talk about implications or at least equivalences the core of any logic? Maybe, but if you do algebraic logic, you don't need to worry about such questions as long as the resulting math is interesting and still sufficiently closely related to logic.

To make the example more extreme, let's remove truth and falsehood (i.e. the requirement that $0$ and $1$ exist) from the Heyting algebra. We can get back truth from $1=(a\to a)$, but if we use the tenary operation $t$ instead of implication, then there is no way to get back truth (or implication). I hope even Doug Spoonwood agrees that a logic without truth is dubious from a logical perspective!

Edit 20.08.16 I just noticed that the example even allows to illustrate a situation where things don't fit together. For a partial function $p:X\to Y$, we have $p^{-1}(A\cap B)=p^{-1}(A)\cap p^{-1}(B)$, $p^{-1}(A\cup B)=p^{-1}(A)\cup p^{-1}(B)$, and $p^{-1}(A-B)=p^{-1}(A)-p^{-1}(B)$. But $p^{-1}(Y)=X$ is only true if $p$ is a total function. If we interpret falsehood $0$ as the empty set, truth $1$ as the entire space $X$ (or $Y$), and $\land$ as intersection $\cap$, or $\lor$ as union $\cup$, and minus as minus, then nearly all operations of classical logic (including the tenary operation $t=t'$) are preserved under inverse partial functions, except for truth (and implication/negation).

---

Categorical logic would be another approach which might seem to offer even more freedom than the algebraic approach. However, it is much more difficult to find your own way there. How long would you take on your own to realize that often the categorial product must be ignored in favor of the bifunctor of a monoidal category? Or you have a nice correspondence between topological spaces and intuitionistic logic, but somehow the category of topological spaces has too many deficiencies, and you don't really know how to best fix those! Add to that the general burden to become sufficiently familiar with category theory in the first place.

Answer 2

The term "algebraic logic" is ambiguous and can have several interpretations. One such interpretation was proposed by Paul Halmos who was indeed advocating a rather radical scheme of algebraisation of logic in terms of polyadic algebras. The main advantage of his approach is to gain on the philosophical battleground, as a propping-up of naive set-theoretic realism. A recent article summarizes the issue as follows.

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms.