minimal logic

Minimal logic, introduced by I. Johansson in 1936, is intuitionistic logic without the *ex falso quodlibet* rule $\bot \vdash A$. It may also be defined by starting with Gentzen's sequent calculus for classical logic (with $\neg$ but not $\bot$) and restricting to sequents $\Gamma \vdash \Delta$ where $\Delta$ must contain *exactly one* formula.

Because it is not the case here that anything follows from a contradiction, minimal logic is a paraconsistent logic.

Models of minimal logic are cartesian closed categories, optionally with finite $n$-ary coproducts for $n\gt 1$. Under the Curry-Howard correspondence minimal logic corresponds to simply-typed lambda calculus, which is roughly speaking another way of saying that both are forms of the internal logic of cartesian closed categories.

Finn, I'm unsure of what exactly is supposed to be ‘minimal logic’. Does this unreadable reference match your understanding? To wit: two connectives ($\Rightarrow$ and $\bot$, with $\neg{p}$ defined as $p \Rightarrow \bot$), two axiom schemes ($p \Rightarrow q \Rightarrow p$ and $(p \Rightarrow q \Rightarrow r) \Rightarrow (p \Rightarrow q) \Rightarrow p \Rightarrow r$) and modus ponens. Note in particular no characterisation of $\bot$ whatsoever (which strikes me as a little too minimal).

I think that I thought that minimal logic did not have $\neg$ or $\bot$ at all; that seems like a more reasonable amount of minimality to me. Also, it seems that your first paragraph fits the above reference better, while your last fits better the version with no negation at all. But I may be missing something. —Toby Bartels

Mike: I’ve definitely seen “minimal logic” used to mean logic that has $\bot$ and $\neg$ but no rule $\bot\vdash A$.

*Toby*: Possibly I was just thinking of the wrong thing when I created the links here; it doesn't seem so interesting anymore. (^_^) Well, no point changing it now.

*Finn*: I too have seen minimal logic presented using $\neg$ and $\bot$. I don’t have my van Heijenoort with me, but I think Hilbert gave a definition of such a system in the early 1920s. Johansson was the first to call it ‘minimal logic’, although I can’t remember whether he included negation or $\bot$.

I think the two approaches, with and without $\bot$, are equivalent anyway, in the absence of *ex falso quodlibet*, because ’$\bot$’, if it’s included, is just another proposition. The only reason I would include it is for continuity with presentations of classical or intuitionistic logic. Then the difference between them seems to be in how they handle $\bot$.

(That said, I would take the sequent-calculus presentations to be canonical, because they make it more obvious that the differences are structural, rather than logical. But I don’t know how to draw inference rules on the wiki.)

*Toby*: Actually, I find the sequent-structure distinction between intuitionistic and classical (and presumably minimal) logics rather arbitrary. After all, there are sequents with exactly one formula on the right that are valid classically but not intuitionistically or valid intuitionistically but not minimally; it's just that Gentzen chose the axiomatic ones to avoid those.

Actually, as I like dependent type theory, I tend to always think of logical judgements as having arbitrary left-hand sides but always a single item on the right. Of course, dependent type theory does lend itself particularly well to intuitionistic logic, so perhaps that's unfair. Maybe I ought to get more into the spirit of the sequent calculus to understand why the differences between these three logics *should* be understood as structural.

Mike has some inference rules here.

*Finn*: Thanks for the link. I’ll have a go at typesetting some inference rules at some point.

I don’t think the distinction between different kinds of sequent is arbitrary at all – it seems to me to be fundamental. Of course, a sequent like $\vdash P\vee\neg P$ has exactly one formula on the right, but in proving it you need to pass through $\vdash P,\neg P$. You need the same sequent to prove the double negation law too, so I like to think of these axioms as really being artefacts of the extra structural freedom of classical logic.

I don’t understand you when you say ‘Gentzen chose the axiomatic [sequents] to avoid [well-formed but invalid sequents]’. The axiom sequents are just those that have the same formula $P$ both on the left and on the right – are you thinking of derived ones like $P\wedge Q\vdash P$? They are not essential. It may be a matter of taste, but I certainly find sequent calculus to be far more elegant than natural deduction, precisely because it draws a very clear distinction between logical rules and structural rules.

If you’re interested, my TCD technical report is all about the Curry–Howard correspondence for classical logic, and it includes (what I hope is) a fairly intuitive explanation of sequent calculus and associated term notations for minimal and classical logic. It’s available here.

*Toby*: Thanks, that looks interesting. As for Gentzen's choice and the arbitrariness, you must have seen that LK can be presented in slightly different forms and that the little choices here make a big difference when you restrict to sequents with so many formulas on a given side.

For example, is the rule for disjunction on the left

$\frac{\Gamma, A \vdash \Delta \qquad \Sigma, B \vdash \Pi}
{\Gamma, \Sigma, A \vee B \vdash \Delta, \Pi}$

as on Wikipedia or

$\frac{\Gamma, A \vdash \Delta \qquad \Gamma, B \vdash \Delta}
{\Gamma, A \vee B \vdash \Delta}$

as in your report? By Wikipedia's version, there aren't enough such intuitionistic sequents (and so their definition of $LJ$ needs an exception to fix this) and no such minimal sequents, while your version allows them. On the other hand, if you include corresponding rules for $\bot$ (which are surely required for elegance), then Wikipedia would have

$\frac{}
{\bot \vdash}$

which is (correctly) an intuitionistic but not minimal sequent, while you would have

$\frac{}
{\Gamma, \bot \vdash \Delta}$

which could be a minimal sequent even though you wouldn't want it.

Anyway, I'll read your report more closely than is needed just to see how you write LK. (^_^)

*Finn*: What you’re talking about here is the distinction between ‘multiplicative’ rules (Wikipedia) and ‘additive’ ones (me (and Gentzen :)), as in linear logic. In the presence of contraction and weakening on *both* sides these are equivalent, of course, but the restriction on intuitionistic and minimal sequents is ‘really’ about restricting these structural rules on the right – only one weakening for LJ, none for (what I call) LM and no right contraction for either.

You’re right that the multiplicative versions don’t work well for these subsystems of LK – but I think the choice is far from arbitrary, as linear logic shows. That is, the choice of additive versus multiplicative inference rules seems to be intimately related to the structural aspects of the logic.

As for $\Gamma,\bot\vdash\Delta$ as an axiom, that just isn’t included in minimal logic. (By the way, your point above was well taken, that it would seem better to leave $\bot$ out of minimal logic entirely, but I think most people like having $\neg$ in their logics, even if it’s just $\to$ in disguise)

*Toby*: Well, I can't accept that minimal logic is really about exactly one item on the right while intuitionistic logic is really about at most one item on the right, not if you have to ignore $\bot$ to make that work. But I think that I now have a better understanding of how intuitionistic logic is (in one sense) really about having one item on the right (and using additive rules).

*Finn*: Maybe a better way to think of this – and I’m probably not explaining things very well – is to say that $\bot$ and an empty RHS are really the same thing (just like $\top$ and an empty context), so that leaving the *ex falso quodlibet* schema out of minimal logic is (morally) the same as forbidding right weakening. Does that help? I still often get confused by sequent calculus; as the case of linear logic shows, it’s a surprisingly subtle formalism.

*Toby*: I don't think that they're morally equivalent at all; the difference between these is exactly the difference between structural and logical changes. You might as well say that $\vee$ is the same as a multiple RHS so that intuitionistic logic shouldn't have it (or at least shouldn't have the left-hand rule for it), but that's clearly not correct.

*Finn*: Sorry, I wasn’t very clear. What I meant was that the empty RHS in LJ corresponds to $\bot$ in NJ, so you can think of the extra axioms of NJ and NK as ‘shadows’ of the extra structural rules of LJ and LK.

You’re right that the distinction between $\bot$ and the empty RHS is exactly the logical/structural distinction, and what I’m saying really is that I prefer sequent calculus because (I think) it shows that the differences between classical, intuitionistic and minimal logic are (or can be viewed as) structural, rather than logical. I suppose I’m thinking of these systems in terms of their interpretation as term calculi – there you find that the structural differences translate nicely into restrictions on the ability of a term to manipulate its ‘continuation’, whereas the corresponding differences for terms based on natural deduction are unobvious to say the least.

*Toby*: Well, I don't think that I was very clear either. Let me put it this way:

Even assuming that we write LK additively, there are still some variations possible. Both you and Wikipedia (and probably Gentzen, although I've never read the original) leave out $\top$ and $\bot$, but one could also include them on the same footing as $\wedge$ and $\vee$. I think that it's obvious that it's best to include them; indeed, this goes to the heart of one of my strongest intuitions about mathematics.

If one does so and then restricts to sequents with exactly one formula on the right, then one gets intuitionistic logic, not minimal logic. And if such a restriction leads one to remove the axioms for $\bot$ in a shadowing manner, then it also ought to lead one to remove $\vee$, which will not get one the right result either. So while you've convinced me that seeing intuitionistic logic as a *structural* weakening of classical logic really is an important way to look at it (and not just an artefact of some specific representation), I'm far from convinced about minimal logic.

Incidentally, here's another way that I've thought of that you can see the relationships between these logics. If you start with LK (including $\top$ and $\bot$) and then remove $\neg$, then you get intuitionistic logic. (If you then go on to remove $\bot$, then you get the version of minimial logic without negation.) This view still pleases me more, but I'll try to like the structural view too. (^_^)

*Finn*: You make some good points here, and I should admit that I still haven’t grokked the logical/structural distinction in fullness, especially in the case of $\bot$ and $\vee$ in LJ. As I said, in my sort of work the structural side of things comes through more strongly, but I’m not trying to convert anyone (and I hope it didn’t come across that way). I suppose whatever way you look at this hierarchy of logics, there are some pleasing patterns to be found, even if they’re not as straightforward as we might like!

You’ve certainly given me food for thought. I’ll ponder this for a bit and maybe try to distil some of the above wisdom into the sequent calculus entry.

(By the way, Gentzen’s original paper (auf Deutsch) is available on the Springer-Verlag website, linked to at sequent calculus. If you don’t have Szabo’s translation handy it’s worth a look, even if your German is as bad as mine :).

*Toby*: Actually, I think that you *should* try to convert me, if you can, to the view that a structural variation is a good way to view intuitionistic logic (as you pretty much have) or minimal logic (which you haven't done yet). Not to the view that other ways are bad or even that the structural way is best; but the more that I understand certain points of view, well, the more that I understand!

Revised on April 22, 2009 20:46:34
by Toby Bartels
(71.104.234.95)