Thought for the Evening: Modal Operators with Multiple Arguments
In modal logic we most commonly deal with Box (strong modality) and Diamond (weak modality) as operators that each take one argument. Thus
So, for instance, these would be something like "Necessarily, p" and "Possibly, p", or else "Always p" and "Sometimes p", or however we are defining Box and Diamond. And of course these can be negated to get Not-Possibly, or what have you. But there's nothing that really requires us to hold that you can only have Boxes and Diamonds that take one argument. You can make perfect sense of a nullary modal operator, taking no arguments at all:
This is in fact pretty much how Top or Verum (⊤, T, or 1) and Bottom or Falsum (⊥, F, or 0) work in certain logical systems; Top, which represents tautology or always-true, is a Box and Bottom, which represents impossibility or never-true, is a Not-Diamond.
One can also have modal operators that take more than one argument. Two, for instance:
(The parentheses are optional, of course, but they are helpful at times if you are dealing with arguments that can be negated.) There is in fact one form of modal logic that is well known that is binary in this way: mereology. If, for instance, you have two things, a and b, that overlap,
O(a,b) or aOb,
depending on your conventions, the overlap functions here as a binary Diamond. Its behavior is a little more complicated than the unary Diamond we usually associate with modal logic (you have to be more careful with how you handle negations, for instance), but the only complications are those arising from having two arguments rather than just one, and not with how the operator itself works. One can even think of overlap in terms of possibility, if one wishes: if a and b overlap, then it is possible for something to be in both a and b.
While there is not as much work done on it, another instance of a binary Diamond would be a compossibility operator -- instead of "Possibly, p" you'd have "p and q are possible together", but we'd obviously still be dealing with possibilities; we'd just have added another argument. Modal logics of compossibility would have analogues in mereologies; indeed, if you think about it, one way to think about compossibility is to think of it by a metaphor of overlapping possibilities.
Nothing in overlap or compossibility requires that you only have binary operators for them. You can have three things that overlap, or four, or a hundred trillion, and likewise you can have multiple things that are all compossible together, as many as you please.
Various Links of Interest
* It's been known for some time that there is a considerable amount of corruption in the hierarchy of the Catholic Church. Events of the past few years, though, have increasingly suggested that it went much farther than previously thought. At this point I think there's a good argument that the episcopacy of the past fifty to seventy years has been at least as corrupt as it has ever been in the past thousand years, which is saying something given some of the low points. The most recent revelation, of course, has been the release of the grand jury document on clerical sex abuse in six dioceses of Pennsylvania, which serve about one and a half million Catholics. It is harrowing reading -- nearly nine hundred pages, describing about three hundred priests preying on over a thousand victims, each case determined to be at least probable given the evidence. Some of the cases are stomach-wrenching, involving rape of children, desecration of the sacraments, and cover-up. And justice is not going to be found: the report doesn't go past 2000, most of the actual instances deal with things whose time limit under the statute of limitations has already ended, and two-thirds of the priests involved are already dead. But what has really, I think, sparked a fury is that a number of bishops who would have had to have been complicit with the cover-ups have been presenting themselves to the public for years now as especially tough on sex abuse cases -- most notably Cardinal Wuerl, who was Bishop of Pittsburgh toward the end of the nightmare. Wuerl has, of course, disputed this and has said he does not intend to resign, which is rather ridiculous because at 77 he has already submitted resignation to the Holy See (all bishops have to do so at the age of 75); the Holy See just hasn't accepted it yet. Wuerl's case is made all the worse by the fact that he also claims to have known nothing of the evils perpetrated by his successor in the Washington DC see, McCarrick, despite the fact that it is difficult to see how he could not have known. And the bishops in general seem, unfortunately, to be hunkering down, which is not the right move, because words do not convey the sheer simmering fury I have seen among lay Catholics over this. It's increasingly clear that the bishops' feet are going to have to be held to the fire.
In any case, two posts of note on the whole thing:
The Czar of Muscovy at Gormogons
Darwin at DarwinCatholic
(Incidentally, some Catholics have proposed refusing to give to diocesan appeals until the bishops do something. This would, unfortunately, do nothing; diocesan appeals are basically taxes imposed on parishes, and the appeal to the laity is essentially asking them to pay for it so it doesn't have to come out of the parish operating budget. The diocese always gets its money, no matter how much or how little the laity give to an appeal. The only question is whether the parish will have to cut budgetary corners to cough it up or not.)
* How amateur sleuths finally tracked down the burial place of William Blake
* Anthony Madrid, Pop Songs in English, Written by Native Speakers of Swedish
* I have recently been watching the anime show, Cells at Work!, available online at Crunchyroll, and have been greatly enjoying it. The subs have the usual problem of English subtitles for anime, in that they convey the false impression that the Japanese swear at the drop of a hat, but I've seen worse, and the episodic stories about anthropomorphized cells in the metropolis of the human body are quite fun.
* Riccardo Sabbino, Ibn Sina's Logic, at the SEP
* The works of Aquinas online in Latin and English, in a fairly nice format.
Frances Mossiker, Napoleon and Josephine
Thomas Joseph White, The Incarnate Lord: A Thomistic Study in Christology
Emmanuel Falque, The Wedding Feast of the Lamb
Rosamund Hodge, Endless Water, Starless Sky
Jules Verne, The Giant Raft