Thought for the Evening: Plurative Modalities
Modal operators, as we often study them, are divided into two major kinds: strong (or Box) and weak (or Diamond). Strong modalities are necessity-like: necessary, always, everywhere, obligatory, known. Weak modalities are possibility-like: possible, sometimes, somewhere, permissible. In practice, we often use a third, intermediate between them, which is sometimes just called the null modality, and means (more or less) what is true without any modification. And this is a giveaway, I suppose, since the null is effectively like a strong modality when compared with a weak modality and like a weak modality when compared with a strong modality. Strong modality and weak modality are not really two groups of different kinds; they are comparative. You see this further in multimodal systems; 'believed' may be a strong modality in a doxastic system but a weak modality in an epistemic system.
In short, there are many intermediate modal operators between absolute necessity and absolute possibility. This is in fact what we would expect on other grounds. Even if we stay with a given kind of modality -- say, temporal modalities -- we can weaken the strong modal operator or strengthen the weak modal operator to get a new modal operator. In possible world semantics, we analogize all other modal operators to quantifiers -- strong modal operators all work like the universal quantifier, weak modal operators all work like the existential quantifier. But you can have different quantifiers as well. Quantity that is weaker than universal quantity but stronger than particular quantity goes by different names in different systems, but one of the names used is 'plurative'. Examples of markers of plurative quantity are things like 'most' or 'few'. Consider the following basic plurative syllogism:
Most of those who will study hard will pass.
Most students will study hard.
Therefore, some students will pass.
This is a valid argument. That's somewhat peculiar, since if we treat 'most' like 'some', we shouldn't get a conclusion (two particular premises), and if we treat 'most' like 'all', we can get a much stronger conclusion (since we could get a universal conclusion), but what we find is that we can get a conclusion, just not a universal one. There's a reason why syllogistic primarily focuses on universal and particular quantity, because these are special in several ways. If we look at distribution, for instance, they both have nice, clean distribution rules associated with quantity: universal propositions distribute the subject, particular propositions don't. If we try to pin down the distribution rule for 'most', we get something more like (this is only approximate, since it depends on some assumptions about how distribution works): the subject term is distributed when, and only when, the subject term is the minor term (or would be the minor term if the conclusion were converted). In short, you can't tell how the quantity works until you know the role of the proposition in the argument. That's a much messier rule. But it's also notable that we can, in fact, give a distribution rule, and, despite its greater messiness, it is one that is practically useful in real argumentative contexts. Similar issues come up when we look at other logical properties.
Plurative modal operators would be modal operators that correspond to plurative quantifiers -- they are about what happens in most (or few) possible worlds, if we use a possible world framework. We can take any kind of Box and 'plurativize' it. The most obvious cases are with temporal modalities, since in ordinary English we often use words that correspond to plurative temporal operators: most of the time, usually, often, almost always. But we can do it with other cases as well, without any real difficulty, even though we don't always have straightforward English terms for them all. Not only is there 'necessary', there is 'nearly necessary'; not only is there 'everywhere', there is 'most places' or 'almost everywhere'; not only is there 'obligatory', there is 'nearly obligatory' or 'usually to be done'. And the same sort of thing can be done from the other side, strengthening weak modalities that correspond to 'some' or 'at least one' so that they correspond to 'at least a few'.
Links of Interest
* John Lawless, Against Acceptance Theories of Social Norms (PDF)
* John Plaice, Pierre Louis Maupertuis and the Principle of Least Action, at "Fiat Lux"
* David M. Berry, Toward a Genealogy of the Concept of "Plate Glass Universities, at "Stunlaw"
* Takuya Niikawa, Consciousness Aesthetics (PDF)
* Estrada González Luis & Romero Rodríguez Christian, How we learned to stop worrying and love tonk (PDF)
* John Hawks, How archaeologists are missing Pleistocene cultures
* Nirmalya Kajuri, An Ode to the Spherical Cow, at "The Spacetime Beat"
Currently Reading
In Book
Maurice Leblanc, The Confessions of Arsene Lupin
Ian Mortimer, The Time Traveler's Guide to Medieval England
Frank M. Turner, John Henry Newman: The Challenge to Evangelical Religion
Brian Kemple, Linguistic Signification
Marc Morris, The Norman Conquest
In Audiobook
Umberto Eco, The Name of the Rose
Brandon Sanderson, The Final Empire
Jim Butcher, Storm Front
Robert Jordan, The Eye of the World
Dorothy L. Sayers, Clouds of Witness
