I want to write an inadequate appreciation of Saul Kripke, a great analytical philosopher. The Guardian has an obituary. I start from an example of a pratical use of his work. Chin and Older (2011) use modal logic to specify and reason about the security properties of (computer) systems. One wants to be able to assert who (or what) has access to certain data and who can grant access. Certain states of a system should never arrive. A Kripke structure, as presented in Chin and Older (2011), is a three-tuple consisting of: A set W, known as the set of all possible worlds. An interpretation I, that maps each proposition to a subset of W. Informally, I(p) is the set of worlds in which the proposition p is true. A function J that maps the name of each principal to a set of ordered pairs
Robert Vienneau considers the following as important:
This could be interesting, too:
Robert Vienneau writes Utility Maximization A Tautology?
Stavros Mavroudeas writes Ένα παλιό αλλά επίκαιρο κείμενο για την απάτη της «αξιολόγησης»
Lars Pålsson Syll writes For as long as it takes
Joel Eissenberg writes Distinguishing science from pseudoscience
I start from an example of a pratical use of his work. Chin and Older (2011) use modal logic to specify and reason about the security properties of (computer) systems. One wants to be able to assert who (or what) has access to certain data and who can grant access. Certain states of a system should never arrive.
A Kripke structure, as presented in Chin and Older (2011), is a three-tuple consisting of:
- A set W, known as the set of all possible worlds.
- An interpretation I, that maps each proposition to a subset of W. Informally, I(p) is the set of worlds in which the proposition p is true.
- A function J that maps the name of each principal to a set of ordered pairs of possible worlds. Consider J(A), the set of ordered pairs (W1, W2) mapped to by this function. This is intended to be such that if the current world is W1, A believes the world might be W2.
For the above definition to be completely formal, one needs to specify the language for propositions p. In propositional logic, operators such as "and", "or", and "not" combine atomic propositions. Predicate calculus introduces such notions as "for all" and "there exists". Model theory is used to characterize semantics, to specify models in which a set of sentences is true. I guess a Kripke structure is a kind of structure, as structures are defined in model theory. Modal logic uses the apparatus of model theory to characterize propositions as "necessarily true", "possibly true", and so on. The textbook I have available for model theory, which I cannot get very far into, does not go into this.
Misleading talk, some of that by Kripke, about possible worlds raises the question of "transworld identification". Consider a proposition p(a) about an individual a. How do we identify a across possible worlds? If one uses the name Nixon to identify the winner of the 1968 U.S. presidential election, are we not requiring one to call Humphrey "Nixon" in some possible worlds? Amusingly, Kripke also raises the question of why we might say that 9 is necessarily greater than 7, but be unwilling to say that the number of planets is neccessarily greater than 7. In the current world, the number of planets is now eight. Pluto is now, by stipulation, not a planet. Kripke argues, in Naming and Necessity that Frege and Russell had mistaken theories about reference.
I will probably not be able to wrap my head around the idea of necessary contingent truths in a couple of weeks. Consider the length of a certain rod in Paris back when that was the definition of a meter. It was a necessary truth that a meter is that length. In some possible worlds, the room it is stored in might possibly be hotter. A meter would still be a meter in all possible worlds, according to Kripke, but the length of the standard meter might possibly be different.
I find Kripke (1982) to be easier to understand, maybe. I know that Kripke says that he is presenting an argument from Wittgenstein, not his own. If I try to orally summarize it, I tend to bring in Goodman's "grue" and "bleen", Putnam's "twater" on twin earth, or Wittgenstein's beetle in a box. The topic seems to be how words mean, not epistemology or ontology. How do you know that when talking to people they do not mean "grue" when they say "green"? Even more troublesome, how do you know you did not mean "grue" every time you said "green" in the past? By postulation in the argument, there does not seem to be any empirical fact one can point to. Meaning is not in an individual's mind.References
- Shiu-Kai Chin and Susan Older. 2011. Access Control, Security, and Trust: A Logical Approach. CRC Press.
- Nelson Goodman. 1983. Fact, Fiction, and Forecast, 4th ed. Harvard University Press.
- Wilfrid Hodges. 1997. A Shorter Model Theory. Cambridge University Press.
- Saul A. Kripke. 1980. Naming and Necessity, 2nd ed. Harvard University Press.
- Saul A. Kripke. 1982. Wittgenstein on Rules and Private Language: An Elementary Exposition. Harvard University Press.
- Hilary Putnam. 1983. Reason, Truth, and History. Cambridge University Press.