Kripke's Contributions to Modal Logic PY4617 The Philosophy of Saul Kripke Week Eleven

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1 Kripke's Contributions to Modal Logic PY4617 The Philosophy of Saul Kripke Week Eleven Background: proofs and models for a propositional language Distinguish a proof-system for a propositional formal language from the formal semantics (or model-theory) for that language A proof-system, e.g. Lemmon-style natural deduction, consists of a purely syntactic set of definitions, operations, and rules. It's syntactic in that it's entirely defined in terms of strings of symbols in the language, not in terms of the interpretation of those symbols. Say that an argument from the premises in a set X to a conclusion C is provable in our system iff the rules of our proof-system allows us to construct a proof of C from X. We state the latter fact by writing: X C. A model-theory for the language provides a formal way to think about the language's interpretation. Informally, we take p, q, etc. to express propositions and we take '&', 'v', etc. to correspond to English's 'and', 'or', etc. But why have a more formal interpretation? One purpose of formal semantics is to provide a precise way to investigate whether our proofsystem provides a model of good argumentation. Without any interpretation of our language, the proof-system is just a set of rules that tells you when you can draw a bunch on little shapes who cares? In doing logic, we care about valid arguments. A formal semantics helps us connect the proof-system to validity An argument is valid iff it's impossible for its premises to be true while its conclusion is false. So we see that for the notion of validity to apply to our formal propositional language, we at least need to take each sentence-letter to have possible truth-values. Possible truth-values are explained in terms of arbitrary assignments of truth-values to all sentence-letters (evaluations). These are our models for the propositional language. In each model, the truth-values of complex sentences are determined recursively according to the truth-tables for the connectives. An argument from the premises in a set X to a conclusion C in the formal language is then valid iff there is no model in which the premises are True while the conclusion is False. We state the latter fact by writing: X C. Now we can precisely investigate the question: Does our proof system let us construct proofs for all and only the arguments that are valid in our formal language? It's a fact about standard proof-systems for propositional logic, provable by doing some meta-theory, that if X C, then X C. In other words, if the proof-system allows you to construct a proof, then the argument is valid. We say: the proof-system is sound. It's also a fact about standard proof-systems for propositional logic, provable by doing some meta-theory, that if X C, then X C. In other words, if an argument is valid, the proof-system allows you to construct a proof of it. We say: the proof-system is complete. So, using the formal semantics provided by our models for propositional logic, it becomes a precise mathematical fact that a given proof-system is sound and complete. That wouldn't be the case if we had only an informal interpretation of the language. Pg. 1 of 5 E. Glick 22/04/2013

2 Background: proofs and models for a quantificational language Things become a bit more complicated when we turn to predicate logic, where we now have quantifiers, variables, and so on. What is it for a proof-system for predicate logic to be sound and complete? We retain the definition of validity in terms of truth-preservation in all models: X C iff there is no model on which the premises are True while the conclusion is False. But now we need a more complex notion of a model! We don't want to simply assign truth-values to sentences like ( x)rx, because we want that sentence's truth to be determined by whether Ra, Rb, Rc, etc for every object. We also want the truth of Ra (e.g.) to be determined by the interpretations of the predicate R and the name a. We take a model M for predicate logic to be a pair <D,I> consisting of a domain of individuals D and an interpretation function I. D might be, for instance, {Alice, Jane, Bieber, Fred, } Intuitively, I just tells you what names what, and what the predicates apply to. More formally, I assigns members of D to names and assigns n-tuples (ordered pairs or triples or etc.) of members of D to n-place predicate letters, for each n. So, e.g., we might stipulate that I(a)=Alice and that I(b)=Bieber, etc. If T is a two-place predicate, I(T) might be {<Alice,Bieber>, <Bieber,Jane>, } We call that set of tuples I(T) the extension of T in the model. Informally, imagine T meaning taller than, so that the pairs in the extension are just the pairs of individuals such that the first is taller than the second. Now, how is the truth of a formula determined in a model? First, a clause for closed (containing no free variables) atomic formulae: Where P n is an n-place predicate and c 1 c n are n names such that I(c 1 )=a 1, I(c 2 )=a 2, etc.: Pc 1...c n is T in the model iff <a 1 a n > I(P) Since we now have formulas containing free variables, and free variables aren't names for particular things in D, we only determine truth-values for, say, Pxy, relative to the possible objects that x and y range over relative to assignments of objects to the variables. E.g.: Pxy is T in <D,I> relative to an assignment of a to x and b to y iff <a, b> I(P). xpxy is T in <D,I> relative to an assignment of b to y iff for every o in D, Pxy is T in <D,I> relative to an assignment of o to x and b to y. xpxy is T in <D,I> relative to an assignment of b to y iff for some o in D, Pxy is T in <D,I> relative to an assignment of o to x and b to y. We use the same recursive strategy as earlier to determine truth-values for truth-functional compounds in predicate logic. In general, if a formula is closed, its truth-value in a model will be independent of any assignments of objects to variables. It is standard to say that a closed formula is T simpliciter iff it is T relative to every assignment of objects to variables. Now we have a notion of truth in a model for closed formulas, and using our earlier definition of validity, we can investigate soundness and completeness for proof-systems. Fact: For all standard proof-systems for predicate logic, X C iff X C. Propositional modal logic before Kripke: One main motivation for modern investigations of modal logics was the so-called paradoxes of the material conditional (the truth-functional conditional of propositional logic). Pg. 2 of 5 E. Glick 22/04/2013

3 E.g., according to now-standard logics (largely derived from Russell and Whitehead's Principia Mathematica), from any formula p, q p follows, and from p, p q follows. C.I. Lewis took these paradoxes to show that the logic of the material conditional doesn't capture the real notion of implication. He sought a better account of implication. His proposal: p implies q iff (p q), where ' ' means necessarily. This raised new questions: How do we come up with a logical system that incorporates this new symbol ' ', and the accompanying symbol ' ' for possibility? Lewis proceeded to explore a number of different sets of rules and axioms for proof-systems for modal logic, with the resulting systems names S1-S5. Some key candidate rules and axioms: (N): If p, then p (K): (p q) ( p q) ('K' for Kripke! Also known as 'Distribution'.) (M): p p (4): p p (5): p p In exploring the different combinations of axioms, the rules of the game were somewhat unclear. Intuitive judgments were the only real test of the appropriateness of various theorems. Logicians wanted more clear standards for determining the correct modal logic. One of the complaints raised against modal logic was that no formal semantics existed for modal logic as existed for non-modal predicate logic. (Fitch, 8) A general concern was that perhaps there were different kinds of possibility or necessity, with different logics appropriate to them. Why not just use the same models for modal logic as we used for propositional logic, and say p is true in M iff necessarily, p is true in M? Because everything true in M is necessarily true in M, it being a mathematical fact about functions etc. (Burgess, 123) Kripke's contributions to propositional modal logic: Kripke's most influential achievements in modal logic were (a) to devise a clear and simple model-theory, and (b) to prove completeness results for a number of proof-systems. A model for modal logic is now a tuple <G, K, R, φ>. (Now called a Kripke model.) K is a set. Relative to each of member of K, φ assigns a truth-value to every sentence-letter. The truth-values of truth-functional compounds are determined recursively. G is a member of K. It has the special property that a formula is true in the model iff it's true at G. (E.g., where p is atomic, p is true in the model iff φ(g,p)=t.) Informally, Kripke says we can think of K as a set of worlds (better: possible states of the world), and think of G as the actual world. R is a relation between worlds that holds when the second world is visible from the first. The most interesting clause in the definition of truth is for the modal operators: p is true at a world H in the model iff for every world H' such that R(H,H'), p is true at H'. So p is true in the model simpliciter iff for every world H' such that R(G,H'), p is true at H'. For other clauses, see Burgess pg. 124 (in Berger) Now we can stick with our claim that an argument from X to C is valid iff there's no model in which all members of X are true and C is false. Notice that validity concerns truth in all models, not truth at all elements of K. And truth of a formula p in a model is a matter of p's truth at all elements of K, not a matter of p's truth in all alternative models. This is one respect in which Kripke differed from earlier writers such as R. Carnap. Pg. 3 of 5 E. Glick 22/04/2013

4 Let K be the proof-system resulting from adding (N) and (K) to basic propositional logic. Then one of Kripke's results is this: K is sound and complete for the class of all Kripke models. (Note that there is no restriction on R in this class of models. In some of the models, R might hold between no worlds, in others between every pair of worlds.) Kripke realized, and went on to prove, that other proof-systems for modal logic are sound and complete with respect to other classes of models defined by specifying restrictions on R. System T is K plus (M). Definition: X T C iff C is provable from the members of X in T. Definition: X T C iff in every model in which R is reflexive (for all H in the model, R(H,H) holds), if the members of X are true, so is C. Fact: X T C iff X T C. System S4 is K plus (M) and (4). Definition: X S4 C iff in every model in which R is transitive and reflexive, if the members of X are true, so is C. Fact: X S4 C iff X S4 C. System S5 is K plus (M) and (5). Definition: X S5 C iff in every model in which R is symmetric, transitive and reflexive, if the members of X are true, so is C. Fact: X S5 C iff X S5 C. Kripke thus revealed an interesting correspondence between properties of the R relations in the models and the various axioms one might adopt in one's proof-system. Quantified modal logic before Kripke: It's fair to say that the move from propositional modal logic to quantified modal logic introduces a significant degree of complication, and it faced greater skepticism. Quine was perhaps the staunchest critic, accusing modal logic of resting on use /mention confusions. (For a summary, see Hughes, pg. 82. For details, see Quine's Three Grades of Modal Involvement among other essays.) Quine thought necessity, if it made sense at all, had to be understood in terms of analyticity. ' p' really just meant that the sentence 'p' was analytic. On this way of thinking, ' ' should be a symbol of our meta-language, used to talk about sentences, not a symbol of the object language. Predicate logic then looks very confused. What could it mean to say that x Px? For every object, the sentence 'Px' is analytic?? Kripke eventually answered the concern by distinguishing metaphysical possibility from a priority and analyticity. A more technical concern was that extant versions of modal predicate logic had as theorems the Barcan formula and converse Barcan formula (named after Ruth Barcan Marcus). Where Px is any formula containing free variable x: If x Px then xpx. And if xpx, then x Px. Both formulae seem implausible. Suppose the only things in the actual world are a 1...a n, and then let Px to be x=a 1 or or x=a n. Then the first principle tells us that there couldn't be anything other than what there actually is! Now for the converse principle. Take Px to be y y=x. Then since in every world, everything there exists, it would follow that everything in this world also exists in every other world! Kripke's contributions to quantified modal logic: Kripke's major contribution here was to extend his models for propositional modal logic to the quantified systems, to provide a semantics that avoided the Barcan formulae, and to prove Pg. 4 of 5 E. Glick 22/04/2013

5 soundness and completeness results for quantificational proof-systems. A model is now a tuple <G, K, R, ψ, φ>. The new component, ψ, assigns a domain of individuals to each member of K informally, the individuals that exist in that world. Some key ideas are these: The domains are allowed to differ from world to world. For any world H, even for tuples of individuals that don't exist in H, Φ specifies whether or not those tuples are in the extensions of predicates at H. But when we evaluate a quantified claim at a world H, the quantifiers range only over individuals that exist in H. Here's a counterexample to the Barcan formula: Let K={G,H}, let ψ(g)={a} and ψ(h)={a,b}, and let Φ(P,G)={<a>} and Φ(P,H)={<a>}. Now every object at G is in Φ(P,G) and in Φ(P,H), so x Px is true at G and hence true simpliciter in this model. Here's a counterexample to the converse Barcan: Let K={G,H}, let ψ(g)={a,b} and ψ(h)={b}, and let Φ(P,G)={<a,b>} and Φ(P,H)={<b>}. Now xpx comes out true at every world, so xpx is true. But since <a> is not in Φ(P,H), it's not the case that for every object in G, that object is in Φ(P,H). So x Px is false at G and hence false simpliciter in this model. Things to think about as you read: A common way to do semantics is in terms of complete assignments, functions that assign objects to every variable of the language. (See Tarski, e.g..) Then you define satisfaction of formulae by assignments in a model, and define truth as satisfaction by all assignments. Burgess explains truth in a Kripke model in terms of satisfaction, but doesn't employ complete assignments. Does this matter? Can you see a way to restate the semantic clauses in terms of complete assignments? Kripke does not mention the notion of satisfaction at all. Does this matter? Can you see a way to restate his semantics in terms of satisfaction? Kripke asks (pg 85) should Φ(P(x),H) be given a truth-value when x is assigned a value in the domain of some other world H', and not in the domain of H? For instance, Ephraim might be in the domain of G but not H. If we assign Ephraim to x, should P(x) be given a truth-value at H? What is your answer to that question, and why? It may be easier to think about names rather than variables. Should 'P(a)' be false at every world whose domain doesn't include a? Truth-valueless? What if P is just a predicate for self-identity, so that P(a) is a=a? Is that false at a world that doesn't contain a? What if P expresses the property of existence, as when P(a) is y y=a? When reading Kripke, try to understand the technical bits of pages 84 and 87. This is difficult but important. Feel free to pose on the blog any questions about the technical material. Pg. 5 of 5 E. Glick 22/04/2013