January 22, 2013
Modal logic is, among other things, the logic of possibility and necessity. Its history goes back at least to Aristotle s discussion of modal syllogisms in the Prior Analytics. But modern discussions begin with the works of C. I. Lewis, who, starting in 1912, developed some systems of modal logic, calling them S1, S2, etc. He focused primarily on the conditional connective, seeking something that would be more faithful to natural language conditionals.
Three Traditions Twentieth-century work in modal logic forms three clusters: the syntactic tradition, stemming from Lewis; the algebraic tradition, stemming from Tarski; and the model-theoretic tradition, stemming from Kripke. In this course we focus mostly on the last cluster.
Connectives Since the work of Hintikka in the early 1960s, modal logic has expanded to include many other non-truth-functional connectives, including epistemic notions (knowledge, belief, warranted belief, etc.), tenses, deontic operators
Senses of Modality Additionally, there are many different senses of necessity and possibility we might want to represent in a modal logic:
Senses of Modality Additionally, there are many different senses of necessity and possibility we might want to represent in a modal logic: logical necessity, analytic or verbal necessity, conceptual necessity, metaphysical necessity, nomological necessity, epistemic necessity, etc. We want to start by defining a modal logic that is weak enough to have promise for providing a framework for representing all of these notions.
Functional Classical propositional logic is functionally complete: any truth function at all can be expressed in its language. Say that an n-ary truth function is a map from n truth values (in classical logic, 0 or 1) to truth values. There are 4 singulary truth functions, 16 binary truth functions, 256 ternary truth functions, and, in general, for any n, 2 2n truth functions. Every one of them is expressible in terms of,,,, and. In fact, all are definable in terms of just and, or in terms of and, or in terms of and.
Proof To prove this, we can think of a truth table defining a truth function, and associate a formula with each row on which the function yields the value truth. (If it never does, define it as p p.) Say the function comes out true on the row a 1...a n. The associated formula for a row is a conjunction with p i or p i as a conjunct depending on whether p i receives 1 or 0 on that row. The formula representing the truth function is simply the disjunction of all those associated conjunctions. That formula requires only,, and.
Reducing the set To see that we don t need both conjunction and disjunction, we can use DeMorgan s Laws to define one in terms of the other: p q is equivalent to ( p q), and p q is equivalent to ( p q). Moreover, p q is equivalent to p q, and p q is equivalent to (p q). So, negation and conjunction, negation and disjunction, and negation and the conditional suffice to define all truth functions.
Sheffer functions Two binary truth functions, called Sheffer functions, suffice all by themselves: nor ( ) and nand (/).
Sheffer functions p (p p) p q ((p p) (q q))
Sheffer functions p (p/p) p q ((p/q)/(p/q))
Fixed points It is easy to see that our basic set of connectives would not be functionally complete without negation; truth would be a fixed point. That tells us something important about Sheffer connectives: they must take all 1s into 0, and all 0s into 1. This in turn makes it easy to show that nand and nor are the only two binary Sheffer functions.
Non-truth-functional Connectives There are many connectives that are not truth-functional. Among singulary connectives, think of it is possible that, it is necessary that, it is probable that, it ought to be the case that, it used to be the case that, John believes that, Mary knows that, Ralph fears that, Holly hopes that, etc. Among binary connectives, think of when, while, since, until, because, explains, Sam would prefer that... rather than..., etc.
Non-truth-functional Connectives Modal logic, the logic of possibility and necessity, takes non-truth-functional connectives as its central subject matter. It thus has no choice but to go beyond the resources of classical propositional logic.
Possible Worlds The key concept of modal logic, from a model-theoretic point of view, is that of a possible world. Think of a possible world as a way the world could be. There are serious metaphysical issues about what possible worlds are, and serious epistemological issues about how we have knowledge of them. For the moment, however, let s put those aside, and focus on the logic.
Possible Worlds
Possible Worlds
Possible Worlds
Possible Worlds
Possible Worlds
Possible Worlds
History Possible worlds were introduced into contemporary discussion by Kripke s 1959 paper A Theorem in. The idea comes from Leibniz, who thought of necessity as truth in all possible worlds. That in turn is inspired by Suarez, who thought of possibilities as ideas in the mind of God.
Accessibility Kripke contributes the insight that not all worlds might be accessible that is, relevant to modal propositions at a given world. He introduces an accessibility relation, allowing us to make sense of the idea that worlds might differ in the worlds relevant to modal truths. (Suarez had explicitly held that all worlds were relevant to modal judgments in all worlds, since God could survey all the ideas in His mind simultaneously.)
Possible Worlds
Accessibility On the Suarez-Leibniz conception, there are six possible modalities (ways a proposition might be true or false), reflected in the six modalities in Kant s Table of the Logical Forms of Judgment: truth (A), falsehood ( A), possibility ( A), impossibility ( A = A), necessity ( A), and contingency ( A = A).
Modal Square of Opposition
Modal Square of Opposition
Rules We add the operators and and the syntactic rule: If A is a formula, so are A and A. An interpretation < W, R, v > is a triple consisting of a nonempty set W of worlds, a binary relation R of accessibility on W, and an assignment function v from each world-propositional parameter pair to {0, 1}. v w (p) = 1 means that v assigns p truth at world w.
Truth clauses The truth conditions for truth-functional connectives remain unchanged, except for the addition of a subscript for worlds. The truth clauses for possibility and necessity: v w ( A) = 1 w (wrw v w (A) = 1) v w ( A) = 1 w (wrw v w (A) = 1) Necessity is truth in all accessible worlds.
Implication We define implication in terms of interpretations and worlds: X = A iff for every interpretation I =< W, R, v > and every world w W, if v w (B) = 1 for all B X, then v w (A) = 1. Abbreviating: X = A I w W v w (X) = 1 v w (A) = 1.
Tableau rules: Necessity A, i irj A, j
Tableau rules: Negated Necessity A, i irj A, j
Necessity and possibility Since we can think of A as equivalent to A, these allow us to characterize rules for possibility as well:
Tableau rules: Possibility A, i irj A, j
Tableau rules: Negated Possibility A, i irj A, j
The proof of the soundness of our tableau rules is almost identical to that for propositional logic. We need only replace talk of assignments with talk of interpretations, add relativization to worlds, and add arguments for the rules for and.
Theorem X A X = A.
Proof Proof: Assume that it is not the case that X = A. Then there is an interpretation I =< W, R, v > and world w W such that v w (B) = 1 for all B X but v w (A) = 0, so v w ( A) = 1. We show that X A. Consider a completed tableau with the initial list X, A. I is faithful to that initial list.
Inductive Proof Given the soundness lemma, which we will state and prove in a moment, we can show that I is faithful to an open branch on the completed tableau by induction on the number of rules applied to complete the tableau.
Base case Base: No rules need to be applied. Then the initial list is itself a completed tableau, and I is faithful to it.
Inductive step Inductive step: Assume that I is faithful to an open branch up to the nth rule application. By the lemma that rules preserve fidelity I is faithful to at least one resulting branch.
Fidelity So, I is faithful to a branch on the completed tableau. If it were closed, no assignment could be faithful to it, since no assignment makes both a formula and its negation true. So, the branch is open, and thus X A.
Lemma The real work is done in the Lemma. Rules preserve fidelity. Say that I =< W, R, v > is faithful to branch b iff there is a function f : N W such that if A, i is on b, v f(i) (A) = 1, and if irj is on b, then f(i)rf(j). Such an f shows I to be faithful to b.
Lemma Lemma: If I is faithful to b, and a tableau rule is applied to b, I is faithful to at least one resulting branch.
Proof of Lemma Proof: Assume that I is faithful to b, to which a tableau rule is applied. There are as many cases as rules.
Conjunction Case 1: The rule for A B, i is applied. Then A, i and B, i both appear on the branch. Since v f(i) (A B) = 1, by the truth definition v f(i) (A) = v f(i) (B) = 1. So, I is faithful to the resulting branch.
Negated Conjunction Case 2: The rule for (A B), i is applied. Then two branches result, one with A, i, and one with B, i. By the truth definition, since v f(i) ( (A B)) = 1, v f(i) ( A) = 1 or v f(i) ( B) = 1. Thus, I is faithful to at least one of the resulting branches.
Necessity Case 3: The rule for A, i is applied. Then, for every irj on b, f(i)rf(j). Since I is faithful to b, v f(i) ( A) = 1, so v f(j) (A) = 1. I is thus faithful to the extension of the branch.
Possibility Case 4: The rule for A, i is applied. Then we extend b to include irj and A, j for some new j. Since I is faithful to b, v f(i) ( A) = 1, so, by the truth clause for, there is a world w such that f(i)rw and v w (A) = 1. Let f be the same as f on all i j, letting f (j) = w. f shows that I is faithful to b, since f and f agree on everything on b. Since f (i)rf (j) and v f (j)(a) = 1, f shows I to be faithful to the extended branch.
The completeness proof also follows closely the structure of the proof for classical propositional logic. Any open branch b with world indices 0,..., n induces an interpretation I =< W, R, v >, where W = {w i : i occurs on b}, R = {< w i, w j >: irj is on b}, and, for any propositional parameter p such that p, i is on b, v wi (p) = 1; if p, i is on b, then v wi (p) = 0. (Assign any other parameters any values you like.)
Strategy Strategy for completeness proof: Open branch Set of literals Induced assignment Countermodel
theorem : X = A X A.
Proof Proof: Again we prove the contrapositive. Suppose that X A. Then there is no completed closed tableau with X, A as initial list. So, every completed tableau with X, A as initial list has at least one open branch b.
Induced interpretations Any open branch b, moreover, induces an interpretation I. In a moment we will prove the completeness lemma, stating that every interpretation induced by a branch is faithful to it. That is, we prove that, if you get the sentence letters right, you get everything right. Given that lemma, we can conclude that I is faithful to b. But b has the initial list X, A, so every member of X, as well as A, is on b. Thus, v w0 makes every member of X true but A false. So, X does not imply A.
lemma lemma: If b induces I, I is faithful to b.
Proof of lemma Proof: Let b induce I. I is faithful to b iff, for every A, i on b, v wi (A) = 1 and, for every irj on b, w i Rw j. The latter is part of what it means for b to induce i. We show the former by induction on the complexity of A, that is, the number of connectives in A. Let A, i be on b.
Base case Base: A is a propositional parameter. Then v wi (A) = 1, since b induces I.
Inductive step Inductive step: Assume that v wj (A) = 1 for every A, j on b with fewer than n connectives. Assume that A has n connectives. There are twelve cases to consider. Those not involving modal connectives proceed much as they do in propositional logic:
Negation Case 1: A = p. Since b induces I, v wi (p) = 0, so, by the truth definition, v wi (A) = 1.
Conjunction Case 2: A = (B C). By the tableau rules, if A, i, that is, B C, i is on b, then so are B, i and C, i. But they contain fewer connectives than A, so, by inductive hypothesis, v wi (B) = v wi (C) = 1. By the truth definition, therefore, v wi (A) = 1.
Negated Conjunction Case 3: A = (B C). By the tableau rules, if A, i, that is, (B C), i is on b, then either B, i or C, i is on b. Both B and C have fewer connectives than A, so, by the inductive hypothesis, v wi ( B) = 1 or v wi ( C) = 1. By the truth definition, then, v wi (B) = 0 or v wi (C) = 0, so v wi (B C) = 0, and v wi (B C) = 1.
Necessity The modal cases are more interesting. Case 4: A = B. By the tableau rules, if B, i is on b, then, for every j such that irj is on b, B, j is on b. Since B has fewer connectives than A, by the inductive hypothesis, if w i Rw j, v wj (B) = 1. But then by the truth definition v wi ( B) = 1.
Possibility Case 5: A = B. By the tableau rules, if B, i is on b, then there is a j such that irj and B, j are on b. Since B has fewer connectives than A, by the inductive hypothesis, w i Rw j and v wj (B) = 1. But then by the truth definition v wi ( B) = 1.
Which inferences are valid in this logic? And what additional inferences should be valid, on various conceptions of modality? The general strategy is to place conditions on the relation R to obtain stronger modal logics.
Interdefinability In this logic itself, we have the interdefinability of possibility and necessity, which follows from the interdefinability of the quantifiers: A A A A
Interdefinability Since being a valid formula is a matter of being true at all worlds in all models, we have But notice that although = A = A = A = A we also have = A / = A
Strict Conditional C.I. Lewis defined the strict conditional in terms of necessity: A B (A B) The strict conditional in basic modal logic is extremely weak. It deviates from the material conditional in many respects: A B, A =/ B A B, B =/ A B =/ A B A =/ A B
Strict Conditional (A B) C =/ (A C) (B C) (A B) (C D) =/ (A D) (C B) (A B) =/ A =/ (A B) (B A)