First-Degree Entailment

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1 March 5, 2013

2 Relevance Logics Relevance logics are non-classical logics that try to avoid the paradoxes of material and strict implication: p (q p) p (p q) (p q) (q r) (p p) q p (q q) p (q q)

3 Counterintuitive? Many philosophers, beginning with Hugh MacColl (1908), have claimed that these theses are counterintuitive. Relevance logicians object that, in each of them, the antecedent seems irrelevant to the consequent a property shared by the classically valid inferences that correspond to the conditionals above.

4 Variable Sharing One might say that what is wrong with the above is that is that the antecedents and consequents (or premises and conclusions) are on completely different topics. There is a formal principle that relevant logicians apply to force theorems and inferences to stay on topic: the variable sharing principle. No formula A B should be provable if A and B do not have at least one propositional parameter in common. Similarly, no inference can be valid if the premises and conclusion do not share at least one propositional parameter.

5 Syntax and Semantics This is a syntactic principle, which goes only part way toward capturing the central idea of relevance logic. As Anderson, Belnap, and others have developed the syntactic approach, the key idea is to keep track of the use of the premises of an inference.

6 Explosion A central problem is the problem of explosion: contradictions, in classical logic, entail anything even something completely unrelated to the propositions involved in the contradiction. But that seems implausible. Naive set theory is inconsistent, but it s possible to work with it so long as one stays away from the contradictions. No one would say that unions of sets are not always sets in naive set theory, merely on the ground that such an assertion follows from Russell s paradox.

7 Disjunctive Syllogism If we look closely at explosion, we see that it depends on two principles: 1. A 2. A 3. A B (Disjunction Introduction) 4. B (Disjunctive Syllogism)

8 Disjunctive Syllogism Relevance logicians tend to reject DS. It might seem more plausible, intuitively, to reject Disjunction Introduction, for it introduces the irrelevant element B. But A B does share vocabulary with A, and it is hard to give a semantics for disjunction that makes Disjunction Introduction invalid. Nevertheless various people have explored this and other options. For some time, the approach to relevance logic was chiefly syntactic; as with modal logic, the semantics came later. There are now several different semantics for relevance logic even for the fairly simple system FDE.

9 LP LP is a three-valued logic that is identical to K3, except that D = {1, i}. It amounts to K3 with an altered definition of validity: An argument is valid if and only if, if its conclusion is false, at least one premise must be false. In LP, we might think of i as representing not indeterminacy, is neither true nor false, but inconsistency, is both true and false.

10 LP The law of excluded middle is valid in LP. In fact, the logical truths of LP are just those of classical logic. But LP is nonclassical when applied to arguments. Contradictions do not imply everything. (Assign p and p i, and q 0.) Also, modus ponens fails. (Let p and p q be i and q 0.)

11 RM3 To regain modus ponens, we can change the truth conditions for the conditional, letting it be i iff both components are i. This yields RM3.

12 Post s Cyclic Negation

13 Post s Cyclic Negation One way to do this is to add Post s cyclic negation function: Table: Post s Cyclic Negation A A 1 i i 0 0 1

14 K3+ Call this logic K3+. K3+ is functionally complete. Unlike K3, it has valid formulas, such as A A, A A A, and (A A A).

15 Truth-value Gaps K3+ seems to need a stronger notion of implication If we allow for truth-value gaps, implication in the sense of truth preservation seems too weak Contrast three connectives: negation, strong negation, and Post s cyclic negation

16 Negations A A A A i i i These are true in exactly the same circumstanceswhen A is false But they are not intersubstitutable salva veritate; their negations aren t equivalent

17 Equivalence So, are A, A, and A equivalent? In one sense, yes: they imply one another in the truth-preservation sense In another, no: they have different semantic values; they arent intersubstitutable, even in extensional contexts

18 Concepts of Implication X = A iff No model of X makes A false (rules out 1 0) Every model of X is a model of A (rules out 1 0 and 1 i) Any model making A false makes something in X false (rules out 1 0 and i 0) All of the above (rules out 1 0, 1 i, i 0)

19 No decline Suppose we go with the last, all of the above, no decline in truth value conception for X A Then equivalence does guarantee substitutability salva veritate But that concept can t be explained in terms of preservation of truth or designated value

20 K3+ with What logic do we get if we use K3+ with that conception of implication? The,, fragment is FDE, This is a way to get FDE with three values, retaining valuations as functions It relies upon the fact that FDE = K3 LP But FDE does not fit the standard definition of a many-valued logic; one cannot isolate a set of designated values

21 Four Truth Values Suppose we want to allow for the possibility of truth value gluts and truth value gaps, recognizing that they are different. We might also want to reflect the epistemic state of an agent or a database: we might have information that p is true, or false; we might have no information about p; or we might have information that p is true and that p is false.

22 Nuel Belnap

23 Four Truth Values Let s introduce four values, 1, 0, n, and b, as Belnap does in his Useful 4-valued Logic. Belnap defines negation, conjunction, and disjunction for such a logic. It turns out to be close to classical logic, but it is paraconsistent.

24 Four Truth Values Belnap s idea is that a natural logic for information systems, including human beings, should have four values: I have information that the proposition is true (1), I have information that it is false (0), I have information that it is true and information that it is false (b), and I have no information about its truth value (n). We might, as Belnap does, construct on this basis a four-valued logic.

25 Four Truth Values Consider a lattice with 1 as top element and 0 as bottom, with n and b in between. Then we can define v(a B) = glb(v(a), v(b)); v(a B) = lub(v(a), v(b)). v( A) = 1 v(a) = 0; v( A) = 0 v(a) = 1; v( A) = n v(a) = n; v( A) = b v(a) = b. Take 1 and b as designated values.

26 FDE Negation This gives us the truth tables: A A 1 0 b b n n 0 1

27 FDE Conjunction 1 b n b n 0 b b b 0 0 n n 0 n

28 FDE Disjunction 1 b n b 1 b 1 b n 1 1 n n 0 1 b n 0

29 FDE The logic that results is new. But suppose that an interpretation assigns nothing b; that is, suppose that there are no truth value gluts. Then the logic that results is exactly K3. Suppose, alternatively, that we assume that there are no truth value gaps, so that nothing gets value n. Then the logic that results is exactly LP. If there are neither gaps nor gluts, the result is classical logic. So, this logic, which we may call (FDE), is a normal logic.

30 J. Michael Dunn

31 Relational Valuations Alternatively, we might think of a valuation not as a function from propositions to truth values, as we have up to now, but instead as a relation between propositions and the truth values {0, 1}. A function assigns each element of its domain one and only one value; a relation may assign no value (thus producing a truth value gap) or more than one value (thus producing a truth value glut).

32 FDE Interpretations Let s begin with three connectives,,, and, and define A B as A B. An FDE-interpretation is a relation ρ relating propositional parameters to truth values {0, 1}. We extend this to the entire language by means of the truth clauses:

33 FDE Truth Clauses A Bρ1 Aρ1 and Bρ1 A Bρ0 Aρ0 or Bρ0 A Bρ1 Aρ1 or Bρ1 A Bρ0 Aρ0 and Bρ0 Aρ1 Aρ0 Aρ0 Aρ1

34 Normality These are the obvious correlates of classical truth conditions. Plainly, then, FDE is normal, agreeing with classical logic whenever the propositional parameters involved have exactly one truth value.

35 Entailment As with any many-valued logic, there are choices to be made concerning the concepts of entailment, validity, etc. FDE stays with truth preservation. Say that Xρ1 iff, for all B X, Bρ1: X = A (Xρ1 Aρ1).

36 Tableau Rules Entries have the forms A, + or A,, representing Aρ1 and A ρ0, respectively. Tableaux begin with an initial list X, +, A,. Rules for disjunctions and conjunctions are as expected. Those for negated conjunctions and disjunctions are DeMorgan s laws; there are also rules for double negation.

37 Tableau Rules Define the rules as in Priest (except that I have played out the DeMorgan rules, which actually stresses the parallel between them and the corresponding classical rules):

38 Tableau Rules A, + A, + A, A,

39 Tableau Rules A B, + A, + B, + A B, A, B,

40 Tableau Rules A B, + A, + B, + A B, A, B,

41 Tableau Rules A B, + A, + B, + A B, A, B,

42 Tableau Rules (A B), + A, + B, + (A B), A, B,

43 Tableau Rules (A B), + (A B), A, + B, + A, B,

44 Tableau Rules (A B), + A, + B, + (A B), A, B,

45 Tableau Rules Branches close if they have nodes of the form A, + and A,. Open branches determine countermodels. These rules are sound and complete for FDE.

46 Exclusion We can derive tableau systems for K3 and LP by adding new constraints and rules. Suppose that an interpretation obeys Exclusion: for no p, pρ1 and pρ0. Then there are no truth value gluts. So, count a branch as closed if there are nodes of the form A, +, A, +. This yields sound and complete tableaux for K3.

47 Exhaustion Suppose that an interpretation obeys Exhaustion: for all p, pρ1 or pρ0. Then there are no truth value gaps. So, count a tableau branch as closed if it contains nodes of the form A,, A,. This yields a sound and complete system for LP.

48 FDE Valid All K3 interpretations are FDE interpretations (those that obey Exclusion); all LP interpretations are also FDE interpretations (those that obey Exhaustion). So, FDE is a proper sublogic of K3 and of LP.

49 FDE Valid That means, of course, that FDE, like K3, has no valid formulas. But K3 and FDE aren t as radical a departure from classical logic as that makes it appear. Most classical rules and inference patterns survive in K3 and FDE. All these, for example, hold in both (taking A B as A B):

50 FDE Valid Double Negation: A A Conjunction Exploitation: A B = A; A B = B Conjunction Introduction: A, B = A B Disjunction Introduction: A = A B; B = A B

51 FDE Valid Absorption: A B = A (A B) Tautology: A A A; A A A DeMorgan s Laws: (A B) A B; (A B) A B Exportation: A (B C) (A B) C

52 FDE Valid Contraposition: A B B A Commutativity: A B B A; A B B A Associativity: A (B C) (A B) C; A (B C) (A B) C Distribution: A (B C) (A B) (A C); A (B C) (A B) (A C)

53 K3 but not FDE These, in addition, are valid in K3: Disjunction Exploitation (Proof By Cases): A B, A C, B C = C Constructive Dilemma: A B, A C, B D = C D Disjunctive Syllogism: A B, A = B; A B, B = A

54 K3 but not FDE Modus Ponens: A B, A = B Modus Tollens: A B, B = A Hypothetical Syllogism: A B, B C = A C Another Proof By Cases: A B, A B = B Explosion: A, A = B

55 FDE and Classical Logic This is interesting, for this is a complete set of inference rules for classical propositional logic! Or, at least, it almost is: the only thing missing is a tautology, such as A A or A A.

56 Richard Routley

57 Routley and Routley (1972) adopt a new strategy for the semantics of relevance logic, treating negation as an intensional operator. Classical negation defines the truth of A at w solely in terms of what holds or fails to hold at w. Routley semantics defines it in terms of what happens at another world.

58 Star Worlds Assume that each world w has a star world w such that A is true at w if A is false at w. If w = w, this is classical. It isn t easy to see what the star world is van Benthem (1979) considers it solely a technical trick without any philosophical motivation, and Priest seems to agree.

59 Star Worlds Dunn (1993) gives perhaps the best understanding of it: w is the maximal world the world containing the most information compatible with w. So, we might think of w as the completion of w. This makes it strange, however, to think of w as being the same as w. Dunn characterizes it as follows: roughly speaking, A is true in a world w iff A is false in every world w compatible with w. The compatibility relation must be taken to be symmetric, directed, and convergent.

60 Star Worlds

61 Gaps and Gluts Another possible interpretation is this. Imagine that we take Belnap s four-valued logic as our understanding of FDE. Take w to be like w, except that n and b are swapped. That is, all truth value gaps in w become truth value gluts in w, and all truth value gluts in w become truth value gaps in w. The semantics for negation then make sense.

62 In Routley semantics, every propositional parameter is assigned 1 or 0 in each world. (Think of 0 as representing, not false, but simply not true.) So, we want, for truth value gaps, both p and p to get the value 0. For gluts, we want both p and p to get 1.

63 Gaps and Gluts Consider a sentence that might plausibly get a glut, say, The sentence in italics in this paragraph is false. (Call it p.) We don t want to say that p is true in w iff p is false in w; since p gets 1 at w, p would get 0, and we would not have a glut after all. But consider w, which swaps gluts and gaps. p gets 0 there, and so does p (for p gets 1 at w = w). The Routley truth condition gives the right result; p gets 1 at w because p gets 0 at w.

64 Gaps and Gluts Consider a sentence that gets a truth value gap in w, say, The present King of France is bald. (Let s call that sentence p.) How do we interpret The present King of France is not bald? We don t want to say that it gets 1 in w iff p gets 0 in w, because then it would get 1, and we would no longer have a truth value gap. If p and p are both 0 in w, however, then, swapping gaps and gluts, both are 1 in w. So, p gets 0 at w, for p gets 1 at w.

65 Routley interpretations A Routley interpretation is a triple < W,, v >, where W is a set of worlds, * a function from W to W such that w = w, and v assigns each propositional parameter 1 or 0 at each world. Truth clauses for conjunction and disjunction are classical. For negation: v w ( A) = 1 v w (A) = 0. Validity is truth preservation in all worlds in all models. (NB: v w ( A) = 1 v w (A) = 0 v w (A) = 0.)

66 Disjunctive Syllogism FDE is paraconsistent; it escapes the problem of explosion. Contradictions do not imply anything. Disjunctive syllogism also fails. Once one realizes that truth value gluts are possible, it is easy to see why.

67 Disjunctive Syllogism Suppose that p is a truth value glut, so pρ1 and pρ0. (For Routley semantics, p and p are both 1 at w; for Belnap, p gets b.) Then (p q)ρ1, but it does not follow that qρ1; nothing at all follows about the truth value of q.

68 Disjunctive Syllogism One may object that we need DS for ordinary reasoning; indeed, as Sextus Empiricus noted, dogs seem to use it! But it is perfectly fine as long as the situation in which we find ourselves is consistent. Only inconsistent information can lead to a failure. That suggests an analogy with nonmonotonic reasoning; we may ordinarily operate with rules that are sound so long as our information is consistent, but fall back to stricter rules when consistency is in question.

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