A four-valued frame semantics for relevant logic

Transcription

1 A four-valued frame semantics for relevant logic Takuro Onishi (Kyoto University) Kyoto Workshop on Dialetheism and Paraconsistency October

2 Background Negation as negative modal operator Do sen 1986; Dunn 1999; Restall 1999; Berto 2015 etc. Dualist view of negation (2015) Negation as impossibility and unnecessity: defined by binary relations of compatibility and exhaustiveness Definition of Routley star in terms of twin binary relations Implicationalization of relevant negation (2016) Definition of twin binary relations in terms of twin ternary relations Application to four-valued semantics for relevant logic

3 1 Two plans in the semantics of relevant logic 2 Routley s American plan 3 Making sense of star

4 Fallacies of relevance B A B A B B A A B A A B (A B) (B A) A B B Classically valid even if there is no connection between the content of A and B

5 Australian plan Routley-Meyer semantics (1972) Implication: ternary relation R on a frame. x = A B y, z : Rxyz & y = A z = B. Negation: the Routley star. x = A x = A. Criticized for lack of intuitive interpretation (Copeland 1979; van Benthem 1979, etc.)

6 American plan Four-valued semantics (Dunn 1966, 1976; Belnap 1977) A B is true A is true and B is true A B is false A is false or B is false A B is true A is true or B is true A B is false A is false and B is false A is true A is false A is false A is true Classical semantic clauses (esp. for negation) Truth and falsity are independent Weak in treatment of implication

7 Four-valued frame semantics Integration of two plans: Elegant explanation of implication by ternary relation Classical semantic clause for negation (avoiding Routley star) Routley 1984; Priest and Sylvan 1992; Restall 1995; Mares Based on Routley s version, I present a simpler and conceptually less problematic one.

8 1 Two plans in the semantics of relevant logic 2 Routley s American plan 3 Making sense of star

9 Four-valued frame semantics In American plan completed (Routley 1984). x = + A B x = + A and x = + B x = A B x = A or x = B x = + A B x = + A or x = + B x = A B x = A and x = B x = + A x = A x = A x = + A Truth and falsehood relativized to states. Classical clause for negation preserved.

10 Truth and falsity of implication Twin ternary relations for truth and falsehood of implication: x = + A B y, z : Rxyz & y = + A z = + B x = A B y, z : Syxz & y = B & z = A (the order of arguments in Syxz changed from Routley s)

11 Frame for the basic relevant logic B Definition A B-frame is a structure 0, 0, U, V, K, R, S such that: K: set of states 0 U K: set of regular states 0 V K: set of coregular states R, S K 3 with a few conditions. A B-model is a B-frame equipped with a twin valuation = +, =.

12 Validity A is 2-valid in a frame if 0 = + A and 0 = A for any valuation = +, =. Theorem (Routley 1984) A is 2-valid in all frames if and only if A is provable in B.

13 Positive extensions A B.B C.A C { u : Rxyu & Ruwz v : Ryvz & Rxwv corresponds u : Sxuy & Suwz v : Sxzv & Svwy A.A B B { Rxyz Ryxz corresponds Sxyz Sxzy etc. Axioms defines or corresponds to structural constraints on ternary relations.

14 Negative extensions A B. B A (If A is true then B is true) implies (if B is false then A is false)? Structural constraints on ternary relations are not enough. Cross-over between truth and falsity is required.

15 Options (1) Doubling of -rules: x = + A B y, z :Rxyz (similarly for the falsity condition.) (2) Star imitation x : (3) Mixture of (1) and (2) x = + p x = p { y = + A z = + B y = B z = A

16 Doubling alone is not sufficient to model Reductio (A A A), Assertion (A (A B).A B), etc. Star imitation required. But their mixture causes nightmarish complication. Star imitation without doubling looks promising. But the point of four-valued setting was to avoid the use of star-like operation. Let us make sense of star operation in terms of ternary relations.

17 1 Two plans in the semantics of relevant logic 2 Routley s American plan 3 Making sense of star

18 Compatibility and exhaustiveness x = + p x = p (1) x = + p x = p: x and x are compatible. (2) x = + p x = p: x and x are jointly exhaustive

19 Negation as impossibility and the star postulate x = A y : xcy y = A. A is impossible at x iff it is not true at any state y that is compatible with x. Under symmetry of C and the star postulate x y : xcy & z : xcz z y, Routley star is definable and is de Morgan negation (Mares 1995, 2004; Restall 2000).

20 Negation as unnecessity x = A y : xey & y = A. A is unnecessary at x iff it is not true at some state that is jointly exhaustive with x. The axiom A A makes and de Morgan negation (under the assumption of symmetry of C and E). This corresponds to the dualist star postulate.

21 Dualist star postulate (Onishi 2015) The star state x is characterized as the maximally compatible and minimally exhaustive mate state of x. E x E C x C x y : xcy & xey & ( z : xcz z y & xez y z).

22 Routley star can be understood as a special case of the twin binary relations What is it that two states are compatible with each other or jointly exhaustive?

23 Regular and coregular state Recall the complete theorem: A is provable 0 = + A and 0 = A in any frame. Regular state 0 supports every logical truth Coregular state 0 never makes a logical error

24 Information-theoretic interpretation +A B x +A y R z +B [T]he combination of the pieces of information in x and y... is a piece of information in z. (Dunn and Restall 2002)

25 Fusion (intensional conjunction) +A x +B y R z +A B z = + A B x, y : Rxyz & x = + A & y = + B

26 Defining compatibility x y R 0 Any combination of information from x and y is not a logical error / does not cause any logical clash. x and y are (logically) compatible. xcy def = Rxy0 (or u V : Rxyu)

27 Fission (intensional disjunction) A B := ( A B) +A B x y S z +A or +B ( x = A B y, z : Sxyz y = + A or z = + B )

28 Defining exhaustiveness +A B 0 +A y S or z +B Suppose A B is a logical truth, i.e., logically A or B. In S0yz, y and z jointly exhaust the logical possibilities. yez def = S0yz (or u U : Suyz)

29 Constraint on valuation xcy def = Rxy0 xey def = S0xy We assume C and E are symmetric (for simplicity): (Sym) xcy ycx xey yex The following constraint on valuation should be natural: (C) (E) xcy & x = + p y = p xey & x = + p y = p

30 Extension of (C) and (E) Proposition Suppose that a B-frame F satisfies: (GS1) (GS2) (GS3) (GS4) Sxyz & ycw = Rxwz; Rxwz = y : ycw & Sxyz; Rxyz & xew = Syzw; Syzw = x : xew & Rxyz. Then the constraints (C) and (E) on a valuation on F extend to arbitrary formulas. (GS for Generalized Star postulate)

31 Star is definable Lemma (dualist star postulate) In a frame satisfying (Sym) and (GS1,2) or (GS3,4), the dualist star postulate holds: E x E C y C x y : xcy & xey & ( z : xcz z y & xez y z).

32 Define x to be that y. Then, Moreover, x y y x and x = x. Syzx Rxyz Sxy z. Extendability of (C) and (E) is immediately proved by these equivalences. And they are summarized as: x = + A x = A, because x is compatible and jointly exhaustive with x.

33 Negative extensions A B. B A corresponds Rxyz Rxz y and Sxyz Syz x A A A corresponds Rxx x and Sxxx etc.

34 What is the logic with GS? Definition A GS-frame is a B-frame that satisfies (GS1-4) and (Sym). A GS-model is a GS-frame equipped with a twin valuation = +, = that satisfies (C) and (E). Proposition The logic defined by the class of GS-models is equivalent to the one defined by the class of two-valued B-models that satisfies Rxyz Ryz x.

35 What GS is We say that A 2-entails B (A B) if x = + A x = + B and x = B x = A for any x in any model. GS1-4 corresponds to the two-way rule: A B C A B f C (GS) where f is the false constant such that 0 = f and 0 = + f. A B + C A + B f C (GS1) A + B f C A B + C (GS2) A B C A B f C (GS3) A B f C A B C (GS4)

36 What GS is A B C (GS) A B f C is a special case of classical rule for implication: A B D C A B D C or X, B D, Y X B D, Y ( R )

37 What GS does Definability of Routley star Extendability of (C) and (E) Implicationalization of negation: A A f

38 Conclusion Modification of Routley s American plan Twin ternary relations Generalized star postulate