Vague and Uncertain Entailment: Some Conceptual Clarifications

Transcription

1 Conditionals, Counterfactuals, and Causes in Uncertain Environments Düsseldorf, May 2011 Vague and Uncertain Entailment: Some Conceptual Clarifications Chris Fermüller TU Wien, Austria

2 Overview How does Fuzzy Logic fit in? What is uncertainty? and how to measure it? Vagueness and degrees of truth how to understand Fuzzy Logic? Deductive (t-norm based) Fuzzy Logic degree based notions of entailment A case study Giles s game justifying degrees of truth via uncertainty Uncertainty about degrees of truth interval based logics and their problems Rescue from Giles? justifying truth functions over intervals

3 Models of reasoning in uncertain environments NB: Fuzzy Logic researchers use the term uncertainty in a very brood sense but always connected to mathematical models Baoding Liu: Uncertainty Theory Some chapter titles: probability theory credibility theory trust theory fuzzy random theory random fuzzy theory bifuzzy theory rough fuzzy theory Klir/Wiermann: Uncertainty-Based Information Some chapter titles: probability theory fuzzy set theory fuzzy measure theory evidence theory possibility theory

4 Stanford Encyclopedia of Philosophy: James Hawthorne (from the Inductive Logic article) Some Prominent Approaches for Representing Uncertain Inferences :

5 Vagueness and Fuzzy Logic NB: Zadeh distinguished fuzziness from vagueness: Fuzziness: gradedness of predication (membership) and truth Vagueness: fuzziness + unspecificity The following tenets of a FL based approach to vagueness are widely but by no means universally accepted by now: FL is not a theory of vagueness, but a component of one FL is not a poor man s Ersatz for probability, DS-theory, etc. FL provides a mathematical toolkit for many applications truth degrees do not necessarily entail truth functionality rich and growing literature on deriving FL from first principles one should distinguish between FLw and FLn

6 Interlude Is there a logic of vagueness?

7 Interlude Is there a logic of vagueness? I claim: no!

8 Interlude Is there a logic of vagueness? I claim: no! Or rather: the problem is ill-posed.

9 Interlude Is there a logic of vagueness? I claim: no! Or rather: the problem is ill-posed. One can define all kinds of logics that in principle are candidates for modeling (aspects of) reasoning under vagueness. Judging the adequateness and usefulness of these models is not a matter of comparing (theory) neutral data with predictions. Modeling may mean different things. I take (fuzzy) logicians to emphasize the aspect of constructing formal structures that support robust reasoning. NB: It is unreasonable to expect unique solutions to this task.

10 Interlude Is there a logic of vagueness? I claim: no! Or rather: the problem is ill-posed. One can define all kinds of logics that in principle are candidates for modeling (aspects of) reasoning under vagueness. Judging the adequateness and usefulness of these models is not a matter of comparing (theory) neutral data with predictions. Modeling may mean different things. I take (fuzzy) logicians to emphasize the aspect of constructing formal structures that support robust reasoning. NB: It is unreasonable to expect unique solutions to this task. As to the potential relevance for philosophy and psychology: Compare Keith Stenning & Michiel van Lambalgen s concept of reasoning to an interpretation.

11 Deductive Fuzzy Logic (FLn) Hájek s design choices 1. [0, 1] as set of truth values 2. truth functionality 3. conjunction is modeled by a continuous t-norm : [0, 1] 2 [0, 1]: (x y) z = x (y z) (associative) x y = y z (commutative) x 1 = x (unit element) x y implies x z y z (non-decreasing) 4. implication is the (unique!) residuum of : x y = def sup{z x z y} 5. negation: x = def x 0

12 A rich and well behaved Meta-Mathematics of Fuzzy Logic results from these design choices (cf. [Petr Hájek 98]). Three fundamental continuous t-norms and their residua: Logic L( ) x y x y, for x > y Lukasiewicz L: max(0, x + y 1) 1 x + y Gödel G: min(x, y) y Product P: x y y/x All continuous t-norms are ordinal sums of these three. L is the only logic of type L( ), where is continuous. Entailment (F 1,..., F n = G) can be defined in different ways! v(g) = 1 whenever v(f i ) = 1 for all 1 i n inf 1 i n v(f i ) v(g) v(f 1 )... v(f n ) v(g) Graded entailment: (e.g.) degree(f 1,..., F n = G) = inf 1 i n v(f i ) v(g)

13 Giles s dialogue foundations for approximate reasoning Meaning of connectives specified by dialogue rules (Lorenzen): Let X/Y stand for me/you or for you/me X asserts attack by Y answer by X A B A B A B? A or B (X chooses) A B l? or r? (Y chooses) A or B (accordingly) A & B? A and B Note: A abbreviates A. where is indefensible Answer ( quit ) is allowed! (= Giles s principle of limited liability only relevant for &) Dialogue states: [A 1,..., A m B 1,..., B n ] (multisets) To obtain a logic we additionally need winning conditions for atomic states regulations defining admissible runs of a game

14 ad: winning conditions Let the players bet on the truth of their (atomic) claims! (Yes/no-)experiments that may be dispersive decide. I pay 1 to you for each of my false atomic assertions, if you agree to do the same for your atomic assertions A final states [p 1,..., p m q 1,..., q n ] results in a pay-off of ( m n ) p i q j for me i=1 j=1 p... risk value = (subjective) probability of no as result for p ad: regulations Constraints on runs of a dialogue like the following suffice: (R ) if you attack my assertion of A B by claiming A, then I have to assert also B at some state Each formula is attacked at most once. No particular regulation for the order of moves is required!

15 Giles s Theorem: F 1,..., F n = L G iff for every risk value assignment I have a strategy for avoiding expected loss in dialogues starting with my assertion of G and your s of F 1,..., F n. F 1,..., F n = L G... v(f 1 ) L... L v(f n ) v(g) for all v Beyond Giles: generalizations of the game to G, P, and CHL close connections to analytic proof systems: logical rules can be seen as dialogue rules in particular: proofs in a uniform hypersequent calculus for L, G, P correspond to success strategies

16 Modeling imprecise knowledge of truth degrees Esteva, Garcia-Calvés, Godo [IMPU 92, IJUFKS 1(94)]: This kind of imprecision can be handled by means of an interval-based calculus: the more imprecision there is, the larger intervals are. = bilattice structures Int(L),,, N (here always: L = [0, 1]) Int(L)... set of closed intervals over L + the empty interval truth ordering : [a, b] [c, d] iff a c and b d imprecision ordering : subinterval relation N... involutative negator, antitone w.r.t. both orderings EGG define t-norm based truth functions for logical connectives: Let v(a) = [a l, a h ], v(b) = [b l, b h ] t-norm based strong conjunction: v(a&b) = [a l b l, a h b h ] similar for (weak) lattice conjunction and lattice disjunction implication: v(a B) = [a h b l, a l b h ]

17 Problems (1) The truth function for is no longer a residuum of the truth function for & = Cornelis, Deschrijver, Kerre (and others) define: v(a B) = [(a l b l ) (a h b h ), a h b h ] This results in a residuated lattice, containing the original lattice as a substructure. But a different undesired feature arises with this implication [Hajek, unpublished]: Sharpenings don t preserve (graded) truth!

18 Problems (2) The involved impreciseness is explicitly declared to arise from an incompleteness of knowledge of the real degree of truth. At his point EGG follow like many others Belnap [1977]. However (cf. similar criticism by Didier Dubois): Belnap s interpretation of bilattice based truth functions is incoherent! Consider Belnap s bilattice FOUR= {0,,, 1}, t, k, N Intended interpretation: 0 / 1... (definitely) false / true (={}) / (={0, 1})... no / inconsistent information truth ordering : 0 t [ / ] t 1 knowledge ordering : k [0 / 1] k = no coherent truth functions over {0,,, 1} are possible, if,, are to retain their usual (classical) meaning! E.g., consider A A and B A, where the semantic status (knowledge) of A and B is equal, namely empty: {} (or {0, 1}).

19 (Like for probability:) partial knowledge about degrees of truth cannot be propagated truth functionally from atomic propositions to disjunctions and conjunctions! = The interval v(f ) calculated via truth functions does not reflect our best knowledge about the real value of F, given the partial information about atoms that is coded in the interval assignment v. Example: No knowledge about the real value of atom p. Consequently: v(p) = [0, 1], and also v( p) = [0, 1]. According to EEG: v(p p) = [max(0, 0), max(1, 1)] = [0, 1] But the real value of p p can never be below 0.5! Observation by Godo and Esteva: At least v(f ) provides valid (but sub-optimal) lower and upper bounds for the unknown real value of F. Can we say more about the Godo/Esteva bounds?

20 (a) We use a Giles/Lorenzen-style dialogue game to reduce our logically complex assertions to atomic ones. (b) For each of assertion of atom p we have to pay between 1 h(p) and 1 l(p), where v = [l(p), h(p)]. Possible interpretation: [l(p), h(p)] is the range of plausible answers of competent speakers to questions about the degree of truth of p. We have to pay for the falsity of an assertion of p following these answers. Note: different answers are expected for each occurrence of p. Claim: Given v, F evaluates to [l, h] according to EGG implies: An optimal strategy for F w.r.t. some v-compatible risk value assignment, results in an expected loss of at most 1-l, but at least 1-h.

21 Can we do better? Cautious vs. bold reasoning Let the intervals v(p) = [l(p), h(p)] represent plausible ranges of truth degrees for atomic propositions. Evaluating final state [p 1,..., p m q 1,..., q n ] for me in : w.r.t. ( average loss (assuming centered distributions): n i=1 1 (h(q m ) i) l(q i ))/2) ( i=1 1 (h(p i) l(p i ))/2 ( n optimistic: i=1 i)) ( 1 h(q m ) i=1 1 l(p i) ( n pessimistic: i=1 i)) ( 1 l(q m ) i=1 1 h(p i) Strategies minimizing average loss lead back to the Giles s original characterization of Lukasiewicz logic. Definition: W.r.t. interval assignment v, I play v-cautiously(v-boldly) if I minimize my pessimistically (optimistically) calculated loss.

22 Characterization via dialogue strategies Claim. The following are equivalent: Given v, F evaluates to [l, h] according to EGG. Optimal strategies for playing v-cautiously bound my maximum loss by 1-h and optimal strategies for v-boldly bound my minimum loss by 1-l. Moreover, these bounds are optimal. In other words: the sketched Giles-style dialogue + betting is an adequate evaluation game for the logic defined by EGG. Note: Like in (e.g.) Kleene s 3-valued logic there are no tautologies (without truth constants) if [1, 1] is taken as designated truth value for interval logics. However, a non-trivial entailment relation is characterized by the game!

23 Conclusions Vagueness can be seen as a particular form of uncertainty. But this does not entail an epistemic theory of vagueness. Fuzzy Logic provides different entailment relations based on degrees of truth. Resulting models of reasoning are formal constructions, intended as tools, rather than images of coherent reasoning with vague predicates and propositions. Giles s game is an example of a model that justifies or it least elucidates degree based, truth functional connectives. Giles s game may also help to correct misguided interpretations of truth value intervals as representing epistemic uncertainty about an ill-known degree of truth, A rich research agenda is attached to our case study : other versions of interval based logics relation to (e.g.) Shapiro s Vagueness in Context winning strategies as analytic proofs...