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Pollock's Theory of Defeasible Reasoning Pollock’s Theory of Defeasible Reasoning Jonathan Weisberg University of Toronto Northern Institute of Philosophy June 18, 2010 Pollock Weisberg Motivations Graphs Outline Semantics Paradoxes 1 Motivations Worries Other 2 Inference Graphs References 3 Pollock’s Semantics 4 Success Stories: The Paradoxes of Acceptance 5 Worries 6 Other Topics Pollock Weisberg Motivations Graphs Semantics Paradoxes Worries Other Motivations References Motivations Pollock Weisberg 1 Problematic Bayesian Idealizations Motivations Graphs 1 Computational Demands 2 Semantics Storage Demands 3 Logical Omniscience Paradoxes Worries 2 Representing Sequential Reasoning Other 3 Paradoxes of Acceptance References 1 The Lottery Paradox 2 The Lottery Paradox Paradox 3 The Preface Paradox 4 The Epistemological Role of Non-Doxastic States 5 Dissatisfaction with Standard Non-Monotonic Logics Problematic Bayesian Idealizations Pollock É Computational Demands: a Bayesian agent updates all her Weisberg probabilities with each new piece of evidence. Motivations É Computationally demanding, often wasteful. Graphs É At odds with our actual reasoning. Semantics É Storage Demands: a Bayesian agent stores a real number for Paradoxes each conditional belief, a “combinatorial nightmare” Worries (Pollock 2008). Other É Suppose an agent has 300 beliefs. References É The number of conditional probabilities of the form 90 p(A B1 ...Bn) that must be stored is about 10 . 90j É 10 > the number of particles in the universe. É Logical Omniscience: a Bayesian agent assigns probability 1 to all logical truths, but we surely can’t and don’t. Pollock (2008) advertises his framework as avoiding the first two problems. É I’m advertising it as avoiding the last. Representing Sequential Reasoning Pollock Weisberg Motivations A lot of our reasoning appears to be sequential, in two ways: Graphs É Collecting reasons. Semantics Paradoxes É Deploying reasons. Worries Bayesianism, DST, ranking theory, etc. all ignore this reality. Other É As a result, they may fail to acknowledge beliefs that are References justified despite not taking account of all the evidence. É If other cognitive demands (pragmatic or epistemic) rationally interrupt a train of reasoning, you may be justified in believing the conclusions drawn so far. Paradoxes of Acceptance Pollock Weisberg Motivations Pollock’s treatment of the paradoxes of acceptance respects the Graphs following desiderata. Semantics Preface: you are justified in believing the claims in your Paradoxes É Worries book. Other É Lottery: you are not justified in believing your ticket will References win. É Conjunction: if you are justified in believing A and B, you are justified in believing A&B. This package is very hard to come by. The Epistemological Role of Non-Doxastic States Pollock Weisberg On many epistemological views, non-doxastic states play a role Motivations in justifying beliefs: Graphs É Perceptual states Semantics Paradoxes É Memories Worries É Module outputs Other On some views, non-doxastic states alone justify: References É Pollock, Pryor On others, they do so in conjunction with background beliefs: É Vogel, White? But formal epistemologies almost never address the justificatory role of non-doxastic states. Dissatisfaction with Standard Non-Monotonic Logics Pollock Weisberg Motivations Graphs Pollock’s reasons for dissatifaction with other non-monotonic Semantics formalisms vary from case to case: Paradoxes Worries É Too limited Other É Implausible results References É Off-topic For a survey, see (Pollock 1995: 104-9). Pollock Weisberg Motivations Graphs Semantics Paradoxes Worries Inference Graphs Other References Epistemic States Pollock Weisberg In Pollock’s system, an agent’s epistemic state is represented by Motivations an inference graph. Graphs É Nodes: reasons and the propositions they bear on. Semantics É Directed edges: relations of support and defeat. Paradoxes Worries Example: Other References F B P Defeat: Rebutting vs. Undercutting Pollock Pollock acknowledges two kinds of defeaters: Weisberg 1 Rebutters: R is a rebutting defeater of P if it is a reason for Motivations P. : Graphs 2 Undercutters: U is an undercutting defeater of P as a reason Semantics for Q if it is a reason for (P wouldn’t be true unless Q Paradoxes were true). : Worries É The negated conditional is symbolized P Q. ⊗ Other So the previous example is properly represented: References F B F ⊗ B P Example: Rebutting Defeat Pollock Weisberg Motivations Example: Pam says that Robert will be at the party, whereas Graphs Qbert says he won’t be: Semantics Paradoxes Worries R R : Other References P Q Inference Rules Pollock Weisberg Motivations Where do the arrows come from? That is, when is one thing a Graphs reason for another? Semantics É Pollock proposes a number of inference rules in various Paradoxes writings, but does not pretend to have a complete list. Worries Other É The methodology: propose rules that seem plausible and References test them on numerous examples. É Finding a list of complete rules that yield sensible results is a major burden of the theory. É Compare the Bayesian’s task of specifying rationality constraints on priors: Reflection, PP, Indifference, etc. Inference Rules: Some Examples1 Pollock Perceptual Justification x’s appearing R is a defeasible reason for Weisberg believing that x is R. (Pollock 1971, 1974) Motivations Temporal Projection Believing P@t is a defeasible reason for Graphs believing P@(t + ∆t), the strength of the reason Semantics being a monotonic decreasing function of ∆t (for Paradoxes appropriate P). (Pollock 2008) Worries Discontinuity Defeat P@t1 is an undercutting defeater for the Other : inference by Temporal Projection from P@t0 to References P@t2. (ibid) Statistical Syllogism If r > 1=2 then Fc & p(G F) r is a prima facie reason for Gc, the strengthj of≥ the reason being a monotonic increasing function of r. (Pollock 1990,1995) Subproperty Defeat Hc & p(G F&H) = p(G F) is an undercutting defeaterj for6 the Statisticalj Syllogism. 1NB: these are simplified glosses, omitting important qualifiers and details. Initial Nodes: Perception Pollock Inference rules tell us how to introduce new nodes into the Weisberg graph given initial nodes, but where do initial nodes come from? Motivations É In other words, what can be used as a reason without Graphs appeal to a supporting reason? Semantics Pollock is surprisingly brief on this point. Paradoxes Worries É Formally, we just help ourselves to a set of premises: input. Other É He does say, References “Epistemic reasoning starts with premises that are input to the reasoner. In human beings, these are provided by perception.” (Pollock 1995: 39) “Perception provides the premises in input from which epistemic cognition reasons forward [. ]” (ibid: 47) Initial Nodes: Further Candidates Pollock Should other things be included in input too? Weisberg É Existing (justified) beliefs Motivations Graphs É Memory states Semantics É Outputs of non-perceptual modules Paradoxes Fortunately, we can explore the formalism and many of its Worries applications without answering this question. Other But it does raise important, tricky questions about what an References É inference graph is supposed to represent. É An agent’s epistemic state at a time: the reasons and inferences she is currently aware of? É A record of her reasoning over time: all the reasons and inferences she has taken account of in her lifetime? É The framework’s appeal may depend heavily on our choice here. Pollock Weisberg Motivations Graphs Semantics Paradoxes Worries Other Pollock’s Semantics References Defeat Statuses Pollock We want to be able to figure out what beliefs are justified given Weisberg the reasons and inferences taken into account so far. Motivations É We want an algorithm for assigning the statuses defeated Graphs and undefeated to nodes in a given graph. Semantics Paradoxes Using to symbolize defeated and + to symbolize undefeated, − Worries we want results like: Other References + − − − F B F R R ⊗ : + + + + B P P Q Semantics: A First Attempt Pollock Weisberg Definition: D-initial Node Motivations A node is D-initial iff neither it nor any of its ancestors are Graphs Semantics termini of a defeat link. Paradoxes Then here’s a plausible, first attempt: Worries (1) D-initial nodes are undefeated. Other References (2) If the immediate ancestors of node A are undefeated, and all nodes defeating it are defeated, then A is undefeated. (3) If A has a defeated immediate ancestor, or there is an undefeated node that defeats A, then A is defeated. This proposal gets the right results for Tweety and other simple examples. A Problem: Collective Defeat Pollock Weisberg But it does poorly in cases of “collective defeat”, like our example of conflicting testimony. Motivations Graphs É The only assignments consistent with (1)–(3) are: Semantics Paradoxes + + Worries − − R R R R Other : : References + + + + P Q P Q É Both are counterintuitive and unjustifiably anti-symmetric. Another Problem: Self-Defeat Pollock Weisberg If we assign to Q, we violate (2): − Motivations Graphs − − Semantics R P Q Paradoxes ⊗ Worries Other References − Q + P Another Problem: Self-Defeat Pollock Weisberg If we assign + to Q, we violate (3): Motivations Graphs + + Semantics R P Q Paradoxes ⊗ Worries Other + References Q + P Partial & Maximal Assignments Pollock These complications (and others) motivate a more sophisticated Weisberg approach: Motivations Definition: Partial Status Assignments Graphs Semantics A partial status assignment assigns + and to at least some − Paradoxes nodes and satisfies: Worries (P1) All D-initial nodes are undefeated. Other (P2) A is undefeated iff the immediate ancestors of A are References undefeated, and all nodes defeating A are defeated. (P3) A is defeated iff A has a defeated immediate ancestor, or there is an undefeated node that defeats A. Definition: Maximal Status Assignment A status assignment is maximal iff it is partial and is not contained in any larger partial assignment. The Final Proposal Pollock Weisberg Motivations Proposal: Supervaluation Graphs Semantics A node is undefeated iff every maximal status assignment gives it Paradoxes a +; otherwise it is defeated.
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