Justification Logic Who? Natalia Kotsani - based on the work of S. Artemov (The Logic of Justification, 2008) When? 24 October 2012 Table of contents 1. Introduction An example What is Justification Logic Why Justification Logic Historically Basics Structural Assumptions Design Decisions 2. Principles Involved The Logic of Proofs Gettier examples Gettier examples Goldman's reliabilism Lehrer and Paxson's indefeasibility condition Introduction - An example An agent receives a product of two very large primes integers. In what sense does the agent know those primes? In what sense does the agent know all the derivable formulas regardless to the complexity of their justifications? These and similar problems could be approached in a framework of reasoning about knowledge justifications vs the traditional F . Introduction - What is Justification Logic is an epistemic logic that explicitly includes justifications for knowledge by using a construct to formalize statements t is a justification of F the evidence term t can be viewed as an informal justification or a formal mathematical proof evidence terms are built by means of operations that correspond to the axioms of a modal logic system Introduction - Why Justification Logic behind each epistemic modal logic, there is a system of justifications simple and compound reasoning about epistemic justification comparing different pieces of evidence pertaining to the same fact measuring the complexity of justifications, which leads to a coherent theory of logical omniscience renderind a novel, evidence-based mechanism of truth-tracking which seems to be a key ingredient of the analysis of knowledge provide a tool for analyzing assumptions and to ensure that we draw correct conclusions filling the gap between epistemic logic and mainstream epistemology Introduction - Historically Knowledge as Justified True Belief [Plato, ∼400 BC] A first approach to modal syllogistic [Aristotle, ∼350 BC] Logic of Proofs (LP) [Godel¨ , 1933] KF (F is known) & BF (F is believed), F is known ∼ F holds in all epistemically possible situations [Von Wright, 1951] Kripke semantics for Modal Logic [Kripke 1959] Epistemic Logic [Fagin, 1995] The absence of justification logic remained an impediment to both formalizing the Brouwer−Heyting−Kolmogorov semantics of proofs and providing a long-anticipated exact provability semantics for S4 and intuitionistic logic [Artemov, 1999] Introduction - Basics t : F t is a justification for F t is accepted by agent as a justification of F F satisfies conditions t we do not directly analyze what it means for t to justify F beyond the format t : F , but rather attempt to characterize this relation axiomatically Introduction - Structural Assumptions justifications are abstract objects which have structure agents do not loose or forget justifications apply the laws of classical logic and accept their conclusions operations sum "+" pooling evidence together without performing any epistemic action (concatenation of proofs), s + t = s ∗ t application "·" performs one epistemic action, a one-step deduction according to the Modus Ponens rule, s · t = s ∗ t ∗ pG1q ∗ ::: ∗ pGnq pX q denotes the Godel¨ number of X & Gi are all formulas from t for which there is a formula F ! Gi from s Introduction - Design Decisions For this installment of Justification Logic we 1 limit our attention to propositional and quantifier-free systems of Justification Logic 2 build our systems on the simplest base: classical Boolean logic 3 treat both partial and factive (justifications are sufficient for an agent to conclude truth) justifications 4 consider the case of one agent only for whom no formula is provably justified Principles Involved - The Logic of Proofs justifications are represented by proof polynomials (justifications terms) 1 proof variables x, y, z, . 2 proof constants a, b, c, . 3 binary operation application "·" 4 binary operation sum (union, choice) "+" 5 unary operation proof checker "!" formulas of LP are those of propositional classical logic (by means of logical connectives) augmented by the formation rule: if t is a proof polynomial and F is a formula, then t : F is again a formula Principles Involved - The Logic of Proofs LP contains the axioms of classical propositional logic and the rule of Modus Ponens along with the axioms of application s :(F ! G)(t : F ! (s · t): G) sum s : F ! (s + t): F t : F ! (s + t): F proof checker t : F !!t :(t : F ) reflection t : F ! F in addition to the usual logical properties, enjoys the internalization If ` F , then there is a proof polynomial p such that property ` p : F Principles Involved - LP vs Modal Logic Theorem each theorem of LP becomes a theorem of S4 if all the terms are replaced with the modality ; and vice versa, each theorem of S4 can be transformed into a theorem of LP if the occurencies of modality are replace with suitable evidence terms [S. Artemov, 2001] S4 axioms LP axioms (F ! G) ! (F ! G) s :(F ! G)(t : F ! (s · t): G) s : F ! (s + t): F t : F ! (s + t): F A ! A t : F !!t :(t : F ) A ! A t : F ! F LP is an explicit counterpart of the modal logic S4 (reflexive &) this process is called realization (theorem) Principles Involved - Gettier examples Knowledge ,? Justified True Belief Gettier (1963) described two situations, Case I and Case II, that were supposed to provide examples of justified true beliefs which should not be considered knowledge. Gettier's problem has attracted a range of "epistemological solutions" which consist of trying to find alternative analyses of knowledge (e.g. JTB+G approaches: Justified True Belief + "no-Gettier-problem condition") Principles Involved - Gettier examples CASE I Suppose that Smith and Jones have applied for a certain job. And suppose that Smith has strong evidence for the proposition: (d) Jones is the man who will get the job, and Jones has ten coins in his pocket. Proposition (d) entails: (Suppose that Smith sees this entailment.) (e) The man who will get the job has ten coins in his pocket. But unknown to Smith, he himself, not Jones, will get the job. And, unknown to Smith, he himself has ten coins in his pocket. proof Smith accepts (e) on the grounds of (d). Then, all of the following are true: (e) is true Smith believes that (e) is true Smith is justified in believing that (e) is true Smith does not know that (e) is true Principles Involved - Gettier examples CASE II Suppose that Smith has strong evidence for the following: (f) Jones owns a Ford. Let us imagine, that Smith has another friend, Brown, of whose whereabouts he is totally ignorant. Smith selects three place names quite at random and constructs the following three propositions entailed by (f): (Suppose that Smith sees this entailment.) (g) Either Jones owns a Ford, or Brown is in Boston. (h) Either Jones owns a Ford, or Brown is in Barcelona. (i) Either Jones owns a Ford, or Brown is in Brest-Litovsk. But imagine now that two further conditions hold. Jones does not own a Ford, but is at present driving a rented car. The place mentioned in (h) happens really to be where Brown is. proof Imagine that Smith proceeds to accept (g), (h), and (i). Then Smith does not know that (h) is true, even though (h) is true, Smith does believe that (h) is true, and Smith is justified in believing that (h) is true. Principles Involved - Gettier examples Gettier uses a version of the epistemic closure principle, closure of justification under logical consequence: if Smith is justified in believing P for some t, t : F and Smith deduces Q from P P ! Q then Smith is justified in believing Q t : Q, for some t Such a rule holds for the Logic of Proofs, as well as for all other Justification Logic systems, and it is a combination of: Internalization Rule If ` F , then there is a proof polynomial p such that ` p : F Application Axiom s :(F ! G)(t : F ! (s · t): G) and the use of Modus Ponens twice. Principles Involved - Goldman's reliabilism Knowledge ,? Justified True Belief Goldman (1967) offered the fourth condition to be added to the Justified True Belief definition of knowledge, according to which: "A subjects belief is justified only if the truth of a belief has caused the subject to have that belief, and for a justified true belief to count as knowledge, the subject must also be able to correctly reconstruct (mentally) that causal chain." A situation t justifies F for some t only if F is true, which provides the Factivity Axiom for knowledge-producing justifications: Factivity Axiom t : F ! F Principles Involved - Factive Justifications The Factivity Axiom is assumed for factive justifications justifications yield the truth of a belief, e.g. systems JT, LP, JT45 but it is not assumed for general justification systems e.g. systems J, J4, J45, JD45 We can assume that the causal chain leading from the truth of F to a justified belief that F manifests itself in the Principle of Internalization which holds for many Justification Logic systems: Internalization Rule If F is valid, then one could construct a justification p such that p:F is valid. Internalization is usually represented in an equivalent form (in the presence of the Completeness Theorem) as a meta-rule. The algorithm which builds a justified belief p:F from a strong evidence (proof) of the validity of F seems to be an instance of Goldmans 'causal chain'. Principles Involved - Lehrer and Paxson's indefeasibility condition Lehrer and Paxson (1969) offered the following 'indefeasibility condition': "There is no further truth which, had the subject known it, would have defeated [subjects] present justification for the belief." The further truth here could refer to a possible update of the subjects database, or some possible-worlds situation, and so forth: these readings lie outside the scope of our language of Boolean Justification Logic.