CUNY Graduate Center Logic Colloquium 2015, Helsinki
Objectives 1. We show that the intuitionstic view of knowledge as the result of verification supports the paradigm Justified True Belief yields Knowledge and the paradigmatic Russell and Gettier counterexamples fail intuitionistically. 2. We present Intuitioinistic Epistemic Logic (developed by Artemov & Protopopescu in 2014).
Outline JTB account of knowledge The intuitionistic analysis of Russell s and Gettier s examples JTB yields Knowledge, intuitionistically Findings so far Brouwer-Heyting-Kolmogorov Semantics and Knowledge Intuitionistic Epistemic Logic Models for IEL Intuitionistic beliefs IEL findings
JTB account of knowledge Edmund Gettier, 1963: Various attempts have been made in recent years to state necessary and sufficient conditions for someone s knowing a given proposition. The attempts have often been such that they can be stated in a form similar to the following (a reference to Plato): S knows that P iff i P is true, ii S believes that P, and iii S is justified in believing that P.
Gettier s Case II (a shortened version) Let us suppose that Smith has strong evidence for Smith constructs the proposition: F. Jones owns a Ford. H. Either Jones owns a Ford, or Brown is in Barcelona entailed by (F) though Smith has no idea where Brown is. But imagine now that Jones does not own a Ford, and entirely unknown to Smith, Brown is in Barcelona. Then Smith does not know that (H) is true, even though i (H) is true, ii Smith does believe that (H) is true, and iii Smith is justified in believing that (H) is true.
Gettier s Case II - logic structure F = Jones owns a Ford B = Brown is in Barcelona Then F is justified, and B is true, which makes F B both justified and true. However, the obvious mismatch of justification and truth does not allow us to conclude that Smith knows F B. Hence Justified True Belief (JTB) does not yield Knowledge.
Russell example Bertrand Russell, 1912, gave a different example: If a man believes that the late Prime Minister s last name began with a B, he believes what is true, since the late Prime Minister was Sir Henry Campbell Bannerman 1. But if he believes that Mr. Balfour was the late Prime Minister, he will still believe that the late Prime Minister s last name began with a B, yet this belief, though true, would not be thought to constitute knowledge. B = the late Prime Minister s last name began with a B In the scenario when a man concludes B believing that Mr. Balfour is the late Prime Minister, B is JTB, but not knowledge. 1 In 1912 the British Prime Minister was Herbert Henry Asquith, who succeeded Henry Campbell Bannerman in 1908, who succeeded Arthur James Balfour in 1905.
Intuitionistic analysis of Russell example with full awareness Intuitionistically, a proposition is true if proved. Since B is true, there should be a proof/truthmaking evidence of B. Providing the agent is aware of the aforementioned proof of B, and assuming that a proof is sufficient ground for knowledge, we conclude that the agent knows B, but the reason for this knowledge has nothing to do with the erroneous belief that Mr. Balfour was the late Prime Minister (which therefore is a mere distraction). Intuitionistically Russell s case is not a counterexample to JTB yields Knowledge.
Russell example with limited awareness Since B is true, there is a proof/truthmaking evidence of B. Suppose the agent w is not aware of this evidence and forms his belief on the basis of his erroneous evidence instead. Then we cannot make a case that w is aware that B is true. Naturally, w does not know B but since the true condition from JTB is not met, this is no longer a counterexample for JTB yields Knowledge.
Russell example, two layer analysis A more sophisticated (and more adequate) analysis. Again, since B is true, there should be a proof/truthmaking evidence of B. Suppose the agent w is not aware of this evidence, but the omniscient observer o has a complete picture, in particular, o is aware of the proof of B. Then for w, the truth condition is not met, hence no JTB and no knowledge of B; for o, a truthmaking evidence for B is available, hence o knows B (for the reasons having nothing to do with the erroneous Balfour reasoning of w). For no agent is JTB yields Knowledge violated.
Gettier cases, similar analysis F is justified, and B is true, hence F B is both justified and true. Again, since B is true, there should be a proof/truthmaking evidence of B. Smith is not aware of this evidence, but has a strong evidence for F. The omniscient observer o has a complete picture, in particular, o is aware of the proof of B. Then For Smith, the truth condition is not met, Smith is not aware that B is true hence Smith is not aware that F B is true, no JTB for Smith. For o, a truthmaking evidence for B is available, hence o knows B and F B (for the reasons having nothing to do with Smith s evidence for F ). In neither case is JTB yields Knowledge violated.
JTB yields Knowledge, intuitionistically We argue that not only Russell and Gettier scenarios fail to provide counter-examples to JTB yields Knowledge ( ) in the intuitionistic setting, but ( ) holds intuitionistically, with a proper reading of JTB. truth should be understood as supported by a conclusive proof/truthmaking justification p; justified should mean that this justification p is available to the agent and is recognized by the agent as such; belief should mean agent s belief on the basis of this particular justification p, and not some extraneous evidence.
Does Knowledge yield JTB? The question of whether Knowledge yields JTB intuitionistically is less clear since intuitionistic epistemic framework allows knowledge of A which does not necessarily yield a specific proof of A or awareness of such a proof. Unless the aforementioned intuitionistic JTB conditions are relaxed in an appropriate way, it appears that JTB as is constitutes a sufficient, but not necessary set of conditions for knowledge. In particular, if a proposition A is verified without providing its explicit proof, the intuitionistic truth condition for A is not met.
Findings 1. Tracking the evidence and awareness is the key to this analysis. A significant portion of the confusion about Russell and Gettier examples is caused by using fake evidence as a distraction, and hiding truthmaking evidence. 2. The second ingredient is the intuitionistic postulate that the truth is relative to the epistemic state of an agent: if a truthmaking evidence for A is not available to the agent, there are no reasons to expect this agent to be aware that A is true. For such an agent, the truth value of A is not determined.
Intuitionistic knowledge as verification Together with Williamson and other authors we adopt a view of intuitionistic knowledge as a result of verification by trusted means that do not necessarily produce an explicit proof of what is verified. This distinguishes constructive knowledge from constructive truth and the requirement for the latter turns out to be more stringent than the requirements for the former. Another inspirational example: verification in ITT. In type theory, propositions are types of a special kind and for each type A we may consider Voevodsky s truncated proposition ( squash type, monotype, or bracket type ) inh(a) stating that type A is inhabited, i.e. has a proof. Truncated types may be inhabited (true), without providing a specific witness/inhabitant of A itself. We can naturally interpret KA as inh(a) and consider this as a core semantics of intuitionstic knowledge.
Brouwer-Heyting-Kolmogorov Semantics A proposition A is true if there is a proof of it, and false if we can show that the assumption that there is a proof of A leads to a contradiction. Truth for logical connectives is defined by the following clauses: a proof of A B consists of a proof of A and a proof of B a proof of A B consists in giving either a proof of A or a proof B a proof of A B consists in a construction which given a proof of A returns a proof of B A is an abbreviation for A, and is a proposition that has no proof.
Incorporating constructive knowledge We add an epistemic (knowledge) operator K to our language: what should be the intended semantics of KA? We hereby suggest an epistemic BHK clause for knowledge operator K: a proof of KA is a conclusive evidence that A has a proof (not necessarily by delivering a proof of A explicitly).
Awareness issue Traditional intuitionism assumes that proofs are available to the agent. Heyting says: In the study of mental mathematical constructions, to exist must be synonymous with to be constructed. This position is coherent with the views of Dummett and others. Prawitz and Martin-Löf on the other hand, assume that proofs are timeless, Platonic, entities, and truth is the existence of a proof. The principles of IEL are compatible with either of these positions. If BHK proofs are assumed to be available to the agent, then KA can be read as A is known. If proofs are Platonic entities, not necessarily available to the knower, then KA is read as A can be known under appropriate conditions. To keep things simple, in our exposition we follow the former, more traditional, understanding.
Awareness and Knowability An interesting question is what happens when awareness and knowledge do not match: BHK proofs are platonic entities of which the agent can be unaware, but knowledge is actual rather then potential. Then co-reflection A KA is no longer valid since A can be true without the agent being aware of this fact: in such situation we cannot claim that A is actually known/verified by the agent. However, a more caucious knowability principle holds: A KA, where stands for some kind of metaphysical possibility.
Truth and knowledge in classical logic Take the principle: KA A. Both the intuitionist and the classicist hold the equivalence of A is true with A is known to be counter-intuitive, so both have to deny at least one direction. The classicist denies co-reflection A KA, which is false for classical truth, classical knowledge, and classical implication, and endorses the factiviy of knowledge, a.k.a. reflection KA A.
Constructive truth yields knowledge We argue that from the BHK view of truth and implication, it follows that the intuitionist should endorse the constructivity of truth, a.k.a co-reflection A KA.
Proofs are verifications The idea that proof yields knowledge is practically constitutive of the concept of proof because proofs are a special and most strict kind of verification. That proofs are taken to be verifications is a matter of the ordinary usage of the term which understands a proof as an argument that establishes the validity of a proposition (Encyclopedia Britannica). It is also a fairly universal view in mathematics. Within computer science, this concept is the cornerstone of a big and vibrant area of verification in which one of the key purposes of computer-aided proofs is for the verification of the propositions in question. Amongst intuitionists, the idea of a constructive proof is often treated as simply synonymous with verification.
Proof-checking yields validity of co-reflection Proof-checking is generally a feasible operation, routinely implemented in a standard computer-aided proof package. Certainly, in whatever sense we consider a proof to be possible, or to exist, the proof-checking of this proof is always possible or exists in the same sense. Having checked a proof, we have determined that it is indeed a verification and hence we have a proof that the proposition is proved, hence verified, i.e. known. This is the sense in which according to the BHK reading, A KA states that given a proof of A, one can construct a proof of KA.
Intuitionistic failure of classical reflection Taking into account that verification does not necessarily yield proofs, we argue that the factivity of knowledge in the form of the reflection principle KA A should be resisted as a general principle of intuitionistic epistemic logic. It certainly fails in the core type theoretical semantics: a witness of a truncated type inh(a) does not provide a witness for A itself.
Reflection fails in broader contexts We argue for a broader application of intuionistic epistemic logic outside its core mathematical semantics. We will now list a number of paradigmatic situations with a meaningful notion of verification in which the corresponding reflection principle in its straightforward classical form fails.
Example: Zero-knowledge protocols A class of cryptographic protocols, normally probabilistic, by which the prover can convince the verifier that a given statement is true, without conveying any additional information apart from the fact that that statement is true.
Example: Testimony of an authority Take Fermat s Last Theorem. For the educated mathematician it can be claimed as known, but most mathematicians could not produce a proof of it. More generally, it is legitimate to claim to know a theorem when one understands its content, can use it in one s reasoning, and trust that it has been verified by other mathematicians, without being in a position to produce or recite the proof.
Example: Highly probable truth Suppose there is a computerized probabilistic verification procedure, which is constructive in nature, that supports a proposition A with a cosmologically small probability of error. Hence, its result satisfies the strictest practical criteria for truth. Any reasonable agent accepts this certification as adequate justification of A, hence A is known. Moreover, observing the computer program to terminate with success, we have a proof that KA. However, we do not have a proof of A in the sense required by BHK.
Example: Existential generalization Somebody stole your wallet in the subway. 2 You have all the evidence for this: the wallet is gone, your backpack has a cut at the corresponding pocket, but you have no idea who did it. You definitely know xs(x), where S(x) stands for x stole my wallet, so K( xs(x)) holds. If intuitionistic knowledge would yield proof, you would have a constructive proof q of xs(x). However, a constructive proof of the existential sentence xs(x) requires a witness a for x and a proof b that S(a) holds. You are nowhere near meeting this requirement. 2 One could easily produce a mathematical version of this example.
Example: Classified sources Reports from highly reliable sources which remain anonymous for a variety of reasons, e.g. intelligence data, are a common source of knowledge without checkable proof.
Classical reflection fails intuitionistically: summary The intuitionistic validity of reflection requires a uniform procedure which, given a proof of KA, returns a proof of A itself. Since we allow that KA does not necessarily produce specific proofs, factivity of knowledge KA A fails. There is no uniform procedure that can take any adequate, non-proof verification of A and return a proof of A.
Classical reflection is just too strong Nevertheless reflection is often taken to be practically definitive of knowledge, especially from a constructive standpoint. An obvious intention is that reflection expresses the idea that only true propositions can be known and that false propositions cannot be known. However, in an intuitionistic setting, using reflection as the truth condition for knowledge is a mistake.
False propositions cannot be known The truth condition for knowledge can be alternatively expressed in other ways: 1. (KA A) 2. A KA 3. KA A 4. (KA A) 5. K. 1 4 are classically equivalent to reflection = KA A, but intuitionistically, 1 5 are strictly weaker than KA A.
Correct expression of intuitionistic activity Which of 1 5 best expresses the truth condition on knowledge intuitionistically? Our answer is ANY, since, in the presence of co-reflection, 1 5 are all equivalent 3. So, we pick 2 which is conceptually as close to the classical reflection as possible and is classically equivalent to it: A KA. Intuitionistic factivity is the contrapositive of the classical reflection and is classically equivalent to the latter. 3 Note that in the absence of co-reflection, 1 4 are equivalent, but 5 is the weakest.
Intuitionistic vs. Classical Knowledge We can sum up the difference between classical and intuitionistic principles of knowledge as: KA A A KA Classical Knowledge endorse reject Intuitionistic Knowledge replaced by endorse A KA
Intuitionistic Epistemic Logic IEL Given the above discussion, we define a system of intuitionistic epistemic logic, IEL, incorporating a BHK version of knowledge. The language is that of intuitionistic propositional logic augmented with the propositional operator K. Axioms 1. Axioms of propositional intuitionistic logic 2. K(A B) (KA KB) normality 3. A KA co-reflection 4. A KA factivity Rules Modus Ponens
Some properties In IEL, 1. The rule of K-necessitation, A KA, is derivable. 2. IEL is a normal modal logic. 3. The Deduction Theorem holds. 4. Uniform Substitution holds. 5. Positive and Negative Introspection are valid KP KKP, KP K KP.
The following are all theorems of IEL 1. (KA A); 2. KA A; 3. (KA A); 4. K.
Models for IEL Definition 1. A model for IEL is a quadruple < W, R, E, > such that: 1. < W, R, > is an intuitionistic model; 2. E is a binary relation of W s.t. 2.1 E(u) is non-empty; 4 2.2 E R, i.e., E(u) R(u) for any state u; 2.3 R E E, i.e., urv yields E(v) E(u). 3. is an evaluation function such that u KA iff v A for all v E(u). 4 R(u) and E(u) denote the R- and the E-successors, respectively, of u W.
Informal reading of IEL-models To represent K, for a given world u, there is an audit set of possible worlds E(u) in which verifications, though not necessarily strict proofs, could possibly occur. Knowledge, hence, is truth in any audit set, i.e., no matter when and how an audit occurs, it should confirm A. The correctness of audits is reflected in the principle that audit sets are nonempty, hence no false statements can be verified.
Properties of IEL-models Since E(u) does not necessarily contain u, the truth of KA at u does not guarantee that A holds at u. Therefore, KA A does not necessarily hold. The condition if urv, then E(v) E(u) corresponds to the Kripkean ideology that R denotes the discovery process, and that things become more and more certain in the process of discovery. As the set of epistemic possibilities shrink, so do audit sets. If R(u) = {u}, the audit set E(u) is also {u} and hence coincides with R(u). At such leaf worlds u, u KA A for all A s and intuitionistic evaluation behaves classically. In the epistemic case, at leaf worlds the reflexivity of K a typical classical epistemic principle holds.
Lemma 2 (Monotonicity). For each formula A, if u A and urv then v A. Theorem 3 (Soundness and Completeness). IEL A iff IEL A. Theorem 4. IEL Kp p. Consider the following model: 1R2, R is reflexive and transitive, E(1) = E(2) = {2}, p is atomic and 2 p. Clearly, 1 Kp and 1 p. E 1 E 2 R p
Theorem 5 (Reflection for negated formulas). IEL K A A. Proof. 1. A K A - assumption; 2. A KA - co-reflection; 3. KA K A - from 1, 2; 4. K(A A) - from 3; 5. K - from 4; 6. K - theorem of IEL; 7. - from 5, 6; 8. (A K A) - from 1 7; 9. K A A.
In IEL knowledge and negation commute. Theorem 6. IEL KA K A. Proof. follows by K A A and A KA. : 1. KA - hypothesis; 2. A KA - co-reflection; 3. KA A - from 2; 4. A - from 2 and 3; 5. A K A - co-reflection; 6. K A - from 4 and 5.
Theorem 7. The rule KA A is admissible. Theorem 8 (Disjunction Property). If IEL A B then either IEL A or IEL B. IEL has a weak disjunction property for verifications. Corollary 9. If K(A B) then either KA or KB.
In the presence of co-reflection, each of the alternatives to reflection are equivalent. In the absence of co-reflection we get the following hierarchy, from strongest to weakest. KA A (KA A) (KA A) (KA A) ( A KA) K
Logics of intuitionsitic beliefs One could imagine a verification procedure which is not necessarily factive, in which case we can speak about intuitionistic belief rather than knowledge. The corresponding logic is IEL which is IEL without the factivity axiom. IEL itself may be regarded as the logic of consistent beliefs since it is easy to show that IEL = IEL + K. The propositional modal language does not distinguish intuitionistic consistent belief and intuitionistic knowledge and we need more expressive languages to separate these two notions.
IEL findings A KA is intuitionistically valid which is a brief formal representation of the principle justified true belief yields knowledge. KA A is a distinctly classical principle of knowledge, not valid intuitionistically. A KA is the adequate form of factivity for intuitionistic knowledge. IEL codifies this view of knowledge. IEL is also the basic logic of intuitionistic beliefs.
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