Logical Omnipotence and Two notions of Implicit Belief

The most widespread models of rational reasoners (the model based on modal epistemic logic and the model based on probability theory) exhibit the problem of logical omniscience. The most common strategy for avoiding this problem is to interpret the models as describing the explicit beliefs of an ideal reasoner, but only the implicit beliefs of a real reasoner. I argue that this strategy faces serious normative issues. In this paper, I present the more fundamental problem of logical omnipotence, which highlights the normative content of the problem of logical omniscience. I introduce two developments of the notion of implicit belief (accessible and stable belief ) and use them in two versions of the most common strategy applied to the problem of logical omnipotence

Dynamic Epistemic Logic and Logical Omniscience

Epistemic logics based on the possible worlds semantics suffer from the problem of logical omniscience, whereby agents are described as knowing all logical consequences of what they know, including all tautologies. This problem is doubly challenging: on the one hand, agents should be treated as logically non-omniscient, and on the other hand, as moderately logically competent. Many responses to logical omniscience fail to meet this double challenge because the concepts of knowledge and reasoning are not properly separated. In this paper, I present a dynamic logic of knowledge that models an agent’s epistemic state as it evolves over the course of reasoning. I show that the logic does not sacrifice logical competence on the altar of logical non- omniscience

Reasoning about Rational, but not Logically Omniscient Agents

We propose in the paper a new solution to the so-called Logical Omniscience Problem of epistemic logic. Almost all attempts in the literature to solve this problem consist in weakening the standard epistemic systems: weaker systems are considered where the agents do not possess the full reasoning capacities of ideal reasoners. We shall argue that this solution is not satisfactory: in this way omniscience can be avoided, but many intuitions about the concepts of knowledge and belief get lost. We shall show that axioms for epistemic logics must have the following form: if the agent knows all premises of a valid inference rule, and if she thinks hard enough, then she will know the conclusion. To formalize such an idea, we propose to \dynamize' epistemic logic, that is, to introduce a dynamic component into the language. We develop a logic based on this idea and show that it is suitable for formalizing the notion of actual, or explicit knowledge

No Rationality Through Brute-Force

All reasoners described in the most widespread models of a rational reasoner exhibit logical omniscience, which is impossible for finite reasoners (real reasoners). The most common strategy for dealing with the problem of logical omniscience is to interpret the models using a notion of beliefs different from explicit beliefs. For example, the models could be interpreted as describing the beliefs that the reasoner would hold if the reasoner were able reason indefinitely (stable beliefs). Then the models would describe maximum rationality, which a finite reasoner can only approach in the limit of a reasoning sequence. This strategy has important consequences for epistemology. If a finite reasoner can only approach maximum rationality in the limit of a reasoning sequence, then the efficiency of reasoning is epistemically (and not only pragmatically) relevant. In this paper, I present an argument to this conclusion and discuss its consequences, as, for example, the vindication of the principle 'no rationality through brute-force'

Providence and Mystery: from Open Theism to New Approaches

In the recent debate on Christian theism, the position called Open Theism (OT) tries to solve the dilemma of omniscience and human freedom. In OT, the key word of the human-divine relationship is "risk": in his relationship with us, God is a risk-taker in that he adapts his plan to human decisions and to the situations that arise from them. "Risk" is the fundamental characteristic of any true love relationship. According to OT, God has no exhaustive knowledge of how humans will use their will, and the divine plan for this world is not seen as fixed for eternity. OT distinguishes between meticulous providence and general providence and denies that the former can exist. After illustrating these positions and a particular view of OT called essential kenosis, I highlight some of their weaknesses and conclude by asking whether the concept of mystery (at least in some of its possible interpretations: I outline four "solutions") can enable a reconciliation between classical theism and OT. By applying an approach to the notion of mystery usually connected to the Trinity, I show that the dilemma of omniscience, human freedom and providence does not compromise the plausibility of theism

Providence and Mystery: from Open Theism to New Approaches

In the recent debate on Christian theism, the position called Open Theism (OT) tries to solve the dilemma of omniscience and human freedom. In OT, the key word of the human-divine relationship is "risk": in his relationship with us, God is a risk-taker in that he adapts his plan to human decisions and to the situations that arise from them. "Risk" is the fundamental characteristic of any true love relationship. According to OT, God has no exhaustive knowledge of how humans will use their will, and the divine plan for this world is not seen as fixed for eternity. OT distinguishes between meticulous providence and general providence and denies that the former can exist. After illustrating these positions and a particular view of OT called essential kenosis, I highlight some of their weaknesses and conclude by asking whether the concept of mystery (at least in some of its possible interpretations: I outline four "solutions") can enable a reconciliation between classical theism and OT. By applying an approach to the notion of mystery usually connected to the Trinity, I show that the dilemma of omniscience, human freedom and providence does not compromise the plausibility of theism

On the epistemic foundations of agent theories

We argue that none of the existing epistemic logics can adequately serve the needs of agent theories. We suggest a new concept of knowledge which generalizes both implicit and explicit knowledge and argue that this is the notion we need to formalize agents in Distributed Artificial Intelligence. A logic of the new concept is developed which is formally and practically adequate in the following sense: first, it does not suffer from any kind of logical omniscience. Second, it can account for the intuition that agents are rational, though not hyper-rational. Third, it is expressive enough. The advantages of the new logic over other formalisms is demonstrated by showing that none of the existing systems can fulfill all these requirements simultaneously

Ideal Reasoners don’t Believe in Zombies

The negative zombie argument concludes that physicalism is false from the premises that p ∧¬q is ideally negatively conceivable and that what is ideally negatively conceivable is possible, where p is the conjunction of the fundamental physical truths and laws and q is a phenomenal truth (Chalmers 2002; 2010). A sentence φ is ideally negatively conceivable iff φ is not ruled out a priori on ideal rational reflection. In this paper, I argue that the negative zombie argument is neither a priori nor conclusive. First, I argue that the premises of the argument are true only if there exists an adequate finite ideal reasoner R that believes ◊(p ∧ ¬q) on the basis of not believing p→q on a priori basis. Roughly, a finite reasoner is a reasoner with cognitive limitations (e.g. finite memory). I argue that R is finite only if R reasons nonmonotonically and only approach ideal reflection at the limit of a reasoning sequence. This would render the argument nonconclusive. Finally, I argue that, for some q, R does not believe ◊(p ∧ ¬q) on the basis of not believing p→q on a priori basis (e.g. for q =‘something is conscious’). This would render the choice of an adequate q dependent on empirical information (and the argument a posteriori). I conclude that the negative zombie argument (and, maybe, all zombie arguments) is neither a priori nor conclusive