Logical Omniscience

I want to model a finite, fallible cognitive agent who imagines that p in the sense of mentally representing a scenario – a configuration of objects and properties – correctly described by p. I propose to capture imagination, so understood, via variably strict world quantifiers, in a modal framework including both possible and so-called impossible worlds. The latter secure lack of classical logical closure for the relevant mental states, while the variability of strictness captures how the agent imports information from actuality in the imagined non-actual scenarios. Imagination turns out to be highly hyperintensional, but not logically anarchic.
§1 sets the stage and impossible worlds are quickly introduced in §2. §3 proposes to model imagination via variably strict world quantifiers. §4 introduces the formal semantics. §5 argues that imagination has a minimal mereological structure validating some logical inferences. §6 deals with how imagination under-determines the represented contents. §7 proposes additional constraints on the semantics, validating further inferences. §8 describes some welcome invalidities. §9 examines the effects of importing false beliefs into the imag- ined scenarios. Finally, §10 hints at possible developments of the theory in the direction of two-dimensional semantics.

abstRaCt: En este trabajo abordaremos el problema del determinismo en la acción humana y de su relación con la omnisciencia (divina o natural). Argumentare-mos sobre la inextricable conexión que existe entre ambos conceptos y el papel que desempeñan en nuestra comprensión de la libertad. Si, como intentaremos mostrar, la omnisciencia implica determinismo, no hay espacio para la libertad en su sentido más profundo. Semejante constatación nos obliga a redefinir la noción de libertad para que resulte compatible con las leyes del universo y con las leyes del pensamiento.

In this paper, I investigate whether we can use a world-involving framework to model the epistemic states of non-ideal agents. The standard possible-world framework falters in this respect because of a commitment to logical omniscience. A familiar attempt to overcome this problem centers around the use of impossible worlds where the truths of logic can be false. As we shall see, if we admit impossible worlds where "anything goes'' in modal space, it is easy to model extremely non-ideal agents that are incapable of performing even the most elementary logical deductions. A much harder, and considerably less investigated challenge is to ensure that the resulting modal space can also be used to model moderately ideal agents that are not logically omniscient but nevertheless logically competent. Intuitively, while such agents may fail to rule out subtly impossible worlds that verify complex logical falsehoods, they are nevertheless able to rule out blatantly impossible worlds that verify obvious logical falsehoods. To model moderately ideal agents, I argue, the job is to construct a modal space that contains only possible and non-trivially impossible worlds where it is not the case that "anything goes''. But I prove that it is impossible to develop an impossible-world framework that can do this job and that satisfies certain standard conditions. Effectively, I show that attempts to model moderately ideal agents in a world-involving framework collapse to modeling either logical omniscient agents, or extremely non-ideal agents.

The purpose of this paper is to consider the explanatory resources that Robert Brandom‟s distinction between acknowledged and consequential commitments affords in relation to the problem of logical omniscience. With this distinction the importance of the doxastic perspective under consideration for the relationship between logic and norms of reasoning is emphasized, and it becomes possible to handle a number of problematic cases discussed in the literature without thereby incurring a commitment to revisionism about logic.

Formal epistemologists often claim that our credences should be representable by a probability function. Complete probabilistic coherence, however, is only possible for ideal agents, raising the question of how this requirement relates to our everyday judgments concerning rationality. One possible answer is that being rational is a contextual matter, that the standards for rationality change along with the situation. Just like who counts as tall changes depending on whether we are considering toddlers or basketball players, perhaps what counts as rational shifts according to whether we are considering ideal agents or creatures more like ourselves. Even though a number of formal epistemologists have endorsed this type of solution, I will argue that there is no way to spell out this contextual account that can make sense of our everyday judgments about rationality. Those who defend probabilistic coherence requirements will need an alternative account of the relationship between real and ideal rationality.

According to certain normative theories in epistemology, rational-ity requires us to be logically omniscient. Yet this prescription clashes with our ordinary judgments of rationality. How should we resolve this tension? In this paper, I focus particularly on the logical omni-science requirement in Bayesian epistemology. Building on a key insight by Ian Hacking (1967), I develop a version of Bayesianism that permits logical ignorance. This includes: (i) an account of the syn-chronic norms that govern a logically ignorant individual at any given time; (ii) an account of how we reduce our logical ignorance by learning logical facts and how we should update our credences in response to such evidence; and (iii) an account of when logical ignorance is irrational and when it isn't. At the end, I explain why the requirement of logical omniscience remains true of ideal agents with no computational, processing, or storage limitations.

We propose a dynamic hyperintensional logic of belief revision for non-omniscient agents, reducing the logical omniscience phenomena affecting standard doxastic/epistemic logic as well as AGM belief revision theory. Our agents don't know all a priori truths; their belief states are not closed under classical logical consequence; and their belief update policies are such that logically or necessarily equivalent contents can lead to different revisions. We model both plain and conditional belief, then focus on dynamic belief revision. The key idea we exploit to achieve non-omniscience focuses on topic- or subject matter-sensitivity: a feature of belief states which is gaining growing attention in the recent literature.

In this paper I discuss Hintikka's system of epistemic and doxastic logic, challenging his idea of constistency, revised as defensibility, by means of Kripke's famous 'A puzzle about belief'. In this way, I consider the possibility for a rational agent to bear contradictory beliefs, given that the agent lacks some empirical information about some identity relations. Doing so, I try to show that, even if Hintikka's logic can escape the problem of logical omniscience, in order not to derive contradictory beliefs for the agent we should attribute him a complete empirical knowledge. Otherwise, we should postulate a perfect natural language in which either a proper name needs no translation or two expressions show 'by themselves' that they share the same referent.

An organizing theme of the dissertation is the issue of how to make philosophical theories useful for scientific purposes. An argument for the contention is presented that it doesn’t suffice merely to theoretically motivate one’s theories, and make them compatible with existing data, but that philosophers having this aim should ideally contribute to identifying unique and hard to vary predictions of their theories.
This methodological recommendation is applied to the ranking-theoretic approach to conditionals, which emphasizes the epistemic relevance and the expression of reason relations as part of the semantics of the natural language conditional. As a first step, this approach is theoretically motivated in a comparative discussion of other alternatives in psychology of reasoning, like the suppositional theory of conditionals, and novel approaches to the problems of compositionality and accounting for the objective purport of indicative conditionals are presented.
In a second step, a formal model is formulated, which allows us to derive quantitative predictions from the ranking-theoretic approach, and it is investigated which novel avenues of empirical research that this model opens up for.
Finally, a treatment is given of the problem of logical omniscience as it concerns the issue of whether ranking theory (and other similar approaches) makes too idealized assumptions about rationality to allow for interesting applications in psychology of reasoning. Building on the work of Robert Brandom, a novel solution to this problem is presented, which both opens up for new perspectives in psychology of reasoning and appears to be capable of satisfying a range of constraints on bridge principles between logic and norms of reasoning, which would otherwise stand in a tension.

World semantics for relevant logics include so-called non-normal or impossible worlds providing model-theoretic counterexamples to such irrelevant entailments as (A ∧ ¬A) → B, A → (B∨¬B), or A → (B → B). Some well-known views interpret non-normal worlds as information states. If so, they can plausibly model our ability of conceiving or representing logical impossibilities. The phenomenon is explored by combining a formal setting with philosophical discussion. I take Priest’s basic relevant logic N4 and extend it, on the syntactic side, with a representation operator, (R), and on the semantic side, with particularly anarchic non-normal worlds. This combination easily invalidates unwelcome “logical omniscience” inferences of standard epistemic logic, such as belief-consistency and closure under entailment. Some open questions are then raised on the best strategies to regiment (R) in order to express more vertebrate kinds of conceivability.

DRAFT The most widespread model of a rational reasoner (the model based in modal epistemic logic) is inadequate for describing nite reasoners (e.g. humans). The reasoners described in the model exhibit logical omniscience and nite reasoners cannot be logically omniscient. This inadequacy is the problem of logical omniscience. The most common strategy for avoiding the problem of logical omniscience is to interpret the model using a notion of beliefs dierent from explicit beliefs (e.g. implicit beliefs). In this paper, I survey two versions of this strategy: one using accessible beliefs; another using stable beliefs. Both strategies avoid the problem in classical settings, but they dier in an essential feature: while the second considers the reasoning sequence that would result in the relevant beliefs, the rst considers only whether this reasoning is possible (in principle). This dierence has consequences in nonmonotonic settings, where the strategy based in accessible beliefs exhibits the problem of logical omnipotence: the reasoners described in the model may have inferential capacities higher than any nite reasoner, but the model is insensitive to that.

The notion of an ideal reasoner has several uses in epistemology. Often, ideal reasoners are used as a parameter of (maximum) rationality for finite reasoners (e.g. humans). However, the notion of an ideal reasoner is normally construed in such a high degree of idealization (e.g. infinite/unbounded memory) that this use is unadvised. In this
dissertation, I investigate the conditions under which an ideal reasoner may be used as a parameter of rationality for finite reasoners. In addition, I present and justify the research program of computational epistemology, which investigates the parameter of maximum
rationality for finite reasoners using computer simulations.

In chapter 1, I investigate the use ideal reasoners for stating the maximum (and minimum) bounds of rationality for finite reasoners. I propose the notion of a strictly ideal reasoner which coincides with the notion of maximum rationality. The notion of a strictly ideal reasoner is relative to a logic and a notion of beliefs (explicit, implicit, etc). I argue that, for some relevant logics, a finite reasoner may only approach maximum rationality at the limit of a reasoning sequence (stable beliefs) 1 . In chapter 2, I investigate the use of ideal reasoners in the zombie argument against physicalism (Chalmers, 2010). This notion is used in the principle that ideal negative conceivability entails possibility.
The conclusion is that the zombie argument is neither an a priori nor a conclusive argument against physicalism. In chapter 3, I investigate the notion of maximum (and minimum) epistemic rationality for finite reasoners. Epistemic rationality is often related to maximizing true beliefs and minimizing false beliefs. I argue that most of the existing models of maximum epistemic rationality have problems in dealing with blindspots and propose a model in terms of the maximization of a function g, which evaluates sets of beliefs regarding truth/falsehood. However, function g may only be maximized at the limit of a reasoning sequence. In chapter 4, I argue that if maximum (epistemic) rationality for finite reasoners must be understood in terms of the limit of a reasoning sequence, then issues about the computational complexity of reasoning are relevant to epistemology. Then I propose the research program of computational epistemology, which uses computer simulations for investigating maximum (epistemic) rationality for
finite reasoners and considers the computational complexity of reasoning. In chapter 5, I provide an example of an investigation in computational epistemology. More specifically, I compare two models of maximum rationality for situations of uncertain reasoning: theory of defeasible reasoning (Pollock, 1995) and Bayesian epistemology (Joyce, 2011).

with this paper, we intend to clarify some of the central notions that are commonly used in contemporary developments of Epistemic Logic, which lack a proper theoretical foundation. We want to follow the steps of some prominent epistemologists and epistemic logicians, who advocate for a correlation between their respective fields of study. We will proceed with a first small step that will consist in adapting one contemporary version of Epistemic Internalism to the framework of Awareness Logic, such that the key concepts in this logic can be sustained by an epistemological view, which, in turn, can work as a theoretical foundation for Awareness Logic.

(pre-publication version) The main goal of this paper is to investigate what explanatory resources Robert Brandom’s distinction between acknowledged and consequential commitments affords in relation to the problem of logical omniscience. With this distinction the importance of the doxastic perspective under consideration for the relationship between logic and norms of reasoning is emphasized, and it becomes possible to handle a number of problematic cases discussed in the literature without thereby incurring a commitment to revisionism about logic. One such case in particular is the preface paradox, which will receive an extensive treatment. As we shall see, the problem of logical omniscience not only arises within theories based on deductive logic; but also within the recent paradigm shift in psychology of reasoning. So dealing with this problem is important not only for philosophical purposes but also from a psychological perspective.

All reasoners described in the most widespread models of a rational reasoner exhibit
logical omniscience, what 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 model could be interpreted as describing the beliefs that the reasoner would hold if the reasoner were able reason indefinitely (stable beliefs). Then the model would describe maximum rationality, what a finite reasoner can only approach in the limit of a reasoning sequence. This strategy has important consequences to 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 section 1, I present
an argument to this conclusion. In section 2, I discuss the consequences of this conclusion, as, for example, the vindication of the principle 'no rationality through brute-force'.

Questo lavoro nasce dalla lettura di un libro di Nicla Vassallo1 e da un fraintendimento. In un capitolo di Non annegare, di cui sono debitore, si affrontava il tema dello statuto epistemico dell’inconscio sollevando in poche righe molti dei problemi che si sono tentati di affrontare nelle prossime pagine.
Il fraintendimento è invece una pura fantasticheria, nasce dal provare a leggere il termine inconscio i maniera innaturalmente letterale. In un senso freudiano non sono cosciente di ciò che ho rimosso ma in realtà so, tuttavia in un senso più lato non sono conscio nemmeno di tutto quello che non so o non credo; perché non ne ho mai sentito parlare, ne credo il contrario o perché è impossibile. L’esercizio dunque è non immaginare più l’inconscio come la soffitta della nostra mente, quanto piuttosto tutto ciò che ne è al di fuori.
A partire da ciò si è tentato di confrontare tutto ciò che avevamo ottenuto; nel primo capitolo ci si è domandati quale sia il rapporto tra la conoscenza della conoscenza e la conoscenza del fatto di ignorare e come ciò si possa declinare nel caso delle nostre credenze. Si sono esplorati nei successivi due capitoli tutti i modi in cui si può immaginare di credere qualcosa senza credere di crederla, ossia tutti i modi in cui si possa credere implicitamente. Infine negli ultimi due capitoli si è tentata una collocazione del catalogo appena ottenuto dal punto di vista logico e filosofico.

In this paper we design a new logical system to explicitly model the different deductive reasoning steps of a boundedly rational agent. We present an adequate system in line with experimental findings about an agent's reasoning limitations and the cognitive effort that is involved. Inspired by Dynamic Epistemic Logic, we work with dynamic operators denoting explicit applications of inference rules in our logical language. Our models are supplemented by (a) impossible worlds (not closed under logical consequence), suitably structured according to the effect of inference rules, and (b) quantitative components capturing the agent's cognitive capacity and the cognitive costs of rules with respect to certain resources (e.g. memory, time). These ingredients allow us to avoid problematic logical closure principles, while at the same time deductive reasoning is reflected in our dynamic truth clauses. We finally show that our models can be reduced to awareness-like plausibility structures that validate the same formulas and a sound and complete axiomatization is given with respect to them.

Beliefs have been formally modelled in the last decades using doxastic logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but they seem to commit us to model agents that are logically omniscient (they believe every classical tautology) and perfect reasoners (their beliefs are closed under classical deductive closure). Thus, this model does not seem to be appropriate to model non-ideal agents, that have resource limitations that can prevent them from attaining such ...