In Tractatus 5.132 Wittgenstein argues that inferential justification depends solely on the understanding of the premises and conclusion, and is not mediated by any further act. On this basis he argues that Frege’s and Russell’s rules of inference are “senseless” and “superfluous”. This line of argument is puzzling, since it is unclear that there could be any viable account of inference according to which no such mediation takes place. I show that Wittgenstein’s rejection of rules of inference (...) can be motivated by taking account of his holistic construal of the relation between inference and understanding. (shrink)

In the theory of meaning, it is common to contrast truth-conditional theories of meaning with theories which identify the meaning of an expression with its use. One rather exact version of the somewhat vague use-theoretic picture is the view that the standard rules of inference determine the meanings of logical constants. Often this idea also functions as a paradigm for more general use-theoretic approaches to meaning. In particular, the idea plays a key role in the anti-realist program of Dummett (...) and his followers. In the theory of truth, a key distinction now is made between substantial theories and minimalist or deflationist views. According to the former, truth is a genuine substantial property of the truth-bearers, whereas according to the latter, truth does not have any deeper essence, but all that can be said about truth is contained in T-sentences (sentences having the form: ‘P’ is true if and only if P). There is no necessary analytic connection between the above theories of meaning and truth, but they have nevertheless some connections. Realists often favour some kind of truth-conditional theory of meaning and a substantial theory of truth (in particular, the correspondence theory). Minimalists and deflationists on truth characteristically advocate the use theory of meaning (e.g. Horwich). Semantical anti-realism (e.g. Dummett, Prawitz) forms an interesting middle case: its starting point is the use theory of meaning, but it usually accepts a substantial view on truth, namely that truth is to be equated with verifiability or warranted assertability. When truth is so understood, it is also possible to accept the idea that meaning is closely related to truth-conditions, and hence the conflict between use theories and truth-conditional theories in a sense disappears in this view. (shrink)

A certain type of inference rules in modal logics, generalizing Gabbay's Irreflexivity rule, is introduced and some general completeness results about modal logics axiomatized with such rules are proved.

This paper discusses the account of alethic modality as presented by Wilfrid Sellars in his earlier work from 1947 to 1958. Its aim is twofold. First, I discuss Sellars' analysis by exploring its historical relationship to Carnap's account of modality. I argue that Carnap's early syntactic treatment of modality profoundly influenced Sellars' own so-called ‘regulist' account of modality in terms of rules of inference. Furthermore, it is suggested that Sellars' lesser-known possible worlds analysis was influenced by Carnap's later semantic (...) account of modality. The second aim of the paper is a critical one. I raise a number of objections to Sellars’ account of modality and argue that his account confronts some serious limitations. I also argue that Sellars does not provide a satisfying explanation of the relationship between his regulist account and his possible worlds analysis. I offer such an explanation by arguing that his regulist account should be understood as a pragmatic account of what is conveyed by the use of modal sentences, whereas his possible worlds analysis is a semantic account of the asserted content of modal statements. That is, the regulist account and the possible worlds analysis have different explanatory targets. (shrink)

Elizabeth Karger has suggested an interpretation of Ockham's theory of the modes of common personal supposition ("TM") according to which the purpose of TM is to provide certain distinctions that Ockham will use in formulating a unified theory of immediate inference among certain kinds of sentences. Karger presents a single, powerful rule of inference that incorporates TM distinctions and that is meant to codify Ockham's theory of immediate inference. I raise an objection to Karger's rule, (...) thereby calling into doubt the interpretation of Ockham that is based on attributing that rule to him. (shrink)

The “Game of the Rule” is easy enough: I give you the beginning of a sequence of numbers (say) and you have to figure out how the sequence continues, to uncover the rule by means of which the sequence is generated. The game depends on two obvious constraints, namely (1) that the initial segment uniquely identify the sequence, and (2) that the sequence be non-random. As it turns out, neither constraint can fully be met, among other reasons because (...) the relevant notion of randomness is either vacuous or undecidable. This may not be a problem when we play for fun. It is, however, a serious problem when it comes to playing the game for real, i.e., when the player to issue the initial segment is not one of us but the world out there, the sequence consisting not of numbers (say) but of the events that make up our history. Moreover, when we play for fun we know exactly what initial segment to focus on, but when we play for real we don’t even know that. This is the core difficulty in the philosophy of the inductive sciences. (shrink)

It is often argued that inferential role semantics (IRS) entails semantic holism as long as theorists fail to answer the question about which inferences, among the many, are meaning-constitutive. Since analyticity, as truth in virtue of meaning, is a widely dismissed notion in indicating which inferences determine meaning, it seems that holism follows. Semantic holism is often understood as facing problems with the stability of content and many usual explanations of communication. Thus, we should choose between giving up IRS, to (...) avoid these holistic entailments, and defending holism against this charge, to rescue IRS. I try to pursue the second goal by analyzing certain patterns of counterfactual reasoning. Wilfrid Sellars and Robert Brandom claim that, to defend IRS, content-constitutive inferences are those counterfactually robust. While it is difficult to assess the goodness of such a view, it nonetheless entails that counterfactually non-robust inferences (which I call “modally ruled out inferences”) are not content-constitutive. If this is true, and if we take certain remarks about the grasp of concepts on board, there is a way to restrict the scope of the holism entailed by IRS to the extent of reshaping problems with the stability of content. (shrink)

Some believe that all arguments make an implicit “inference claim” that the conclusion is inferable from the premises (e.g., Bermejo-Luque, Grennan, the Groarkes, Hitchcock, Scriven). I try to show that this is confused. An act of arguing arises because an inference can be attributed to us, not a meta-level “inference claim” that would make the argument self-referential and regressive. I develop six (other) possible explanations of the popularity of the doctrine that similarly identify confusions.

In a series of papers over the past twenty years, and in a new book, Igor Douven has argued that Bayesians are too quick to reject versions of inference to the best explanation that cannot be accommodated within their framework. In this paper, I survey their worries and attempt to answer them using a series of pragmatic and purely epistemic arguments that I take to show that Bayes’ Rule really is the only rational way to respond to your (...) evidence. (shrink)

A background assumption of this paper is that the repertoire of inference schemes available to humanity is not fixed, but subject to change as new schemes are invented or refined and as old ones are obsolesced or abandoned. This is particularly visible in areas like health and environmental sciences, where enormous societal investment has been made in finding ways to reach more dependable conclusions. Computational modeling of argumentation, at least for the discourse in expert fields, will require the possibility (...) of modeling change in a stock of schemes that may be applied to generate conclusions from data. We examine Randomized Clinical Trial, an inference scheme established within medical science in the mid-20th Century, and show that its successful defense by means of practical reasoning allowed for its black-boxing as an inference scheme that generates (and warrants belief in) conclusions about the effects of medical treatments. Modeling the use of a scheme is well-understood; here we focus on modeling how the scheme comes to be established so that it is available for use. (shrink)

It has been common wisdom for centuries that scientific inference cannot be deductive; if it is inference at all, it must be a distinctive kind of inductive inference. According to demonstrative theories of induction, however, important scientific inferences are not inductive in the sense of requiring ampliative inference rules at all. Rather, they are deductive inferences with sufficiently strong premises. General considerations about inferences suffice to show that there is no difference in justification between an (...) class='Hi'>inference construed demonstratively or ampliatively. The inductive risk may be shouldered by premises or rules, but it cannot be shirked. Demonstrative theories of induction might, nevertheless, better describe scientific practice. And there may be good methodological reasons for constructing our inferences one way rather than the other. By exploring the limits of these possible advantages, I argue that scientific inference is neither of essence deductive nor of essence inductive. (shrink)

Lewis Carroll’s 1895 paper, 'What the Tortoise Said to Achilles' is widely regarded as a classic text in the philosophy of logic. This special issue of 'The Carrollian' publishes five newly commissioned articles by experts in the field. The original paper is reproduced, together with contemporary correspondence relating to the paper and an extensive bibliography.

Defenders of Inference to the Best Explanation claim that explanatory factors should play an important role in empirical inference. They disagree, however, about how exactly to formulate this role. In particular, they disagree about whether to formulate IBE as an inference rule for full beliefs or for degrees of belief, as well as how a rule for degrees of belief should relate to Bayesianism. In this essay I advance a new argument against non-Bayesian versions of (...) IBE. My argument focuses on cases in which we are concerned with multiple levels of explanation of some phenomenon. I show that in many such cases, following IBE as an inference rule for full beliefs leads to deductively inconsistent beliefs, and following IBE as a non-Bayesian updating rule for degrees of belief leads to probabilistically incoherent degrees of belief. (shrink)

I examine and resolve an exegetical dichotomy between two main interpretations of Peirce’s theory of abduction, namely, the Generative Interpretation and the Pursuitworthiness Interpretation. According to the former, abduction is the instinctive process of generating explanatory hypotheses through a mental faculty called insight. According to the latter, abduction is a rule-governed procedure for determining the relative pursuitworthiness of available hypotheses and adopting the worthiest one for further investigation—such as empirical tests—based on economic considerations. It is shown that the Generative (...) Interpretation is inconsistent with a fundamental fact of logic for Peirce—i.e., abduction is a kind of inference—and the Pursuitworthiness Interpretation is flawed and inconsistent with Peirce’s naturalistic explanation for the possibility of science and his view about the limitations of classical scientific method. Changing the exegetical locus classicus from the logical form of abduction to insight and economy of research, I argue for the Unified Interpretation according to which abduction includes both instinctive hypotheses-generation and rule-governed hypotheses-ranking. I show that the Unified Interpretation is immune to the objections raised successfully against the Generative and the Pursuitworthiness interpretations. (shrink)

In our thought, we employ rules of inference and belief-forming methods more generally. For instance, we (plausibly) employ deductive rules such as Modus Ponens, ampliative rules such as Inference to the Best Explanation, and perceptual methods that tell us to believe what perceptually appears to be the case. What explains our entitlement to employ these rules and methods? This chapter considers the motivations for broadly internalist answers to this question. It considers three such motivations—one based on simple cases, (...) one based on a general conception of epistemic responsibility, and one based on skeptical scenarios. The chapter argues that none of these motivations is successful. The first two motivations lead to forms of internalism—Extreme Method Internalism and Defense Internalism—that are too strong to be tenable. The third motivation motivates Mental Internalism (Mentalism), which does not fit with plausible accounts of entitlement. (shrink)

In their Introduction to Logic, Copi and Cohen suggest that students construct a formal proof by "working backwards from the conclusion by looking for some statement or statements from which it can be deduced and then trying to deduce those intermediate statements from the premises. What follows is an elaboration of this suggestion. I describe an almost mechanical procedure for determining from which statement(s) the conclusion can be deduced and the rules by which the required inferences can be made. This (...) method is designed to forestall the quandary in which many beginners find themselves: not knowing how to get started. (shrink)

We are justified in employing the rule of inference Modus Ponens (or one much like it) as basic in our reasoning. By contrast, we are not justified in employing a rule of inference that permits inferring to some difficult mathematical theorem from the relevant axioms in a single step. Such an inferential step is intuitively “too large” to count as justified. What accounts for this difference? In this paper, I canvass several possible explanations. I argue that (...) the most promising approach is to appeal to features like usefulness or indispensability to important or required cognitive projects. On the resulting view, whether an inferential step counts as large or small depends on the importance of the relevant rule of inference in our thought. (shrink)

Legal systems often rule that people own objects in their territory. We propose that an early-developing ability to make territory-based inferences of ownership helps children address informational demands presented by ownership. Across 6 experiments (N = 504), we show that these inferences develop between ages 3 and 5 and stem from two aspects of the psychology of ownership. First, we find that a basic ability to infer that people own objects in their territory is already present at age 3 (...) (Experiment 1). Children even make these inferences when the territory owner unintentionally acquired the objects and was unaware of them (Experiments 2 and 3). Second, we find that between ages 3 and 5, children come to consider past events in these judgments. They move from solely considering the current location of an object in territory-based inferences, to also considering and possibly inferring where it originated (Experiments 4 to 6). Together, these findings suggest that territory-based inferences of ownership are unlikely to be constructions of the law. Instead, they may reflect basic intuitions about ownership that operate from early in development. (shrink)

This essay advances and develops a dynamic conception of inference rules and uses it to reexamine a long-standing problem about logical inference raised by Lewis Carroll’s regress.

Many classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. Thus, (...) supervaluationists can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the consequence relation that preserves truth in a model of the non-normal modal logic _NT_. Finally, we extend our approach to a first-order setting and show that supervaluationism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented. (shrink)

The present PhD thesis is concerned with the question whether good reasoning requires that the subject has some cognitive grip on the relation between premises and conclusion. One consideration in favor of such a requirement goes as follows: In order for my belief-formation to be an instance of reasoning, and not merely a causally related sequence of beliefs, the process must be guided by my endorsement of a rule of reasoning. Therefore I must have justified beliefs about the relation (...) between my premises and my conclusion. -/- The rationality of a belief often depends on whether it is rightly connected to other beliefs, or more generally to other mental states —the states capable of providing a reason to holding the belief in question. For instance, some rational beliefs are connected to other beliefs by being inferred from them. It is often accepted that the connection implies that the subject in some sense ‘takes the mental states in question to be reason-providing’. But views on how exactly this is to be understood differ widely. They range from interpretations according to which ‘taking a mental state to be reason-providing’ imposes a mere causal sustaining relation between belief and reason-providing state to interpretations according to which one ‘takes a mental state to be reason-providing’ only if one believes that the state is reason-providing. The most common worry about the latter view is that it faces a vicious regress. In this thesis a different but in some respects similar interpretation of ‘taking something as reason-providing’ is given. It is argued to consist of a disposition to react in certain ways to information that challenges the reason-providing capacity of the allegedly reason-providing state. For instance, that one has inferred A from B partly consists in being disposed to suspend judgment about A if one obtains a reason to believe that B does not render A probable. The account is defended against regress-objections and the suspicion of explanatory circularity. (shrink)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the (...) categoricity problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning. (shrink)

In his essay ‘Transparency, Belief, Intention’, Alex Byrne (2011) argues that transparency—our ability to form beliefs about some of our intentional mental states by considering their subject matter, rather than on the basis of special psychological evidence—involves inferring ‘from world to mind’. In this reply I argue that this cannot be correct. I articulate an intuitive necessary condition for a pattern of belief to count as a rule of inference, and I show that the pattern involved in transparency (...) does not meet that condition. As a result, I conclude that transparency does not involve inference. (shrink)

Lewis Carroll’s 1895 paper “Achilles and the Tortoise” showed that we need a distinction between rules of inference and premises. We cannot, on pain of regress, treat all rules simply as further premises in an argument. But Carroll’s paper doesn’t say very much about what rules there must be. Indeed, it is consistent with what Carroll says there to think that the only rule is -elimination. You might think that modern Bayesians, who seem to think that the only (...) rule of inference they need is conditionalisation, have taken just this lesson from Carroll. But obviously nothing in Carroll’s argument rules out there being other rules as well. (shrink)

There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, $P(\textit{if } A \textit{ then } B)$, is the conditional probability of $B$ given $A$, $P(B|A)$. We identify a conditional which is such that $P(\textit{if } A \textit{ then } B)= P(B|A)$ with de Finetti's conditional event, $B|A$. An objection to making this identification in the past was that it appeared unclear how to form compounds and iterations of (...) conditional events. In this paper, we illustrate how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations. We interpret the compounds and iterations as conditional random quantities which, given some logical dependencies, may reduce to conditional events. We show how the inference to $B|A$ from $A$ and $B$ can be extended to compounds and iterations of both conditional events and biconditional events. Moreover, we determine the respective uncertainty propagation rules. Finally, we make some comments on extending our analysis to counterfactuals. (shrink)

Inferentialism is a theory in the philosophy of language which claims that the meanings of expressions are constituted by inferential roles or relations. Instead of a traditional model-theoretic semantics, it naturally lends itself to a proof-theoretic semantics, where meaning is understood in terms of inference rules with a proof system. Most work in proof-theoretic semantics has focused on logical constants, with comparatively little work on the semantics of non-logical vocabulary. Drawing on Robert Brandom’s notion of material inference and (...) Greg Restall’s bilateralist interpretation of the multiple conclusion sequent calculus, I present a proof-theoretic semantics for atomic sentences and their constituent names and predicates. The resulting system has several interesting features: (1) the rules are harmonious and stable; (2) the rules create a structure analogous to familiar model-theoretic semantics; and (3) the semantics is compositional, in that the rules for atomic sentences are determined by those for their constituent names and predicates. (shrink)

According to epistemic two-dimensionalism, every expression is associated with two kinds of meaning: a primary intension (a “Fregean” component) and a secondary intension (a “Russellian” component). While the rst kind of meaning lines up with the speaker’s abilities to pick out referents of correctly employed expressions in hypothetical scenarios, the second kind of meaning is a version of what standard semanticists call “semantic content”—a kind of content which does not pivot on speaker abilities. Despite its conciliatory temperament, epistemic two-dimensionalism has (...) come under recent attack. It has been alleged that it is bound to attribute to speakers a priori identifying knowledge of the referents of correctly employed terms, and bound also to reject valid rules of inference such as exportation: P(a) → λx[P(x)](a) (see, e.g., Soames 2005. (shrink)

I discuss a neglected solution to the skeptical problem introduced by Lewis Carroll’s “What the Tortoise Said to Achilles” (1895) in terms of a self-citational inferential license. I then consider some responses to this solution. The most significant response on behalf of the skeptic utilizes the familiar distinction between two ways of accepting a rule: as action-guiding and as a mere truth. I argue that this is ultimately unsatisfactory and conclude by opting for an alternative conception of rules as (...) representations of behavior deployed for various purposes, some theoretical and others practical. This alternative conception does not allow the skeptical problem to get off the ground. (shrink)

In this essay we collect and put together a number of ideas relevant to the under- standing of the phenomenon of creativity, confining our considerations mostly to the domain of cognitive psychology while we will, on a few occasions, hint at neuropsy- chological underpinnings as well. In this, we will mostly focus on creativity in science, since creativity in other domains of human endeavor have common links with scientific creativity while differing in numerous other specific respects. We begin by briefly (...) introducing a few basic notions relating to cognition, among which the notion of ‘concepts’ is of basic relevance. The myriads of concepts lodged in our mind constitute a ‘conceptual space’ of an enormously complex structure, where con- cepts are correlated by beliefs that are themselves made up of concepts and are as- sociated with emotions. The conceptual space, moreover, is perpetually in a state of dynamic evolution that is once again of a complex nature. A major component of the dynamic evolution is made up of incessant acts of inference, where an inference occurs essentially by means of a succession of correlations among concepts set up with beliefs and heuristics, the latter being beliefs of a special kind, namely, ones relatively free of emotional associations and possessed of a relatively greater degree of justification. Beliefs, along with heuristics, have been described as the ‘mind’s software’, and con- stitute important cognitive components of the self-linked psychological resources of an individual. The self is the psychological engine driving all our mental and physical activity, and is in a state of ceaseless dynamics resulting from one’s most intimate ex- periences of the world accumulating in the course of one’s journey through life. Many of our psychological resources are of a dual character, having both a self-linked and a shared character, the latter being held in common with larger groups of people and imbibed from cultural inputs. We focus on the privately held self-linked beliefs of an individual, since these are presumably of central relevance in making possible inductive inferences – ones in which there arises a fundamental need of adopting a choice or making a decision. Beliefs, decisions, and inferences, all have the common link to the self of an individual and, in this, are fundamentally analogous to free will, where all of these have an aspect of non-determinism inherent in them. Creativity involves a major restructuring of the conceptual space where a sustained inferential process eventually links remote conceptual domains, thereby opening up the possibility of a large number of new correlations between remote concepts by a cascading process. Since the process of inductive inference depends crucially on de- cisions at critical junctures of the inferential chain, it becomes necessary to examine the basic mechanism underlying the making of decisions. In the framework that we attempt to build up for the understanding of scientific creativity, this role of decision making in the inferential process assumes central relevance. With this background in place, we briefly sketch the affect theory of decisions. Affect is an innate system of response to perceptual inputs received either from the exter- nal world or from the internal physiological and psychological environment whereby a positive or negative valence gets associated with a perceptual input. Almost every sit- uation faced by an individual, even one experienced tacitly, i.e., without overt aware-ness, elicits an affective response from him, carrying a positive or negative valence that underlies all sorts of decision making, including ones carried out unconsciously in inferential processes. Referring to the process of inferential exploration of the conceptual space that gener- ates the possibility of correlations being established between remote conceptual do- mains, such exploration is guided and steered at every stage by the affect system, analogous to the way a complex computer program proceeds through junctures where the program ascertains whether specified conditions are met with by way of generating appropriate numerical values – for instance, the program takes different routes, depending on whether some particular numerical value turns out to be positive or negative. The valence generated by the affect system in the process of adoption of a choice plays a similar role which therefore is of crucial relevance in inferential processes, especially in the exploration of the conceptual space where remote domains need to be linked up – the affect system produces a response along a single value dimension, resembling a number with a sign and a magnitude. While the affect system plays a guiding role in the exploration of the conceptual space, the process of exploration itself consists of the establishment of correlations between concepts by means of beliefs and heuristics, the self-linked ones among the latter having a special role in making possible the inferential journey along alternative routes whenever the shared rules of inference become inadequate. A successful access to a remote conceptual domain, necessary for the creative solution of a standing problem or anomaly – one that could not be solved within the limited domain hitherto accessed – requires a phase of relatively slow cumulative search and then, at some stage, a rapid cascading process when a solution is in sight. Representing the conceptual space in the form of a complex network, the overall process can be likened to one of self-organized criticality commonly observed in the dynamical evolution of complex systems. In order that inferential access to remote domains may actually be possible, it is necessary that restrictions on the exploration process – necessary for setting the context in ordinary instances of inductive inference – be relaxed and a relatively free exploration in a larger conceptual terrain be made possible. This is achieved by the mind going into the default mode, where external constraints – ones imposed by shared beliefs and modes of exploration – are made inoperative. While explaining all these various aspects of the creative process, we underline the supremely important role that analogy plays in it. Broadly speaking, analogy is in the nature of a heuristic, establishing correlations between concepts. However, analo- gies are very special in that these are particularly effective in establishing correlations among remote concepts, since analogy works without regard to the contiguity of the concepts in the conceptual space. In establishing links between concepts, analogies have the power to light up entire terrains in the conceptual space when a rapid cas- cading of fresh correlations becomes possible. The creative process occurs within the mind of a single individual or of a few closely collaborating individuals, but is then continued by an entire epistemic community, eventually resulting in a conceptual revolution. Such conceptual revolutions make pos- sible the radical revision of scientific theories whereby the scope of an extant theory is broadened and a new theoretical framework makes its appearance. The emerging theory is characterized by a certain degree of incommensurability when compared with the earlier one – a feature that may appear strange at first sight. But incommen- surability does not mean incompatibility and the apparently contrary features of the relation between the successive theories may be traced to the multi-layered structureof the conceptual space where concepts are correlated not by means of single links but by multiple ones, thereby generating multiple layers of correlation, among which some are retained and some created afresh in a conceptual restructuring. We conclude with the observation that creativity occurs on all scales. Analogous to correlations being set up across domains in the conceptual space and new domains being generated, processes with similar features can occur within the confines of a domain where a new layer of inferential links may be generated, connecting up sub- domains. In this context, insight can be looked upon as an instance of creativity within the confines of a domain of a relatively limited extent. (shrink)

In this paper I discuss the rule of inference proposed by Kuipers under the name of Inference to the Best Theory. In particular, I argue that the rule needs to be strengthened if it is to serve realist purposes. I further describe a method for testing, and perhaps eventually justifying, a suitably strengthened version of it.

A priori theories of justification of logic based on meaning often lead to trouble, in particular to issues concerning circularity. First, I present Boghossian’s a prioriview. Boghossian maintains the rule-circular justifications from a conceptual role semantics. However, rule-circular justifications are problematic. Recently, Boghossian (Boghossian, 2015) has claimed that rules should be thought of as contents and contents as abstract objects. In this paper, I discuss Boghossian’s view. My argumentation consists of three main parts. First, I analyse several arguments (...) to show that in fact, Boghossian’s inferentialist solution is not fully satisfying. Second, I discuss the matter further, if one accepts that basic logical rules are constitutive of meaning, that is, they constitute the logical concepts and the content of a rule is an abstract object, then abstract objects — like, for example, rules — could be constitutive of meaning. The question is whether conceptual priority is in the judgment or in the object and what theory of content is pursued. Grasping content as a matter of knowing how a word or concept behaves in inferences is not completely explicative. Finally, I contend that rules come to exist as a result of certain kinds of mental action. These actions function as constitutive norms. Logical rules are not abstract objects but ideal. What one construes as norms or rules of content may involve idealization, but this is because we share a language. (shrink)

This paper critically examines currently influential transparency accounts of our knowledge of our own beliefs that say that self-ascriptions of belief typically are arrived at by “looking outward” onto the world. For example, one version of the transparency account says that one self-ascribes beliefs via an inference from a premise to the conclusion that one believes that premise. This rule of inference reliably yields accurate self-ascriptions because you cannot infer a conclusion from a premise without believing the (...) premise, and so you cannot infer from a premise that you believe the premise unless you do believe it. I argue that this procedure cannot be a source of justification, however, because one can be justified in inferring from p that q only if p amounts to strong evidence that q is true. This is incompatible with the transparency account because p often is not very strong evidence that you believe that p. For example, unless you are a weather expert, the fact that it will rain is not very strong evidence that you believe it will rain. After showing how this intuitive problem can be made precise, I conclude with a broader lesson about the nature of inferential justification: that beliefs, when justified, must be underwritten by beliefs, when justified, must be underwritten by evidential relationships between the facts or propositions which those beliefs represent. (shrink)

ABSTRACT: A traditional objection to inferentialism states that not all inferences can be meaning-constitutive and therefore inferentialism has to comprise an analytic-synthetic distinction. As a response, Peregrin argues that meaning is a matter of inferential rules and only the subset of all the valid inferences for which there is a widely shared corrective behaviour corresponds to rules and so determines meaning. Unfortunately, Peregrin does not discuss what counts as “widely shared”. In the paper, I argue for an empirical plausibility of (...) Peregrin’s proposal. The aim of the paper is to show that we can find examples of meaning-constitutive linguistic action, which sustain Peregrin’s response. The idea is supported by examples of meaning modulation. If Peregrin is right, then we should be able to find specific meaning modulations in which a new meaning is publicly available and modulated in such a way that it has a potential to be widely shared. I believe that binding modulations – a specific type of meaning modulations – satisfy this condition. (shrink)

This tutorial is for beginners wanting to learn the basics of propositional logic; the simplest of the formal systems of logic. Leslie Allan introduces students to the nature of arguments, validity, formal proofs, logical operators and rules of inference. With many examples, Allan shows how these concepts are employed through the application of three different methods for proving the formal validity of arguments.

Counterfactuals are somewhat tolerant. Had Socrates been at least six feet tall, he need not have been exactly six feet tall. He might have been a little taller—he might have been six one or six two. But while he might have been a little taller, there are limits to how tall he would have been. Had he been at least six feet tall, he would not have been more than a hundred feet tall, for example. Counterfactuals are not just tolerant, (...) then, but bounded. This paper presents a surprising paradox: If counterfactuals are tolerant and bounded, then we can prove a flat contradiction using natural rules of inference. Something has to go then. But what? (shrink)

The categoricity problem for a system of logic reveals an asymmetry between the model-theoretic and the proof-theoretic resources of that logic. In particular, it reveals prima facie that the proof-theoretic instruments are insufficient for matching the envisaged model-theory, when the latter is already available. Among the proposed solutions for solving this problem, some make use of new proof-theoretic instruments, some others introduce new model-theoretic constrains on the proof-systems, while others try to use instruments from both sides. On the proof-theoretical side, (...) two main approaches for solving the categoricity problem for propositional classical logic consist in the enforcement of the formal systems of this logic by introducing rules of inference of a new kind. A multiple-conclusions logic allows rules of inference with more than one conclusion, while a bilateralist system contains rules of inference formulated in terms of assertion and rejection. Both these approaches reveal interesting formal features of logical reasoning. This paper analyses some of the advantages and limitations of these two approaches as solutions for a full formalization of logic, i.e., for a categorical formalization. (shrink)

In the chapter “Information and Content” of their Impossible Worlds, Berto and Jago provide us with a semantic account of information in deductive reasoning such that we have an explanation for why some, but not all, logical deductions are informative. The framework Berto and Jago choose to make sense of the above-mentioned idea is a semantic interpretation of Sequent Calculus rules of inference for classical logic. I shall argue that although Berto and Jago’s idea and framework are hopeful, their (...) definitions do not capture what is intended. This is so because the definitions are solely based on the logical complexity of an argument and they fail to capture the richness of the non- logical content of that argument. Then I will suggest some amendments to address this problem. Finally, I will extend the application of the definitions to first-order logic. It shall be observed that in some classical deductions, applying contraction may lead to hiding some epistemic impossibilities. This happens when formulae which contribute different propositions to the proof get contracted at the quantified level. (shrink)

A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out (...) of the paradox. (shrink)

A system of logic usually comprises a language for which a model-theory and a proof-theory are defined. The model-theory defines the semantic notion of model-theoretic logical consequence (⊨), while the proof-theory defines the proof- theoretic notion of logical consequence (or logical derivability, ⊢). If the system in question is sound and complete, then the two notions of logical consequence are extensionally equivalent. The concept of full formalization is a more restrictive one and requires in addition the preservation of the standard (...) meanings of the logical terms in all the admissible interpretations of the logical calculus, as it is proof-theoretically defined. Although classical first-order logic is sound and complete, its standard formalizations fall short to be full formalizations since they allow non-intended interpretations. This fact poses a challenge for the logical inferentialism program, whose main tenet is that the meanings of the logical terms are uniquely determined by the formal axioms or rules of inference that govern their use in a logical calculus, i.e., logical inferentialism requires a categorical calculus. This paper is the first part of a more elaborated study which will analyze the categoricity problem from its beginning until the most recent approaches. I will first start by describing the problem of a full formalization in the general framework in which Carnap (1934/1937, 1943) formulated it for classical logic. Then, in sections IV and V, I shall discuss the way in which the mathematicians B.A. Bernstein (1932) and E.V. Huntington (1933) have previously formulated and analyzed it in algebraic terms for propositional logic and, finally, I shall discuss some critical reactions Nagel (1943), Hempel (1943), Fitch (1944), and Church (1944) formulated to these approaches. (shrink)

What accounts for how we know that certain rules of reasoning, such as reasoning by Modus Ponens, are valid? If our knowledge of validity must be based on some reasoning, then we seem to be committed to the legitimacy of rule-circular arguments for validity. This paper raises a new difficulty for the rule-circular account of our knowledge of validity. The source of the problem is that, contrary to traditional wisdom, a universal generalization cannot be inferred just on the (...) basis of reasoning about an arbitrary object. I argue in favor of a more sophisticated constraint on reasoning by universal generalization, one which undermines a rule-circular account of our knowledge of validity. (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

“What is the structure of thought?” is as central a question as any in cognitive science. A classic answer to this question has appealed to a Language of Thought (LoT). We point to emerging research from disparate branches of the field that supports the LoT hypothesis, but also uncovers diversity in LoTs across cognitive systems, stages of development, and species. Our letter formulates open research questions for cognitive science concerning the varieties of rules and representations that underwrite various LoT-based systems (...) and how these variations can help researchers taxonomize cognitive systems. (shrink)

The question of whether the Pyrrhonist adheres to certain logical principles, criteria of justification, and inference rules is of central importance for the study of Pyrrhonism. Its significance lies in that, whereas the Pyrrhonist describes his philosophical stance and argues against the Dogmatists by means of what may be considered a rational discourse, adherence to any such principles, criteria, and rules does not seem compatible with the radical character of his skepticism. Hence, if the Pyrrhonist does endorse them, one (...) must conclude that he is inconsistent in his outlook. Despite its import, the question under consideration has not received, in the vast literature on Pyrrhonism of the past three decades, all the attention it deserves. In the present paper, I do not propose to provide a full examination of the Pyrrhonist’s attitude towards rationality, but to focus on the question of whether he endorses the law of non-contradiction (LNC). However, I will also briefly tackle the question of the Pyrrhonist’s outlook on both the canons of rational justification at work in the so-called Five Modes of Agrippa and the logical rules of inference. In addition, given that the LNC is deemed a fundamental principle of rationality, determining the Pyrrhonist’s attitude towards it will allow us to understand his general attitude towards rationality. (shrink)

Although the Four Colour Theorem is passe, we give an elementary pre-formal proof that transparently illustrates why four colours suffice to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal 4-coloured planar map M. We note that such a pre-formal proof of the Four Colour Theorem highlights the significance of differentiating between: (a) Plato's knowledge as justified true belief, which seeks a formal proof in a first-order mathematical language in order (...) to justify a belief as true; and (b) Piccinini's knowledge as factually grounded belief, which seeks a pre-formal proof, in Pantsar's sense, in order to justify the axioms and rules of inference of a first-order mathematical language which can, then, formally prove the belief as justifiably true under a well-defined interpretation of the language. (shrink)

In recent years, the e ffort to formalize erotetic inferences (i.e., inferences to and from questions) has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and regular erotetic implication. While an attempt has been made to (...) axiomatize the former in a sequent system, there is currently no proof theory for the latter. Moreover, the extant axiomatization of erotetic evocation fails to capture its defeasible character and provides no rules for introducing or eliminating question-forming operators. In contrast, our calculus encodes defeasibility conditions on sequents and provides rules governing the introduction and elimination of erotetic formulas. We demonstrate that an elimination theorem holds for a version of the cut rule that applies to both declarative and erotetic formulas and that the rules for the axiomatic account of question evocation in IEL are admissible in our system. (shrink)

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined (...) in terms of a \-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints. (shrink)

This essay presents a point of view for looking at `free will', with the purpose of interpreting where exactly the freedom lies. For, freedom is what we mean by it. It compares the exercise of free will with the making of inferences, which usually is predominantly inductive in nature. The making of inference and the exercise of free will, both draw upon psychological resources that define our ‘selves’. I examine the constitution of the self of an individual, especially the (...) involvement of personal beliefs, personal memories, affects, emotions, and the hugely important psychological value-system, all of which distinguish the self of one individual from that of another. The foundational position that adopted in this essay is that all psychological processes are correlated with corresponding ones involving large scale neural aggregates in the brain, communicating with one another through wavelike modes of excitation and de-excitation. Of central relevance is the value-network around which the affect system is organized, the latter, in turn, being the axis around which is assembled the self, with all its emotional correlates. The self is a complex system. I include a brief outline of what complexity consists of. In reality all systems are complex, for complexity is ubiquitous, and certain parts of nature appear to us to be ‘simple’ only in certain specific contexts. It is in this background that the issue of determinism is viewed in this essay. Instead of looking at determinism as a grand principle entrenched in nature independent of our interpretation of it, I look at our ability to explain and to predict events and phenomena around us, which is made possible by the existence of causal links running in a complex course that set up correlations between diverse parts of nature, in this way putting the stamp of necessity on these events. However, the complexity of systems limits our ability to explain and to predict to within certain horizons defined by contexts. Our ability to explain and predict in matters relating to acts of free will is similarly limited by the operations of the self that remain hidden from our own awareness. The aspects of necessity and determinism appear to us in the form of reason and rationality that explain and predict only within a limited horizon, while the rest depends on the complex operation of self-linked psychological resources, where the latter appear as contingent in the context of the exercise of free will. The hallmark of complex systems is the existence of amplifying factors that operate as destabilizing ones, along with inhibiting or stabilizing factors as well that generally limit the destabilizing influences to local occurrences, while preserving the global integrity of a system. This complex interplay of destabilizing and stabilizing influences lead to the possibility of an enormous number of distinct modes of behavior that appear as emergent phenomena in complex systems. Looking at the particular case of the self of an individual that guide her actions and thoughts, it is the operation of our emotions, built around the psychological value-system, that provide for the amplifying and inhibiting factors mentioned above. The operation of these self-linked factors stamps our actions and thoughts as contingent ones that do not fit with our concepts of reason and rationality. And this is what provides the basis of our idea of free will. Free will is not ‘free’ in virtue of exemption from causal links running through all our self-based processes - ones that remain hidden from our awareness, but is free of what is perceived to be ‘reason and rationality’ based on knowledge and the common pool of beliefs and principles associated with a shared world-view. When we speak of the choice involved in an act of exercise of free will what we actually refer to is the commonly observed fact that various different individuals of a similar disposition respond differently when placed under similar circumstances. In other words, free will is ‘free’ precisely because it is not subject to constraints of a commonly accepted and shared set of principles, beliefs, and values. While it is possible that, in an exercise of free will, the self-linked psychological resources are brought into action when a number of alternatives are presented to the self by the operation of the ingredients of a commonly shared world-view, it seems likely that, that set of alternatives is not of primary relevance in the final response that the mind produces, since the latter is primarily a product of the operation of the self-based psychological resources. What is of greater relevance here is the operation of emotion-driven processes, guided by the psychological value-system based on the activity of the so-called reward-punishment network in the brain. These processes lead the individual to a response that appears to be free from the shackles of determinism precisely because their mechanisms, which are hidden from us, do not conform to commonly accepted and shared rules and principles. In contrast, inductive inference is a process that is based on the ‘cognitive face’ of self, where the self-based psychological resources play a supportive role to the commonly shared world-view of an individual. There is never any freedom from the all-pervading causal links representing correlations among all objects, entities, and events in nature. In the midst of all this, the closest thing to freedom that we can have in our life comes with self-examination and self-improvement. The possibility of self-examination appears in the form of specific conjunctions between our complex self-processes and the ceaseless changes of scenario in our external world. This actually makes the emergent phenomenon of self-examination a matter of chance, but one that keeps on appearing again and again in our life. Once realized, self examination creates possibilities that would not be there in the absence of it, and these possibilities include the enhancement of further self-enrichment and further diversity in the exercise of our free will. (shrink)

Research into the neural correlates of consciousness (NCC) aims to identify not just those brain areas that are NCC, but also those that are not. In the received method for ruling out a brain area from being an NCC, this is accomplished by showing a brain area’s content to be consistently absent from subjects’ reports about what they are experiencing. This paper points out how this same absence can be used to infer that the brain area’s content is cognitively inaccessible, (...) in which case we would expect its content to be absent from subjects’ reports whether its content is (phenomenally) conscious or not. If so, such reports cannot count as evidence against that brain area being an NCC, and the received method fails. An alternative method (one suggested in Block, 2007) is considered. (shrink)