The thesis that probabilities of conditionals are conditional probabilities has putatively been refuted many times by so-called ‘triviality results’, although it has also enjoyed a number of resurrections. In this paper I assault it yet again with a new such result. I begin by motivating the thesis and discussing some of the philosophical ramifications of its fluctuating fortunes. I will canvas various reasons, old and new, why the thesis seems plausible, and why we should care about its fate. I (...) will look at some objections to Lewis’s famous triviality results, and thus some reasons for the pursuit of further triviality results. I will generalize Lewis’s results in ways that meet the objections. I will conclude with some reflections on the demise of the thesis—or otherwise. (shrink)

This paper investigates the link between informational consequence and credence. I first suggest a natural constraint, namely that informational consequence should preserve certainty: on any rational credence distribution, when the premises of an informational inferences have credence 1, the conclusion also has credence 1. Then I show that the certainty‐preserving constraint leads to triviality. In particular, the following three claims are incompatible: (i) informational consequence is extensionally distinct from classical consequence; (ii) informational inferences preserve certainty; (iii) credences obey (a (...) subset of) classical Bayesian constraints. The proof is straightforward, but the theoretical implications are substantial. Informational theorists need to either give up the idea that credence applies to epistemic discourse, or develop a nonclassical theory of credence and credal update. Moreover, the result also suggests that there is a connection between informational consequence and triviality results, including classical triviality results like Lewis's. (shrink)

Inquiry into the meaning of logical terms in natural language (‘and’, ‘or’, ‘not’, ‘if’) has generally proceeded along two dimensions. On the one hand, semantic theories aim to predict native speaker intuitions about the natural language sentences involving those logical terms. On the other hand, logical theories explore the formal properties of the translations of those terms into formal languages. Sometimes, these two lines of inquiry appear to be in tension: for instance, our best logical investigation into conditional connectives may (...) show that there is no conditional operator that has all the properties native speaker intuitions suggest if has. Indicative conditionals have famously been the source of one such tension, ever since the triviality proofs of both Lewis (1976) and Gibbard (1981) established conclusions which are in prima facie tension with ordinary judgments about natural language indicative conditionals. In a recent series of papers, Branden Fitelson has strengthened both triviality results (Fitelson 2013, 2015, 2016), revealing a common culprit: a logical schema known as IMPORT-EXPORT. Fitelson’s results focus the tension between the logical results and ordinary judgments, since IMPORT-EXPORT seems to be supported by intuitions about natural language. In this paper, we argue that the intuitions which have been taken to support IMPORT-EXPORT are really evidence for a closely related, but subtly different, principle. We show that the two principles are independent by showing how, given a standard assumption about the conditional operator in the formal language in which IMPORT-EXPORT is stated, many existing theories of indicative conditionals validate one, but not the other. Moreover, we argue that once we clearly distinguish these principles, we can use propositional anaphora to show that IMPORT-EXPORT is in fact not valid for natural language indicative conditionals (given this assumption about the formal conditional operator). This gives us a principled and independently motivated way of rejecting a crucial premise in many triviality results, while still making sense of the speaker intuitions which appeared to motivate that premise. We suggest that this strategy has broad application and an important lesson: in theorizing about the logic of natural language, we must pay careful attention to the translation between the formal languages in which logical results are typically proved, and natural languages which are the subject matter of semantic theory. (shrink)

A pervasive and influential argument appeals to trivial truths to demonstrate that the aim of inquiry is not the acquisition of truth. But the argument fails, for it neglects to distinguish between the complexity of the sentence used to express a truth and the complexity of the truth expressed by a sentence.

A triviality result for what Lewis called “the Desire by Necessity Thesis” and Broome : 265–267, 1991) called “the Desire as Expectation Thesis” is presented. The result shows that this thesis and three other reasonable conditions can be jointly satisfied only in trivial cases. Some meta-ethical implications of the result are discussed. The discussion also highlights several issues regarding Lewis ’ original triviality result for “the Desire as Belief Thesis” that have not been properly understood in the literature.

The strongest possible Lewisian triviality result for the indicative conditional is proven.

According to physicalism, everything is physical, namely there are no entities (or no more restricted sorts of entities) that are not physical. In this paper, I shall examine the truth of this thesis by presenting a triviality objection against physicalism that is somehow similar to the one advanced against presentism. Firstly, I shall distinguish between two different definitions of the physical (roughly, every entity is physical-1 iff it has some feature F, such as impenetrability or exact spatio-temporal location, while (...) every entity is physical-2 iff it is accepted by some ideal, true and complete physical theory) and between unrestricted and restricted versions of physicalism (according to the former ones, physicalism is true for every entity while, according to the latter ones, it is true only with regard to some restricted domain of entities). Secondly, I shall argue that physicalists have to deal with six different problems: the triviality of some versions of physicalism, the content-indeterminacy of the physical and the justification of the “faith” according to which we will formulate some ideal, true and complete physical theory (given the definition of the physical-2), the restricted domain problem (so that restricted versions of physicalism seem not to exclude the existence of seemingly non-physical entities), the (possible and plausible) incompatibility between the two different definitions of the physical, the extension of the physical investigation problem. (shrink)

Suppose that people in the affluent countries can easily save the lives of the starving needy in poor countries. Then, three points seem to follow. First, it is wrong for these people not to make the easy rescue . Second, it is wrong to stop making the easy rescue even if they have made many rescues already . Third, if we accept the first two points, the demands of morality are super-extreme. That is, people have to keep making trivial sacrifices (...) until there is no more trivial sacrifices to make . Here, predictably, our commonsense will resist the third point. However, I argue in this paper that, since the first two points are harder to refute, we have to accept the third point anyway. (shrink)

In this paper we explain that the paraconsistent logic LP promoted by Graham Priest can only be supported by trivial dialetheists, i.e., those who believe that all sentences are dialetheias.

I present two Triviality results for Kratzer's standard “restrictor” analysis of indicative conditionals. I both refine and undermine the common claim that problems of Triviality do not arise for Kratzer conditionals since they are not strictly conditionals at all.

We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T. We show that while being a trivial geometric theory, inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, the NIP and the NTP2. In particular, we show that T is strongly minimal, supersimple of SU‐rank 1, has the NIP or the NTP2 exactly when has these properties. We show that (...) if T is superrosy of thorn rank 1, then so is, and that the converse holds if T satisfies acl = dcl. (shrink)

I defend a new account of constitutive essence on which an entity’s constitutively essential properties are its most fundamental, nontrivial necessary properties. I argue that this account accommodates the Finean counterexamples to classic modalism about essence, provides an independently plausible account of constitutive essence, and does not run into clear counterexamples. I conclude that this theory provides a promising way forward for attempts to produce an adequate nonprimitivist, modalist account of essence. As both triviality and fundamentality in the account (...) are understood in terms of grounding, the theory also potentially has important implications for the relation between essence and grounding. (shrink)

In recent years, a number of theorists have claimed that beliefs about probability are transparent. To believe probably p is simply to have a high credence that p. In this paper, I prove a variety of triviality results for theses like the above. I show that such claims are inconsistent with the thesis that probabilistic modal sentences have propositions or sets of worlds as their meaning. Then I consider the extent to which a dynamic semantics for probabilistic modals can (...) capture theses connecting belief, certainty, credence, and probability. I show that although a dynamic semantics for probabilistic modals does allow one to validate such theses, it can only do so at a cost. I prove that such theses can only be valid if probabilistic modals do not satisfy the axioms of the probability calculus. (shrink)

The purpose of this paper is to reassess some mathematical examples in Plato’s dialogues which at a first glance may appear to be nothing more than trivial puzzles. In order to provide the necessary background for this analysis, I shall begin by sketching a brief overview of Plato’s mathematical passages and discuss the criteria for aptly selecting them. Second, I shall explain what I mean by ‘mathematical examples,’ and reflect on their function in light of the discussion on παραδείγματα outlined (...) in the Sophist and the Statesman. Against this framework, I shall move to a close examination of specific examples drawn from the Theaetetus (154c–55d), the Republic (523c–24d), and the Phaedo (96d–97b, 100e–101d). By placing these examples in the broader context of pre-Euclidean mathematics, I shall show that their mathematical content is often less elementary than might appear at first sight. Moreover, by placing emphasis on the specific philosophical concerns that motivated their introduction, I shall argue that the examples are not merely nonsensical jokes. Even if their illustrative purpose is not always immediately clear, and even if they can sound playful and bizarre, they in fact fulfil an important function: by virtue of their power to generate wonder or confusion, they serve as exercises and act as a trigger with respect to the difficult philosophical issues they are intended to clarify. (shrink)

After indicating why this is needed, the paper proves a non-triviality result for paraconsistent theory containing arithmetic, naive truth and denotation predicates, and descriptions. The result is obtained by dualising a construction of Kroon. Its most notable feature is that there is a trivial object- one that has every property.

As it stands, Frege’s semantic theory does not provide sufficient general keys for interpreting assertible sentences at different levels of complexity, such as (a) counterfactual-modal-intensional sentences and (b) extensional sentences. It is possible to devise that general key by adding non-classical parameters such as possible worlds, which allow sentences with a high degree of non-extensional complexity to occupy a trivial place in Tarski’s hierarchy as T-scheme-eligible substitutes. The paper argues that finding these non-false (if true) conditions under which complex propositions (...) can be trivially treated as eligible T-scheme substitutes is not trivial. It is a challenge that requires several consistency improvements to deal with various competing extensions of the predicate “true”. We will conclude that Frege-Tarski’s semantic conception (enriched by non- classical presuppositions), misrepresents that challenge. It distorts the problem we face in daily practice to strengthen our assertion systems, plan successful assertion strategies, and protect our assertions from semantic value reversals. The true challenge for us is pragmatic and cannot confuse (1) complicated scientific and empirical conditions of assertiveness with (2) the rewardable scoring-conditions under which one learns to use sentences in his native language. (shrink)

According to contextualism, a gender term such as ‘woman’ does not invariantly refer to a specific social or biological kind. Instead, gender terms have different extensions depending on the context of utterance. Contextualism accommodates that speakers are perfectly able to use gender terms in very different ways and still be coherent and successful in their communicative exchanges. However, while the flexibility of contextualism is its primary asset, it has also turned out to be its potential demise. The worry is that (...) the view not only makes trans-including claims true but also allows that trans-excluding claims can be true and therefore, does justice to the claims of trans people only in a trivial sense. This paper defends the view that contextualists can respond to this worry by showing why trans-excluding claims are often morally problematic even in contexts where they are true. Contextualists are well-equipped to say that when speakers insist on using gender terms in trans-excluding ways, they engage in a meta-linguistic negotiation about how gender terms ought to be used – where using them trans-excluding is treated as normatively superior. This constitutes a kind of epistemic injustice. (shrink)

Bypassing Lewis’ Triviality Results. A Kripke-Style Partial Semantics for Compounds of Adams’ Conditionals Alberto Mura University of Sassari Abstract According to Lewis’ Triviality Results (LTR), conditionals cannot satisfy the equa­tion (E) P(C if A) = P(C | A), except in trivial cases. Ernst Adams (1975), however, provided a probabilistic semantics for the so-called simple conditionals that also sat­isfies equation (E) and provides a probabilistic counterpart of logical consequence (called p-entailment). Adams’ probabilistic semantics is coextensive to Stalnaker­Thomason’s (1970) and (...) Lewis’ (1973) semantics as far as simple conditionals are concerned. A theorem, proved in McGee 1981, shows that no truth-functional many-valued logic allows a relation of logical consequence coextensive with Adams’ p-entailment. This paper presents a modified modal (Kripke-style) version of de Finetti’s se­mantics that escapes McGee’s result and provides a general truth-conditional se­mantics for indicative conditionals. It agrees with Adams’ logic and is not affected by LTR. The new framework encompasses and extends Adams’ probabilistic se­mantics (APS) to compounds of conditionals. A generalised set of axioms for prob­ability over the set of tri-events is provided, which coincide with the standard axioms over the set of the two-valued ordinary sentences. -/- Keywords: Conditionals, Probability logic, de Finetti, Tri-events, Adams’ logic, Stal­naker’s thesis, Partial logic, Lewis’ triviality results, Ramsey test. (shrink)

CM-triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions aboutCM-triviality, asking in particular whether a structure of finite rank, which is “coordinatized” byCM-trivial types of rank 1, is itselfCM-trivial. (...) (Actually Wagner worked in a slightly more general context, adapting the definitions to a certain “local” framework, in which algebraic closure is replaced byP-closure, forPsome family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness andCM-triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the “right” notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy. (shrink)

We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey, Griffiths and LaForte.

We study the connection between Schnorr triviality and genericity. We show that while no 2-generic is Turing equivalent to a Schnorr trivial and no 1-generic is tt-equivalent to a Schnorr trivial, there is a 1-generic that is Turing equivalent to a Schnorr trivial. However, every such 1-generic must be high. As a corollary, we prove that not all K-trivials are Schnorr trivial. We also use these techniques to extend a previous result and show that the bases of cones of (...) Schnorr trivial Turing degrees are precisely those whose jumps are at least 0". (shrink)

“Triviality arguments” against functionalism in the philosophy of mind hold that the claim that some complex physical system exhibits a given functional organization is either trivial or has much less content than is usually supposed. I survey several earlier arguments of this kind, and present a new one that overcomes some limitations in the earlier arguments. Resisting triviality arguments is possible, but requires functionalists to revise popular views about the “autonomy” of functional description.

In this study, we used Terrance L. Furin’s concept of transformational countering to examine six Holocaust education resources designed to address Holocaust trivialization and antisemitism. While academic comparisons are useful to helping students understand challenging content, superficial Holocaust analogies for political purposes are identified as Holocaust trivialization. Given that the current context consists of a growing white nationalist movement characterized by authoritarianism, antisemitism, and conspiracy theories, we enter the study understanding that Holocaust trivialization and far-right extremism must be understood together. (...) We, therefore, examined the type of support teachers receive from Holocaust and human rights organizations in countering extremism and Holocaust trivialization. After analyzing the six resources, our findings demonstrate that resources produced to counter Holocaust trivialization become quickly outdated given the rapid evolution of extremism. Also, findings indicate that teachers receive stronger support in countering extremism and Holocaust trivialization from organizations that focus broadly on supporting human rights compared to the organizations that focus exclusively on Holocaust education. (shrink)

Conditional probability is often used to represent the probability of the conditional. However, triviality results suggest that the thesis that the probability of the conditional always equals conditional probability leads to untenable conclusions. In this paper, I offer an interpretation of this thesis in a possible worlds framework, arguing that the triviality results make assumptions at odds with the use of conditional probability. I argue that these assumptions come from a theory called the operator theory and that the (...) rival restrictor theory can avoid these problematic assumptions. In doing so, I argue that recent extensions of the triviality arguments to restrictor conditionals fail, making assumptions which are only justified on the operator theory. (shrink)

This paper challenges defenders/advocates of the semantic-completeness route towards gluts to explain, in simple and plausible terms, why the ‘trivializer paradox’, framed in terms of closure relatives on theories, fails to undermine their argument.

This paper clarifies the relationship between the Triviality Results for the conditional and the Restrictor Theory of the conditional. On the understanding of Triviality proposed here, it is implausible—pace many proponents of the Restrictor Theory—that Triviality rests on a syntactic error. As argued here, Triviality arises from simply mistaking the feature a claim has when that claim is logically unacceptable for the feature a claim has when that claim is unsatisfiable. Triviality rests on a semantic (...) confusion—one which some semantic theories, but not others, are prone to making. On the interpretation proposed here, Triviality Results thus play a theoretically constructive role in the project of natural language semantics. (shrink)

A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null ${\Sigma^0_1}$ (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if ${{\bf h'} \geq_T {\bf 0''}}$ , then h (...) contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees. (shrink)

I argue that the trivial neuron doctrine as characterized by Gold & Stoljar is not trivial; it appears to be inconsistent with property dualism as well as some forms of functionalism and externalism. I suggest that the problem is not so much with the particular way in which Gold & Stoljar draw the distinction as with the unruliness of the distinction itself. Their failure to see this may be why they misunderstand the views of the Churchlands.

A possible way out to Kripke’s Puzzle About Belief could start from the rejection of the notion of epistemic transparency. Epistemic transparency seems, indeed, irremediably incompatible with an externalist conception of mental content. However, Brandom’s inferentialism could be considered a version of externalism that allows, at least in some cases, to save the principle of transparency. Appealing to a normative account of the content of our beliefs, from the inferentialist’s standpoint, it is possible to state that a content is transparent (...) when name-components of that content are a priori associated with some application conditions and, at the same time, reflection alone provides an a priori access to those application conditions, with no need of any empirical investigation. Nevertheless, such requirements are only met in trivial cases. The aim of this paper is to argue that some application conditions of that sort, albeit trivial, can be ontologically ampliative. As a result, the related contents can be regarded as transparent in a substantial and rich way. (shrink)

This paper outlines a non-reductive counterfactual account of grounding along interventionist lines, and uses the account to argue that taking grounding seriously requires ascribing non-trivial truth-conditions to a range of counterpossible counterfactuals. This result allows for a diagnosis of a route to scepticism about grounding, as deriving at least in part from scepticism about non-trivial counterpossible truth and falsity.

We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.

Opponents of the computational theory of mind have held that the theory is devoid of explanatory content, since whatever computational procedures are said to account for our cognitive attributes will also be realized by a host of other ‘deviant’ physical systems, such as buckets of water and possibly even stones. Such ‘triviality’ claims rely on a simple mapping account of physical implementation. Hence defenders of CTM traditionally attempt to block the trivialization critique by advocating additional constraints on the implementation (...) relation. However, instead of attempting to ‘save’ CTM by constraining the account of physical implementation, I argue that the general form of the triviality argument is invalid. I provide a counterexample scenario, and show that SMA is in fact consistent with empirically rich and theoretically plausible versions of CTM. This move requires rejection of the computational sufficiency thesis, which I argue is scientifically unjustified in any case. By shifting the ‘burden of explanatory force’ away from the concept of physical implementation, and instead placing it on salient aspects of the target phenomenon to be explained, it’s possible to retain a maximally liberal and unfettered view of physical implementation, and at the same time defuse the triviality arguments that have motivated defenders of CTM to impose various theory-laden constraints on SMA. (shrink)

We study perfectly trivial theories, 1-based theories, stable varieties, and their mutual interaction. We give a structure theorem for the models of a complete perfectly trivial stable theory without DOP: any model is the algebraic closure of a nonforking regular tree of elements. We also give a structure theorem for stable varieties, all of whose completions have NDOP. Such a variety is a varietal product of an affine variety and a combinatorial variety of an especially simple form.

Presentism is usually understood as the thesis that only the present exists whereas the rival theory of eternalism is usually understood as the thesis that past, present, and future things are all equally real. The significance of this debate has been threatened by the so-called triviality objection, which allegedly shows that the presentist thesis is either trivially true or obviously false: Presentism is trivially true if it is read as saying that everything that exists now is present, and it (...) is obviously false if read as saying that everything that has existed, exits or will exist is present. If eternalism is taken as the negation of presentism, it is also either trivially false or obviously true. In this paper, I try to respond to the triviality objection on behalf of presentism. In second section, I will examine how the argument proceeds. In third section, I will reflect on three possible ways to respond but will argue that none of them succeeds in giving a satisfactory solution. I will then try to clarify the core idea of presentism and to suggest that if we characterise presentism accurately, the problem will disappear. In fourth section, I will offer a plausible definition of presentism and will show how it can avoid the triviality objection and demonstrate why it is advantageous to accept the version of presentism I offer. (shrink)

Suppose you’re certain that a claim—say, ‘Frida is tall’—does not have a determinate truth value. What attitude should you take towards it? This is the question of the cognitive role of indeterminacy. This paper presents a puzzle for theories of cognitive role. Many of these theories vindicate a seemingly plausible principle: if you are fully certain that A, you are rationally required to be fully certain that A is determinate. Call this principle ‘Certainty’. We show that Certainty, in combination with (...) some minimal side premises, entails a very implausible claim: whenever you’re certain that it’s indeterminate whether A, it is rationally required that you reject A. This is a surprising result, which requires abandoning at least some intuitive views about indeterminacy and cognitive role. (shrink)

Presentism is typically characterised as the thesis that everything is present, and therefore there are no dinosaurs or Martian presidential inaugurations. Putting aside the vexed question of exactly what it is to be present in this context, this thesis seems quite straightforward. However, a number of authors—such as Merricks, Lombard, Meyer, Tallant and Sakon —have argued that Presentism so characterised is either trivially true or false even by Presentist lights. This is the so-called Triviality Argument against Presentism. In this (...) paper I show that three of the four premises of the Triviality Argument are plausibly false. I conclude that Presentists have nothing to fear from the Triviality Argument. (shrink)

EveryK-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.

In this paper I argue that there are some sentences whose truth makes no demands on the world, being trivially true in that their truth-conditions are trivially met. I argue that this does not amount to their truth-conditions being met necessarily: we need a non-modal understanding of the notion of the demands the truth of a sentence makes, lest we be blinded to certain conceptual possibilities. I defend the claim that the truths of pure mathematics and set theory are trivially (...) true, and hence accepting their truth brings no ontological commitment; I further defend the claim that the truths of applied mathematics and set theory do not demand the existence of numbers or sets. While the notion of a demand must not be reduced to anything modal, I nonetheless argue that sentences that are trivially true must also be necessary, lest we violate a very weak version of the principle that truth depends on the world. I further argue that all necessary truths are trivially true, lest we admit unexplained necessities. I end by showing one important consequence of this: I argue that if there are truthmakers for intrinsic predications, they must be states of affairs rather than tropes. (shrink)

According to Tarski, a theory of truth for a language L is a theory which logically implies for each sentence S of L a sentence of the form:S is true-in-L if and only if p,where rS1 is replaced by a canonical description of a sentence of L and rp1 is replaced by that sentence if L is contained in the metalanguage or by a translation of S if it is not so contained. Tarski constructed consistent and finitely axiomatized theories of (...) truth for various formal languages and showed how to explicitly define ‘is true in L’ within these theories. We all agree that Tarski's theories of truth have enormous philosophical significance, but there is much less agreement on precisely what that significance consists in. (shrink)

PCCP is the much discussed claim that the probability of a conditional A → B is conditional probability. Triviality results purport to show that PCCP – as a general claim – is false. A particularly interesting proof has been presented in (Hájek, 2011), who shows that – even if a probability distribution P initially satisfied PCCP – a rational update can produce a non-PCCP probability distribution. We argue that the notion of rational update in this argumentation is construed in (...) much too broad a way. In order to make the argumentation precise, we discuss the general rules for modeling conditionals in probability spaces and present formalized version(s) of PCCP and of minimal assumptions concerning the appropriate spaces. Using the introduced apparatus we give a detailed analysis of Hájek’s (2011) triviality proof and show that it is based on an application of revision rules which allow one to construct probability distributions violating not only PCCP, but also fundamental properties of conditionals. This means that they do not really provide arguments against PCCP, properly formalized. We also discuss a Dutch Book argument which shows that the updated belief system is not coherent. This gives an additional, strong argument against accepting the update rules. We also discuss the Converse Dutch Book theorem and argue, that even if the produced probability measure seems to violate it, it cannot serve as the counterexample, as it is not an appropriate model for conditionals. Ultimately, we show that important arguments against PCCP fail. (shrink)

In the present paper I analyse the modularity thesis and, more specifically, the thesis of domain-specificity of processing. I argue that this thesis is not trivial only under the assumption of a variety of processes which differ from each other at the implementation level; otherwise, the variety of cognitive processes can only be explained as emergent on the basic mechanism of associative activation in that it operates on domain-specific representations, which is something that no one would deny. But that assumption (...) is untenable: there are no other processes than associative activation (and inhibition) at the implementation level. Any claim to the contrary is the result of a conceptual confusion between two senses of “associative”: a behavioural one, relative to which there are cognitive processes that exceed the ability to code elementary spatio-temporal contingencies, and one that lies instead at the implementation level. Since the assumption of a plurality of processes at the implementation level is untenable, the only viable interpretation of modularism (as far as domain-specificity is concerned) is a trivial one. By this I do not mean that the thesis is devoid of any content. However, its content is scarcely debatable, and far less thrilling than the debate has suggested so far. (shrink)

Let T be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf {Mod}(T)$ to be homotopically equivalent to $\mathbb {CP}^\infty $. In particular, $\pi _2(\lvert \mathsf {Mod}(T)\rvert )=\mathbb {Z}$, which answers a question in our previous work. The computation is based on Connes’ cycle category $\Lambda $.