Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some (...) class='Hi'>probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

Contrary to what is stated in Lemma 7.1 of [8], it is shown that some Boolean algebras of finitary logic admit finitely additive probabilities that are not σ-additive. Consequences of Lemma 7.1 are reconsidered. The concept of a C-σ-additive probability on B (where B and C are Boolean algebras, and $\mathscr{B} \subseteq \mathscr{C}$ ) is introduced, and a generalization of Hahn's extension theorem is proved. This and other results are employed to show that every S̄(L)-σ-additive probability on s̄(L) (...) can be extended (uniquely, under some conditions) to a σ-additive probability on S̄(L), where L belongs to a quite extensive family of first order languages, and S̄(L) and s̄(L) are, respectively, the Boolean algebras of sentences and quantifier free sentences of L. (shrink)

Propositions are the referents of the ‘that’-clauses that appear in the direct object positions of typical ascriptions of assertion, belief, and other binary cognitive relations. In that sense, propositions are the objects of those cognitive relations. Propositions are also the semantic contents (meanings, in one sense ) of declarative sentences, with respect to contexts. They are what sentences semantically express, with respect to contexts. Propositions also bear truth-values. The truth-value of a sentence, in a context, is the truth-value (...) of the proposition that it semantically expresses, in that context. This much is common ground among many (but not all) philosophers. I accept other claims about propositions that are more controversial. Propositions (I hold) are Russellian: they are structured entities whose constituents include individuals, properties, and relations. The contribution of a proper name to the proposition that a sentence semantically expresses (in a context) is the referent of that name. Thus, the semantic content of ‘Bill Clinton’ is Bill Clinton himself, and the semantic content of ‘Bill Clinton smokes’ is a proposition whose constituents are Bill Clinton and the property of smoking (ignoring tense, as I shall do from here on). Such 1 singular propositions are among the objects of belief, assertion, and other cognitive relations. This combination of a Millian view about proper names with a Russellian theory of propositions might appropriately be called ‘Millian Russellianism’, or ‘MR’ for short. David Chalmers, in his stimulating paper “Probability and Propositions,” defines a closely related view, Referentialism, as follows (see also the penultimate paragraph of his introduction). Referentialist views say that insofar as beliefs are about individuals (such as Nietzche), the objects of these belief are determined by those individuals. On one such view, the objects of belief are Russellian propositions composed from the individuals and properties that one’s belief is about.. (shrink)

We prove a 0-1 law for the fragment of second order logic SO over parametric classes of finite structures which allow only one unary atomic type. This completes the investigation of 0-1 laws for fragments of second order logic defined in terms of first order quantifier prefixes over, e.g., simple graphs and tournaments. We also prove a low oscillation law, and establish the 0-1 law for Σ14 without any restriction on the number of unary types.

The starting point of this paper is the empirically determined ability to reason in natural language by employing probable sentences. A sentence is understood to be logically probable if its schema, expressed as a formula in the language of classical propositional calculus, takes the logical value of truth for the majority of Boolean valuations, i.e., as a logically probable formula. Then, the formal system P is developed to encode the set of these logically probable formulas. Based on natural semantics, (...) a strong completeness theorem for P is proved. Alternative notions of consequence for logically probable sentences are also considered. (shrink)

I will describe the logics of a range of conditionals that behave like conditional probabilities at various levels of probabilistic support. Families of these conditionals will be characterized in terms of the rules that their members obey. I will show that for each conditional, →, in a given family, there is a probabilistic support level r and a conditional probability function P such that, for all sentences C and B, 'C → B' holds just in case P[B | (...) C] ≥ r. Thus, each conditional in a given family behaves like conditional probability above some specific support level. (shrink)

APA PsycNET abstract: This is the first volume of a two-volume work on Probability and Induction. Because the writer holds that probability logic is identical with inductive logic, this work is devoted to philosophical problems concerning the nature of probability and inductive reasoning. The author rejects a statistical frequency basis for probability in favor of a logical relation between two statements or propositions. Probability "is the degree of confirmation of a hypothesis (or conclusion) on the (...) basis of some given evidence (or premises)." Furthermore, all principles and theorems of inductive logic are analytic, and the entire system is to be constructed by means of symbolic logic and semantic methods. This means that the author confines himself to the formalistic procedures of word and symbol systems. The resulting sentence or language structures are presumed to separate off logic from all subjectivist or psychological elements. Despite the abstractionism, the claim is made that if an inductive probability system of logic can be constructed it will have its practical application in mathematical statistics, and in various sciences. 16-page bibliography. (PsycINFO Database Record (c) 2016 APA, all rights reserved). (shrink)

According to ethical neo-expressivism, all declarative sentences, including those used to make ethical claims, have propositions as their semantic contents, and acts of making an ethical claim are properly said to express mental states, which (if motivational internalism is correct) are intimately connected to motivation. This raises two important questions: (i) The traditional reason for denying that ethical sentences express propositions is that these were thought to determine ways the world could be, so unless we provide an analysis (...) of ethical terms in terms of natural properties, are we not committed to the Moorean conclusion that ethical sentences describe sui generis “nonnatural” ways the world could be? (ii) If we reject the claim that motivational attitudes constitute any part of the semantics of ethical sentences, are we not committed to denying the intuition behind motivational internalism after all? This chapter argues for negative answers to both of these questions. (shrink)

This thesis focuses on expressively rich languages that can formalise talk about probability. These languages have sentences that say something about probabilities of probabilities, but also sentences that say something about the probability of themselves. For example: (π): “The probability of the sentence labelled π is not greater than 1/2.” Such sentences lead to philosophical and technical challenges; but can be useful. For example they bear a close connection to situations where ones confidence in (...) something can affect whether it is the case or not. The motivating interpretation of probability as an agent's degrees of belief will be focused on throughout the thesis. -/- This thesis aims to answer two questions relevant to such frameworks, which correspond to the two parts of the thesis: “How can one develop a formal semantics for this framework?” and “What rational constraints are there on an agent once such expressive frameworks are considered?”. (shrink)

Dummettian anti-realism repudiates the realist's notion of verification-transcendent truth. Perhaps the most crucial element in the Dummettian attack on realist truth is the critique of so-called realist semantics, which assigns verification-transcendent truth-conditions as the meanings of (some) sentences. The Dummettian critique charges that realist semantics cannot serve as an adequate theory of meaning for a natural language, and that, consequently, the realist conception of truth must be rejected as well. In arguing for this, Dummett and his followers have appealed (...) to a certain conception of linguistic knowledge. This paper examines closely the appeal to speakers' knowledge of linguistic meaning, its force and limitations. (shrink)

The fact that the standard probabilistic calculus does not define probabilities for sentences with embedded conditionals is a fundamental problem for the probabilistic theory of conditionals. Several authors have explored ways to assign probabilities to such sentences, but those proposals have come under criticism for making counterintuitive predictions. This paper examines the source of the problematic predictions and proposes an amendment which corrects them in a principled way. The account brings intuitions about counterfactual conditionals to bear on the (...) interpretation of indicatives and relies on the notion of causal (in)dependence. (shrink)

This paper presents a formalism that combines useful properties of both logic and probabilities. Like logic, the formalism admits qualitative sentences and provides symbolic machinery for deriving deductively closed beliefs and, like probability, it permits us to express if-then rules with different levels of firmness and to retract beliefs in response to changing observations. Rules are interpreted as order-of-magnitude approximations of conditional probabilities which impose constraints over the rankings of worlds. Inferences are supported by a unique priority ordering (...) on rules which is syntactically derived from the knowledge base. This ordering accounts for rule interactions, respects specificity considerations and facilitates the construction of coherent states of beliefs. Practical algorithms are developed and analyzed for testing consistency, computing rule ordering, and answering queries. Imprecise observations are incorporated using qualitative versions of Jeffrey's rule and Bayesian updating, with the result that coherent belief revision is embodied naturally and tractably. Finally, causal rules are interpreted as imposing Markovian conditions that further constrain world rankings to reflect the modularity of causal organizations. These constraints are shown to facilitate reasoning about causal projections, explanations, actions and change. (shrink)

How does deductive logic constrain probability? This question is difficult for subjectivistic approaches, according to which probability is just strength of (prudent) partial belief, for this presumes logical omniscience. This paper proposes that the way in which probability lies always between possibility and necessity can be made precise by exploiting a minor theorem of de Finetti: In any finite set of propositions the expected number of truths is the sum of the probabilities over the set. This is (...) generalized to apply to denumerable languages. It entails that the sum of probabilities can neither exceed nor be exceeded by the cardinalities of all consistent and closed (within the set) subsets. In general any numerical function on sentences is said to be logically coherent if it satisfies this condition. Logical coherence allows the relativization of necessity: A function p on a language is coherent with respect to the concept T of necessity iff there is no set of sentences on which the sum of p exceeds or is exceeded by the cardinality of every T-consistent and T-closed (within the set) subset of the set. Coherence is easily applied as well to sets on which the sum of p does not converge. Probability should also be relativized by necessity: A T-probability assigns one to every T-necessary sentence and is additive over disjunctions of pairwise T-incompatible sentences. Logical T-coherence is then equivalent to T-probability: All and only T-coherent functions are T-probabilities. (shrink)

We present a coherence-based probability semantics for (categorical) Aristotelian syllogisms. For framing the Aristotelian syllogisms as probabilistic inferences, we interpret basic syllogistic sentence types A, E, I, O by suitable precise and imprecise conditional probability assessments. Then, we define validity of probabilistic inferences and probabilistic notions of the existential import which is required, for the validity of the syllogisms. Based on a generalization of de Finetti's fundamental theorem to conditional probability, we investigate the coherent probability propagation (...) rules of argument forms of the syllogistic Figures I, II, and III, respectively. These results allow to show, for all three Figures, that each traditionally valid syllogism is also valid in our coherence-based probability semantics. Moreover, we interpret the basic syllogistic sentence types by suitable defaults and negated defaults. Thereby, we build a knowledge bridge from our probability semantics of Aristotelian syllogisms to nonmonotonic reasoning. Finally, we show how the proposed semantics can be used to analyze syllogisms involving generalized quantifiers. (shrink)

This paper continues our work on a coherence-based probability semantics for Aristotelian syllogisms (Gilio, Pfeifer, and Sanfilippo, 2016; Pfeifer and Sanfilippo, 2018) by studying Figure III under coherence. We interpret the syllogistic sentence types by suitable conditional probability assessments. Since the probabilistic inference of P|S from the premise set {.

Brian Barry's Culture and Equality is probably the most powerful liberal egalitarian critique of multiculturalism addressing the pathologies of recognizing difference of ethnicity, religion, race, and culture. In this essay, I examine Barry's approach to the law, which underpins his theory of egalitarianism to determine whether it is enough — as Barry thinks it is — to insist on either applying the same law for everyone so that exemptions are foreclosed in general, or repealing the law since the case for (...) its existence is not justified. I find that Barry's effort is inadequate. Because the conditions for exemptions are not specified, exemptions are merely defensible, not just. Using the headscarf controversy in France to illustrate why Barry's approach backfires, I argue how enforcing the same law for all leads to undermining the very politics of redistribution that Barry champions. (shrink)

Numerous studies have examined the acoustic correlates of sentential stress and its underlying linguistic functionality. However, the mechanism that connects stress cues to the listener's attentional processing has remained unclear. Also, the learnability versus innateness of stress perception has not been widely discussed. In this work, we introduce a novel perspective to the study of sentential stress and put forward the hypothesis that perceived sentence stress in speech is related to the unpredictability of prosodic features, thereby capturing the attention of (...) the listener. As predictability is based on the statistical structure of the speech input, the hypothesis also suggests that stress perception is a result of general statistical learning mechanisms. To study this idea, computational simulations are performed where temporal prosodic trajectories are modeled with an n-gram model. Probabilities of the feature trajectories are subsequently evaluated on a set of novel utterances and compared to human perception of stress. The results show that the low-probability regions of F0 and energy trajectories are strongly correlated with stress perception, giving support to the idea that attention and unpredictability of sensory stimulus are mutually connected. (shrink)

A stereotype is a belief or claim that a group of people has a particular feature. Stereotypes are expressed by sentences that have the form of generic statements, like “Canadians are nice.” Recent work on generics lends new life to understanding generics as statements involving probabilities. I argue that generics (and thus sentences expressing stereotypes) can take one of several forms involving conditional probabilities, and these probabilities have what I call a naturalness requirement. This is the natural (...) class='Hi'>probability theory of stereotypes. Each of the two components of the theory entails a family of fallacies that contributes to the spurious reinforcement of stereotypes: inferential slippage within and between the different generic forms, and inferential slippage from facts about frequencies of group traits to beliefs about natural propensities or dispositions of groups. Empirical research suggests that we often commit these fallacies. Moreover, this theory can referee a vitriolic debate between some psychologists, who hold that stereotypes are always false and stereotyping is always wrong, and other psychologists, who hold that stereotypes are often accurate and stereotyping is often reasonable. (shrink)

The question of whether humans represent grammatical knowledge as a binary condition on membership in a set of well-formed sentences, or as a probabilistic property has been the subject of debate among linguists, psychologists, and cognitive scientists for many decades. Acceptability judgments present a serious problem for both classical binary and probabilistic theories of grammaticality. These judgements are gradient in nature, and so cannot be directly accommodated in a binary formal grammar. However, it is also not possible to simply (...) reduce acceptability to probability. The acceptability of a sentence is not the same as the likelihood of its occurrence, which is, in part, determined by factors like sentence length and lexical frequency. In this paper, we present the results of a set of large-scale experiments using crowd-sourced acceptability judgments that demonstrate gradience to be a pervasive feature in acceptability judgments. We then show how one can predict acceptability judgments on the basis of probability by augmenting probabilistic language models with an acceptability measure. This is a function that normalizes probability values to eliminate the confounding factors of length and lexical frequency. We describe a sequence of modeling experiments with unsupervised language models drawn from state-of-the-art machine learning methods in natural language processing. Several of these models achieve very encouraging levels of accuracy in the acceptability prediction task, as measured by the correlation between the acceptability measure scores and mean human acceptability values. We consider the relevance of these results to the debate on the nature of grammatical competence, and we argue that they support the view that linguistic knowledge can be intrinsically probabilistic. (shrink)

This evenhanded treatment addresses the decades-old dispute among probability theorists, asserting that both statistical and inductive probabilities may be treated as sentence-theoretic measurements, and that the latter qualify as estimates of the former. Beginning with a survey of the essentials of sentence theory and of set theory, the author examines statistical probabilities, showing that statistical probabilities may be passed on to sentences, and thereby qualify as truth-values. An exploration of inductive probabilities follows, demonstrating their reinterpretation as estimates of (...) truth-values. Each chapter is preceded by a summary of its contents. Illustrations and footnotes elucidate definitions, theorems, and technicalities. 1962 ed. (shrink)

People’s attitudes towards social norms play a crucial role in understanding group behavior. Norm psychology accounts focus on processes of norm internalization that influence people’s norm following attitudes but pay considerably less attention to social identity and group identification processes. Social identity theory in contrast studies group identity but works with a relatively thin and instrumental notion of social norms. We argue that to best understand both sets of phenomena, it is important to integrate the insights of both approaches. Social (...) status, social identity, and social norms are considered separate phenomena in evolutionary accounts. We discuss assumptions and views that support this separation, and suggest an integrated view of our own. We argue that we should be open to the early origins of human social complexity, and conjecture that the longer that the human social world involved multi-level societies the more probable it is that norm psychology and social identity interacted in rich ways. (shrink)

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1,..., N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, (...) by considering asymptotic conditional probabilities. As shown by Liogon'kii [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable. (shrink)

This paper rejects a traditional epistemic interpretation of conditional probability. Suppose some chance process produces outcomes X, Y,..., with probabilities P(X), P(Y),... If later observation reveals that outcome Y has in fact been achieved, then the probability of outcome X cannot normally be revised to P(X|Y) ['P&Y)/P(Y)]. This can only be done in exceptional circumstances - when more than just knowledge of Y-ness has been attained. The primary reason for this is that the weight of a piece of (...) evidence varies with the means by which it is provided, so knowledge of Y-ness does not have uniform impact on the probability of X. A better updating of the probability of X is provided by P(X|Y*), where Y* is not an outcome of the chance process being observed, but the sentence 'the outcome Y has been observed', an 'outcome' of the subsequent observation process. This alternative formula is widely endorsed in practice, but not well recognized in theory, where the oversight has generated some unsatisfactory consequences. (shrink)

This paper begins with a problem stemming from Hume regarding credences about credences. Suppose one has a credence of .95 in p, and suppose one assesses the credence to be such. But suppose one’s second-order credence in this assessment is less than 1. Then, by a standard conditionalization rule, one’s credence in p becomes less than .95. Moreover, such “erosion” can iterate by considering one’s, third-, fourth-, fifth-order credences, etc. (In light of this, some have rejected higher-order credences; however, it (...) is argued that Lewis’ “Principal Principle” forbids this.) The paper then offers a partial solution by describing circumstances under which a person has reason to assign credence 1 to a first-order credence. The solution takes the form of a “transparency” view of self-knowledge from Parent (2017, chapter 8). Briefly, it is argued that when one “reflex-like” utters a sentence “The probability of p is n”, then as a matter of psychological law, the utterance is an expression of one’s own judgment. The Principal Principle then necessitates that one’s credence in p is n, assuming one is being rational. Roughly, upon perceiving such an utterance, one can thereby have a reason to assign a second-order credence of 1. (shrink)

This paper explores the interaction of well-motivated (if controversial) principles governing the probability conditionals, with accounts of what it is for a sentence to be indefinite. The conclusion can be played in a variety of ways. It could be regarded as a new reason to be suspicious of the intuitive data about the probability of conditionals; or, holding fixed the data, it could be used to give traction on the philosophical analysis of a contentious notion—indefiniteness. The paper outlines (...) the various options, and shows that ‘rejectionist’ theories of indefiniteness are incompatible with the results. Rejectionist theories include popular accounts such as supervaluationism, non-classical truth-value gap theories, and accounts of indeterminacy that centre on rejecting the law of excluded middle. An appendix compares the results obtained here with the ‘impossibility’ results descending from Lewis ( 1976 ). (shrink)

In this study we investigate the influence of reason-relation readings of indicative conditionals and ‘and’/‘but’/‘therefore’ sentences on various cognitive assessments. According to the Frege-Grice tradition, a dissociation is expected. Specifically, differences in the reason-relation reading of these sentences should affect participants’ evaluations of their acceptability but not of their truth value. In two experiments we tested this assumption by introducing a relevance manipulation into the truth-table task as well as in other tasks assessing the participants’ acceptability and (...) class='Hi'>probability evaluations. Across the two experiments a strong dissociation was found. The reason-relation reading of all four sentences strongly affected their probability and acceptability evaluations, but hardly affected their respective truth evaluations. Implications of this result for recent work on indicative conditionals are discussed. (shrink)

Among the various meanings in which the word ‘probability’ is used in everyday language, in the discussion of scientists, and in the theories of probability, there are especially two which must be clearly distinguished. We shall use for them the terms ‘probability1’ and ‘probability2'. Probability1 is a logical concept, a certain logical relation between two sentences ; it is the same as the concept of degree of confirmation. I shall write briefly “c” for “degree of confirmation,” and (...) “c” for “the degree of confirmation of the hypothesis h on the evidence e“; the evidence is usually a report on the results of our observations. On the other hand, probability2 is an empirical concept; it is the relative frequency in the long run of one property with respect to another. The controversy between the so-called logical conception of probability, as represented e.g. by Keynes, and Jeffreys, and others, and the frequency conception, maintained e.g. by v. Mises and Reichenbach, seems to me futile. These two theories deal with two different probability concepts which are both of great importance for science. Therefore, the theories are not incompatible, but rather supplement each other. (shrink)

In standard Bayesian probability revision, the adoption of full beliefs (propositions with probability 1) is irreversible. Once an agent has full belief in a proposition, no subsequent revision can remove that belief. This is an unrealistic feature, and it also makes probability revision incompatible with belief change theory, which focuses on how the set of full beliefs is modified through both additions and retractions. This problem in probability theory can be solved in a model that (i) (...) lets the codomain of the probability function be a hyperreal-valued rather than the real-valued closed interval [0, 1], and (ii) identifies the full beliefs as the propositions whose probability is either 1 or infinitesimally smaller than 1. In this model, changes in the probability function will result in changes in the set of full beliefs (belief set), which constitutes a submodel that can be conceived as the “tip of the iceberg” within the larger model that also contains beliefs on lower levels of probability. The patterns of change in the set of full beliefs in this modified Bayesian model coincides with the corresponding pattern in a slightly modified version of AGM revision, which is commonly conceived as the gold standard of (dichotomous) belief change. The modification only concerns the marginal case of revision by an inconsistent input sentence. These results show that probability revision and dichotomous belief change can be unified in one and the same framework, or – if we so wish – that belief change theory can be subsumed under a modified version of probability revision that allows for iterated change and for the removal of full beliefs. (shrink)

The paper outlines an analysis of token causation . It contains reasons for the choice of event tokens and a list of properties of causal relations. After defining the notion event it is shown, in which form classical philosophical problems arise and how they can be solved: Causality is interpreted as an second order property of first order relations. Temporal order can be induced from causal order. Token causation does not involve neither any kind of necessity, nor probability. Causal (...) laws are generalizations on sentences about token causation by quantifying over event types. Two explications of Causal law are suggested, differing in the ability of allowing for exceptions. (shrink)

Computational models of infant word‐finding typically operate over transcriptions of infant‐directed speech corpora. It is now possible to test models of word segmentation on speech materials, rather than transcriptions of speech. We propose that such modeling efforts be conducted over the speech of the experimental stimuli used in studies measuring infants' capacity for learning from spoken sentences. Correspondence with infant outcomes in such experiments is an appropriate benchmark for models of infants. We demonstrate such an analysis by applying the (...) DP‐Parser model of Algayres and colleagues to auditory stimuli used in infant psycholinguistic experiments by Pelucchi and colleagues. The DP‐Parser model takes speech as input, and creates multiple overlapping embeddings from each utterance. Prospective words are identified as clusters of similar embedded segments. This allows segmentation of each utterance into possible words, using a dynamic programming method that maximizes the frequency of constituent segments. We show that DP‐Parse mimics American English learners' performance in extracting words from Italian sentences, favoring the segmentation of words with high syllabic transitional probability. This kind of computational analysis over actual stimuli from infant experiments may be helpful in tuning future models to match human performance. (shrink)

Alfred Tarski’s work on truth has played such a central role in the discourse on truth that most coming to it for the first time have probably already heard a great deal about what is said there. Unfortunately, since the work is largely technical and Tarski was only tan- gentially philosophical, a certain incautious assimilation dominates many philosophical discussions of Tarski’s ideas, and so, examining Tarski on the concept of truth is in many ways an act of unlearning. -/- In (...) this paper I focus on those key ideas in Tarski’s work that have made a lasting impact on the philosophical discourse. These are the notions of T-sentence, Convention T, Tarskian truth definition, and Tarski’s general limiting theses on the expressibility and definability of truth. Though these ideas are in name familiar, we will seek in this essay to uncover and remove certain widespread misunderstandings of each. Tarski’s name also features prominently in discussions of the liar paradox, so we will take time out to explain Tarski’s connection to this ancient puzzle. (shrink)

In 2001, Le Bars proved that there exists an existential monadic second-order sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence.

In lieu of an abstract, here is a brief excerpt of the content:48 SOME CQUHSEL ON HUMEAN RELATIONS In a paper published eight years ago I tried to bring out a neglected feature of Hume's theory of relations, namely the difference between philosophical and natural re1 2. lations. Now Ijnlay, without referring to my work, has expanded some of its themes in an extremely interesting and, I think, important way. At least he has made me rethink the whole distinction between (...) philosophical and natural relations and, what goes with relation theory, Hume's analytic-synthetic distinction. The strengths of his paper are, embarrassingly, the weaknesses of my own. I did not articulate clearly the connection between relation theory in Hume and the analytic-synthetic distinction. I shall turn to these strengths presently. My focus in this paper, however, will be on a central weakness in Imlay 's; and the point, it turns out, is one I gave full credence to earlier. Imlay does not distinguish what in Hume can and must be distinguished if we are to make any sense (let alone full sense) of his theory of causation. The distinction is be3 tweer, necessity and certainty. In trying to explicate Hume's distinction between intuitive and demonstrative inference on the one hand, and probability on the other, Imlay finally reaches an impasse. For, if the absence of the corresponding impression does not deprive mathematical necessity of... existence, why should it do so in the case of causal necessity? This, as Imlay recognizes, is at least related to standard criticisms of Hume, most forcibly articulated by Prichard. Imlay fortunately does not recommend—as Prichard does — throwing out the baby with the bath. If I understand him correctly, what he advocates is that we simply admit that Hume's attempts to ground necessity are inadequate. Before proceeding to details, let me issue two brief caveats. 49 First. It is of course not surprising that Hume fails to make a clear analytic-synthetic distinction. Has anyone? Lacking the logical tools of a Russell, or the mathematical understanding of a Leibniz, Hume labors under insuperable difficulties which, in many instances, are selfcreated. He is not clear as to whether necessity attaches to 'objects' or ideas (let alone sentences or propositions); he confuses, by invoking pseudo-psychological criteria involving conceivability, analytic with synthetic a priori truths; he has what many, including myself, consider to be a ludicrous theory about mathematics and mathematical objects. With these horrors to work with it is small wonder that necessity never gets grounded. Indeed, it is miraculous that the discussion even gets off the ground. But, Prichard's sneeringly just comments notwithstanding, both he and Imlay miss something, and I hope to show precisely what that is. Second. Hume is notoriously obscure in his discussion of what we would call ordinary physical objects. Sometimes (when he is in a psychologist or even ordinaryperson mood) he speaks as if physical objects cause impressions. Sometimes (when in the philosopher's closet) he speaks as if it makes no sense to speak of anything but impressions and collections of them, ordinary physical objects being relegated to the fictions of the vulgar. His theory of relations rather obviously suffers from the same disease; it is not clear what the relata are. Are impressions related temporally? Spatially? Is it relations between objects 'out there' of which he speaks when he discusses causation? Imlay is quite aware of the confusion ; he just lets it go. For now, I shall more or less follow suit. Later, as we shall see, one's choice makes a difference. In the long run, I think any coherent view of Hume will have to treat his analysis of physical objects in a way similar to Berkeley's · I. Hume, Prichard, and Imlay At first glance it would seem easy to allay at least Prichard' s fears. What, after all, is the connection between Hume's theory of relations, his attempt at distinguishing the necessary from the probable or possible, and his theory of causation? Hume claims that causation is a relation that, unlike some others, eg., resemblance, cannot be 'seen' to hold between its relata. But all of us believe that causal connections are necessary. Why do we all believe what... (shrink)

Himbert of Garda was a little-known Franciscan theologian who studied at Paris around 1320 and probably served as Francis of Meyronnes’ secretary. His commentary on the Sentences provides precious insights on the development of Franciscan thought at Paris, connecting Francis of Meyronnes’ refined presentations of doctrine with raw academic debates between bachelors and masters in the Faculty of Theology. An appendix presents Himbert’s discussion of intrinsic degrees in Book I d.36, and both redactions of his treatment of the formal (...) distinction in the context of the divine attributes in Book I d.8, as well as parallel texts from Francis of Meyronnes. A second appendix provides a full question- and citation-list for Himbert’s commentary on the Sentences. (shrink)

We study the model-theoretic aspects of a probability logic suited for talking about measure spaces. This nonclassical logic has a model theory rather different from that of classical predicate logic. In general, not every satisfiable set of sentences has a countable model, but we show that one can always build a model on the unit interval. Also, the probability logic under consideration is not compact. However, using ultraproducts we can prove a compactness theorem for a certain class (...) of weak models. (shrink)

A careful presentation of the foundations of probability theory, containing many valuable innovations. Two accounts of probability are adduced: probability as a measure on the subsets of a probability set, and as a measure on the sentences of a formal language. The book stresses connections between these two accounts; of particular interest is its thesis that statistical probabilities may be regarded as estimates of inductive probabilities.—R. H. T.

In a recent Analysis article, Quentin Smith argues that classical theism is inconsistent with certain consequences of Stephen Hawking's quantum cosmology.1 Although I am not a theist, it seems to me that Smith's argument fails to establish its conclusion. The purpose of this paper is to show what is wrong with Smith's argument. According to Smith, Hawking's cosmological theory includes what Smith calls "Hawking's wave function law." Hawking's wave function law (hereafter, "HL") apparently has, among its consequences, the following claim. (...) (1) The unconditional probability that a universe like this one - i.e., a universe with the metric hij and matter field Φ - should begin to exist is 95%.2 Smith then argues that the theist who accepts HL must also accept that the following sentence was once true.3.. (shrink)

Denić ( 2018, 2019, To appear ) observes that the availability of distributive inferences—for sentences with disjunction embedded in the scope of a universal quantifier—depends on the size of the domain quantified over as it relates to the number of disjuncts. Based on her observations, she argues that probabilistic considerations play a role in the computation of implicatures. In this paper we explore a different possibility. We argue for a modification of Denić’s generalization, and provide an explanation that is (...) based on intricate logical computations but is blind to probabilities. The explanation is based on the observation that when the domain size is no larger than the number of disjuncts, universal and existential alternatives are equivalent if distributive inferences are obtained. We argue that under such conditions a general ban on ‘fatal competition’ (Magri 2009a, b, Spector 2014 ) is activated, thereby predicting distributive inferences to be unavailable. (shrink)

The difficulty lies in the two words at the end of the first and last lines of the extract, cortus being manifestly corrupt, and pontus inappropriate. In 276 it is fairly clear that ‘the wind’ should be the subject of the sentence, and Markland's uentus is now printed by most editors: in 271, in spite of a variety of conjectures, editors are mostly agreed that ‘the sea’ should be the object, and Marullus' pontum is usually accepted. The sense is thus (...) rectified, but there remains a serious palaeographical difficulty, neither of the corruptions pontum > cortus, uentus > pontus being very probable in itself or paralleled in the Lucretian MSS. It has been suggested, both by Professor Reid and by Professor Merrill , that the last words of the two lines have been interchanged. If this theory can be accepted, it at once gives us pontum in 271: the nominative pontus is only an attempt to adjust it to its new context in 276. We should then be left with cortus at the end of 276, and I suggest that the obvious correction is then corus. (shrink)

In this essay, I critically analyze Wittgenstein’s dispensation with “ = ” in a correct concept-script. I argue inter alia (a) that in the Tractatus the alleged pseudo-character of sentences containing “ = ” or = -sentences remains largely unexplained and propose how it could be explained; (b) that at least in some cases of replacing = -sentences with equivalent identity-sign free sentences the use of the notion of a translation seems inappropiate; (c) that in the (...) Tractatus it remains unclear how identity of the object as that which is expressed by identity of the sign should be understood specifically; (d) that there are = -sentences which have no obvious equivalent in Wittgenstein’s novel notation; (e) that Wittgenstein’s adherence to (non-relational) identity, although he dispenses with “ = ”, is probably motivated by his desire to ensure that the expressive power of an identity-sign free concept-script of first-order is on a par with standard first-order logic containing “ = ”. In the concluding section, I critically discuss some claims in Lampert and Säbel (The Review of Symbolic Logic, 14, 1–21, 2021) and defend Wehmeier’s account of pseudo-sentences in the Tractatus (2012) against the objections they raise. (shrink)

The chapter aims at reconstructing Gregory of Rimini's view on universals in absence of the full-bodied treatment Gregory himself promised in his work. According to Gregory, there is nothing universal outside the mind, and universal concepts are made-up on the basis of prior cognitions. In absence of Gregory's explicit statements of the matter, I argue that these concepts must most probably be qualities in the mind that are really distinct from acts of cognition and remain in the mind even if (...) they are not targeted by an actual act of cognition. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:31 Hunie on Intuitive and Demonstrative Inference This paper is divided into four sections. The first section deals with Hume's attempt to resolve a dilemma concerning the objects of intuitive and demonstrative inference. In the second section I try to show that his resolution of the dilemma is hard to reconcile with his phenomenalist doctrine of the origin of ideas. In the third section I examine tne meaning of (...) "intuition" in Hume. Finally, in the fourth section I compare his view of mathematical inference with his view of causal inference, a kind of inference to which he tends to assimilate the former. What are the objects of intuitive and demonstrative inference in Hume supposed to be? In the Treatise they are co-extensive with four kinds of philosophical relations resemblance, contrariety, degrees in quality and proportions in quantity or numbers. The distinguishing characteristic of the group, according to Hume, is that they all depend entirely on the ideas which we compare together. Tne example he gives is that of a triangle 2 whose three angles are equal to two right ones. And this presumably is at the same time an example of a relation of proportion in quantity or numbers. Needless to say, what cries out for explanation here, apart from the notion of comparison with which we shall deal later, is the phrase "depend entirely upon the ideas." Hume himself does not provide the explanation. Consequently, a certain amount of speculation becomes inevitable. One fairly plausible although, as we shall see, not entirely satisfactory interpretation of the phrase in question would associate it with a thesis concerning the reducibility of relations between exemplars of properties to relations between the properties themselves. Thus it may 32 be argued that the equality holding between the three angles of a triangle and two right angles is reducible to an equality holding between certain of their properties. The properties would be the 180° that each set of angles exemplifies. Indirect evidence for this interpretation is provided by Hume's account of the relations of contiguity and distance, species presumably of relations of time and place, which do not, according to him, depend upon the ideas wnich are compared together. Indeed, these relations may be changed by "an alteration of their place, without any change on the objects themselves or on their 3 ideas." "The place" in its turn "depends on a hundred 4 different accidents which cannot be foreseen by the mind." Leaving aside Hume's failure or refusal to draw a clear distinction here between ideas and objects which probably has to do with his belated realization that only the latter have places and are contiguous to or distant from one another, one notes his insistence that an object's place is not an essential property of the object. For his use of the word "accident" in the above-quoted passage has very little to do with its ordinary use and everything to do with its logical one where it is contrasted with "essential." Furthermore, the sentence immediately preceding the one containing that passage gives some indication of what Hume takes its logical use to be : A determinate property is an accidental one if loss of that property does not affect the object's identity or, as he would put it, the object itself. Nor is the accidental nature of the property anything but reinforced by its dependence logical or otherwise upon other accidental properties which like all such properties require an appeal to observation and induction in order to determine whether they are exemplified by that object. The loss of an essential property, on the other hand, would affect the object's identity because such properties are contained in the very definition of the object. And it suffices to understand the definition - Hume's foreseeing by the 33 7 mind comes in here - in order to determine that the property is exemplified by the object. How does this account provide indirect evidence for our interpretation of Hume? The answer is relatively straightforward. According to that interpretation, relations which depend entirely on the ideas which we compare together are tnose where relations between exemplars of properties are reducible to relations between... (shrink)

What are the objects of knowledge, belief, probability, apriority or analyticity? For at least some of these properties, it seems plausible that the objects are sentences, or sentence-like entities. However, results from mathematical logic indicate that sentential properties are subject to severe formal limitations. After surveying these results, I argue that they are more problematic than often assumed, that they can be avoided by taking the objects of the relevant property to be coarse-grained (“sets of worlds”) propositions, and (...) that all this has little to do with the choice between operators and predicates. (shrink)

We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.