An information recovery problem is the problem of constructing a proposition containing the information dropped in going from a given premise to a given conclusion that folIows. The proposition(s) to beconstructed can be required to satisfy other conditions as well, e.g. being independent of the conclusion, or being “informationally unconnected” with the conclusion, or some other condition dictated by the context. This paper discusses various types of such problems, it presents techniques and principles useful in solving them, and it (...) develops algorithmic methods for certain classes of such problems. The results are then applied to classical number theory, in particular, to questions concerning possible refinements of the 1931 Gödel Axiom Set, e.g. whether any of its axioms can be analyzed into “informational atoms”. Two propositions are “informationally unconnected” [with each other] if no informative (nontautological) consequence of one also follows from the other. A proposition is an “informational atom” if it is informative but no information can be dropped from it without rendering it uninformative (tautological). Presentation, employment, and investigation of these two new concepts are prominent features of this paper. (shrink)

We study from the proof complexity perspective the proof search problem : •Is there an optimal way to search for propositional proofs?We note that, as a consequence of Levin’s universal search, for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal proof search algorithm exists without restricting proof systems iff a p-optimal proof system exists.To characterize precisely the time proof search algorithms need for individual formulas (...) we introduce a new proof complexity measure based on algorithmic information concepts. In particular, to a proof system P we attach information-efficiency function $i_P$ assigning to a tautology a natural number, and we show that: • $i_P$ characterizes time any P-proof search algorithm has to use on $\tau $,•for a fixed P there is such an information-optimal algorithm,•a proof system is information-efficiency optimal iff it is p-optimal,•for non-automatizable systems P there are formulas $\tau $ with short proofs but having large information measure $i_P$.We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for $i_P$ where no super-polynomial size lower bounds are known. We also point out connections of the new measure with some topics in proof complexity other than proof search. (shrink)

Propositional attitude verbs, such as think and want, have long held interest for both theoretical linguists and language acquisitionists because their syntactic, semantic, and pragmatic properties display complex interactions that have proven difficult to fully capture from either perspective. This paper explores the granularity with which these verbs’ semantic and pragmatic properties are recoverable from their syntactic distributions, using three behavioral experiments aimed at explicitly quantifying the relationship between these two sets of properties. Experiment 1 gathers a measure of 30 (...) propositional attitude verbs’ syntactic distributions using an acceptability judgment task. Experiments 2a and 2b gather measures of semantic similarity between those same verbs using a generalized semantic discrimination task and an ordinal scale task, respectively. Two kinds of analyses are conducted on the data from these experiments. The first compares both the acceptability judgments and the semantic similarity judgments to previous classifications derived from the syntax and semantics literature. The second kind compares the acceptability judgments to the semantic similarity judgments directly. Through these comparisons, we show that there is quite fine-grained information about propositional attitude verbs’ semantics carried in their syntactic distributions—whether one considers the sorts of discrete qualitative classifications that linguists traditionally work with or the sorts of continuous quantitative classifications that can be derived experimentally. (shrink)

I will show that the semantic structure of a new imperfect-information propositional logic can be described in terms of extensive forms of semantic games. I will discuss some ensuing properties of these games such as imperfect recall, informational consistency, and team playing. Finally, I will suggest a couple of applications that arise in physics, and most notably in quantum theory and quantum logics.

The notion of a proposition as a set of possible worlds or states occupies central stage in probability theory, semantics and epistemology, where it serves as the fundamental unit both of information and meaning. But this fact should not blind us to the existence of prospects with a different structure. In the paper I examine the use of random variables—in particular, proposition-valued random variables— in these fields and argue that we need a general account of rational attitude formation with respect (...) to them. (shrink)

This paper deals with various substructural propositional logics, in particular with substructural subsystems of Nelson's constructive propositional logics N– and N. Doen's groupoid semantics is extended to these constructive systems and is provided with an informational interpretation in terms of information pieces and operations on information pieces.

Scott Soames has recently argued that traditional accounts of propositions as n-tuples or sets of objects and properties or functions from worlds to extensions cannot adequately explain how these abstract entities come to represent the world. Soames’ new cognitive theory solves this problem by taking propositions to be derived from agents representing the world to be a certain way. Agents represent the world to be a certain way, for example, when they engage in the cognitive act of predicating, (...) or cognizing, an act that takes place during cognitive events, such as perceiving, believing, judging and asserting. On the cognitive theory, propositions just are act types involving the act of predicating and certain other mental operations. This theory, Soames argues, solves not only the problem of how propositions come to represent but also a number of other difficulties for traditional theories, including the problem of de se propositions and the problems of accounting for how agents are capable of grasping propositions and how they come to stand in the relation of expression to sentences. I argue here that Soames’ particular version of the cognitive theory makes two problematic assumptions about cognitive operations and the contents of proper names. I then briefly examine what can count as evidence for the nature of the constituents of the cognitive operation types that produce propositions and argue that the common nature of cognitive operations and what they operate on ought to be determined empirically in cross-disciplinary work. I conclude by offering a semantics for cognitive act types that accommodates one type of empirical evidence. (shrink)

// tl;dr A Proposition is a Way of Thinking // -/- This chapter is about type-theoretic approaches to propositional content. Type-theoretic approaches to propositional content originate with Hintikka, Stalnaker, and Lewis, and involve treating attitude environments (e.g. "Nate thinks") as universal quantifiers over domains of "doxastic possibilities" -- ways things could be, given what the subject thinks. -/- This chapter introduces and motivates a line of a type-theoretic theorizing about content that is an outgrowth of the recent literature on epistemic (...) modality, according to which contentful thought is broadly "informational" in its nature and import. The general idea here is that an object of thought is not a way *the world* could be, but rather a way *one's perspective* could be (with respect to a relevant representational question). I will spend the middle part of this chapter motivating and developing a version of this strategy that is, I’ll argue, well-suited to explaining clear phenomena concerning the attribution of perspectival attitudes -- in particular, attitudes towards loosely information-sensitive propositions -- with which extant approaches struggle. My overarching goal here will be to motivate a distinctive version of the "informational" approach -- the "Flexible Types" approach, which is based on the theory proposed in Charlow (2020). According to the Flexible Types approach, propositional attitude verbs are quantifiers over sets of possibilities, but a possibility is a type-flexible notion -- sometimes a possible world, sometimes a perspective, sometimes a set of possible worlds, sometimes a set of perspectives. -/- After introducing the Flexible Types approach, this chapter circles back to more traditional concerns for the analysis of propositions as types of possibilities -- Frege's Puzzle and the problem of Logical Omniscience. Here too the Flexible Types approach bears fruit. Although there are certainly significant differences -- I note some in the concluding section -- the gist of this theory is Hinitkkan or Lewisian in spirit (if not quite in letter). We can make progress on addressing the challenges for the analysis of propositional content in terms of types of possibilities, through empirically driven refinement of our notion of what kind of thing a "doxastic possibility" is. (shrink)

Book Review: Samuel Lebens (2017) "Bertrand Russell and the Nature of Propositions: a History and Defense of the Multiple Relation Theory of Judgement".

According to rationalists, synthetic a priori propositions convey new knowledge, whereas analytic propositions are non-informative or vacuous conceptual truths. However, as we argue in this article, each a priori proposition is necessarily true because of its semantic constituents and the way they are combined, and hence can be transformed into its equivalent analytic form. So each synthetic a priori proposition conveys only non-informative conceptual truths like analytic propositions.

Skyrms, building on the work of Dretske, has recently developed a novel information-theoretic account of propositional content in simple signalling systems. Information-theoretic accounts of content traditionally struggle to accommodate the possibility of misrepresentation, and I show that Skyrms’s account is no exception. I proceed to argue, however, that a modified version of Skyrms’s account can overcome this problem. On my proposed account, the propositional content of a signal is determined not by the information that it actually carries, but by the (...) information that it would carry at the nearest separating equilibrium of the underlying evolutionary dynamics. I show that this amended account yields reasonable ascriptions of false propositional content in a well-known formal model of the evolution of communication , and close with a discussion of the serious but perhaps not insuperable difficulties we face in applying the account to examples of signalling in the real world. (shrink)

Reprinted in Philosophy of Religion: An Anthology, Wadsworth 2015, 6th edition, eds Michael Rea and Louis Pojman. What is propositional faith? At a first approximation, we might answer that it is the psychological attitude picked out by standard uses of the English locution “S has faith that p,” where p takes declarative sentences as instances, as in “He has faith that they’ll win”. Although correct, this answer is not nearly as informative as we might like. Many people say that there (...) is a more informative answer. They say that, at the very least, propositional faith requires propositional belief. More precisely, they say that faith that p requires belief that p or that it must be partly constituted by belief that p. This view is common enough; call it the Common View. I have two main aims in this paper: (i) to exhibit the falsity of the Common View and the paucity of reasons for it, and (ii) to sketch a more accurate and comprehensive account of what propositional faith is. (shrink)

A framework for pragmatic analysis is proposed which treats discourse as a game, with context as a scoreboard organized around the questions under discussion by the interlocutors. The framework is intended to be coordinated with a dynamic compositional semantics. Accordingly, the context of utterance is modeled as a tuple of different types of information, and the questions therein — modeled, as is usual in formal semantics, as alternative sets of propositions — constrain the felicitous flow of discourse. A requirement (...) of Relevance is satisfied by an utterance (whether an assertion, a question or a suggestion) iff it addresses the question under discussion. Finally, it is argued that the prosodic focus of an utterance canonically serves to reflect the question under discussion (at least in English), placing additional constraints on felicity in context. (shrink)

The five English words—sentence, proposition, judgment, statement, and fact—are central to coherent discussion in logic. However, each is ambiguous in that logicians use each with multiple normal meanings. Several of their meanings are vague in the sense of admitting borderline cases. In the course of displaying and describing the phenomena discussed using these words, this paper juxtaposes, distinguishes, and analyzes several senses of these and related words, focusing on a constellation of recommended senses. One of the purposes of this paper (...) is to demonstrate that ordinary English properly used has the resources for intricate and philosophically sound investigation of rather deep issues in logic and philosophy of language. No mathematical, logical, or linguistic symbols are used. Meanings need to be identified and clarified before being expressed in symbols. We hope to establish that clarity is served by deferring the extensive use of formalized or logically perfect languages until a solid “informal” foundation has been established. Questions of “ontological status”—e.g., whether propositions or sentences, or for that matter characters, numbers, truth-values, or instants, are “real entities”, are “idealizations”, or are “theoretical constructs”—plays no role in this paper. As is suggested by the title, this paper is written to be read aloud. -/- I hope that reading this aloud in groups will unite people in the enjoyment of the humanistic spirit of analytic philosophy. (shrink)

Kaplan (drawing on Montague and Prior, inter alia) made explicit the idea of world and time neutral propositions, which bear truth values only relative to world and time parameters. There was then a debate over the role of time. Temporalists sided with Kaplan in maintaining time neutral propositions with time relative truth values, while eternalists claimed that all propositions specify the needed time information and so bear the same truth value at all times. But there never was (...) much of a parallel debate over the role of worlds. Let contingentism be the view (parallel to temporalism) that sides with Kaplan in maintaining world neutral propositions with world relative truth values, and let necessitarianism be the view (parallel to eternalism) that propositions specify the needed world information and so bear the same truth value at all worlds. This is the story of how the debate between the contingentists and the necessitarians might begin. (shrink)

Soames's cognitive propositions are strings of acts to be performed by an agent, such as predicating a property of an individual. King takes these structured propositions to task for proliferating too easily. King's objection is based on an example that purports to show that three of Soames's propositions are really just one proposition. I translate the informally stated propositions King attributes to Soames into the intensional λ-calculus. It turns out that they are all β-equivalent to the (...) proposition King claims Soames's three propositions are identical to. I argue on philosophical grounds against identifying β-equivalent propositions. The reason is that β-conversion obliterates too many of the procedural distinctions that are central to an act-based theory such as Soames's and which are worth preserving. In fact, β-expansion allows the addition of a fifth proposition that highlights additional procedural distinctions and propositional structure. The welcome conclusion is that we have five procedurally distinct, if equivalent, propositions. (shrink)

Information-theoretic approaches to formal logic analyse the "common intuitive" concept of propositional implication (or argumental validity) in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; an argument is valid if the conclusion contains no information beyond that of the premise-set. This paper locates information-theoretic approaches historically, philosophically and pragmatically. Advantages and disadvantages are identified by examining such approaches in (...) themselves and by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyse validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. (shrink)

The dominant account of propositions holds that they are structured entities that have, as constituents, the semantic values of the constituents of the sentences that express them. Since such theories hold that propositions are structured, in some sense, like the sentences that express them, they must provide an answer to what I will call Soames’ Question: “What level, or levels, of sentence structure does semantic information incorporate?”. As it turns out, answering Soames’ Question is no easy task. I (...) argue in this paper that the two most promising ways of answering it, the Logical Form Account and the LF Account, are both unsatisfactory. This result casts doubt on the very idea that propositions are structured. (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)

This special issue of GPS collects 11 papers (and a long introduction), by leading philosophers and young researchers, which tackle more or less from close the topic of propositions by trying to provide the reader with a cross-section of the ongoing debate in this area. The raised issues range over the semantics, the ontology, the epistemology, and the philosophy of mathematics and stimulate the reader to reflect on crucial problems such as the following: are propositions objects? In the (...) positive case, what kind of objects are they? Can they be grasped by cognitive creatures such as we are? When can we say that two people entertain the same proposition? Have propositions any role to play in speech act theory? Even though the notion of proposition has received considerable attention in the past philosophical debate, it is still of great interest, in particular in connection with the attacks which have recently been launched against it in the theory of language. The volume, which is equipped with a long and detailed introduction that supplies the young reader with useful background information on the different stances in the debate, could prove useful also for didactic purposes. (shrink)

Propositional content arises from the practice of signaling with information transfer when a signaling process settles into some sort of a pattern, and eventually what we call meaning or propositional content crystallizes out. We give an evolutionary account of this process.

[Edited from Conclusion section:] We have looked at various challenging issues to do with getting connectionism to cope with high-level cognitive activities such a reasoning and natural language understanding. The issues are to do with various facets of generalization that are not commonly noted. We have been concerned in particular with the special forms these issues take in the arena of propositional attitude processing. The main problems we have looked at are: (1) The need to construct explicit representations of generalizations, (...) not just generalize correctly to individual cases; (2) The need to be able to match two or more complex short-term information structures, to enable rapid generalization from recent examples rather than from long-term memories; (3) The need to represent and reason with anomalous combinations of concepts; (4) The need to perform embedded reasoning. This presents special problems for systems using non-concatenative representations (as in mainstream connectionist approaches). We also touched on vague quantification in attitude report complements. Neither this topic nor that of analogies between short-term structures (point 2) has been adequately addressed in the symbolic framework, let alone in connectionism. -/- The opportunities and problems covered are put forward as things worth being optimistic about or pessimistic about, respectively. They are not put forward as decisive arguments for or against connectionism. The hope is that this chapter contributes to a greater understanding of the connectionist/symbolist gap by presenting some unusual issues and by throwing new light on some well known ones. (shrink)

Do Russellian propositions have their constituents as parts? One reason for thinking not is that if they did, they would generate apparent counterexamples to plausible mereological principles. As Frege noted, they would be in tension with the transitivity of parthood. A certain small rock is a part of Etna but not of the proposition that Etna is higher than Vesuvius. So, if Etna were a part of the given proposition, parthood would fail to be transitive. As William Bynoe has (...) noted (speaking of facts rather than propositions), they would seem to violate certain supplementation principles. Consider the singular proposition, concerning identity, that it is identical with itself. Given the relevant form of Russellianism, this proposition would have identity as a proper part, but it would not have any parts disjoint from identity, and indeed it would not have even a single pair of disjoint parts, in violation of various supplementation principles. This chapter offers a unified solution to the problems about transitivity and supplementation. One key ingredient in the solution is the view that parthood is a four-place relation expressed by ‘x at y is a part of z at w’. Another key ingredient is the view that the semantic contents of predicates and sentential connectives have ‘slots’ or ‘argument positions’ in them. (Both ingredients are independently motivated elsewhere.) Four-place analogues of the transitivity and supplementation principles are set out, and it is argued that these are not threatened by the examples from Frege and Bynoe. (shrink)

An important objection to sententialist theories of attitude reports is that they cannot accommodate the principle that one cannot know that someone believes that p without knowing what it is that he believes. This paper argues that a parallel problem arises for propositionalist accounts that has gone largely unnoticed, and that, furthermore, the usual resources for the propositionalist do not afford an adequate solution. While non-standard solutions are available for the propositionalist, it turns out that there are parallel solutions that (...) are available for the sententialist. Since the difficulties raised seem to show that the mechanism by which sentential complements serve to inform us about attitudes and about sentence meaning does not depend on their referring to propositions, this casts doubt on whether talk of propositions should retain a significant theoretical role in the enterprise of understanding thought, language and communication. (shrink)

Our online interaction with information-systems may well provide the largest arena of formal logical reasoning in the world today. Presented here is a critique of the foundations of Logic, in which the metaphysical assumptions of such 'closed world' reasoning are contrasted with those of traditional logic. Closed worlds mostly employ a syntactic alternative to formal language namely, recording data in files. Whilst this may be unfamiliar as logical syntax, it is argued here that propositions are expressed by data stored (...) in files which are essentially non-linguistic and so cannot be expressed by simple formulae F(a), with the inference-rules normally used in Logic. Hence, the syntax of data may be said to define a fundamentally new kind of logical form for simple propositions. In this way, the logic of closed systems is shown to be non-classical, differing from traditional logic in its truth-conditions, inferences and metaphysics. This paper will be concerned mainly with how the reference and certain inferences in such a closed system differ metaphysically from classical logic. (shrink)

This paper concerns the extent to which uncertain propositional reasoning can track probabilistic reasoning, and addresses kinematic problems that extend the familiar Lottery paradox. An acceptance rule assigns to each Bayesian credal state p a propositional belief revision method B p , which specifies an initial belief state B p (T) that is revised to the new propositional belief state B(E) upon receipt of information E. An acceptance rule tracks Bayesian conditioning when B p (E) = B p|E (T), for (...) every E such that p(E) > 0; namely, when acceptance by propositional belief revision equals Bayesian conditioning followed by acceptance. Standard proposals for uncertain acceptance and belief revision do not track Bayesian conditioning. The "Lockean" rule that accepts propositions above a probability threshold is subject to the familiar lottery paradox (Kyburg 1961), and we show that it is also subject to new and more stubborn paradoxes when the tracking property is taken into account. Moreover, we show that the familiar AGM approach to belief revision (Harper, Synthese 30(1-2):221-262, 1975; Alchourrón et al., J Symb Log 50:510-530, 1985) cannot be realized in a sensible way by any uncertain acceptance rule that tracks Bayesian conditioning. Finally, we present a plausible, alternative approach that tracks Bayesian conditioning and avoids all of the paradoxes. It combines an odds-based acceptance rule proposed originally by Levi (1996) with a non-AGM belief revision method proposed originally by Shoham (1987). (shrink)

The Feferman–Vaught theorem provides a way of evaluating a first order sentence \(\varphi \) on a disjoint union of structures by producing a decomposition of \(\varphi \) into sentences which can be evaluated on the individual structures and the results of these evaluations combined using a propositional formula. This decomposition can in general be non-elementarily larger than \(\varphi \). We introduce a “tree” generalization of the prenex normal form (PNF) for first order sentences, and show that for an input (...) sentence in this form having a fixed number of quantifier alternations, a Feferman–Vaught decomposition can be obtained in time elementary in the size of the sentence. The sentences in the decomposition are also in tree PNF, and further have the same number of quantifier alternations and the same quantifier rank as the input sentence. We extend this result by considering binary operations other than disjoint union, in particular sum-like operations such as join, ordered sum and NLC-sum, that are definable using quantifier-free translation schemes. (shrink)

Information-theoretic approaches to formal logic analyze the "common intuitive" concepts of implication, consequence, and validity in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; one given proposition is a consequence of a second if the latter contains all of the information contained by the former; an argument is valid if the conclusion contains no information beyond that of the (...) premise-set. This paper locates information-theoretic approaches historically, philosophically, and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves and by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyze validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. (shrink)

In formal semantics based on modern type theories, some sentences may be interpreted as judgements and some as logical propositions. When interpreting composite sentences, one may want to turn a judgemental interpretation or an ill-typed semantic interpretation into a proposition in order to obtain an intended semantics. For instance, an incorrect judgement $$a:A$$ may be turned into its propositional form $$\textsc {is}(A,a)$$ and an ill-typed application p(a) into $$\textsc {do}(p,a)$$, so that the propositional forms can take part in logical (...) compositions that interpret composite sentences, especially those that involve negations and conditionals.In this paper, we propose an operator not that facilitates such a transformation. Introducing not axiomatically, with five axiomatic laws to govern its behaviour, we shall use it to define $$\textsc {is}$$ and $$\textsc {do}$$ and give examples to illustrate its use in semantic interpretation. The introduction of not into type theories is logically consistent – this is justified by showing that not can be defined by means of the heterogeneous equality $$\textrm{JMeq}$$ so that all of the axiomatic laws for not become provable. Therefore, since the extension with $$\textrm{JMeq}$$ preserves logical consistency, so does the extension with not. We shall also study conditions under which $$\textsc {is}$$ and $$\textsc {do}$$ operators can be used safely without the risk of over-generation. (shrink)

In discussing propositional quantifiers we have considered two kinds of variables: variables occupying the argument places of connectives, and variables occupying the argument places of predicates.We began with languages which contained the first kind of variable, i.e., variables taking sentences as substituends. Our first point was that there appear to be no sentences in English that serve as adequate readings of formulas containing propositional quantifiers. Then we showed how a certain natural and illuminating extension of English by prosentences did provide (...) perspicuous readings. The point of introducing prosentences was to provide a way of making clear the grammar of propositional variables: propositional variables have a prosentential character — not a pronominal character. Given this information we were able to show, on the assumption that the grammar of propositional variables in philosopher's English should be determined by their grammar in formal languages (unless a separate account of their grammar is provided), that propositional variables can be used in a grammatically and philosophically acceptable way in philosophers' English. According to our criteria of well-formedness Carnap's semantic definition of truth does not lack an ‘essential’ predicate - despite arguments to the contrary. It also followed from our account of the prosentential character of bound propositional variables that in explaining propositional quantification, sentences should not be construed as names.One matter we have not discussed is whether such quantification should be called ‘propositional’, ‘sentential’, or something else. As our variables do not range over (they are not terms) either propositions, or sentences, each name is inappropriate, given the usual picture of quantification. But we think the relevant question in this context is, are we obtaining generality with respect to propositions, sentences, or something else?Because people have argued that all bound variables must have a pronominal character, we presented and discussed in the third section languages in which the variables take propositional terms as substituends. In our case we included names of propositions, that-clauses, and names of sentences in the set of propositional terms. We made a few comparisons with the languages discussed in the second section. We showed among other things how a truth predicate could be used to obtain generality. In contrast, the languages of the second section, using propositional variables, obtain generality without the use of a truth predicate. (shrink)

Semantic information is usually supposed to satisfy the veridicality thesis: p qualifies as semantic information only if p is true. However, what it means for semantic information to be true is often left implicit, with correspondentist interpretations representing the most popular, default option. The article develops an alternative approach, namely a correctness theory of truth (CTT) for semantic information. This is meant as a contribution not only to the philosophy of information but also to the philosophical debate on the nature (...) of truth. After the introduction, in Sect. 2, semantic information is shown to be translatable into propositional semantic information (i). In Sect. 3, i is polarised into a query (Q) and a result (R), qualified by a specific context, a level of abstraction and a purpose. This polarization is normalised in Sect. 4, where [Q + R] is transformed into a Boolean question and its relative yes/no answer [Q + A]. This completes the reduction of the truth of i to the correctness of A. In Sects. 5 and 6, it is argued that (1) A is the correct answer to Q if and only if (2) A correctly saturates Q by verifying and validating it (in the computer science’s sense of “verification” and “validation”); that (2) is the case if and only if (3) [Q + A] generates an adequate model (m) of the relevant system (s) identified by Q; that (3) is the case if and only if (4) m is a proxy of s (in the computer science’s sense of “proxy”) and (5) proximal access to m commutes with the distal access to s (in the category theory’s sense of “commutation”); and that (5) is the case if and only if (6) reading/writing (accessing, in the computer science’s technical sense of the term) m enables one to read/write (access) s. Sect. 7 provides some further clarifications about CTT, in the light of semantic paradoxes. Section 8 draws a general conclusion about the nature of CTT as a theory for systems designers not just systems users. In the course of the article all technical expressions from computer science are explained. (shrink)

We present an informational view of classical propositional logic that stems from a kind of informational semantics whereby the meaning of a logical operator is specified solely in terms of the information that is actually possessed by an agent. In this view the inferential power of logical agents is naturally bounded by their limited capability of manipulating “virtual information”, namely information that is not implicitly contained in the data. Although this informational semantics cannot be expressed by any finitely-valued matrix, it (...) can be expressed by a non-deterministic 3-valued matrix that was first introduced by W.V.O. Quine, but ignored by the logical community. Within the general framework presented in [21] we provide an in-depth discussion of this informational semantics and a detailed analysis of a specific infinite hierarchy of tractable approximations to classical propositional logic that is based on it. This hierarchy can be used to model the inferential power of resource-bounded agents and admits of a uniform proof-theoretical characterization that is half-way between a classical version of Natural Deduction and the method of semantic tableaux. (shrink)

Abstract: According to the Veridicality Thesis, information requires truth. On this view, smoke carries information about there being a fire only if there is a fire, the proposition that the earth has two moons carries information about the earth having two moons only if the earth has two moons, and so on. We reject this Veridicality Thesis. We argue that the main notions of information used in cognitive science and computer science allow A to have information about the obtaining of (...) p even when p is false. (shrink)

The idea is that, in a wide range of contexts, utterances of the sentences in (a) in each case will communicate the assumption in (b) in each case (or something closely akin to it, there being a certain amount of contextually governed variation in the speaker's propositional attitude and so the scope of the negation). These scalar inferences are taken to be one kind of (generalized) conversational implicature. As is the case with pragmatic inference quite generally, these inferences are defeasible (...) (cancellable), which distinguishes them from entailments, and they are nondetachable, which distinguishes them from conventional implicatures. The core idea is that the choice of a weaker element from a scale of elements ordered in terms of semantic strength (that is, numbers of entailments) tends to implicate that, as far as the speaker knows, none of the stronger elements in the scale holds in this instance. The pattern is quite clear in (1) and (2), where the weak/strong alternatives are some/all and five/six respectively. In the case of (3), the stronger expression must be intelligent and good-hearted which entails intelligent; what Y's utterance implicates is that Mary does not have the two properties: intelligence and good-heartedness, so that, given the proposition expressed (Mary is intelligent) it follows, deductively, that she is not good-hearted, in Y's opinion. The example in (4) involves a scale inversion due to the negation, so that the weak/strong alternatives are not necessarily/not possibly; the negation which the scalar inference generates creates a double negation, which is eliminated giving possibly. (shrink)

Informational semantics were first developed as an interpretation of the model-theory of substructural (and especially relevant) logics. In this paper we argue that such a semantics is of independent value and that it should be considered as a genuine alternative explication of the notion of logical consequence alongside the traditional model-theoretical and the proof-theoretical accounts. Our starting point is the content-nonexpansion platitude which stipulates that an argument is valid iff the content of the conclusion does not exceed the combined content (...) of the premises. We show that this basic platitude can be used to characterise the extension of classical as well as non-classical consequence relations. The distinctive trait of an informational semantics is that truth-conditions are replaced by information-conditions. The latter leads to an inversion of the usual order of explanation: Considerations about logical discrimination (how finely propositions are individuated) are conceptually prior to considerations about deductive strength. Because this allows us to bypass considerations about truth, an informational semantics provides an attractive and metaphysically unencumbered account of logical consequence, non-classical logics, logical rivalry and pluralism about logical consequence. (shrink)

This paper combines propositional dynamic logic ) with propositional inquisitive logic ). The result of this combination is a logical system \ that conservatively extends both \ and \, and, moreover, allows for an interaction of the question-forming operator from \ with the structured modalities from \. We study this system from a semantic as well as a syntactic point of view. These two perspectives are linked via a completeness proof, which also shows that \ is decidable.

Contemporary epistemology has assumed that knowledge is represented in sentences or propositions. However, a variety of extensions and alternatives to this view have been proposed in other areas of investigation. We review some of these proposals, focusing on (1) Ryle's notion of knowing how and Hanson's and Kuhn's accounts of theory-laden perception in science; (2) extensions of simple propositional representations in cognitive models and artificial intelligence; (3) the debate concerning imagistic versus propositional representations in cognitive psychology; (4) recent treatments (...) of concepts and categorization which reject the notion of necessary and sufficient conditions; and (5) parallel distributed processing (connectionist) models of cognition. This last development is especially promising in providing a flexible, powerful means of representing information nonpropositionally, and carrying out at least simple forms of inference without rules. Central to several of the proposals is the notion that much of human cognition might consist in pattern recognition rather than manipulation of rules and propositions. (shrink)

An information recovery problem is the problem of constructing a proposition containing the information dropped in going from a given premise to a given conclusion that folIows. The proposition(s) to beconstructed can be required to satisfy other conditions as well, e.g. being independent of the conclusion, or being “informationally unconnected” with the conclusion, or some other condition dictated by the context. This paper discusses various types of such problems, it presents techniques and principles useful in solving them, and it (...) develops algorithmic methods for certain classes of such problems. The results are then applied to classical number theory, in particular, to questions concerning possible refinements of the 1931 Gödel Axiom Set, e.g. whether any of its axioms can be analyzed into “informational atoms”. Two propositions are “informationally unconnected” [with each other] if no informative (nontautological) consequence of one also follows from the other. A proposition is an “informational atom” if it is informative but no information can be dropped from it without rendering it uninformative (tautological). Presentation, employment, and investigation of these two new concepts are prominent features of this paper. (shrink)

The aim of this paper1 is to provide a unified semantic analysis for three important readings of the Italian Imperfetto (and Presente): the PROGressive, the HABitual, and the FUTurate reading. To highlight the role of the utterance context in setting the relevant parameters of interpretation, explicit temporal adverbials are left out of the scene and prominence is given to the situations where the context provides the temporal information required to discriminate between alternative readings, by exploiting a single logical form. The (...) paper is organized as follows. After a short presentation of the data (sect. 1), in the second section I discuss some intuitive features of imperfectivity by focusing on the fact that the conclusion of an event or series of events is left open. This indeterminacy with respect to a given perspective point is formalized in the third section of the paper by resorting to a branching time model. In the third section a unitary treatment (based on the derivation of a single logical form) is proposed for the three main readings of tenses such as the Presente and the Imperfetto and for an intriguing side effect that, in some particular circumstances, makes a "perfective" reading possible. Since the context has a key role in this reconstruction of imperfectivity, the last section of the paper is devoted to the consideration of the temporal parameters to which the evaluation of an utterance is relativized. (shrink)

In this paper, we present a propositional logic (called mixed logic) containing disjoint copies of minimal, intuitionistic and classical logics. We prove a completeness theorem for this logic with respect to a Kripke semantics. We establish some relations between mixed logic and minimal, intuitionistic and classical logics. We present at the end a sequent calculus version for this logic.

Semantic information is usually supposed to satisfy the veridicality thesis: p qualifies as semantic information only if p is true. However, what it means for semantic information to be true is often left implicit, with correspondentist interpretations representing the most popular, default option. The article develops an alternative approach, namely a correctness theory of truth (CTT) for semantic information. This is meant as a contribution not only to the philosophy of information but also to the philosophical debate on the nature (...) of truth. After the introduction, in Sect. 2, semantic information is shown to be translatable into propositional semantic information (i). In Sect. 3, i is polarised into a query (Q) and a result (R), qualified by a specific context, a level of abstraction and a purpose. This polarization is normalised in Sect. 4, where [Q + R] is transformed into a Boolean question and its relative yes/no answer [Q + A]. This completes the reduction of the truth of i to the correctness of A. In Sects. 5 and 6, it is argued that (1) A is the correct answer to Q if and only if (2) A correctly saturates Q by verifying and validating it (in the computer science’s sense of verification and validation ); that (2) is the case if and only if (3) [Q + A] generates an adequate model (m) of the relevant system (s) identified by Q; that (3) is the case if and only if (4) m is a proxy of s (in the computer science’s sense of proxy ) and (5) proximal access to m commutes with the distal access to s (in the category theory’s sense of commutation ); and that (5) is the case if and only if (6) reading/writing (accessing, in the computer science’s technical sense of the term) m enables one to read/write (access) s. Sect. 7 provides some further clarifications about CTT, in the light of semantic paradoxes. Section 8 draws a general conclusion about the nature of CTT as a theory for systems designers not just systems users. In the course of the article all technical expressions from computer science are explained. (shrink)

According to contextualist theories in metaethics, when you use a moral term in a context, the context plays an ineliminable part in determining what natural property will be the semantic value of the term. Furthermore, on subjectivist and relativist versions of these views, it is either the speaker's own moral code or her moral community's moral code that constitutes the reference-fixing context. One standard objection to views of this type is that they fail to enable us to disagree in ordinary (...) conversations. In this chapter, I develop a new response to this objection on the basis of Kai von Fintel and Anthony Gillies' notion of proposition clouds. I argue that, because we live in a multicultural society, the conversational contexts we face will fail to disambiguate between all the things we could mean. This is why we can at best put into play proposition clouds when we make moral utterances. All the propositions in such clouds are then available for rejection and acceptance on the behalf of our audiences. The norms of conversation then guide us to make informative contributions to the conversation - accept and reject propositions in a way that leads to co-ordination of action and choice. (shrink)

Scroggs's theorem on the extensions of S5 is an early landmark in the modern mathematical studies of modal logics. From it, we know that the lattice of normal extensions of S5 is isomorphic to the inverse order of the natural numbers with infinity and that all extensions of S5 are in fact normal. In this paper, we consider extending Scroggs's theorem to modal logics with propositional quantifiers governed by the axioms and rules analogous to the usual ones for ordinary quantifiers. (...) We call them Π-logics. Taking S5Π, the smallest normal Π-logic extending S5, as the natural counterpart to S5 in Scroggs's theorem, we show that all normal Π-logics extending S5Π are complete with respect to their complete simple S5 algebras, that they form a lattice that is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N, that they have arbitrarily high Turing-degrees, and that there are non-normal Π-logics extending S5Π. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:“Unauthorized Propositions”The Federalist Papers and Constituent PowerJason Frank (bio)The PEOPLE, who are the sovereigns of the State, possess a power to alter it when and in what way they please. To say otherwise is to make the thing created, greater than the power that created it.—Anonymous, Federal Gazette, March 18, 1789The we of the Constitution’s “We the People” was as much of an artificial construct as the Constitution (...) itself, it was its creature, not its creator.—Sheldon Wolin, The Presence of the PastIn Federalist No. 40, James Madison engaged the vexing question of whether the Philadelphia Convention of 1787 had been properly authorized “to frame and propose” the new Federal Constitution of the United States [83]. Madison explored there “the ground on which the Convention stood” [264], which he identified as the purportedly solid foundation of the people themselves [Morgan 272]. Federalist No. 40 offers a theoretically probing yet avowedly practical reflection on the availability of foundations and on dilemmas of popular authorization in times of constitutional crisis; it may be productively read as an exemplary eighteenth-century American reflection on a state of exception. As such, Federalist No. 40 provides helpful orientation to how the popular constitutionalism of the revolutionary and postrevolutionary years confronted and navigated—if never fully resolved—some of the most insistent paradoxes of modern democratic theory.In No. 40, Madison was preoccupied with the theoretical and historical dilemmas surrounding the eighteenth-century American conception of the people’s constituent power—that is, the ultimate power of the people to alter fundamental law “when and in what way they please.”1 However, in contrast to most contemporary theoretical approaches to constituent power—whether influenced by the political theology of Carl Schmitt or the immanentist optimism of Michael Hardt and Antonio Negri—Madison’s navigation of the attending dilemmas of popular authorization resisted the turn to metaphysical abstraction and the appeal to a transcendent or immanent absolute. In his approach to the problem of constituent power, Madison insisted on the productive irreducibility of institutional mediation and the performative elicitation of a retrospective authorization by the people. While accepting the absence of a firm ground of legal authorization (the unbroken authority of law), Madison also rejected a wholly extralegal or unmediated appeal to the people themselves (the transcendental appeal to spirit beyond the letter of the law). Instead, he endorsed a political navigation of these dilemmas in medias res. Confronting [End Page 103] the dilemma of constituency—how does the people give birth to itself as a self-authorizing subject?—Madison shifted the register of discourse away from the formal and juridical and toward the informally ethicopolitical. Doing so, he was able to affirm a rupture in law while simultaneously enlisting another register of normative continuity. This essay outlines Madison’s productive encounter with these dilemmas in The Federalist, and suggests the relevance of his example for recent debates in democratic theory, too often captivated by the formal rubrics of “norm” and “exception.”1The government which [the Federalists] are so enthusiastically fond of is as yet an ideal phantom, a chimera, a mere theory detested and execrated by every true friend to government.—“One of the People,” Carlisle Gazette, January 9, 1788Madison wrote Federalist No. 40 as a response to widespread and persuasive Anti-Federalist claims that the Philadelphia Convention had exceeded the mandate granted to it by the Annapolis Convention of 1786 and the Congressional Act of 1787, both of which authorized the meeting of the Philadelphia Convention “for the sole and express purpose of revising the Articles of Confederation” [259]. While the Philadelphia Convention was, in Bruce Ackerman’s words, “expressly subordinated to existing institutions and procedures” [Ackerman 43], it had nonetheless produced an entirely new constitutional text and—theoretically more problematic for Madison—an entirely new procedure for its popular ratification [see also Ackerman and Katyal]. Under Article 13 of the Articles of Confederation, constitutional reform could come about only through the unanimous assent of the thirteen state legislatures.2 Under the Philadelphia Convention’s plan, the ratification of the proposed constitution would bypass the constituted power of the state legislatures and appeal instead to the... (shrink)