In this paper I describe the birth of Russell’s notion of a propositionalfunction on 3 May 1902 and its immediate context and implications. In particular, I consider its significance in relation to the development of his views on analysis.
In this paper I describe the birth of Russell’s notion of a propositionalfunction on 3 May 1902 and its immediate context and implications. In particular, I consider its significance in relation to the development of his views on analysis.
Wittgenstein's Tractatus carefully distinguished the concept all from\nthe notion of a truth-function, and thereby from the quantifiers.\nI argue that Wittgenstein's rationale for this distinction is lost\nunless propositional functions are understood within the context\nof his picture theory of the proposition. Using a model Tractatus\nlanguage, I show how there are two distinct forms of generality implicit\nin quantified Tractatus propositions. Although the explanation given\nin the Tractatus for this distinction is ultimately flawed, the distinction\nitself is a genuine one, and the forms of (...) generality that Wittgenstein\nindicated can be seen in the quantified sentences of contemporary\nlogic. (shrink)
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)
This paper considers two reasons that might support Russell’s choice of a ramified-type theory over a simple-type theory. The first reason is the existence of purported paradoxes that can be formulated in any simple-type language, including an argument that Russell considered in 1903. These arguments depend on certain converse-compositional principles. When we take account of Russell’s doctrine that a propositionalfunction is not a constituent of its values, these principles turn out to be too implausible to make these (...) arguments troubling. The second reason is conditional on a substitutional interpretation of quantification over types other than that of individuals. This reason stands up to investigation: a simple-type language will not sustain such an interpretation, but a ramified-type language will. And there is evidence that Russell was tacitly inclined towards such an interpretation. A strong construal of that interpretation opens a way to make sense of Russell’s simultaneous repudiation of propositions and his willingness to quantify over them. But that way runs into trouble with Russell’s commitment to the finitude of human understanding. (shrink)
In the Tractatus Logico-Philosophicus , a name is always a propositionalfunction. Wittgenstein makes a radical shift in the Fregean opposition between saturated and unsaturated entities. Any sentential component which is not itself a sentence is unsaturated. The proposition is therefore a synthesis of propositional functions. The name is just a limit case of propositionalfunction, and as such it can be negated.
Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that (...) depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement. The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, which studies logical operators and connectives that are used to produce complex statements whose truth-value depends entirely on the truth-values of the simpler statements making them up, and in which it is assumed that every statement is either true or false and not both. However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectives that are used to produce statements whose truth-values depend not simply on the truth-values of the parts, but additional things such as their necessity, possibility or relatedness to one another. (shrink)
The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p -instantiations of A . It is also shown that without the admissibility qualification many of the systems considered (...) are semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap’s system E of entailment, extended by the mingle axiom and the Ackermann constant t . The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras. (shrink)
The folk Psychology frames propositional attitudes as fundamental theoretical entities for the construction of a model designed to predict the behavior of a subject. A trivial, such as grasping a pen and writing reveals - something complex - about the behavior. When I take a pen and start writing I do, trivially, because I believe that a certain object in front of me is a pen and who performs a specific function that is, in fact, that of writing. (...) When I believe that the object that stands before me is a pen, I am in relation to "believe" with the propositional content: that in front of me is a pen. Philosophers of the proposition, from Frege onwards, have dedicated their studies to the analysis of what kinds of entities are the propositional attitudes. Jerry Fodor says that now, the proper prediction of the psychology of common sense, can not be questioned and that the propositional attitudes represent the most effective way to describe our behavior. What Fodor says, however, is that propositional attitudes function, but not how they work. Most philosophers interested in the issue, we are dedicated to the search for a theory that can account consistently both a semantics for propositional attitudes, both of these entities that seem to cause the behavior of a rational subject. There are two main paradigms in the theory of the proposition that contributed to the discussion of the propositional attitudes. One is the one that begins with Gottlob Frege, the other with Bertrand Russell. Defenders of Frege argue that the paradigm scrub objects and properties can not be constituents of the propositional content which have a purely conceptual. In other words, the philosophers belonging to the paradigm of Frege, but not all, mean that you can test in a rigorous way the truth conditions of propositional attitudes. Who defends the russellian’s paradigm argues that the propositional content are made by the objects and properties on which propositional attitudes relate. The purpose of this article is not to rebuild - in detail - both paradigms, nor to reconstruct one but, in a sense, my work will be a completely partial objective is to demonstrate how the paradigm is more profitable russell not only to make a coherent semantic theory for propositional attitudes2 but also to predict the behavior of a rational subject thing, completely innovative, given the repeated objections in contemporary literature3. At the end of this paper will be drafted a proposal to build a consistent model to predict the behavior,of a rational agent, based on a referential theory of propositional attitudes. (shrink)
Arguments are given against the thesis that properties and propositional functions are identical. The first shows that the familiar extensional treatment of propositional functions -- that, for all x, if f(x) = g(x), then f = g -- must be abandoned. Second, given the usual assumptions of propositional-function semantics, various propositional functions (e.g., constant functions) are shown not to be properties. Third, novel examples are given to show that, if properties were identified with propositional (...) functions, crucial fine-grained intensional distinctions would be lost. (shrink)
The standard truth-conditional semantics for substitutional quantification, due to Saul Kripke, does not specify what proposition is expressed by sentences containing the particular substitutional quantifier. In this paper, I propose an alternative semantics for substitutional quantification that does. The key to this semantics is identifying an appropriate propositionalfunction to serve as the content of a bound occurrence of a formula containing a free substitutional variable. I apply this semantics to traditional philosophical reasons for interest in substitutional quantification, (...) namely, theories of truth and ontological commitment. (shrink)
Peter Geach has said that Russell's use of ‘propositionalfunction’ is ‘hopelessly confused and inconsistent’. Geach is right, and attempts to say what exactly a Russellian propositionalfunction is, or is supposed to be, are bound to end in frustration. Nevertheless, it may be worthwhile to pursue an account of propositional functions that accommodates a good deal of what Russell says about them and that can provide some of what he expected of them.
The most common account of attitude reports is the relational analysis according towhich an attitude verb taking that-clause complements expresses a two-placerelation between agents and propositions and the that-clause acts as an expressionwhose function is to provide the propositional argument. I will argue that a closerexamination of a broader range of linguistic facts raises serious problems for thisanalysis and instead favours a Russellian `multiple relations analysis' (which hasgenerally been discarded because of its apparent obvious linguistic implausibility).The resulting account (...) can be given independent philosophical motivations within anintentionalist view of truth and predication. (shrink)
I argue for two claims. First I argue against the consensus view that accurate behavioral prediction based on accurate representation of cognitive states, i.e. mind reading , is the sustaining function of propositional attitude ascription. This practice cannot have been selected in evolution and cannot persist, in virtue of its predictive utility, because there are principled reasons why it is inadequate as a tool for behavioral prediction. Second I give reasons that favor an alternative account of the sustaining (...)function of propositional attitude ascription. I argue that it serves a mind-shaping function. Roughly, propositional attitude ascription enables human beings to set up regulative ideals that function to mold behavior so as to make it easier to coordinate with. (shrink)
Determiner phrases embedded under a propositional attitude verb have traditionally been taken to denote answers to implicit questions. For example, 'the capital of Vermont' as it occurs in 'John knows the capital of Vermont' has been thought to denote the proposition which answers the implicit question 'what is the capital of Vermont?' Thus, where 'know' is treated as a propositional attitude verb rather than an acquaintance verb, 'John knows the capital of Vermont' is true iff John knows that (...) Montpelier is the capital of Vermont. The traditional view lost its popularity long ago, because it was thought to rest on the controversial assumption that determiner phases embedded under a propositional attitude verb function semantically in the same way as the corresponding wh -clauses. Here we defend the traditional assumption against objections. We then argue that wh -clauses are not to be given a uniform treatment as indirect questions. When occurring under a propositional attitude verb, wh -clauses are better treated as having a predicate-type semantic value. We conclude by considering some possible objections to the predicate view. (shrink)
Theories that seek to explain the status of psychological states experienced in fictional contexts either claim that those states are special propositional attitudes specific to fictional contexts (make-believe attitudes), or else define them as normal propositional attitudes by stretching the concept of a propositional attitude to include ‘objectless’ states that do not imply constraints such as truth or satisfaction. I argue that the first theory is either vacuous or false, and that the second, by defining the reality (...) of the states in question only nominally, risks having a result similar to the first. Then I put forward an explanation of how propositional attitudes function in fictional contexts which meets the following requirements: (i) does not postulate the existence of attitudes specific to or definitive of fictionality; (ii) does not imply that we transgress our knowledge of the ontological claims of fictions for some attitudes (for example, fear) but not others (belief); (iii) explains how we can adopt normal propositional attitudes towards fictions; (iv) allows explanation of how attitudes adopted during fictional response connect or are relevant to our broader systems of belief and volition. (shrink)
Although ‘glue semantics’ is the most extensively developed theory of semantic composition for LFG, it is not very well integrated into the LFG projection architecture, due to the absence of a simple and well-explained correspondence between glue-proofs and f-structures. In this paper I will show that we can improve this situation with two steps: (1) Replace the current quantificational formulations of glue (either Girard’s system F, or first order linear logic) with strictly propositional linear logic (the quantifier, unit and (...) exponential free version of either MILL or ILL, depending on whether or not tensors are used). (2) Reverse the direction of the standard σ-projection from f-structure to meaning, giving one going from the (atomic nodes of) the glue-proof to the f-structure, rather than from the f-structure to a ‘semantic projection’ which is itself somehow related to the glue-proof. As a side effect, the standard semantic projection of LFG glue semantics can be dispensed with. A result is that LFG sentence structures acquire a level composed of strictly binary trees, constructed out of nodes representing function application and lambda abstraction, with a significant resemblance to external and internal merge in the Minimalist Program. This increased resemblance between frameworks might assist in making useful comparisons. (shrink)
According to the standard definition, a first-order theory is categorical if all its models are isomorphic. The idea behind this definition obviously is that of capturing semantic notions in axiomatic terms: to be categorical is to be, in this respect, successful. Thus, for example, we may want to axiomatically delimit the concept of natural number, as it is given by the pre-theoretic semantic intuitions and reconstructed by the standard model. The well-known results state that this cannot be done within first-order (...) logic, but it can be done within second-order one. Now let us consider the following question: can we axiomatically capture the semantic concept of conjunction? Such question, to be sure, does not make sense within the standard framework: we cannot construe it as asking whether we can form a first-order (or, for that matter, whatever-order) theory with an extralogical binary propositional operator so that its only model (up to isomorphism) maps the operator on the intended binary truth-function. The obvious reason is that the framework of standard logic does not allow for extralogical constants of this type. But of course there is also a deeper reason: an existence of a constant with this semantics is presupposed by the very definition of the framework1. Hence the question about the axiomatic capturability of concunction, if we can make sense of it at all, cannot be asked within the framework of standard logic, we would have to go to a more abstract level. To be able to make sense of the question we would have to think about a propositional ‘proto-language’, with uninterpreted logical constants, and to try to search out axioms which would fix the denotations of the constants as the intended truth-functions. Can we do this? It might seem that the answer to this question is yielded by the completeness theorem for the standard propositional calculus: this theorem states that the axiomatic delimitation of the calculus and the semantic delimitation converge to the same result.. (shrink)
In Logical Forms Part II, Chateaubriand begins the Chapter on “Propositional Logic” by considering the reading of the ‘conditional’ by ‘implies’; in fact he states that:There is a confusion, as a matter of fact, and it runs deep, but it is a confusion in propositional logic itself, and the mathematician’s reading is a rather sensible one.After a careful, erudite analysis of various philosophical viewpoints of logic, Chateaubriand comes to the conclusion that:Pure propositional logic, as just characterized, belongs (...) to ontological logic, and it does not include a theory of deduction as a human activity. This is a part of epistemological logic, and is more closely connected to the applications of pure propositional logic.An implicit assumption in Chateaubriand’s reasoning appears to be that propositions have a timeless status. I will present arguments for the opposite viewpoint which leads to an analysis of Propositional Logic not covered under Chateaubriand’s monograph and perhaps resolves some conflicts therein; much as the conflict between the Intuitionist and Classical Mathematician on whether every function on the Reals is continuous is resolved by the realization that they are talking about different “entities”.Em Logical Forms II, Chateaubriand inicia o capítulo “Lógica Proposi-cional” considerando a leitura do ‘condicional’ como ‘implica’. De fato, ele diz o seguinte:Na verdade, existe uma confusão, e ela é profunda, mas é uma confusão na lógica proposicional ela mesma, e a leitura de um matemático é bastante sensível.Depois de uma análise cuidadosa e erudita dos vários pontos de vista filosóficos da lógica , Chateaubriand chega à conclusão que:A lógica proposicional pura, tal como aqui caracterizada, pertence à lógica ontológica, e não inclui uma teoria da dedução como atividade humana. Isto é parte da lógica epistemológica, e é mais intimamente conectada às aplicações da lógica proposicional.Uma premissa implícita no raciocínio de Chateaubriand parece ser a de que proposições têm um estatuto atemporal. Eu argumentarei em favor da visão oposta, que leva a uma análise da Lógica Proposicional não abordada no texto de Chateaubriand e que talvez resolva alguns conflitos. Muito do conflito entre Intuicionistas e Matemáticos Clássicos sobre se toda função sobre os números reais é contínua é resolvido pela compreensão de que eles estão falando de “entidades” diferentes. (shrink)
In this article I will develop the ﬁrst steps of a wholly general theory of how indexical and reﬂexive pronouns function in propositional attitude ascriptions. This will involve a theory of ascriptions of de se beliefs and de se utterances, which can probably be also generalized so as to apply to ascriptions of other attitudes. It will also involve a theory about the ascriptions of beliefs or other attitudes a person has at a time about what happens then (...) (attitudes de praesente, as they are sometimes called) and the beliefs of a person concerning the one whom he is addressing (which I might call beliefs de recipiente) etc.. The most distinctive aspect of the theory will be that I will argue that many phenomena associated with such ascriptions that are nowadays most often viewed as pragmatic are semantic. I will use a system of symbolic logic to formalize such ascriptions. I will start from David Kaplan’s Logic of Demonstratives and generalize it into a logic I call Doxastic Logic of Demonstratives, DLD. Crucial to the semantics of the logic will be an exact deﬁnition of the adjustments of a character from one context to another. (shrink)
The final portion of the thesis surveys proposals for the introduction of hidden variables into quantum mechanics, proofs of the impossibility of such hidden-variable proposals, and criticisms of these impossibility proofs. And arguments in favour of the partial-Boolean algebra, rather than the orthomodular lattice, formalization of the quantum propositional structures are reviewed. ;As for , each quantum state-induced expectation-function on a P truth-functionally assigns 1 and 0 values to the elements in a ultrafilter and dual ultraideal of P, (...) where in general the union of an ultrafilter and its dual ultraideal is smaller than the entire structure. Thus it is argued that each expectation-function is the quantum analog of a classical semantic mapping, even though the domain where each expectation-function is bivalent and truth-functional is usually a non-Boolean substructure of P. ;In classical propositional logic, propositions form a Boolean algebra, and each semantic mapping assigns the value 1 to the members of a certain subset of the algebra, namely, an ultrafilter, and assigns 0 to the members of the dual ultraideal, where the union of these two subsets is the entire algebra. The propositional structures of classical mechanics are likewise Boolean algebras, so one can straightforwardly provide a classical semantics, which also satisfies . However, quantum propositional structures are non-Boolean, so it is an open question whether a semantics satisfying , and can be provided. ;Von Neumann first proposed that the algebraic structures of the subspaces of Hilbert space be regarded as the propositional structures P of quantum mechanics. These structures have been formalized in two ways: as orthomodular lattices which have the binary operations "and", "or", defined among all elements, compatible and incompatible ; and as partial-Boolean algebras which have the binary operations defined among only compatible elements. In the thesis, two basic senses in which these structures are non-Boolean are discriminated. And two notions of truth-functionality are distinguished: truth-functionality ,) applicable to the Plattices; and truth-functionality ) applicable to both the Plattices and partial-Boolean algebras. Then it is shown how the lattice definitions of "and", "or", among incompatibles rule out a bivalent, truth-functional ,) semantics for any PQM lattice containing incompatible elements. In contrast, the Gleason and Kochen-Specker proofs of the impossibility of hidden-variables for quantum mechanics show the impossibility of a bivalent, truth-functional ) semantics for three-or-higher dimensional Hilbert space PQM structures; and the presence of incompatible elements is necessary but is not sufficient to rule out such a semantics. ;The thesis investigates the possibility of a classical semantics for quantum propositional structures. A classical semantics is defined as a set of mappings each of which is bivalent, i.e., the value 1 or 0 is assigned to each proposition, and truth-functional, i.e., the logical operations are preserved. In addition, this set must be "full", i.e., any pair of distinct propositions is assigned different values by some mapping in the set. When the propositions make assertions about the properties of classical or of quantum systems, the mappings should also be "state-induced", i.e., values assigned by the semantics should accord with values assigned by classical or by quantum mechanics. (shrink)
In this paper we give a new proof of Richardson's theorem : a global function G[MATHEMATICAL DOUBLE-STRUCK CAPITAL A] of a cellular automaton [MATHEMATICAL DOUBLE-STRUCK CAPITAL A] is injective if and only if the inverse of G[MATHEMATICAL DOUBLE-STRUCK CAPITAL A] is a global function of a cellular automaton. Moreover, we show a way how to construct the inverse cellular automaton using the method of feasible interpolation from . We also solve two problems regarding complexity of cellular automata formulated (...) by Durand . (shrink)
The paper is dedicated to the 80th birthday of the outstanding Russian logician A.V. Kuznetsov. It is addressing a history of the ideas and research conducted by him in non-classical and intermediate logics.
Along with offering an historically-oriented interpretive reconstruction of the syntax of PM ( rst ed.), I argue for a certain understanding of its use of propositionalfunction abstracts formed by placing a circum ex on a variable. I argue that this notation is used in PM only when de nitions are stated schematically in the metalanguage, and in argument-position when higher-type variables are involved. My aim throughout is to explain how the usage of function abstracts as “terms” (...) (loosely speaking) is not inconsistent with a philosophy of types that does not think of propositional functions as mind- and language-independent objects, and adopts a nominalist/substitutional semantics instead. I contrast PM’s approach here both to function abstraction found in the typed λ-calculus, and also to Frege’s notation for functions of various levels that forgoes abstracts altogether, between which it is a kind of intermediary. (shrink)
ABSTRACT: An introduction to Stoic logic. Stoic logic can in many respects be regarded as a fore-runner of modern propositional logic. I discuss: 1. the Stoic notion of sayables or meanings (lekta); the Stoic assertibles (axiomata) and their similarities and differences to modern propositions; the time-dependency of their truth; 2.-3. assertibles with demonstratives and quantified assertibles and their truth-conditions; truth-functionality of negations and conjunctions; non-truth-functionality of disjunctions and conditionals; language regimentation and ‘bracketing’ devices; Stoic basic principles of propositional (...) logic; 4. Stoic modal logic; 5. Stoic theory of arguments: two premisses requirement; validity and soundness; 6. Stoic syllogistic or theory of formally valid arguments: a reconstruction of the Stoic deductive system, which consisted of accounts of five types of indemonstrable syllogisms, which function as nullary argumental rules that identify indemonstrables or axioms of the system, and four deductive rules (themata) by which certain complex arguments can be reduced to indemonstrables and thus shown to be formally valid themselves; 7. arguments that were considered as non-syllogistically valid (subsyllogistic and unmethodically concluding arguments). Their validity was explained by recourse to formally valid arguments. (shrink)
The positions of Frege, Russell and Wittgenstein on the priority of complexes over (propositional) functions are sketched, challenging those who take the "judgment centered" aspects of the Tractatus to be inherited from Frege not Russell. Frege's views on the priority of judgments are problematic, and unlike Wittgenstein's. Russell's views on these matters, and their development, are discussed in detail, and shown to be more sophisticated than usually supposed. Certain misreadings of Russell, including those regarding the relationship between propositional (...) functions and universals, are exposed. Wittgenstein's and Russell's views on logical grammar are shown to be very similar. Russell's type theory does not countenance types of genuine entities nor metaphysical truths that cannot be put into words, contrary to conventional wisdom. I relate this to the debate over "inexpressible truths" in the Tractatus. I lastly comment on the changes to Russell's views brought about by Wittgenstein's influence. (shrink)
This paper argues that there is a problem for the justificatory significance of perceptions that has been overlooked thus far. Assuming that perceptual experiences are propositional attitudes and that only propositional attitudes which assertively represent the world can function as justifiers, the problem consists in specifying what it means for a propositional attitude to assertively represent the world without losing the justificatory significance of perceptions—a challenge that is harder to meet than might first be thought. That (...) there is such a problem can be seen by reconsidering and modifying a well-known argument to the conclusion that beliefs cannot be justified by perceptions but only by other beliefs. Nevertheless, the aim of the paper is not to conclude that perceptions are actually incapable of justifying our beliefs but rather to highlight an overlooked problem that needs to be solved in order to properly understand the justificatory relationship between perceptions and beliefs. (shrink)
There has been a long tradition of characterizing man as the animal that is capable of propositional language. However, the remarkable ability of using pictures also only belongs to human beings. Both faculties however depend conceptually on the ability to refer to absent situations by means of sign acts called 'context building'. The paper investigates the combined roles of quasi-pictorial sign acts and proto-assertive sign acts in the situation of initial context building, which, in the context of “concept-genetic” considerations, (...) aims for a philosophical explanation of the anthropological functon of pictures and their relation to imagination. (shrink)
This paper provides a new approach to a family of outstanding logical and semantical puzzles, the most famous being Frege's puzzle. The three main reductionist theories of propositions (the possible-worlds theory, the propositional-function theory, the propositional-complex theory) are shown to be vulnerable to Benacerraf-style problems, difficulties involving modality, and other problems. The nonreductionist algebraic theory avoids these problems and allows us to identify the elusive nondescriptive, non-metalinguistic, necessary propositions responsible for the indicated family of puzzles. The algebraic (...) approach is also used to defend antiexistentialism against existentialist prejudices. The paper closes with a suggestion about how this theory of content might enable us to give purely semantic (as opposed to pragmatic) solutions to the puzzles based on a novel formulation of the principle of compositionality. (shrink)
Consciousness is a mongrel concept: there are a number of very different "consciousnesses." Phenomenal consciousness is experience; the phenomenally conscious aspect of a state is what it is like to be in that state. The mark of access-consciousness, by contrast, is availability for use in reasoning and rationally guiding speech and action. These concepts are often partly or totally conflated, with bad results. This target article uses as an example a form of reasoning about a function of "consciousness" based (...) on the phenomenon of blindsight. Some information about stimuli in the blind field is represented in the brains of blindsight patients, as shown by their correct "guesses," but they cannot harness this information in the service of action, and this is said to show that a function of phenomenal consciousness is somehow to enable information represented in the brain to guide action. But stimuli in the blind field are BOTH access-unconscious and phenomenally unconscious. The fallacy is: an obvious function of the machinery of access-consciousness is illicitly transferred to phenomenal consciousness. (shrink)
§1. Introduction. Although Whitehead and Russell’s Principia Mathematica (hereafter, PM ), published almost precisely a century ago, is widely heralded as a watershed moment in the history of mathematical logic, in many ways it is still not well understood. Complaints abound to the effect that the presentation is imprecise and obscure, especially with regard to the precise details of the ramified theory of types, and the philosophical explanation and motivation underlying it, all of which was primarily Russell’s responsibility. This has (...) had a large negative impact in particular on the assessment of the socalled “no class” theory of classes endorsed in PM. According to that theory, apparent reference to classes is to be eliminated, contextually, by means of higher-order “propositionalfunction”—variables and quantifiers. This could only be seen as a move in the right direction if “propositional functions,” and/or higher-order quantification generally, were less metaphysically problematic or obscure than classes themselves. But this is not the case—or so goes the usual criticism. Years ago, Geach (1972, p. 272) called Russell’s notion of a propositionalfunction “hopelessly confused and inconsistent.” Cartwright (2005, p. 915) has recently agreed, adding “attempts to say what exactly a Russellian propositionalfunction is, or is supposed to be, are bound to end in frustration.” Soames (2008) claims that “propositional functions . . . are more taken for granted by Russell than seriously investigated” (p. 217), and uses the obscurity surrounding them as partial justification for ignoring the no class theory in a popular treatment of Russell’s work (Soames, 2003).1 A large part of the usual critique involves charging Russell with confusion, or at least obscurity, with regard to what a propositionalfunction is supposed to be. Often the worry has to do with the use/mention distinction: is a propositionalfunction, or even a proposition. (shrink)
Bertrand Russell, in the second of his 1914 Lowell lectures, Our Knowledge of the External World, asserted famously that ‘every philosophical problem, when it is subjected to the necessary analysis and purification, is found either to be not really philosophical at all, or else to be, in the sense in which we are using the word, logical’ (Russell 1993, p. 42). He went on to characterize that portion of logic that concerned the study of forms of propositions, or, as he (...) called them, ‘logical forms’. This portion of logic he called ‘philosophical logic’. Russell asserted that ... some kind of knowledge of logical forms, though with most people it is not explicit, is involved in all understanding of discourse. It is the business of philosophical logic to extract this knowledge from its concrete integuments, and to render it explicit and pure. (p. 53) Perhaps no one still endorses quite this grand a view of the role of logic and the investigation of logical form in philosophy. But talk of logical form retains a central role in analytic philosophy. Given its widespread use in philosophy and linguistics, it is rather surprising that the concept of logical form has not received more attention by philosophers than it has. The concern of this paper is to say something about what talk of logical form comes to, in a tradition that stretches back to (and arguably beyond) Russell’s use of that expression. This will not be exactly Russell’s conception. For we do not endorse Russell’s view that propositions are the bearers of logical form, or that appeal to propositions adds anything to our understanding of what talk of logical form comes to. But we will be concerned to provide an account responsive to the interests expressed by Russell in the above quotations, though one clarified of extraneous elements, and expressed precisely. For this purpose, it is important to note that the concern expressed by Russell in the above passages, as the surrounding text makes clear, is a concern not just with logic conceived narrowly as the study of logical terms, but with propositional form more generally, which includes, e.g., such features as those that correspond to the number of argument places in a propositionalfunction, and the categories of objects which propositional.... (shrink)
Russell's involuted path in the development of his theory of logical types from 1903 to 1910-13 is examined and explained in terms of the development in his early philosophy of the notion of a logical subject vis-a-vis the problem of the one and many; i.e., the problem for russell, first, of a class-as-one as a logical subject as opposed to a class as many, and, secondly, of a propositionalfunction as a single and separate logical subject as opposed (...) to existing only in the many propositions that are its values. (shrink)
What in Aristotle corresponds, in whole or (more likely) in part, to our contemporary notion of predication? This paper sketches counterparts in Aristotle's text to our theories of expression and of truth, and on this basis inquires into his treatment of sentences assigning an individual to its kinds. In some recent accounts, the Metaphysics offers a fresh look at such sentences in terms of matter and form, in contrast to the simpler theory on offer in the Categories . I argue (...) that the Metaphysics initiates no change in this regard over the Categories . The point that form is (metaphysically) predicated of matter is a contribution, not to the account of statement predication, but to the analysis of compound material substances. Otherwise put, in our terms Aristotelian form is not - in particular, is not also - a propositionalfunction, but a function from matter to compound material substances. (shrink)
In this paper we show that the usual intuitionistic characterization of the decidability of the propositionalfunction B prop [x : A], i. e. to require that the predicate ∨ ¬ B) is provable, is equivalent, when working within the framework of Martin-Löf's Intuitionistic Type Theory, to require that there exists a decision function ψ: A → Boole such that = Booletrue) ↔ B). Since we will also show that the proposition x = Booletrue [x: Boole] is (...) decidable, we can alternatively say that the main result of this paper is a proof that the decidability of the predicate Bprop [x : A] can be effectively reduced by a function ψ A → Boole to the decidability of the predicate ψ = Booletrue [x : A]. All the proofs are carried out within the Intuitionistic Type Theory and hence the decision function ψ, together with a proof of its correctness, is effectively constructed as a function of the proof of ∨ ¬ B). (shrink)
It might appear to be beyond question, for Christian theism, both that God is omniscient and that omniscience includes knowledge of future truth. For it seems obvious that if P is true, then an omniscient being knows that P. P , in this propositionalfunction, is entirely general, and must therefore include propositions of the form: ‘it will be the case that X ’. If, truly, it will be the case that X , then God knows that truth.
This is a brief report on results reported at length in our paper , made for the purpose of a presentation at the workshop to be held in November 2011 in Cambridge on the Principia Mathematica of Russell and Whitehead ([?], hereinafter referred to brieﬂy as PM ). That paper grew out of a reading of the paper  of Kamareddine, Nederpelt, and Laan. We refereed this paper and found it useful for checking their examples to write our own independent (...) computer type-checker for the type system of PM (), which led us to think carefully about formalization of the language and the type system of PM A modern mathematical logician reading PM ﬁnds that it is not completely formalized in a modern sense. The type theory in particular is inarguably not formalized, as no notation for types is given at all! In PM itself, the only type annotations which appear are occasional numerical indices indicating order; the type notation we use here extends one introduced later by Ramsey. The authors of PM regard the absence of explicit indications of type as a virtue of their system: they call it “systematic ambiguity”; modern computer scientists refer to this as “polymorphism”. The language of PM is also not completely formalized, and it is typographically inconvenient for computer software to which ASCII input is to be given. The notation of PM for abstractions (propositional functions) does not use head binders; the order of the arguments of a complex expression is determined by the alphabetical order of the bound variables. For example ˆa < ˆb is the “less than” relation while ˆb < ˆa is the “greater than” relation (this is indicated by the alphabetical order of the variables). In PM , the fact that a variable is bound in a propositionalfunction is indicated by circumﬂexing it. Variables bound by quantiﬁers are not circumﬂexed. A feature of the notation of , carried over into ours, is that no circumﬂexes are used: notations for propositions and the corresponding propositional functions are identical.. (shrink)
This paper has two aims. First, it aims to provide an adverbial account of the idea of an intransitive self-consciousness and, second, it aims to argue in favor of this account. These aims both require a new framework that emerges from a critical review of Perry’s famous notion of the “unarticulated constituents” of propositional content. First, I aim to show that the idea of an intransitive self-consciousness can be phenomenologically described in an analogy with the adverbial theory of perception. (...) In an adverbial theory of perception, we do not see a blue sense-data, but we see something blue-ly, whereas in intransitive self-consciousness we are not conscious of ourselves when we undergo a conscious experience—instead, we experience something self-consciously. But what does this mean precisely? First, I take transitive self-consciousness to be the first-person operator that prefixes the content of any experience that the subject undergoes, regardless of whether or not the subject is self-referred. Further, I argue that this first-person adverbial way of entertaining a content of any experience in Perry’s revised framework fixes the subject as part of the circumstance of the evaluation of the content of her own experience. We can only evaluate whether the content is veridical of falsidical relative to the subject undergoing the experience. This is referred to here as “self-concernment without self-reference.” When I am absorbed reading a book, I do not self-represent my own experience of reading a book, let alone see myself as a constituent of the content of this experience. Even so, I experience that reading self-consciously in the precise sense that I do belong the circumstance of the evaluation of the selfless content of my experience of reading the book. The content of the experience of reading a book is simply a propositionalfunction, true or false of myself. (shrink)
In the article "Podstawy analizy metodologicznej kanonów Milla"  Jerzy Łoś proposed an operator that refered sentences to temporal moments. Let us look, for example, at a sentence ‘It is raining in Toruń’. From a logical point of view it is a propositionalfunction, which does not have any logical value, unless we point at a temporal context from a fixed set of such contexts. If the sentence was considered today as a description of a state of affairs, (...) it could be true. If it was considered yesterday, it could be false. 1 The operator enables us to connect any sentence p with any temporal context t. Such a complex sentence we read as: a sentence p is realized at a temporal context t (a point of time, an interval of some kind, etc). (shrink)
A brief historical comment on scientific knowledge as Socratic ignorance -- Some critical comments on the text of this book, particularly on the theory of truth Exposition  -- Problem of Induction (Experience and Hypothesis) -- Two Fundamental Problems of the Theory of Knowledge -- Formulation of the Problem -- The problem of induction and the problem of demarcation -- Deductivtsm and Inductivism -- Comments on how the solutions are reached and preliminary presentation of the solutions -- Rationalism and empiricism-deductivism (...) and inductivism -- The possibility of a deductivist psychology of knowledge -- The Problem of Induction -- The infinite regression (Hume's argument) --The inductivist positions -- The Normal-Statement Positions -- The normal-statement positions: naive inductivism, strict positivism and apriorism -- Critique of strict positivism - twofold transcendence of natural laws -- The transcendental method - presentation of apriorism -- Critique of apriorism -- Kant and Fries -- Supplement to the critique of apriorism. (Psychologism and transcendentalism in Kant and Fries.-On the question of the empirical basis.) -- Probability Positions -- The probability positions - subjective belief in probability -- Statements about the objective probability of events -- Probability as an objective degree of validity of universal empirical statements -- One way of more closely defining the concept of the probability of a hypothesis (primary and secondary probability of hypotheses). The concept of simplicity -- The concept of the corroboration of a hypothesis - positivist, pragmatist and probabilistic interpretations of the concept of corroboration -- The infinite regression of probability statements -- Pseudo-Statement Positions -- The pseudo-statement positions: new formulation of the problem -- Natural laws as "instructions for the formation of statements" -- "True - false" or "useful - useless"? Consistent pragmatism --Difficulties of consistent pragmatism -- Tool and schema as purely pragmatic constructs -- Natural laws as propositional functions -- Conventionalism -- The pseudo-statement positions will temporarily be put away: conventionalism -- The three interpretations of axiomatic systems. (The circle of problems surrounding conventionalism) -- Conventionalist implicit and explicit definitions Propositionalfunction and propositional equation -- Conventionalist propositional equations as tautological general implications -- Can axiomatic-deductive systems also be understood as consequence classes of pure propositional functions (of pseudo-statements)? -- The coordinative definitions of empiricism: synthetic general implications -- Conventionalist and empiricist interpretations, illustrated by the example of applied geometry -- Strictly Universal Statements and Singular Statements -- Implication and general implication -- General implication and the distinction between strictly universal and singular statements -- Universal concept and individual concept-class and element -- Strictly universal statements-the problem of induction and the problem of universals -- Comments on the problem of universals -- Back to the Pseudo-Statement Positions -- Return to the discussion of the pseudo-statement positions -- Symmetry or asymmetry in the evaluation of natural laws? -- The negative evaluation of universal statements. Critique of the strictly symmetrical interpretation of pseudo-statements -- An infinite regression of pseudo-statements -- An apriorist pseudo-statement position -- Interpretation of the critique up to this point; comments on the unity of theory and practice -- A last chance for the pseudo-statement positions -- Pseudo-Statement Positions and the Concept of Meaning -- The concept of meaning and logical positivism -- The concept of meaning and the demarcation problem-the fundamental thesis of inductivism -- Critique of the inductivist dogma of meaning -- Fully decidable and partially decidable empirical statements-the antinomy of the knowability of the world. (Conclusion of the critique of the pseudo-statement positions.) -- The dialectical and the transcendental corroboration of the solution -- Is the problem of induction solved? (shrink)
Many people have argued that the evolution of the human language faculty cannot be explained by Darwinian natural selection. Chomsky and Gould have suggested that language may have evolved as the by-product of selection for other abilities or as a consequence of as-yet unknown laws of growth and form. Others have argued that a biological specialization for grammar is incompatible with every tenet of Darwinian theory – that it shows no genetic variation, could not exist in any intermediate forms, confers (...) no selective advantage, and would require more evolutionary time and genomic space than is available. We examine these arguments and show that they depend on inaccurate assumptions about biology or language or both. Evolutionary theory offers clear criteria for when a trait should be attributed to natural selection: complex design for some function, and the absence of alternative processes capable of explaining such complexity. Human language meets these criteria: Grammar is a complex mechanism tailored to the transmission of propositional structures through a serial interface. Autonomous and arbitrary grammatical phenomena have been offered as counterexamples to the position that language is an adaptation, but this reasoning is unsound: Communication protocols depend on arbitrary conventions that are adaptive as long as they are shared. Consequently, language acquisition in the child should systematically differ from language evolution in the species, and attempts to analogize them are misleading. Reviewing other arguments and data, we conclude that there is every reason to believe that a specialization for grammar evolved by a conventional neo-Darwinian process. (shrink)
I argue first that attention is a (maybe the) paradigmatic case of an object-directed, non-propositional intentional mental episode. In addition attention cannot be reduced to any other (propositional or non-propositional) mental episodes. Yet, second, attention is not a non-propositional mental attitude. It might appear puzzling how one could hold both of these claims. I show how to combine them, and how that combination shows how propositionality and non-propositionality can co-exist in a mental life. The crucial move (...) is one away from an atomistic, building block picture to a more holistic, structural picture. (shrink)
What makes it the case that an utterance constitutes an illocutionary act of a given kind? This is the central question of speech-act theory. Answers to it—i.e., theories of speech acts—have proliferated. Our main goal in this chapter is to clarify the logical space into which these different theories fit. -/- We begin, in Section 1, by dividing theories of speech acts into five families, each distinguished from the others by its account of the key ingredients in illocutionary acts. Are (...) speech acts fundamentally a matter of convention or intention? Or should we instead think of them in terms of the psychological states they express, in terms of the effects that it is their function to produce, or in terms of the norms that govern them? In Section 2, we take up the highly influential idea that speech acts can be understood in terms of their effects on a conversation’s context or “score”. Part of why this idea has been so useful is that it allows speech-act theorists from the five families to engage at a level of abstraction that elides their foundational disagreements. In Section 3, we investigate some of the motivations for the traditional distinction between propositional content and illocutionary force, and some of the ways in which this distinction has been undermined by recent work. In Section 4, we survey some of the ways in which speech-act theory has been applied to issues outside semantics and pragmatics, narrowly construed. (shrink)
The past 50 years have seen an accumulation of evidence suggesting that associative learning depends on high-level cognitive processes that give rise to propositional knowledge. Yet, many learning theorists maintain a belief in a learning mechanism in which links between mental representations are formed automatically. We characterize and highlight the differences between the propositional and link approaches, and review the relevant empirical evidence. We conclude that learning is the consequence of propositional reasoning processes that cooperate with the (...) unconscious processes involved in memory retrieval and perception. We argue that this new conceptual framework allows many of the important recent advances in associative learning research to be retained, but recast in a model that provides a firmer foundation for both immediate application and future research. (shrink)