We study elementary second order extensions of the theoryID 1 of non-iterated inductive definitions and the theoryPA Ω of Peano arithmetic with ordinals. We determine the exact proof-theoretic strength of those extensions and their natural subsystems, and we relate them to subsystems of analysis with arithmetic comprehension plusΠ 1 1 comprehension and bar induction without set parameters.

Let be a relational system whereby D is a nonempty set and P1 is an m1-ary relation on D. With we associate the (weak) monadic second-order theory consisting of the first-order predicate calculus with individual variables ranging over D; monadic predicate variables ranging over (finite) subsets of D; monadic predicate quantifiers; and constants corresponding to P1, P2, …. We will often use ambiguously to mean also the set of true sentences of.

Büchi, J. R. The monadic second order theory of [omega symbol]₁.--Büchi, J. R. and Siefkes, D. Axiomatization of the monadic second order theory of [omega symbol]₁.

We show that the weak monadic second order theory of the structure is first order interpretable in its automorphism group.

The paper argues that no second order theory is ontologically commited to anything beyond what its individual variables range over.

Open peer commentary on the article “Constructivist Model Building: Empirical Examples From Mathematics Education” by Catherine Ulrich, Erik S. Tillema, Amy J. Hackenberg & Anderson Norton. Upshot: I address the value of Ulrich et al.’s distinction between three types of second-order models. I conclude that their work contributes to the theorizing of adaptive teaching on the basis of a constructivist stance on knowing and learning.

Aleksandar Ignjatović. Delineating Classes of Computational Complexity via Second Order Theories with Weak Set Existence Principles (I).

Under a weak set-theoretic assumption we interpret second-order logic in the monadic theory of order.

We prove that PTCN cannot be a model of U12. This implies that there exists a first order sentence of bounded arithmetic which is provable in U12 but does not hold in PTCN.

Secrecy and “social cocooning” are critical mechanisms allowing the normalization of corruption within organizations. Less studied are processes of normalization that occur when corruption is an “open secret.” Drawing on an empirical study of Greek public-sector organizations, we suggest that a second-order normalization process ensues among non-corrupt onlookers both inside and beyond the organization. What is normalized at this level is not corruption, but its tolerance, which we disaggregate into agent-focused tolerance and structure-focused tolerance. Emphasizing the importance of (...) non-corrupt bystanders, we claim that second-order normalization helps corruption persist in situations where its presence is openly acknowledged. This adds an important new dimension to normalization theory and we unpack its implications for both future research and practice in this area. (shrink)

We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be (...) in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenon we call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory. (shrink)

Second-order abstract categorial grammars (de Groote in Association for computational linguistics, 39th annual meeting and 10th conference of the European chapter, proceedings of the conference, pp. 148–155, 2001) and hyperedge replacement grammars (Bauderon and Courcelle in Math Syst Theory 20:83–127, 1987; Habel and Kreowski in STACS 87: 4th Annual symposium on theoretical aspects of computer science. Lecture notes in computer science, vol 247, Springer, Berlin, pp 207–219, 1987) are two natural ways of generalizing “context-free” grammar formalisms for string (...) and tree languages. It is known that the string generating power of both formalisms is equivalent to (non-erasing) multiple context-free grammars (Seki et al. in Theor Comput Sci 88:191–229, 1991) or linear context-free rewriting systems (Weir in Characterizing mildly context-sensitive grammar formalisms, University of Pennsylvania, 1988). In this paper, we give a simple, direct proof of the fact that second-order ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for tree-generating hyperedge replacement grammars given by Engelfriet and Maneth (Graph transformation. Lecture notes in computer science, vol 1764. Springer, Berlin, pp 15–29, 2000) can then be used to show that the tree generating power of second-order ACGs is exactly the same as that of hyperedge replacement grammars. (shrink)

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second (...) order number theory is forcible from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

In this paper we explore the connections between the monadic second-order theory of one successor (MSO[<] for short) and the theories of ?-layered structures for time granularity. We first prove that the decision problem for MSO[<] and that for a suitable first-order theory of the upward unbounded layered structure are inter-reducible. Then, we show that a similar result holds for suitable chain variants of the MSO theory of the totally unbounded layered structure (this allows us to (...) solve some decision problems about theories of time granularity left open by Franceschet et al. [FRA 06]). (shrink)

Let H be a proof system for quantified propositional calculus (QPC). We define the Σqj-witnessing problem for H to be: given a prenex Σqj-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq1-witnessing problems for the systems G*1and G1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce and study the systems G*0 and G0, (...) in which cuts are restricted to quantifier-free formulas, and prove that the Σqj-witnessing problem for each is complete for NC1. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G*0 and proving that G0-proofs are TC0-recognizable. We also introduce QPC systems for TC0 and prove witnessing theorems for them. We introduce a finitely axiomatizable second-order system VNC1 of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σb0-CA for uniform NC1. We describe simple translations of VNC1 proofs of all bounded theorems to polynomial size families of G*0 proofs. From this and the above theorem we get alternative proofs of the NC1 witnessing theorems for VNC1 and AID. (shrink)

A second-order preference is a preference over preferences. This paper addresses the role that second-order preferences play in a theory of instrumental rationality. I argue that second-order preferences have no role to play in the prescription or evaluation of actions aimed at ordinary ends. Instead, second-order preferences are relevant to prescribing or evaluating actions only insofar as those actions have a role in changing or maintaining first-order preferences. I establish these claims (...) by examining and rejecting the view that second-order preferences trump first-order preferences. I also examine and reject the view that second-order preferences give additional normative force to an agent’s preferred first-order preferences. I conclude by arguing that second-order preferences should be integrated into an agent’s object-level preference ordering, and by explaining how best to make sense of this integration. (shrink)

Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments. The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably (...) construed, in accordance with an abundant or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational. (shrink)

Psychological game theory, as defined by Colman, is meant to offer a series of solution concepts that should reduce the indeterminacy of orthodox game theory when applied to a series of situations. My main criticism is that, actually, they introduce a second-order indeterminacy problem rather than offering a viable solution. The reason is that the proposed solution concepts are under-specified in their definition and in their scope.

There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set theory plus the axiom of choice (ZFC) is desirable. One advantage of such an axiomatization is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form Vκ, ∈ ∩ (Vκ × Vκ) , for κ (...) a strongly inaccessible ordinal. (shrink)

In this paper a system, RPF, of second-order relevance logic with S5 necessity is presented which contains a defined, notion of identity for propositions. A complete semantics is provided. It is shown that RPF allows for more than one necessary proposition. RPF contains primitive syntactic counterparts of the following semantic notions: (1) the reflexive, symmetrical, transitive binary alternativeness relation for S5 necessity, (2) the ternary Routley-Meyer alternativeness relation for implication, and (3) the Routley-Meyer notion of a prime intensional (...) theory, as well as defined syntactic counterparts, of the semantic notions of a possible world and the Routley-Meyer * operator. (shrink)

After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal κ is supercompact if and only if every Π11 sentence true in a structure (...) M (of any size) containing κ in a language of size less than κ is also true in a substructure m≺M of size less than κ with m∩κ∈κ. (shrink)

A second-order conspiracy (SOC) is a conspiracy that aims to create (and typically also disseminate) a conspiracy theory. Second-order conspiracy theories (SOCT) are theories that explain the occurrence of a given conspiracy theory by appeal to a conspiracy. In this paper I argue that SOC and SOCT are useful and coherent concepts, while also having numerous philosophically interesting upshots (in terms of epistemology, explanation, and prediction). Secondly, I appeal to the nature of two specific (...) kinds of second-order conspiracies to make the case for what has been called ‘local generalism’ (Stamatiadis-Bréhier 2023a). Specifically, I focus on so-called ‘denial industries’ to argue that the structure of these second-order conspiracies allows us to infer non-accidental generalisations about the domain of conspiracy theories. Even though it is true that there is nothing epistemically problematic with the general class of conspiracy theories, there are specific subsets of conspiracy theories that warrant immediate strong suspicion (cf. Dentith 2022). By looking at the intricate mechanisms by which these denial industries operate, we can infer that the conspiracy theories that are produced by them are epistemically unwarranted. I conclude by making some exploratory remarks about what the metaphysics of second-order conspiracies would look like. (shrink)

Second-order quantifier elimination in the context of classical logic emerged as a powerful technique in many applications, including the correspondence theory, relational databases, deductive and knowledge databases, knowledge representation, commonsense reasoning and approximate reasoning. In the current paper we first generalize the result of Nonnengart and Szałas [17] by allowing second-order variables to appear within higher-order contexts. Then we focus on a semantical analysis of conditionals, using the introduced technique and Gabbay’s semantics provided in [10] (...) and substantially using a third-order accessibility relation. The analysis is done via finding correspondences between axioms involving conditionals and properties of the underlying third-order relation. (shrink)

This three-part article reinforces crosscurrents between cybernetician Gordon Pask’s work towards creating responsive machines applied to theater and education, and Vygotsky’s theory, to advance sociohistorical approaches into the internet age. We first outline Pask’s discovery of possibilities of a neoclassical cybernetic framework for human–human, human–machine, and machine–machine conversations. Second, we outline conversation theory as an elaboration of the reconstruction of mental models/concepts by observers through reliance on sociocultural psychological approaches, and apply concepts like the zone of proximal development and (...) perezhivanie to Paskian aesthetic technologies. Third, we interpret Pask’s teaching/learning devices as zones of proximal development, and outline how Paskian algorithms in digital devices like THOUGHTSTICKER have been generalized on today’s internet, supplemented by corporate interests. We conclude that Paskian theory may offer understandings of the roles of internet technologies in transforming human thinking, and suggest (re)designing tools incorporating algorithms that contextually advance conceptual understanding that deviates from current indexing approaches. (shrink)

In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levelsUαVα. The recursive definition of theVα's is:Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory shows thatVω, the first transfinite level of the hierarchy, is a model of all the axioms ofZFwith the exception of the axiom of infinity. And, in general, one finds that ifκis a strongly inaccessible ordinal, thenVκis a model of all of (...) the axioms ofZF. Doubtless, when cast as a first-order theory,ZFdoes not characterize the structures 〈Vκ,∈∩〉 forκa strongly inaccessible ordinal, by the Löwenheim-Skolem theorem. Still, one of the main achievements of [12] consisted in establishing that a characterization of these models can be attained when one ventures into second-order logic. For let second-orderZFbe, as usual, the theory that results fromZFwhen the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-orderZFcan only be satisfied in models of the form 〈Vκ,∈∩〉 forκa strongly inaccessible ordinal. (shrink)

This paper explores the thesis that the concept of cultural techniques should be strictly limited to symbolic technologies that allow for self-referential recursions. Writing enables one to write about writing itself; painting itself can be depicted in painting; films may feature other films. In other words, cultural techniques are defined by their ability to thematize themselves; they are second-order techniques as opposed to first-order techniques like cooking or tilling a field. To illustrate his thesis, Macho discusses a (...) sequence of historical examples, from body signs and death masks to digital code and ID papers. These examples serve to reiterate another basic proposal that is already announced in the paper’s title. The recursive, self-observing qualities of cultural techniques make them a ‘technology of the self’ and thus render them indispensable for the generation, repetition and maintenance of identity. (shrink)

In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) (...) There is no minimum transitive model of KM. (4) There is a minimum β-model of GB+ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB+ETR. (shrink)

Second-order uncertainty, also known as model uncertainty and Knightian uncertainty, arises when decision-makers can (partly) model the parameters of their decision problems. It is widely believed that subjective probability, and more generally Bayesian theory, are ill-suited to represent a number of interesting second-order uncertainty features, especially “ignorance” and “ambiguity”. This failure is sometimes taken as an argument for the rejection of the whole Bayesian approach, triggering a Bayes vs anti-Bayes debate which is in many ways analogous (...) to what the classical vs non-classical debate used to be in logic. This paper attempts to unfold this analogy and suggests that the development of non-standard logics offers very useful lessons on the contextualisation of justified norms of rationality. By putting those lessons to work I will flesh out an epistemological framework suitable for extending the expressive power of standard Bayesian norms of rationality to second- order uncertainty in a way which is both formally and foundationally conservative. Contents. (shrink)

Following [3], we build higher-order models of analysis resembling the frameworks of nonstandard analysis. The models are entirely canonical, constructed without Choice. Weak transfer principles are developed and the models are applied to topology, graph theory, and measure theory. A Loeb-like measure is constructed.

Context: The problems that are most in need of interdisciplinary collaboration are “wicked problems,” such as food crises, climate change mitigation, and sustainable development, with many relevant aspects, disagreement on what the problem is, and contradicting solutions. Such complex problems both require and challenge interdisciplinarity. Problem: The conventional methods of interdisciplinary research fall short in the case of wicked problems because they remain first-order science. Our aim is to present workable methods and research designs for doing second-order (...) science in domains where there are many different scientific knowledges on any complex problem. Method: We synthesize and elaborate a framework for second-order science in interdisciplinary research based on a number of earlier publications, experiences from large interdisciplinary research projects, and a perspectivist theory of science. Results: The second-order polyocular framework for interdisciplinary research is characterized by five principles. Second-order science of interdisciplinary research must: 1. draw on the observations of first-order perspectives, 2. address a shared dynamical object, 3. establish a shared problem, 4. rely on first-order perspectives to see themselves as perspectives, and 5. be based on other rules than first-order research. Implications: The perspectivist insights of second-order science provide a new way of understanding interdisciplinary research that leads to new polyocular methods and research designs. It also points to more reflexive ways of dealing with scientific expertise in democratic processes. The main challenge is that this is a paradigmatic shift, which demands that the involved disciplines, at least to some degree, subscribe to a perspectivist view. Constructivist content: Our perspectivist approach to science is based on the second-order cybernetics and systems theories of von Foerster, Maruyama, Maturana & Varela, and Luhmann, coupled with embodied theories of cognition and semiotics as a general theory of meaning from von Uexküll and Peirce. (shrink)

Second or higher order probabilities have commonly been viewed with scepticism by those working within the realm of probability and decision theory. The aim of the present note is to show how the notion of second order probabilities can add to our understanding of judgmental and decision processes and how the traditional framework of Bayesian decision theory can be extended in a fruitful way by taking such entities into account. Section one consists of a brief account (...) of arguments put forth against higher order probabilities as well as of counterarguments. In order to provide an example of the applicability of second order probabilities a decision theory encompassing such probabilities will be presented in section two. In section three I will try to emphasize the value of second order probabilities for a deeper and more complete understanding of the notion of risk. (shrink)

Larry Wright and others have advanced causal accounts of functional explanation, designed to alleviate fears about the legitimacy of such explanations. These analyses take functional explanations to describe second order causal relations. These second order relations are conceptually puzzling. I present an account of second order causation from within the framework of Eells' probabilistic theory of causation; the account makes use of the population-relativity of causation that is built into this theory.

Context: Philosophy of science is the branch of philosophy that deals with methods, foundations, and implications of science. It is a theory of how to create scientific knowledge. Presently, there is widespread agreement on how to do science, namely conjectures, ideally in the form of a mathematical model, and refutations, testing the model using empirical evidence. Problem: Many social scientists are using a conception of science created for the physical sciences. Expanding philosophy of science so that it more successfully encompasses (...) social systems would create new avenues of inquiry. Two dimensions could be added to philosophy of science: the amount of attention paid to the observer and the amount of impact of a theory on the system described. Method: My approach is to illuminate underlying assumptions. I claim that there are at least three epistemologies and that they can be combined to form a more robust conception of knowledge and of how to do research. There are at least four models and four basic elements being used by scientists. Results: The article identifies the logical propositions underlying second-order science. It suggests strategies for developing second-order science. And it describes several methods that can be used to practice second-order science, including how past theories have not only described but also changed the phenomenon being studied. Implications: The task for members of the scientific community, particularly social scientists, is to practice second-order science and to develop further its theories and methods. A practical implication is to accept methods for acting as well as theories as a contribution to science, since methods explicitly define the role of an observer/ participant. Constructivist content: The paper is an extension of the work of Heinz von Foerster and other second-order cyberneticians. (shrink)

Let S2S [WS2S] espectively be the storn [weak] monadic second order theory of the binary tree T in the language of two successor functions. An S2S-formula whose free variables are just individual variables defines a relation on T (rather than on the power set of T). We show that S2S and WS2S define the same relations on T, and we give a simple characterization of these relations.

We can understand and act upon the beliefs of other people, even when these conflict with our own beliefs. Children’s development of this ability, known as Theory of Mind, typically happens around age 4. Research using a looking-time paradigm, however, established that toddlers at the age of 15 months old pass a non-verbal false-belief task (Onishi and Baillargeon in Science 308:255–258, 2005). This is well before the age at which children pass any of the verbal false-belief tasks. In this study (...) we present a more complex case of false-belief reasoning with older children. We tested second-order reasoning, probing children’s ability to handle the belief of one person about the belief of another person. We find just the opposite: 7-year-olds pass a verbal false-belief reasoning task, but fail on an equally complex low-verbal task. This finding suggests that language supports explicit reasoning about beliefs, perhaps by facilitating the cognitive system to keep track of beliefs attributed by people to other people. (shrink)

The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the second-order theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordinal characterizability and ordinal arithmetic. The broader significance of cardinal characterizability and the relationships between different notions of characterizability are also discussed.

We can understand and act upon the beliefs of other people, even when these conflict with our own beliefs. Children’s development of this ability, known as Theory of Mind, typically happens around age 4. Research using a looking-time paradigm, however, established that toddlers at the age of 15 months old pass a non-verbal false-belief task (Onishi and Baillargeon in Science 308:255–258, 2005). This is well before the age at which children pass any of the verbal false-belief tasks. In this study (...) we present a more complex case of false-belief reasoning with older children. We tested second-order reasoning, probing children’s ability to handle the belief of one person about the belief of another person. We find just the opposite: 7-year-olds pass a verbal false-belief reasoning task, but fail on an equally complex low-verbal task. This finding suggests that language supports explicit reasoning about beliefs, perhaps by facilitating the cognitive system to keep track of beliefs attributed by people to other people. (shrink)

In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether second-order logic can count as logic in some proper sense of logic, and what epistemic status it occupies. More specifically, second-order logic is sometimes taken to be mathematical, a mere notational variant of some fragment of set theory. If this is the case, it might be argued (...) that it does not have the "epistemic innocence" which would be needed for, e.g., foundational programmes in mathematics for which second-order logic is sometimes used. I suggest a Deductivist conception of logic, that characterises logical consequence by means of inference rules, and argue that on this conception second-order logic should count as logic in the proper sense. (shrink)

Nowadays, the power systems are getting more and more complicated because of the delays introduced by the communication networks. The existence of the delays usually leads to the degradation and/or instability of power system performance. On account of this point, the traditional load frequency control approach for power system sketches a destabilizing impact and an unacceptable system performance. Therefore, this paper proposes a new LFC based on adaptive integral second-order sliding mode control approach for the large-scale power system (...) with communication delays. First, a new linear matrix inequality is derived to ensure the stability of whole power systems using Lyapunov stability theory. Second, an AISOSMC law is designed to ensure the finite time reachability of the system states. To the best of our knowledge, this is the first time the AISOSMC is designed for LFC of the LSPSwCD. In addition, the report of testing results presents that the suggested LFC based on AISOSMC can not only decrease effectively the frequency variation but also make successfully less in mount of power oscillation/fluctuation in tie-line exchange. (shrink)

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditionalω. Namely, the equivalence between normal transfinite recursion scheme and newdependent transfinite recursionscheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universeVof sets is (...) treated as the given totality. (shrink)

This dissertation develops an inferentialist theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. From this starting point the dissertation develops a theory of quantification as marking (...) coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators. (shrink)