We study the non-Fregean propositional logic with propositional quantifiers, denoted by $\mathsf{SCI}_{\mathsf{Q}}$. We prove that $\mathsf{SCI}_{\mathsf{Q}}$ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in $\mathsf{SCI}_{\mathsf{Q}}$ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of $\mathsf{SCI}_{\mathsf{Q}}$-sentences. Finally, we present a translation of $\mathsf{SCI}_{\mathsf{Q}}$ into a classical two-sorted first-order logic, and we use the translation to prove some model-theoretic properties of (...) $\mathsf{SCI}_{\mathsf{Q}}$. (shrink)

This paper offers a framework for extending Arnon Avron and Iddo Lev’s non-deterministic semantics to quantified predicate logic with the intent of resolving several problems and limitations of Avron and Anna Zamansky’s approach. By employing a broadly Fregean picture of logic, the framework described in this paper has the benefits of permitting quantifiers more general than Walter Carnielli’s distribution quantifiers and yielding a well-behaved model theory. This approach is purely objectual and yields the semantical equivalence of both α-equivalent formulae (...) and formulae differing only by codenotative terms. Finally, we make a brief excursion into non-deterministic model theory, proving a strong Łoś’ Theorem and compactness for all finitely-valued, non-deterministic logics whose quantifiers have intensions describable in a first order metalanguage. (shrink)

Information‐theoretic complexity metrics, such as Surprisal (Hale, 2001; Levy, 2008) and Entropy Reduction (Hale, 2003), are linking hypotheses that bridge theorized expectations about sentences and observed processing difficulty in comprehension. These expectations can be viewed as syntactic derivations constrained by a grammar. However, this expectation‐based view is not limited to syntactic information alone. The present study combines structural and non‐structural information in unified models of word‐by‐word sentence processing difficulty. Using probabilistic minimalist grammars (Stabler, 1997), we extend expectation‐based models to include (...) frequency information about noun phrase animacy. Entropy reductions derived from these grammars faithfully reflect the asymmetry between subject and object relatives (Staub, 2010; Staub, Dillon, & Clifton, 2017), as well as the effect of animacy on the measured difficulty profile (Lowder & Gordon, 2012; Traxler, Morris, & Seely, 2002). Visualizing probability distributions on the remaining alternatives at particular parser states allows us to explore new, linguistically plausible interpretations for the observed processing asymmetries, including the way that expectations about the relativized argument influence the processing of particular types of relative clauses (Wagers & Pendleton, 2016). (shrink)

This paper aims to show that modeling complex systems inevitably involves non-propositional knowledge and thus the uncertainties associated with the corresponding model predictions cannot be fully quantified. This is exemplified by means of the climate system and climate modeling. The climate system is considered as a paradigm for a complex system, whereby the notion of complexity adopted in this paper is epistemic in nature and does not equate with the technical definition of a complex system as for example used (...) within physics or complexity theory. The epistemic notion of complexity allows to view the climate system as complex with respect to some features, while simple with respect to others. This distinction is of practical significance for political decision making as it allows to treat some climate predictions as (fairly) certain, while acknowledging high uncertainties with others. (shrink)

Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.

The Mares-Goldblatt semantics for quantified relevant logics have been developed for first-order extensions of R, and a range of other relevant logics and modal extensions thereof. All such work has taken place in the the ternary relation semantic framework, most famously developed by Sylvan and Meyer. In this paper, the Mares-Goldblatt technique for the interpretation of quantifiers is adapted to the more general neighbourhood semantic framework, developed by Sylvan, Meyer, and, more recently, Goble. This more algebraic semantics allows one (...) to characterise a still wider range of logics, and provides the grist for some new results. To showcase this, we show, using some non-augmented models, that some quantified relevant logics are not conservatively extended by connectives the addition of which do conservatively extend the associated propositional logics, namely fusion and the dual implication. We close by proposing some further uses to which the neighbourhood Mares-Goldblatt semantics may be put. (shrink)

We consider the theoryT IT of infinite terms. The axioms for the theoryT IT are analogous to the Mal'cev's axioms for the theoryT IT of finite terms whose models are the locally free algebras. Recently Maher [Ma] has proved that the theoryT IT in a finite non singular signature plus the Domain Closure Axiom is complete. We give a description of all the complete extension ofT IT from which an effective decision procedure forT IT is obtained. Our approach considers formulas (...) built up with syntactic terms containing pointers. Using such a technique, the analysis of the theoryT IT can be carried out in analogy with Mal'cev's analysis ofT FT. Our results follow from an effective quantifier elimination procedure for formulas with pointers. (shrink)

A semantics for quantified modal logic is presented that is based on Kleene's notion of realizability. This semantics generalizes Flagg's 1985 construction of a model of a modal version of Church's Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church's Thesis and a variant of a modal set (...) theory due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra's generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott's graph model (for the untyped lambda calculus) which witnesses the failure of the stability of non-identity. (shrink)

Urn models were developed by Veikko Rantala to provide a non-standard semantics for first-order logic in which the domains, over which the quantifiers range, are allowed to vary. Rantala uses game-theoretical semantics in his presentation, and the present paper is a study of urn models from a more classical, truth-conditional point of view. An axiomatic system for urn logic is set out and completeness is proved by the method of maximal consistent sets.

We study definability in terms of monotone generalized quantifiers satisfying Isomorphism Closure, Conservativity and Extension. Among the quantifiers with the latter three properties - here called CE quantifiers - one finds the interpretations of determiner phrases in natural languages. The property of monotonicity is also linguistically ubiquitous, though some determiners like an even number of are highly non-monotone. They are nevertheless definable in terms of monotone CE quantifiers: we give a necessary and sufficient condition for such definability. We further identify (...) a stronger form of monotonicity, called smoothness, which also has linguistic relevance, and we extend our considerations to smooth quantifiers. The results lead us to propose two tentative universals concerning monotonicity and natural language quantification. The notions involved as well as our proofs are presented using a graphical representation of quantifiers in the so-called number triangle. (shrink)

The Joint Non-Trivialization Theorem, two Definability Theorems and the generalized Quantifier Elimination Theorem are proved for J 3-theories. These theories are three-valued with more than one distinguished truth-value, reflect certain aspects of model type logics and can. be paraconsistent. J 3-theories were introduced in the author's doctoral dissertation.

There are logics where necessity is defined by means of a given identity connective: \ is a tautology). On the other hand, in many standard modal logics the concept of propositional identity \ can be defined by strict equivalence \}\). All these approaches to modality involve a principle that we call the Collapse Axiom : “There is only one necessary proposition.” In this paper, we consider a notion of PI which relies on the identity axioms of Suszko’s non-Fregean logic SCI. (...) Then S3 proves to be the smallest Lewis modal system where PI can be defined as SE. We extend S3 to a non-Fregean logic with propositional quantifiers such that necessity and PI are integrated as non-interdefinable concepts. CA is not valid and PI refines SE. Models are expansions of SCI-models. We show that SCI-models are Boolean prealgebras, and vice-versa. This associates non-Fregean logic with research on Hyperintensional Semantics. PI equals SE iff models are Boolean algebras and CA holds. A representation result establishes a connection to Fine’s approach to propositional quantifiers and shows that our theories are conservative extensions of S3–S5, respectively. If we exclude the Barcan formula and a related axiom, then the resulting systems are still complete w.r.t. a simpler denotational semantics. (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)

We study the theory of a Hilbert space H as a module for a unital C*-algebra ${\mathcal{A}}$ from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of ${\mathcal{A}}$ . Finally, we show that there is an homeomorphism (...) between the space of types of norm less than 1 in this model companion, and the space of quasistates of the C*-algebra ${\mathcal{A}}$. (shrink)

Non-epistemic values pervade climate modelling, as is now well documented and widely discussed in the philosophy of climate science. Recently, Parker and Winsberg have drawn attention to what can be termed “epistemic inequality”: this is the risk that climate models might more accurately represent the future climates of the geographical regions prioritised by the values of the modellers. In this paper, we promote value management as a way of overcoming epistemic inequality. We argue that value management can be seriously (...) considered as soon as the value-free ideal and inductive risk arguments commonly used to frame the discussions of value influence in climate science are replaced by alternative social accounts of objectivity. We consider objectivity in Longino's sense as well as strong objectivity in Harding's sense to be relevant options here, because they offer concrete proposals that can guide scientific practice in evaluating and designing so-called multi-model ensembles and, in fine, improve their capacity to quantify and express uncertainty in climate projections. (shrink)

The paper gives a survey of known results related to computational devices (finite and push–down automata) recognizing monadic generalized quantifiers in finite models. Some of these results are simple reinterpretations of descriptive—feasible correspondence theorems from finite–model theory. Additionally a new result characterizing monadic quantifiers recognized by push down automata is proven.

Axiomatization of Gödel-Dummett predicate logics S2G, S3G, and PG, where PG is the weakest logic in which all prenex operations are sound, and the relationships of these logics to logics known from the literature are discussed. Examples of non-prenexable formulas are given for those logics where some prenex operation is not available. Inter-expressibility of quantifiers is explored for each of the considered logics.

There is growing interest in the effects of sports-related repetitive head impacts (RHIs) on athletes’ cognitive capabilities. This study examines the effect of RHIs in data collected from adolescent athletes to estimate the magnitude and longevity of RHIs on sensorimotor and cognitive performance. A non-linear regression model estimated the longevity of RHI effects by adding a half-life parameter embedded in an exponential decay function. A model estimate of this parameter allows the possibility of RHI effects to attenuate over time and (...) introduces a mechanism to study the cumulative effect of RHIs. The posterior distribution of the half-life parameter associated with short-distance headers (<30 m) is centered around 6 days, whereas the posterior distribution of the half-life parameter associated with long-distance headers extends beyond a month. Additionally, the magnitude of the effect of each short header is around 3 times smaller than that of a long header. The results indicate that, on both tasks, response time (RT) changes after long headers are bigger in magnitude and last longer compared to the effects of short headers. Most important, we demonstrate that deleterious effects of long headers extend beyond 1 month. Although estimates are based on data from a relatively short-duration study with a relatively small sample size, the proposed model provides a mechanism to estimate long-term behavioral slowing from RHIs, which may be helpful to reduce the risk of additional injury. Finally, differences in the longevity of the effects of short and long RHIs may help to explain the large variance found between biomechanical input and clinical outcome in studies of concussion tolerance. (shrink)

Given a lovely pair P ≺ M of models of a simple theory T, we study the structure whose universe is P and whose relations are the traces on P of definable (in ℒ with parameters from M) sets in M. We give a necessary and sufficient condition on T (which we call weak lowness) for this structure to have quantifier-elimination. We give an example of a non-weakly-low simple theory.

In this note, we show that the first-order logic IKω is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth IKTω obtained by expanding IKω with transparent truth, because the models are inconsistent with transparent truth. Finally, we show that whether or not this (...) inconsistency can be reproduced in the sequent calculus for IKTω depends on how vacuous quantification is treated. (shrink)

It has been alleged that actualism and quantified modal logic are incompatible. My aim in this dissertation is twofold: to defend thoroughgoing actualism with respect to possible objects, and to present a modified semantics for quantified modal logic that is compatible with such a position. The basic strategy is to draw on the parallels between fictions and possible worlds to develop a hierarchical system of worlds-within-worlds ;Actualists usually take first-order modal statements as being about the right objects, by (...) stipulation. Any actualistically acceptable semantics must extend this approach to higher-order modal statements. "I could have had a brother who was a banker but could have been a pianist" means: there is a first-order world in which I have a banker brother, and there exists within that world a second-order world in which that very brother is a pianist. This eliminates the troubling need to identify non-actual objects across worlds of the same order. ;The concept of worlds-within-worlds is analogous to that of consistent fictions-within-fictions. Independent investigations of each concept reveal that both nesting structures are treelike, irreflexive and intransitive. Despite these restrictions, AA models are relatively tractable, being derivable from Kripke models for by unraveling. They are sound and complete for normal propositional modal logics, and for the quantified modal logic Q1R as long as constants are excluded from the language. ;In Chapter 1, I argue for actualism and against some proposed responses to the iterated modality problem. I then present the idea of nested worlds as a solution. Chapter 2 examines the analogous concept of nested fictions, to shed some light on consistent nesting structures. Chapter 3 returns to modality proper; AA models are defined and defended for both propositional and quantified modal logic. Chapter 4 contains the soundness and completeness results, and ends with a somewhat surprising philosophical argument motivated by AA semantics: unless it is metaphysically necessary that all objects have sufficient essences, the correct logic for metaphysical necessity must be weaker than B. (shrink)

We study the computational complexity of the model checking problem for quantifier-free dependence logic ${(\mathcal{D})}$ formulas. We characterize three thresholds in the complexity: logarithmic space (LOGSPACE), non-deterministic logarithmic space (NL) and non-deterministic polynomial time (NP).

Gödel logics with truth sets being countable closed subsets of the unit real interval containing 0 and 1 are studied under their usual semantics and under the witnessed semantics, the latter admitting only models in which the truth value of each universally quantified formula is the minimum of truth values of its instances and dually for existential quantification and maximum. An infinite system of such truth sets is constructed such that under the usual semantics the corresponding logics have pairwise (...) different sets of tautologies, all these sets being non-arithmetical, whereas under the witnessed semantics all the logics have the same set of tautologies and it is Π2-complete. Further similar results are obtained. (shrink)

We make a proposal for formalizing simultaneous games at the abstraction level of player’s powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of ‘concurrent game logic’ CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.

Computational models of child language development can help us understand the cognitive underpinnings of the language learning process, which occurs along several linguistic levels at once (e.g., prosodic and phonological). However, in light of the replication crisis, modelers face the challenge of selecting representative and consolidated infant data. Thus, it is desirable to have evaluation methodologies that could account for robust empirical reference data, across multiple infant capabilities. Moreover, there is a need for practices that can compare developmental trajectories of (...) infants to those of models as a function of language experience and development. The present study aims to take concrete steps to address these needs by introducing the concept of comparing models with large-scale cumulative empirical data from infants, as quantified by meta-analyses conducted across a large number of individual behavioral studies. We formalize the connection between measurable model and human behavior, and then present a conceptual framework for meta-analytic evaluation of computational models. We exemplify the meta-analytic model evaluation approach with two modeling experiments on infant-directed speech preference and native/non-native vowel discrimination. (shrink)

This paper investigates the question of characterizing first-order LFIs (logics of formal inconsistency) by means of two-valued semantics. LFIs are powerful paraconsistent logics that encode classical logic and permit a finer distinction between contradictions and inconsistencies, with a deep involvement in philosophical and foundational questions. Although focused on just one particular case, namely, the quantified logic QmbC, the method proposed here is completely general for this kind of logics, and can be easily extended to a large family of (...) class='Hi'>quantified paraconsistent logics, supplying a sound and complete semantical interpretation for such logics. However, certain subtleties involving term substitution and replacement, that are hidden in classical structures, have to be taken into account when one ventures into the realm of nonclassical reasoning. This paper shows how such difficulties can be overcome, and offers detailed proofs showing that a smooth treatment of semantical characterization can be given to all such logics. Although the paper is well-endowed in technical details and results, it has a significant philosophical aside: it shows how slight extensions of classical methods can be used to construct the basic model theory of logics that are weaker than traditional logic due to the absence of certain rules present in classical logic. Several such logics, however, as in the case of the LFIs treated here, are notorious for their wealth of models precisely because they do not make indiscriminate use of certain rules; these models thus require new methods. In the case of this paper, by just appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus, some new constructions and crafty solutions to certain non-obvious subtleties are proposed. The result is that a richer extension of model theory can be inaugurated, with interest not only for paraconsistency, but hopefully to other enlargements of traditional logic. (shrink)

A modal logic for translating a sequence of English sentences to a sequence of logical forms is presented, characterized by Kripke models with points formed from input/output sequences, and valuations determined by entailment relations. Previous approaches based (to one degree or another) on Quantified Dynamic Logic are embeddable within it. Applications to presupposition and ambiguity are described, and decision procedures and axiomatizations supplied.

Vectorization of a class of structures is a natural notion in finite model theory. Roughly speaking, vectorizations allow tuples to be treated similarly to elements of structures. The importance of vectorizations is highlighted by the fact that if the complexity class PTIME corresponds to a logic with reasonable syntax, then it corresponds to a logic generated via vectorizations by a single generalized quantifier (Dawar in J Log Comput 5(2):213–226, 1995). It is somewhat surprising, then, that there have been few systematic (...) studies of the expressive power of vectorizations of various quantifiers. In the present paper, we consider the simplest case: the cardinality quantifiers C S. We show that, in general, the expressive power of the vectorized quantifier logic $${{\rm FO}(\{{\mathsf C}_S^{(n)}\, | \, n \in \mathbb{Z}_+\})}$$ is much greater than the expressive power of the non-vectorized logic FO(C S ). (shrink)

Cloud computing (ClC) has become a more popular computer paradigm in the preceding few years. Quality of Service (QoS) is becoming a crucial issue in service alteration because of the rapid growth in the number of cloud services. When evaluating cloud service functioning using several performance measures, the issue becomes more complex and non-trivial. It is therefore quite difficult and crucial for consumers to choose the best cloud service. The user's choices are provided in a quantifiable manner in the current (...) methods for choosing cloud services. Hence, this study attempts to achieve this objective through construction. decision-making framework so-called cloud services evaluator (CSsEv). The main indicator and its sub-indicators are formed as nodes at levels(n) in tree soft sets (TSSs). Thereafter Single Value Neutrosophic Sets (SVNSs) as branch of neutrosophic sets which conjunction with the Multi-Criteria Decision Making (MCDM) technique to facilitate analysis and evaluation process for the available Cloud services providers. Hence, entropy is employed to obtain indicators and sub_indicators weights and Complex Proportional Assessment utilizes these weights to facilitate the decision process of selecting optimal ClSPs. (shrink)

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal (...) systems by Stolpe, 557–575, 2003), but unfortunately that work used only models and not frames. (shrink)

Growth of different body parts in humans is sensitive to different resource constraints that are mediated by parental investment. Parental investment can involve the expenditure of material, cognitive, and emotional resources on offspring. Cranial volume, an important predictor of cognitive ability, appears understudied in this context. We asked whether there are associations between growth and family structure, self-reported estimates for resource availability, and sibling number; and whether these constraints relate to head and body growth in a similar manner. We assessed (...) the associations between parental investment, height, and cranial volume in a cross-sectional study of Estonian children. Height correlated negatively with the number of siblings but this association became negligible in a model controlling for birthweight, parental heights, and mother’s age at birth. Unlike height, cranial volume was unrelated to sibling number, but it was negatively associated with self-reported meat and general resource shortage. Cranial volume was related to family structure and paternal education. Children living with both birth-parents had larger heads than those living in families containing a step-parent. Since these family types did not differ with respect to meat or general resource shortage, our findings suggest that families including both genetic parents provide non-material benefits that stimulate predominantly cranial growth. For the studied developmental period, cranial volume appeared a more sensitive marker of growth constraints than height. The potential of using cranial volume for quantifying physical impact of non-material parental investment deserves further attention. (shrink)

This paper deals with the question of whether uncertainty regarding model structure, especially in climate modeling, exhibits a kind of “chaos.” Do small changes in model structure, in other words, lead to large variations in ensemble predictions? More specifically, does model error destroy forecast skill faster than the ordinary or “classical” chaos inherent in the real-world attractor? In some cases, the answer to this question seems to be “yes.” But how common is this state of affairs? Are there precise mathematical (...) results that can help us answer this question? And is dependence on model structure “sensitive” in that arbitrarily small errors can destroy forecast skill? We examine some efforts in the literature to answer this last question in the affirmative and find them to be unconvincing. (shrink)

A new technique for proving realisability results is presented, and is illustrated in detail for the simple case of arithmetic minus induction. CL is a Gentzen formulation of classical logic. CPQ is CL minus the Cut Rule. The basic proof theory and model theory of CPQ and CL is developed. For the semantics presented CPQ is a paraconsistent logic, i.e. there are non-trivial CPQ models in which some sentences are both true and false. Two systems of arithmetic minus induction are (...) introduced, CL-A and CPQ-A based on CL and CPQ, respectively. The realisability theorem for CPQ-A is proved: It is shown constructively that to each theorem A of CPQ-A there is a formula A *, a so-called “realised disjunctive form of A ”, such that variables bound by essentially existential quantifiers in A * can be written as recursive functions of free variables and variables bound by essentially universal quantifiers. Realisability is then applied to prove the consistency of CL-A, making use of certain finite non-trivial inconsistent models of CPQ-A. (shrink)

In this paper we relax the assumption that the logical constants of ordinary first-order logic be linearly ordered. As a consequence, we shall have formulas involving not only partially ordered quantifiers, but also partially ordered connectives. The resulting language, called the language of informational independence will be given an interpretation in terms of games of imperfect information. The II-logic will be seen to have some interesting properties: It is very natural to define in this logic two negations, weak negation as (...) failure to verify a sentence, and strong negation as the existence of a falsifying strategy: One can express in this logic complete problems of finite structures, like the non-connectedness and 3-colorability of finite graphs, the satisfiability problem for Boolean circuits built up from NAND-gates, etc. (shrink)

The existence of non-standard models of first-order Peano-Arithmetic (PA) threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to the natural number structure is possible on the basis of our best arithmetical theories. The move to logics stronger than FOL is denied to the moderate realist on the grounds that it merely shifts the indeterminacy “one level up” into the meta-theory by, illegitimately, assuming the determinacy of the notions needed to formulate such logics. This paper argues (...) that the challenge can be met. We show how the quantifier “there are infinitely many” can be uniquely determined in a naturalistically acceptable fashion and thus be used in the formulation of a theory of arithmetic. We compare the approach pursued here with Field’s justification of the same device and the popular strategy of invoking a second-order formalism, and argue that it is more robust than either of the alternative proposals. (shrink)

We have two main aims: to construct mathematical models for analysing determinism, causality and supervenience; and then to use these to demonstrate the possibility of constructing an ontic construal of the operation of free will - one requiring both the presentation of genuine alternatives to an agent and their selecting between them in a manner that permits the attribution of responsibility. Determinism is modelled using trans-temporal ontic links between discrete juxtaposed universe states and shown to be distinct from predictability. Causality (...) is defined on a temporal sequence of δ-algebras and quantified using a measure. The measure leads to definitions of causal overdetermination and epiphenomena. Proofs are constructed to demonstrate deterministic universes must carry their properties essentially but not necessarily locally. We argue determinism and causality are separate doctrines. These models and results are marshalled to put the case that a counterfactual construal of ontic choice cannot work. In response we propose ‘immanence’ - a modified form of indeterminism whereby a universe can present choices to its denizens. We prove that beings subsumed within a universe cannot pilot their own actions. We then argue these beings can exercise free will only when selecting between choices inhering within immanent relata. A being is responsible for its selections if and only if it is constituted of a temporally evolving deterministic substructure. Our proposal is novel: it avoids injecting indeterminism into the decision process. Topological models for property supervenience are developed and used to reconstruct standard definitions from the literature. These are then used to demonstrate considerations of supervenience do not affect our arguments. We have demonstrated that a model of the exercise of free will involving both genuine choices and responsibility is possible but can only operate within a non-deterministic universe possessing specific traits. (shrink)

This paper concerns the relationship between transitivity of entailment, omega-inconsistency and nonstandard models of arithmetic. First, it provides a cut-free sequent calculus for non-transitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omega-inconsistent. It then identifies the conditions in McGee for an omega-inconsistent logic as quantified standard deontic logic, presents a cut-free labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, (...) proves omega-inconsistency and shows thus, pace Cobreros et al., that the result in McGee does not rely on transitivity. Finally, it also explains why the omega-inconsistent logics of truth in question do not require nonstandard models of arithmetic. (shrink)

A predicate extension SQHT= of the logic of here-and-there was introduced by V. Lifschitz, D. Pearce, and A. Valverde to characterize strong equivalence of logic programs with variables and equality with respect to stable models. The semantics for this logic is determined by intuitionistic Kripke models with two worlds with constant individual domain and decidable equality. Our sequent formulation has special rules for implication and for pushing negation inside formulas. The soundness proof allows us to establish that SQHT= is a (...) conservative extension of the logic of weak excluded middle with respect to sequents without positive occurrences of implication. The completeness proof uses a non-closed branch of a proof search tree. The interplay between rules for pushing negation inside and truth in the “there” world of the resulting Kripke model can be of independent interest. We prove that existence is definable in terms of remaining connectives. (shrink)

In this study, the impacts of two different “methods” for teaching ethics as part of the religious education in the Swedish upper secondary school were compared by means of a non-randomized controlled trial in two parts, involving 542 students. The question was which “method” had the greatest capacity to generate long-term ethical awareness in the students. The intervention condition consisted of students whose teachers were instructed to teach according to the Three Step Model, a teaching method influenced by research concerning (...) how moral autonomy and ethical awareness could be increased by means of instruction and training. The control condition consisted of students whose teachers were instructed to teach basically as usual but with some added guidelines. During the trial, all students were given a pre-test before the ethics section had started and a post-test 10–12 weeks after it was finished. When quantified and summarized, the results showed an advantage of the intervention condition in measure B but an advantage of the control condition in measure A ; however, the advantage of the intervention condition was clearer and stronger. Even though the intervention students did not experience a stronger development, they appeared to have learned significantly more, not least in terms of procedural knowledge in ethical problem solving. The tentative conclusion is therefore that the Three Step Model is a more effective method for increasing ethical awareness, at least if one defines ethical awareness and measures it the way it was done in this study. (shrink)

The general issue addressed in this dissertation is: what do the models of formal model-theoretic semantics represent? In chapter 2, I argue that those of first-order classical logic represent meaning assignments in possible worlds. This motivates an inquiry into what the interpretations of first-order quantified model logic represent, and in Chapter 3 I argue that they represent meaning assignments in possible universes of possible worlds. A possible universe is unpacked as one way model reality might be. The problem arises (...) here as to how we are to understand the distinction between the actual and the possible as it relates to modal reality. ;Along with the development of the main arguments in Chapters 2 and 3, the dissertation assesses the status of semantic accounts or logical properties and relations. Specifically, what does the model-theoretic account of a logically possible situation add to the syntactic account ? ;Proofs of invalidity in terms of the models of formal semantics do not establish that it is possible for the premises to be true and the conclusion false, since a formal model is merely given by a consistent set of sentences. Unless there is some way to generate a non-formal model from a formal one, such proofs do not really go beyond syntactic notions. The dissertation ends by concluding that there is no way to generate a non-formal model from a formal one without relying on logical intuitions that are syntactical. ;Hence efforts to construct a semantic basis for model logic independent of syntactic commitments are misguided. However, in classical logic the independence of the semantic account from the syntactic one is grounded on the intuition that it is metaphysically possible for there to be a denumerably infinite totality of objects. (shrink)

In this paper, I shall provide a defence of second-order logic in the context of its use in the philosophy of mathematics. This shall be done by considering three problems that have been recently posed against this logic: (1) According to Resnik [1988], by adopting second-order quantifiers, we become ontologically committed to classes. (2) As opposed to what is claimed by defenders of second-order logic (such as Shapiro [1985]), the existence of non-standard models of first-order theories does not establish the (...) inadequacy of first—order axiomatisations (Melia [1995]). (3) In contrast with Shapiro’s suggestion (in his [1985]), second-order logic does not help us to establish referential access to mathematical objects (Azzouni [1994]). As I shall argue, each of these problems can be neatly solved by the second-order theorist. As a result, a case for second-order logic can be made. The first two problems will beconsidered rather briefly in the next section. The rest of the paper is dedicate to a discussion of the third. (shrink)

Vann McGee has recently argued that Belnap’s criteria constrain the formal rules of classical natural deduction to uniquely determine the semantic values of the propositional logical connectives and quantifiers if the rules are taken to be open-ended, i.e., if they are truth-preserving within any mathematically possible extension of the original language. The main assumption of his argument is that for any class of models there is a mathematically possible language in which there is a sentence true in just those models. (...) I show that this assumption does not hold for the class of models of classical propositional logic. In particular, I show that the existence of non-normal models for negation undermines McGee’s argument. (shrink)

Free logics are a family of first-order logics which came about as a result of examining the existence assumptions of classical logic (Hintikka _The Journal of Philosophy_, _56_, 125–137 1959 ; Lambert _Notre Dame Journal of Formal Logic_, _8_, 133–144 1967, 1997, 2001 ). What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free (...) logics reject the claim that names need to denote in (ii). _Positive_ free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, _negative_ free logic treats them as uniformly false, and _neutral_ free logic as taking a third value. There has been a renewed interest in analyzing proof theory of free logic in recent years, based on intuitionistic logic in Maffezioli and Orlandelli (_Bulletin of the Section of Logic_, _48_(2), 137–158 2019 ) as well as classical logic in Pavlović and Gratzl (_Journal of Philosophical Logic_, _50_, 117–148 2021 ), there for the positive and negative variants. While the latter streamlines the presentation of free logics and offers a more unified approach to the variants under consideration, it does not cover neutral free logic, since there is some lack of both clear formal intuitions on the semantic status of formulas with empty names, as well as a satisfying account of the conditional in this context. We discuss extending the results to this third major variant of free logics. We present a series of G3 sequent calculi adapted from Fjellstad (_Studia Logica_, _105_(1), 93–119 2017, _Journal of Applied Non-Classical Logics_, _30_(3), 272–289 2020 ), which possess all the desired structural properties of a good proof system, including admissibility of contraction and all versions of the cut rule. At the same time, we maintain the unified approach to free logics and moreover argue that greater clarity of intuitions is achieved once neutral free logic is conceptualized as consisting of two sub-varieties. (shrink)

The book is a collection of papers and aims to unify the questions of syntax and semantics of language, which are included in logic, philosophy and ontology of language. The leading motif of the presented selection of works is the differentiation between linguistic tokens (material, concrete objects) and linguistic types (ideal, abstract objects) following two philosophical trends: nominalism (concretism) and Platonizing version of realism. The opening article under the title “The Dual Ontological Nature of Language Signs and the Problem of (...) Their Mutual Relations” provides a broad introduction into the problem area connected with this differentiation, while the logic-formal characteristics of the distinction are framed in the work entitled “On the Type-Token Relationships” (Chapter 1). The basic part of the book deals with issues relating to syntax (Chapters 2-4) and semantics of language (Chapters 5-6), as well as pertaining to syntactic-semantic pragmatic questions (Chapters 7-13). Throughout the book, language, categorial language, is characterized syntactically as generated by classical categorial grammar (Chapter 2) and formalized on two opposing levels: as language of expression-tokens (level of tokens) and language of expression-types (level of types). The author’s considerations contained in Chapters 2 and 4 lead to the important philosophical conclusion that in formal-logical syntactic studies on language the assumption that expression-types constitute the primary language layer while expression-tokens make the secondary one, can be neglected; thus, this speaks in favour of the opposing standpoint—the concretistic one—in the ontology of language syntax. In the works “Meaning and Interpretations”, Parts I and II (Chapters 5 and 6), it is underlined, however, that such semantic concepts as: meaning, denotation and interpretation are defined on the types level, yet their formal definitions require making use of notions of the tokens level. The semantic notions introduced in the above-mentioned articles are also used in the following works of the present selection, under the titles: “Three Principles of Compositionality” and “On Metaknowledge and Truth” (Chapters 7 and 8). They formalize two principles of compositionality that are well known in the literature on the subject, deriving from Frege, i.e. those of meaning and of denotation; they are related to the syntactic principle of compositionality which was introduced by the author. All the three principles are, at the same time, three conditions of homomorphism of categorial language algebra into three kinds of non-standard models of language (one syntactic and two semantic ones: intensional and extensional), which allows introducing three definitions of truthfulness into these models. The next two works in the collection, entitled: “On Language Adequacy” and “What is the Sense in Logic and Philosophy of Language” (Chapters 9 and 10) concern adequacy of categorial language syntax along with its dual semantics: intensional and extensional, and categorial compatibility of any of its syntactic categories with two corresponding semantic categories: intensional and extensional, based on the compatibility the syntactic category of each language expression with the ontological category assigned to its denotatum. The well-known problem of categorial compatibility for first-order quantifiers finds its solution in the paper “Categories of First-Order Quantifiers” (Chapter 11). In the work “Logic and Ontology of Language” (Chapter 12), being in a sense a summary of the considerations presented in the preceding chapters of the book, language is treated as an ontological being, characterized in compliance with the logical conception of language proposed by Ajdukiewicz. Application—like throughout the book—of tools of classical logic and set theory has resulted in emergence of a general formal logical theory of syntax, semantics and pragmatics of language, which takes into account duality in the understanding of linguistic expressions as tokens (concretes) and types (abstract objects). In terms that take into account a functional approach to language itself, there comes out an ontological neutrality of logic with respect to existential assumptions relating to the ontological nature of linguistic expressions and their extra-linguistic ontological counterparts. The issues connected with applying logic while explaining the manner of using linguistic tokens and linguistic types to determine notions of language communication are raised and illustrated in the last chapter of the work, bearing the title “A Logical Conceptualization of Knowledge on the Notion of Language Communication”. (shrink)

The -logic of Terwijn is a variant of first-order logic with the same syntax in which the models are equipped with probability measures and the quantifier is interpreted as ‘there exists a set A of a measure such that for each,...’. Previously, Kuyper and Terwijn proved that the general satisfiability and validity problems for this logic are, i) for rational, respectively -complete and -hard, and ii) for, respectively decidable and -complete. The adjective ‘general’ here means ‘uniformly over all languages’. We (...) extend these results in the scenario of finite models. In particular, we show that the problems of satisfiability and validity with respect to finite models in E-logic are, i) for rational, respectively -complete and -complete, and ii) for, respectively decidable and -complete. Although partial results toward the countable case are also achieved, the computability of E-logic over countable models still remains largely unsolved. In addition, most o... (shrink)

In what follows I try to show that Spinoza modelled his project of rational psychology, in some of its major respects, upon Descartes's metaphysics of matter. I argue further that, like Descartes, who paid for the rationalization of the science of matter the price of having to leave out of his description non-quantifiable qualities, so Spinoza left out of his psychology the non-rationalizable aspects of emotions, i.e. whatever in them could not be subsumed under common notions. He therefore was left (...) with the cognitive aspects of emotions, keeping outside of his report the inner feeling which accompanies them. Spinoza's psychology, I claim, disregards any non-cognitive aspect of emotions. (shrink)

This paper argues that investigations into how to best make something often provide researchers with an objectual understanding of their target phenomena. This argument starts with an extended investigation into the non-representational uses of models. In particular, we identify a special sort of “design model” whose aim is to guide the production of phenomena. Clarifying how these design models are evaluated shows that they are evaluated in different ways than representational models. Once the character of design models has been fixed, (...) we argue that grasping design models can provide objectual understanding of phenomena. This argument proceeds through a critical engagement with Dellsén’s ( 2020 ) position that a grasp of a good representational model of dependencies provides objectual understanding of a phenomenon. We agree with Dellsén that this is one way to achieve understanding, but maintain that grasping a good design model is another way to achieve understanding. The paper concludes by considering some important objections to our proposal and also by noting some of the broader questions about understanding and knowledge in both science and engineering. (shrink)

In this study meal sharing is used as a way of quantifying food transfers between households. Traditional food-sharing studies measure the flow of resources between households. Meal sharing, in contrast, measures food consumption acts according to whether one is a host or a guest in the household as well as the movement of people between households in the context of food consumption. Our goal is to test a number of evolutionary models of food transfers, but first we argue that before (...) one tests models of who should receive food one must understand the adaptiveness of food transfers. For the Ye’kwana, economies of scale in food processing and preparation appear to set the stage for the utility of meal sharing. Evolutionary models of meal sharing, such as kin selection and reciprocal altruism, are evaluated along with non-evolutionary models, such as egalitarian exchange and residential propinquity. In addition, a modified measure of exchange balance—proportional balance—is developed. Reciprocal altruism is shown to be the strongest predictor of exchange intensity and balance. (shrink)

Corroborating evidence supports a proposition that is already supported by other initial evidence. It bolsters or confirms the original body of evidence. Corroboration has striking psychological and epistemic force: It potently affects how people do and should assess the target proposition. This essay investigates the distinctive powers of corroborating evidence. Corroboration does not simply increase the quantifiable probability of the adjudicated claim. Drawing on the relevant alternatives framework, I argue that corroboration winnows remaining uneliminated error possibilities. This illuminates the independence, (...) weight, and non-fungibility of corroborating evidence. I compare corroborating evidence to prudential safeguards, like fire doors, that forfend against non-epistemic harms. I thereby sketch a general, non-quantificational model of risk management. Finally, I turn to legal corroboration requirements and the epistemic significance of corroboration for legal proof. (shrink)