Increasingly encompassing models have been suggested for our world. Theories range from generally accepted to increasingly speculative to apparently bogus. The progression of theories from ego- to geo- to helio-centric models to universe and multiverse theories and beyond was accompanied by a dramatic increase in the sizes of the postulated worlds, with humans being expelled from their center to ever more remote and random locations. Rather than leading to a true theory of everything, this trend faces a turning point (...) after which the predictive power of such theories decreases (actually to zero). Incorporating the location and other capacities of the observer into such theories avoids this problem and allows to distinguish meaningful from predictively meaningless theories. This also leads to a truly complete theory of everything consisting of a (conventional objective) theory of everything plus a (novel subjective) observer process. The observer localization is neither based on the controversial anthropic principle, nor has it anything to do with the quantum-mechanical observation process. The suggested principle is extended to more practical (partial, approximate, probabilistic, parametric) world models (rather than theories of everything). Finally, I provide a justification of Ockham's razor, and criticize the anthropic principle, the doomsday argument, the no free lunch theorem, and the falsifiability dogma.", . (shrink)

We introduce the notion of positive strong amalgamation property and we investigate some universal forms and properties of this notion. Considering the close relationship between the amalgamation property and the notion of complete theories, we explore the fundamental properties of positively complete theories, and we illustrate the behaviour of this notion by bringing changes to the language of the theory through the groups theory.

Let T be a complete first-order theory over a finite relational language which is axiomatized by universal and existential sentences. It is shown that T is almost trivial in the sense that the universe of any model of T can be written $F \overset{\cdot}{\cup} I_1 \overset{\cdot}{\cup} I_2 \overset{\cdot}{\cup} \cdots \overset{\cdot}{\cup} I_n$ , where F is finite and I 1 , I 2 ,...,I n are mutually indiscernible over F. Some results about complete theories with ∃∀-axioms over a (...) finite relational language are also mentioned. (shrink)

An example is given of a complete theory with minimal models of arbitrarily large minimality rank.

We distinguish traditional cognition theories from hierarchically complex stacked neural networks that meet many of Newell's criteria. The latter are flexible and can learn anything that a person can learn, by using their mistakes and successes the same way humans do. Shortcomings are due largely to limitations of current technology.

We show 13 stages of the development of tool-use and tool making during different eras in the evolution of Homo sapiens. We used the NeoPiagetian Model of Hierarchical Complexity rather than Piaget's. We distinguished the use of existing methods imitated or learned from others, from doing such a task on one's own.

A sequential, hierarchical stage model of empathy can account for a comprehensive range of empathic behaviors. We provide an illustrative table, “Stages of Empathy,” to demonstrate how increasingly complex empathic behaviors emerge at each stage, beginning with the infant's “automatic empathy” and ending with the advanced adult's “coconstruction of empathetic reality.”.

In the present paper, we continue the research in Zhao (2021, Logic J. IGPL) to develop the Sahlqvist completeness theory for hybrid logic with satisfaction operators and downarrow binders |$\mathcal {L}( @, {\downarrow })$|⁠. We define the class of restricted Sahlqvist formulas for |$\mathcal {L}( @, {\downarrow })$| following the ideas in Conradie and Robinson (2017, J. Logic Comput., 27, 867–900), but we follow a different proof strategy which is purely proof-theoretic, namely showing that for every restricted Sahlqvist formula (...) |$\varphi $| and its hybrid pure correspondence |$\pi $|⁠, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| proves |$\pi $|⁠; therefore, |$\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $| is complete with respect to the class of frames defined by |$\pi $|⁠, using a modified version |$\textsf {ALBA}^{{\downarrow }}_{\textsf {Modified}}$| of the algorithm |$\textsf {ALBA}^{{\downarrow }}$| defined in Zhao (2021, Logic J. IGPL). (shrink)

In this paper, I will give a new characterisation of the spaces of complete theories of pseudofinite fields and of algebraically closed fields with a generic automorphism (ACFA) in terms of the Vietoris topology on absolute Galois groups of prime fields.

The purpose of this paper is to give a purely logical proof of a result of Mostowski [1937] concerning the complete theories of a calculus based on classical propositional logic; and then modestly to generalize it. Mostowski’s result is announced by Tarski on p. 370 of Logic, Semantics, Metamathematics [1956]. (All references to Tarski’s work here are to this book.) Tarski himself provides only a fragment of a proof, and the proof published by Mostowski makes extensive use of topological (...) methods and results. The a proof offered here is undoubtedly longer than Mostowski’s and not by any means independent of it. But it should not be beyond the powers of anyone who has followed assiduously a couple of courses in propositional logic and knows a little set theory. The axiom of choice is assumed, but not the continuum hypothesis. (shrink)

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ (...) -statement with this feature in the sense that any other $\Sigma _1^2 $ -statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment $V_\lambda $ of the universe of sets (for example, one might take $V_\lambda $ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of $V_\lambda $ . If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies¬CH. (shrink)

We argue that autism and psychosis spectrum disorders cannot be conceptualized as polar extremes of mentalizing ability. We raise two main objections: (1) the autistic-psychotic continuum, as conceptualized by the authors, excludes defining features of schizophrenia spectrum: negative symptoms, which correlate more strongly with mentalizing impairments; and (2) little evidence exists for a relationship between mentalizing ability and positive symptoms.

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained (...) by Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space. (shrink)

The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Eršov principle is proven for a theory of valued D-fields of residual characteristic zero.

With quantifier elimination and restriction of language to a binary relation symbol and constant symbols it is shown that countable complete theories having three isomorphism types of countable models are "essentially" the Ehrenfeucht example [4, $\s6$ ].

Summary According to the so-called Starnberger Group1, the amenability of a science to science policy measures, conceptualised by the Group in terms of finalization , depends crucially on conditions intrinsic to the science not invariably present at every stage of its development. Finalization is possible only at junctures where the science in question faces methodologically divergent alternative lines of development. The most significant kind of case depends on the presence of completed , or classical , theories.

The prospects for a complete stochastic theory of microscopic phenomena are considered. The two traditional schools of stochastic physics, the diffusion process school and the zero-point electromagnetic field school, are reviewed. A completely relativistic theory, stochastic field theory, is proposed as an extension of the ideas of these two schools. Within the context of stochastic field theory we present the following new results: an elementary stochastization scheme which produces the zero-point electromagnetic field; a physical interpretation (...) of the mathematical methods developed by Lukosz for calculating zero-point energies; a calculation of the first-order Lamb shift which generalizes that of Welton; a new setting for a finite-temperature theory; and comments on the bag model for quark confinement. (shrink)

Here we take the view that LPC(=) is applicable to structures whose domain is too large to be a set. This is not just a matter of class theory versus set theory, although it can be interpreted as such, and this interpretation is discussed briefly at the end.

For a complete first order theory of Boolean algebras T which has nonisomorphic countable models, we determine the first limit ordinal α = α(T) such that We show that for some and for all other T‘s, A nonprincipal ideal I of B is almost principal, if a is a principal ideal of B} is a maximal ideal of B. We show that the theory of Boolean algebras with an almost principal ideal has complete extensions and characterize (...) them by invariants similar to the Tarski’s invariants. (shrink)

We build a new spectrum of recursive models (SRM(T)) of a strongly minimal theory. This theory is non-disintegrated, flat, model complete, and in a language with a finite structure.

On the set of all first-order complete theories $$T(\sigma )$$ of a language $$\sigma$$ we define a binary operation $$\{\cdot \}$$ by the rule: $$T\cdot S= {{\,\textrm{Th}\,}}(\{A\times B\mid A\models T \,\,\text {and}\,\, B\models S\})$$ for any complete theories $$T, S\in T(\sigma )$$. The structure $$\langle T(\sigma );\cdot \rangle$$ forms a commutative semigroup. A subsemigroup S of $$\langle T(\sigma );\cdot \rangle$$ is called an absorption’s formula definable semigroup if there is a complete theory $$T\in (...) T(\sigma )$$such that$$S=\langle \{X\in T(\sigma )\mid X\cdot T=T\};\cdot \rangle $$. In this event we say that a theory T absorbs S. In the article we show that for any absorption’s formula definable semigroup S the class $${{\,\textrm{Mod}\,}}(S)=\{A\in {{\,\textrm{Mod}\,}}(\sigma )\mid A\models T_0\,\,\text {for some}\,\, T_0\in S\}$$ is axiomatizable, and there is an idempotent element $$T\in S$$ that absorbs S. Moreover, $${{\,\textrm{Mod}\,}}(S)$$ is finitely axiomatizable provided T is finitely axiomatizable. We also prove that $${{\,\textrm{Mod}\,}}(S)$$ is a quasivariety (variety) provided T is an universal (a positive universal) theory. Some examples are provided. (shrink)

Machery proposes that the construct of detracts from research progress. However, ignoring development also detracts from research progress. Developmental research has advanced our understanding of how concepts are acquired and thus is essential to a complete theory. We propose a framework that both accounts for development and holds great promise as a new direction for thinking about concepts.

The evolution of humans required performing increasingly hierarchically complex tasks within multiple domains. Hierarchical complexity increases task by task. Tasks occur within, and differ by, determinable domains, their stages of performance measurable using the Model of Hierarchical Complexity. How well one performs within single and multiple domains is considered to indicate intelligence. Original task-initiation is more difficult than imitational learning and can create new domains. Levels of support reduce task difficulty, increasing performance. Task-performance may be generalized to other domains. Stages (...) of developing tools and empathy are presented to demonstrate domains' roles in the evolution of human intelligence. (shrink)

While quantum theory cannot be described by a local hidden variable model, it is nevertheless possible to construct such models that exhibit features commonly associated with quantum mechanics. These models are also used to explore the question of \-ontic versus \-epistemic theories for quantum mechanics. Spekkens’ toy theory is one such model. It arises from classical probabilistic mechanics via a limit on the knowledge an observer may have about the state of a system. The toy theory for (...) the simplest possible underlying system closely resembles stabilizer quantum mechanics, a fragment of quantum theory which is efficiently classically simulable but also non-local. Further analysis of the similarities and differences between those two theories can thus yield new insights into what distinguishes quantum theory from classical theories, and \-ontic from \-epistemic theories. In this paper, we develop a graphical language for Spekkens’ toy theory. Graphical languages offer intuitive and rigorous formalisms for the analysis of quantum mechanics and similar theories. To compare quantum mechanics and a toy model, it is useful to have similar formalisms for both. We show that our language fully describes Spekkens’ toy theory and in particular, that it is complete: meaning any equality that can be derived using other formalisms can also be derived entirely graphically. Our language is inspired by a similar graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens’ toy bit theory and stabilizer quantum mechanics can be analysed and compared using analogous graphical formalisms. (shrink)

Equational hybrid propositional type theory ) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: Completeness in type theory, The completeness of the (...) first-order functional calculus and Completeness in propositional type theory. More precisely, from and we take the idea of building the model described by the maximal consistent set; in our case the maximal consistent set has to be named, \-saturated and extensionally algebraic-saturated due to the hybrid and equational nature of \. From, we use the result that any element in the hierarchy has a name. The challenge was to deal with all the heterogeneous components in an integrated system. (shrink)

By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued,pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos E of M is said to represent M if it is complete and its cumulative hierarchy, as (...) defined by Fourman and Hayashi, coincides with the usual cumulative hierarchy of M. We show that, although M need not be a complete topos, it has a smallest complete representing subtopos, and we describe this subtopos in terms of definability in M. We characterize, again in terms of definability, those models M whose smallest representing topos is a Grothendieck topos. Finally, we discuss the extent to which a model can be reconstructed when its smallest representing topos is given. (shrink)

We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo–Fraenkel set theory, the completeness of such semantics is equivalent to the Boolean prime ideal theorem. Using a result of McCarty, we conclude that the completeness of Kripke semantics is equivalent, over intuitionistic Zermelo–Fraenkel set theory, to the Law of Excluded Middle plus BPI. Along the way, we also prove the equivalence, over (...) ZF, between BPI and the completeness theorem for Kripke semantics for both first-order and propositional theories. (shrink)

Earlier in this century, many philosophers of science (for example, Rudolf Carnap) drew a fairly sharp distinction between theory and observation, between theoretical terms like 'mass' and 'electron', and observation terms like 'measures three meters in length' and 'is _2° Celsius'. By simply looking at our instruments we can ascertain what numbers our measurements yield. Creatures like mass are different: we determine mass by calculation; we never directly observe a mass. Nor an electron: this term is introduced in order (...) to explain what we observe. This (once standard) distinction between theory and observation was eventually found to be wanting. First, if the distinction holds, it is difficult to see what can characterize the relationship between theory :md observation. How can theoretical terms explain that which is itself in no way theorized? The second point leads out of the first: are not the instruments that provide us with observational material themselves creatures of theory? Is it really possible to have an observation language that is entirely barren of theory? The theory-Iadenness of observation languages is now an accept ed feature of the logic of science. Many regard such dependence of observation on theory as a virtue. If our instruments of observation do not derive their meaning from theories, whence comes that meaning? Surely - in science - we have nothing else but theories to tell us what to try to observe. (shrink)

We show that if T is a trivial uncountably categorical theory of Morley Rank 1 then T is model complete after naming constants for a model.

We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret $@_i$ in propositional and first-order hybrid logic. This means: interpret $@_i\alpha _a$ , where $\alpha _a$ is an expression of any type $a$ , as an expression of type $a$ that (...) rigidly returns the value that $\alpha_a$ receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and (as is usual inhybrid logic) we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logicover Henkin’s logic. (shrink)