Free Semantics

Free Semantics is based on normalized natural deduction for the weak relevant logic DW and its near neighbours. This is motivated by the fact that in the determination of validity in truth-functional semantics, natural deduction is normally used. Due to normalization, the logic is decidable and hence the semantics can also be used to construct counter-models for invalid formulae. The logic DW is motivated as an entailment logic just weaker than the logic MC of meaning containment. DW is the logic focussed upon, but the results extend to MC. The semantics is called ‘free semantics’ since it is disjunctively and existentially free in that no disjunctive or existential witnesses are produced, unlike in truth-functional semantics. Such ‘witnesses’ are only assumed in generality and are not necessarily actual. The paper sets up the free semantics in a truth-functional style and gives a natural deduction interpetation of the meta-logical connectives. We then set out a familiar tableau-style system, but based on natural deduction proof rather than truth-functional semantics. A proof of soundness and completeness is given for a reductio system, which is a transform of the tableau system. The reductio system has positive and negative rules in place of the elimination and introduction rules of Brady’s normalized natural deduction system for DW. The elimination-introduction turning points become closures of threads of proof, which are at the points of contradiction for the reductio system.     

Valuation Semantics for First-Order Logics of Evidence and Truth

This paper introduces the logic QLET_F, a quantified extension of the logic of evidence and truth LET_F, together with a corresponding sound and complete first-order non-deterministic valuation semantics. LET_F is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment (FDE) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘A entails that A behaves classically, ∙A follows from A’s violating some classically valid inferences. The semantics of QLET_F combines structures that interpret negated predicates in terms of anti-extensions with first-order non-deterministic valuations, and completeness is obtained through a generalization of Henkin’s method. By providing sound and complete semantics for first-order extensions of FDE, K3, and LP, we show how these tools, which we call here the method of anti-extensions + valuations, can be naturally applied to a number of non-classical logics.

     

Refutations,Proofs, and Models in the Modal Logic K4

In this paper we study the method of refutation rules in the modal logic K4. We introduce refutation rules with certain normal forms that provide a new syntactic decision procedure for this logic. As corollaries we obtain such results for the following important extensions: S4, the provability logic G, and Grzegorczyk's logic. We also show that tree-type models can be constructed from syntactic refutations of this kind.     

State-of-affairs Semantics for Positive Free Logic

In the following the details of a state-of-affairs semantics for positive free logic are worked out, based on the models of common inner domain–outer domain semantics. Lambert's PFL system is proven to be weakly adequate (i.e., sound and complete) with respect to that semantics by demonstrating that the concept of logical truth definable therein coincides with that one of common truth-value semantics for PFL. Furthermore, this state-of-affairs semantics resists the challenges stemming from the slingshot argument since logically equivalent statements do not always have the same extension according to it. Finally, it is argued that in such a semantics all statements of a certain language for PFL are state-of-affairs-related extensional as well as salva extensione extensional, even though their salva veritate extensionality fails.     

Against logical generalism

The orthodox view of logic takes for granted the central importance of logical principles. Logic, and thus logical reasoning, is to be understood as a system of rules or principles with universal application. Let us call this orthodox view logical generalism. In this paper we argue that logical generalism, whether monist or pluralist, is wrong. We then outline an account of logical consequence in the absence of general logical principles, which we call logical particularism.

     

On the dynamics of institutional agreements

In this paper we investigate a logic for modelling individual and collective acceptances that is called acceptance logic. The logic has formulae of the form A_G:x j{\rm A}_{G:x} \varphi reading ‘if the agents in the set of agents G identify themselves with institution x then they together accept that j{\varphi} ’. We extend acceptance logic by two kinds of dynamic modal operators. The first kind are public announcements of the form x!y{x!\psi}, meaning that the agents learn that y{\psi} is the case in context x. Formulae of the form [x!y]j{[x!\psi]\varphi} mean that j{\varphi} is the case after every possible occurrence of the event x!ψ. Semantically, public announcements diminish the space of possible worlds accepted by agents and sets of agents. The announcement of ψ in context x makes all \lnoty{\lnot\psi} -worlds inaccessible to the agents in such context. In this logic, if the set of accessible worlds of G in context x is empty, then the agents in G are not functioning as members of x, they do not identify themselves with x. In such a situation the agents in G may have the possibility to join x. To model this we introduce here a second kind of dynamic modal operator of acceptance shifting of the form G:x-y{G:x\uparrow\psi}. The latter means that the agents in G shift (change) their acceptances in order to accept ψ in context x. Semantically, they make ψ-worlds accessible to G in the context x, which means that, after such operation, G is functioning as member of x (unless there are no ψ-worlds). We show that the resulting logic has a complete axiomatization in terms of reduction axioms for both dynamic operators. In the paper we also show how the logic of acceptance and its dynamic extension can be used to model some interesting aspects of judgement aggregation. In particular, we apply our logic of acceptance to a classical scenario in judgment aggregation, the so-called ‘doctrinal paradox’ or ‘discursive dilemma’ (Pettit, Philosophical Issues 11:268–299, 2001; Kornhauser and Sager, Yale Law Journal 96:82–117, 1986).     

Cut-Elimination and Quantification in Canonical Systems

Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with unary quantifiers and show that it remains constructive. Then we provide semantics for such canonical systems using 2-valued non-deterministic matrices extended to languages with quantifiers, and prove that the following properties are equivalent for a canonical system G: (1) G admits Cut-Elimination, (2) G is coherent, and (3) G has a characteristic 2-valued non-deterministic matrix.     

A Cut-Free Sequent Calculus for Defeasible Erotetic Inferences

In recent years, the effort to formalize erotetic inferences—i.e., inferences to and from questions—has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and erotetic implication. While an effort has been made to axiomatize the former in a sequent system, there is currently no proof theory for the latter. Moreover, the extant axiomatization of erotetic evocation fails to capture its defeasible character and provides no rules for introducing or eliminating question-forming operators. In contrast, our calculus encodes defeasibility conditions on sequents and provides rules governing the introduction and elimination of erotetic formulas. We demonstrate that an elimination theorem holds for a version of the cut rule that applies to both declarative and erotetic formulas and that the rules for the axiomatic account of question evocation in IEL are admissible in our system.

     

The Tense Logic for Master Argument in Prior’s Reconstruction

In this paper we examine Prior’s reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic K _t 4 plus a new axiom (P): ‘p Λ G p ⊃ P G p’. This formula was used by Prior in his original analysis of Master Argument. (P) is usually added as an extra axiom to an axiomatization of the logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by (P) in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. Presented by Jacek Malinowski     

Free Speech Fallacies as Meta-Argumentative Errors

Free speech fallacies are errors of meta-argument. One commits a free speech fallacy when one argues that since there are apparent restrictions on one’s rights of free expression, procedural rules of critical exchange have been broken, and consequently, one’s preferred view is dialectically better off than it may otherwise seem. Free speech fallacies are meta-argumentative, since they occur at the level of assessing the dialectical situation in terms of norms of argument and in terms of meta-evidential principles of interpreting how and why people follow (or fail to follow) argumentative rules. Our plan here is to begin with a brief explanation of meta-argument and meta-argumentative fallacy. We will then turn to the variety of forms of the free speech fallacy, which we will explain as meta-argumentatively erroneous.

     

Axiomatic world theory: An overview the general theory of evolution in brief

The world, its many subsystems and all their theories, starting with logic, can be reduced to two related functions: a combinatorial system generator and a hamiltonian system organizer. These can be derived, in turn, from an Axiom of Lawfulness, the expansion being guided by pseudo‐category and pseudo‐functor analysis to produce an axiomatic theory of the world or general theory of evolution. Specifically, world evolution is generated by a constrained combinatorial world generator, F:G_(X), deduced from two related axioms: I. The Axiom of World Lawfulness and II. The Axiom of World Constraint Constants, c = c₁, c₂, of primordial physical combinatee (substance), c₁, and physical combinator (motion), c₂.

Axiom I postulates a lawful analysis by an analyzer adhering to appropriate coordinate systems, CS, of a lawful analysand obeying a conservation law, X = X. The analysand consists of a base combinatee (the set and elements), X = {x₁, x₂,… x_n}, and a base combinator, namely, the universal Boolean operator, NOR = NOT + OR. Base combinatee and combinator both have attributes of quantity combinatorially generated by NOR operating on the universal number, 1, and of quality generated by NOR operating on the universal dimensions, MLT (mass, length, time), including the null sets.

Axiom II fixes the base constants, c, = c₁, c₂, thereby converting X to material substance using c₁ and NOR to material motion using c₂. This comprehensive, quality and quantity‐competent foundational science is called Universal Combinatorics. Its elements comprise the logical alphabet or metavector, A = {c, 1, MLT; X, NOR}, where c is obtained from the remaining terms. These give: (1) the attributive pseudo‐functor, F = P_(c,1,MLT), where P is the power set of the indicated attributes, and (2) the logic generator, G_(X), where G = NOR_(NOR). F then maps G_(X) into world evolution, F:G_(X) → world evolution, as follows:

Expanding the abstract generator, F₁:G,_(x), with world constants eliminated, i.e., c = 0, generates Universal Grammar consisting of (1) the substantive content of the abstract science chain running from linguistic grammar to mathematics and logic and (2) a comprehensive epistemology equivalent to an explicit theory of the strategic aspects of the scientific method, including a universal hamiltonian theory structure informally related to a mathematical category. The four epistemological theorems are:

1. I. The Combinatorial System Generator, F:G_(X), (read as “The attributive functor, F, maps the logic generator, G_(X), into world theory” or “The world is an attributive combinatorial function of logic").

2. II. The Hamiltonian System Operating Theorem, h (an abstract theory‐category structure).

3. III. The System Stability Theorem, PI?, where PI is the extremal Performance Index or controlling law.

4. IV. The Intersystem Abstraction Ranking Theorem given by the Attributive Functor/ Function, F.

F₂ admits the world constants, c > 0, to materialize the grammar generator, G_(X), to an homologous concrete Euler combinatorial physical wave generator, namely, the superstring equation of quantum theory, E_(NI) = A(σ,τ), where E is the permutational function, NI, is the set of nonintegers and the solution is the dual amplitude, σ,τ. Expanding generates the elementary particles of nonadaptive physics and, by inference, the substantive content of Universal Physics consisting of three additional primary systems comprising the world, where a primary system is defined as one having a distinct but derivative extremal controlling law:

1. I. Nonadaptive physics and chemistry (harmonic hamiltonian wave systems) : Minimize Action, subject to conservation constraints.

2. II. Adaptive physics or biology (membrane bound duplicating polymer‐copolymer hamiltonian systems) : Maximize Survival, subject to energy constraints.

3. III. Sentient physics or sociopsychology (neuromatrix hamiltonian systems) : Maximize subjective Happiness, subject to survival constraints and

4. IV. Representational physics or language (a symbolic combinatorial routine): Maximizes the Information Gain, subject to happiness constraints.

The world can then be viewed as a perpetual superfluid computer implicitly using the epistemology of Axiom I as a world program to process the physical data base, c > 0, of Axiom II into world evolution. After evolving through Systems I and II, mankind, i.e., System III, evolves as an internal metacomputer which makes the combinatorial program explicit and uses it to put all four primary systems in standard hamiltonian theory (pseudo‐category) format and terminology. This can be viewed as a generalization of the Darwinian variation‐and‐selection theme in which combinatorial‐variation is recursively hamiltonian‐selected thereby incrementing world logic and logic constraints on successive primary systems. Because Universal Physics and Universal Grammar are functor‐related homologous concrete and abstract combinatorial pseudocategories, related by a pseudo‐functor, thus, differing only in the presence and absence, respectively, of the World Constants, c ≥ 0, they constitute, ipso facto, Universal Science (Formal Philosophy, World Evolution, World Unification, Explicit Theory of Everything, ETOE, or Axiomatic World Theory).

QED: Because intricate verified predictions, ranging from particles to personality types, mental disorders, political parties and the abstract sciences, result from a system which is merely expanding to fill its possibility set, it is concluded that the world is lawful and that this means it is an object deterministic but not fully analytically determinable combinatorial system. In the object domain, the world is system‐number complete at four. Dually, in the analytical codomain, understanding of it is approximately complete, as measured by a world information gain function. Hence, the dualistic, analysand‐analyzer world program is finite and has dualistic completion criteria, as required of an involuted program.     

Stability and Paradox in Algorithmic Logic

There is significant interest in type-free systems that allow flexible self-application. Such systems are of interest in property theory, natural language semantics, the theory of truth, theoretical computer science, the theory of classes, and category theory. While there are a variety of proposed type-free systems, there is a particularly natural type-free system that we believe is prototypical: the logic of recursive algorithms. Algorithmic logic is the study of basic statements concerning algorithms and the algorithmic rules of inference between such statements. As shown in [1], the threat of paradoxes, such as the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. The first part of the paper develops a rich collection of inference rules that do not lead to paradox. The second part identifies traditional rules of logic that are paradoxical in algorithmic logic, and so should be viewed with suspicion in type-free logic generally.     

A Sequent Calculus for a Negative Free Logic

This article presents a sequent calculus for a negative free logic with identity, called N. The main theorem (in part 1) is the admissibility of the Cut-rule. The second part of this essay is devoted to proofs of soundness, compactness and completeness of N relative to a standard semantics for negative free logic.     

Hume’s Principle and Axiom V Reconsidered: Critical Reflections on Frege and His Interpreters

In this paper, I shall discuss several topics related to Frege’s paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege’s notion of evidence and its interpretation by Jeshion, the introduction of the course-of-values operator and Frege’s attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik (1884) Frege hardly could have construed Hume’s Principle (HP) as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck’s arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane.     

Enlarged Range of Free Flaps for Phalloplasty in Transsexual Reassignment Surgery

ABSTRACT

Our concept for complete phalloplasty which we applied in 66 female-to-male transsexuals, using free prelaminated osteofasciocutaneous fibula or fasciocutaneous forearm flaps, consisted of the following three operative stages:

1. mastectomy, ovariohysterectomy, urethra lengthening, colpectomy, and neourethra prelamination

2. after 3–6 months, neophallus creation with free sensate and prelaminated osteofasciocutaneous fibula (n = 41) or radial forearm flaps (n = 25)

3. 3–6 months later, urethral connection, neoscrotum formation and testicle prosthesis implantation.

Results: After mastectomy 2 hematoma had to be removed, and twice colpectomy revision was needed because of hematoma. No complications occurred after ovariohysterectomy. Partial flap necrosis took place in 1 patient of the forearm group and total necrosis in 2 patients of the fibula group. Eleven patients presented urethral stricture, and 9 a fistula. In 7 patients an operative stricture expansion was required, and in 6 patients surgical closure of the fistula. Overall patients' satisfaction was excellent.

Conclusions: The applied results demonstrates the effectiveness of such a multistage and interdisciplinary approach for female-to-male transsexual and it shows, that the fibula flap is an equal routine method extending the therapeutical range of gender assignment operations in female-to-male transsexuals.     

An Incomplete Relevant Modal Logic

The relevant modal logic G is a simple extension of the logic RT, the relevant counterpart of the familiar classically based system T. Using the Routley–Meyer semantics for relevant modal logics, this paper proves three main results regarding G: (i) G is semantically complete, but only with a non-standard interpretation of necessity. From this, however, other nice properties follow. (ii) With a standard interpretation of necessity, G is semantically incomplete; there is no class of frames that characterizes G. (iii) The class of frames for G characterizes the classically based logic T.     

A Generic Framework for Adaptive Vague Logics

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some extra set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true, etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided.     

Properties,Powers, and the Subset Account of Realization

According to the subset account of realization, a property, F, is realized by another property, G, whenever F is individuated by a non‐empty proper subset of the causal powers by which G is individuated (and F is not a conjunctive property of which G is a conjunct). This account is especially attractive because it seems both to explain the way in which realized properties are nothing over and above their realizers, and to provide for the causal efficacy of realized properties. It therefore seems to provide a way around the causal exclusion problem. There is reason to doubt, however, that the subset account can achieve both tasks. The problem arises when we look closely at the relation between properties and causal powers, specifically, at the idea that properties confer powers on the things that have them. If realizers are to be ontically prior to what they realize, then we must regard the conferral of powers by properties as a substantive relation of determination. This relation of conferral is at the heart of a kind of exclusion problem, analogous to the familiar causal exclusion problem. I argue that the subset account cannot adequately answer this new exclusion problem, and is for that reason ill‐suited to be the backbone of a non‐reductive physicalism.