Independence-friendly logic without Henkin quantification

We analyze the expressive resources of $$\mathrm {IF}$$ IF logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular $$\mathrm {IF}$$ IF sentences, this amounts to the study of the fragment of $$\mathrm {IF}$$ IF logic which is individuated by the game-theoretical property of action recall (AR). We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using “signalling by disjunction” instead of Henkin or signalling patterns. We also study irregular IF logic (in which requantification of variables is allowed) and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes behave quite differently in finite and infinite models. In particular, we show that, over infinite structures, every irregular prefix is equivalent to a regular one; and we present an irregular prefix which is second order on finite models but collapses to a first-order prefix on infinite models.

Even-If: Sufficient, Yet Unnecessary Conditions for Worldmaking

The scientific method seeks to identify cause-and-effect relationships using experimentation to test hypotheses through correspondence with an objective reality. Hans Vaihinger explicated the role of as-if logic in how science actually works in practice. Milton Friedman built on Vaihinger’s philosophy to argue the same for economics. Here I offer even-if as the logic for effectuation that in turn forms the basis for the entrepreneurial method. Entrepreneurship seeks not merely to test hypotheses but to co-create hypotheses worth reifying. In doing so, effectual entrepreneurship embodying an even-if logic can work with both possibilistic thinking and a variety of aspects of human imagination.

What Would a Phenomenology of Logic Look Like?

The phenomenological movement begins in the Prolegomena to Husserl’s Logical Investigations as a philosophy of logic. Despite this, remarkably little attention has been paid to Husserl’s arguments in the Prolegomena in the contemporary philosophy of logic. In particular, the literature spawned by Gilbert Harman’s work on the normative status of logic is almost silent on Husserl’s contribution to this topic. I begin by raising a worry for Husserl’s conception of ‘pure logic’ similar to Harman’s challenge to explain the connection between logic and reasoning. If logic is the study of the forms of all possible theories, it will include the study of many logical consequence relations; by what criteria, then, should we select one (or a distinguished few) consequence relation(s) as correct? I consider how Husserl might respond to this worry by looking to his late account of the ‘genealogy of logic’ in connection with Gurwitsch’s claim that ‘[i]t is to prepredicative perceptual experience … that one must return for a radical clarification and for the definitive justification of logic’. Drawing also on Sartre and Heidegger, I consider how prepredicative experience might constrain or guide our selection of a logical consequence relation and our understanding of connectives like implication and negation.

Giro dinámico y lógica de la investigación científica

Para presentar la incidencia del giro dinámico en la lógica de la investigación científica, en este artículo comenzamos con una sección que trata de los juegos lógicos como desencadenantes de este giro dinámico en la lógica contemporánea, junto con el programa de dinámica lógica de la información y la interacción. Sucintamente presentamos las principales características de la lógica favorable a la independencia y la semántica juego-teórica (IF-logic y GTS, respectivamente, en Hintikka y Sandu 1997), de la lógica dialógica (Redmond y Fontaine 2011), así como los elementos esenciales de dicho programa. Si bien a partir de cualquiera de estos puntos de vista se cuenta con un elenco de herramientas lógicas para abordar cuestiones más claramente epistemológicas, destacamos el papel de la lógica epistémica dinámica (LED), a la que dedicamos la siguiente sección. Sigue otra en la que entramos en los estudios lógicos de la abducción como uno de los problemas fundamentales de la epistemología contemporánea y, en una nueva sección, en términos de los constructos teóricos de las secciones precedentes, presentamos y explicamos un fenómeno surgido en el campo de la lingüística, el caso del descubrimiento de la lengua amazónica pirahã, que debería ser considerado una anomalía en el marco de la teoría chomskiana.

Non-Normative Logical Pluralism and the Revenge of the Normativity Objection

Logical pluralism is the view that there is more than one correct logic. Most logical pluralists think that logic is normative in the sense that you make a mistake if you accept the premisses of a valid argument but reject its conclusion. Some authors have argued that this combination is self-undermining: Suppose that $\mathcal {L}_{1}$ and $\mathcal {L}_{2}$ are correct logics that coincide except for the argument from Γ to ϕ, which is valid in $\mathcal {L}_{1}$ but invalid in $\mathcal {L}_{2}$. If you accept all sentences in Γ, then, by normativity, you make a mistake if you reject ϕ. In order to avoid mistakes, you should accept ϕ or suspend judgment about ϕ. Both options are problematic for pluralism. Can pluralists avoid this worry by rejecting the normativity of logic? I argue that they cannot. All else being equal, the argument goes through even if logic is not normative.

Multiverse—Too Much or Not Enough?

The aim of this essay is to look at the idea of the multiverse—not so much from the standpoint of physics or cosmology, but rather from a philosophical perspective. The modern story of the multiverse began with Leibniz. Although he treated “other worlds” as mere possibilities, they played an important role in his logic. In a somewhat similar manner, the practice of cosmology presupposes a consideration of an infinite number of universes, each being represented by a solution to Einstein’s equations. This approach prepared the way to the consideration of “other universes” which actually exist, first as an auxiliary concept in discussing the so-called anthropic principle, and then as real universes, the existence of which were supposed to solve some cosmological conundrums. From the point of view of the philosophy of science, the question is: Could the explanatory power of a multiverse ideology compensate for the relaxation of empirical control over so many directly unobservable entities? It is no surprise that appealing to a possibly infinite number of “other universes” in order to explain some regularities in our world would seem “too much” for a self-disciplined philosopher. With no strict empirical control at our disposal, it is logic that must be our guide. Also, what if logic changes from one world to another in the multiverse? Such a possibility is suggested by the category theory. From this point of view, our present concepts of the multiverse are certainly “not enough”. Should this be read as a warning that the learned imagination can lead us too far into the realms of mere possibilities?

Logical Disagreement

This chapter investigates what we disagree about when we disagree about logic, on the assumption that judgments of logical validity are normative. If logic is normative, then the popular anti-realist thesis that there are no normative facts or properties generalizes—it entails that there are no logical facts or properties. When faced with this anti-realism, it is tempting to endorse a pluralist thesis, according to which two people who disagree about the validity of an argument can both say something true. This chapter explores the limitations of three prominent forms of pluralism: contextualism, relativism, and expressivism. It argues that none of these forms of pluralism gives an adequate account of what we disagree about when we disagree about logic.

Kant and the Science of Logic

This book is both a history of philosophy of logic told from the Kantian viewpoint and a reconstruction of Kant’s theory of logic from a historical perspective. Kant’s theory represents a turning point in a history of philosophical debates over the following questions: (1) Is logic a science, instrument, standard of assessment, or mixture of these? (2) If logic is a science, what is the subject matter that differentiates it from other sciences, particularly metaphysics? (3) If logic is a necessary instrument to all philosophical inquiries, how is it so entitled? (4) If logic is both a science and an instrument, how are these two roles related? Kant’s answer to these questions centers on three distinctions: general versus particular logic, pure versus applied logic, pure general logic versus transcendental logic. The true meaning and significance of each distinction becomes clear, this book argues, only if we consider two factors. First, Kant was mindful of various historical views on how logic relates to other branches of philosophy (viz. metaphysics and physics) and to the workings of common human understanding. Second, he first coined “transcendental logic” while struggling to secure metaphysics as a proper “science,” and this conceptual innovation would in turn have profound implications for his mature theory of logic. Against this backdrop, the book reassesses the place of Kant’s theory in the history of philosophy of logic and highlights certain issues that are still debated today, such as normativity of logic and the challenges posed by logical pluralism.

Logical laws

There are at least three different kinds of answer to the question ‘What is a logical law?’ One establishes what it means for something to be a logical law. This answers the semantic question: What is the meaning of ‘logical law’? The second explains what makes something a logical law. This answers the metaphysical question: What is the ground of logical law? The third tells you what the logical laws are. This answers the question: What is the extension of ‘logical law’? Even though logic is often seen as a complete science, the answers to all three questions are disputed. For example, there are at least three different conceptions of what it means for something to be a law of logic. Different conceptions account for logic in terms of necessity, truth in all models, and proof. There are also different answers to the metaphysical question. If truth-preservation is central to logic, then the ground of logic depends on the metaphysics of truth. If logic is a matter of the meanings of terms, then the metaphysics of meaning is important for logic. Unfortunately, there is no widespread agreement on the metaphysics of meaning or truth. Finally, there is no widespread agreement as to what the logical laws are. There are two general disputes here. First, it is not clear what notions count as logical. Does logic contain laws about identity, second-order quantification or modality? Second, given agreement on the scope of logic, there are still questions about the logical laws in that area. Intuitionists, quantum logicians, relevance and paraconsistent logicians each reject things taken as laws by others, even in the language of ‘and’, ‘or’ and ‘not’.