Hannes Leitgeb

This article argues for a (quasi-)Carnapian version of logicism about mathematics: there is a logicist conceptual framework in which (i) all standard mathematical terms are defined by logical terms, and (ii) all standard mathematical theorems are (likely to be) analytic. Along the way, the article explains the historical-philosophical background, how the definitions in (i) are to proceed, what the framework and the semantic notion of analyticity-in-a-framework are like, and why the probabilistic qualification ‘likely to be’ is used in (ii). The upshot is not some logicist epistemic foundationalism about mathematics but the insight that mathematics can be rationally reconstructed as being conceptual, i.e., as coming along with a conceptual framework.

Philosophers often have tried to either reduce "disagreeable" objects or concepts to (more) acceptable objects or concepts. Reduction is regarded attractive by those who subscribe to an ideal of ontological parsimony. But the topic is not just restricted to traditional metaphysics or ontology. In the philosophy of mathematics, abstraction principles, such as Hume's principle, have been suggested to support a reconstruction of mathematics by logical means only. In the philosophy of language and the philosophy of science, the logical analysis of language has long been regarded to be the dominating paradigm, and liberalized projects of logical reconstruction remain to be driving forces of modern philosophy.

Logical inferentialists hold that the meaning of logical operators is given by their rules of inference. Arthur Prior cast doubt on this by introducing rules for his tonk operator that would allow for the derivation of any sentence whatsoever from any sentence whatsoever. The obvious inferentialist reply was to require some constraints on the defining rules, such as conservativeness (Belnap) or harmony (Dummett). In this paper, I will defend and investigate a different constraint: rules define a classical logical operator just in case they translate into an explicit definition in pure classical second-order logic. The right-hand side of this criterion will be found (i) to be philosophically principled in taking the idea of rules as definitions perfectly seriously, (ii) to explain how the semantic meaning of the operators can be determined from their rules, (iii) to be local in a similar sense as harmony, (iv) to validate the intuitionistic natural deduction rules and the intuitionistic/classical sequent calculus rules as defining the classical propositional operators while ruling out Prior’s rules for tonk, (v) to make clear why already the intuitionistic natural deduction rules define the classical meaning of logical operators so long as metavariables are interpreted as expressing classical propositions, (vi) to validate the classical natural deduction rules as analytic, (vii) to entail consistency, and, in the case of propositional operators (not quantifiers), (viii) to be decidable and (ix) to determine precisely those operators to be definable by rules that correspond to truth-functions.

This chapter explains how artificial neural networks may be used as models for reasoning, conditionals, and conditional logic. It starts with the historical overlap between neural network research and logic, it discusses connectionism as a paradigm in cognitive science that opposes the traditional paradigm of symbolic computationalism, it mentions some recent accounts of how logic and neural networks may be combined, and it ends with a couple of open questions concerning the future of this area of research.

We propose a new interpretability method for neural networks, which is based on a novel mathematico-philosophical theory of reasons. Our method computes a vector for each neuron, called its reasons vector. We then can compute how strongly this reasons vector speaks for various propositions, e.g., the proposition that the input image depicts digit 2 or that the input prompt has a negative sentiment. This yields an interpretation of neurons, and groups thereof, that combines a logical and a Bayesian perspective, and accounts for polysemanticity (i.e., that a single neuron can figure in multiple concepts). We show, both theoretically and empirically, that this method is: (1) grounded in a philosophically established notion of explanation, (2) uniform, i.e., applies to the common neural network architectures and modalities, (3) scalable, since computing reason vectors only involves forward-passes in the neural network, (4) faithful, i.e., intervening on a neuron based on its reason vector leads to expected changes in model output, (5) correct in that the model's reasons structure matches that of the data source, (6) trainable, i.e., neural networks can be trained to improve their reason strengths, (7) useful, i.e., it delivers on the needs for interpretability by increasing, e.g., robustness and fairness.

We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for predicates and individual terms of an arbitrary mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory can be given an analytically true reading in the logical framework.

The article recapitulates what logic is about traditionally and works out two roles it has been playing in philosophy: the role of an instrument and of a philosophical discipline in its own right. Using Tarski’s philosophical-logical work as case study, it develops a logical reconstructionist methodology of philosophical logic that extends and refines Rudolf Carnap’s account of explication and rational reconstruction. The methodology overlaps with, but also partially diverges from, contemporary anti-exceptionalism about logic.

The aim of this paper is to argue for a revised and precisified version of the infamous Verifiability Criterion for the meaningfulness of declarative sentences. The argument is based on independently plausible premises concerning probabilistic confirmation and meaning as context-change potential, it is shown to be logically valid, and its ramifications for potential applications of the criterion are being discussed. Although the paper is not historical but systematic, the criterion thus vindicated will resemble the original one(s) in some important ways. At the same time, it will also be more modest insofar as meaningfulness will turn out to be relativized linguistically and probabilistically, and different choices of the linguistic and probabilistic parameters may lead to different verdicts on meaningfulness.

If an agent believes that the probability of E being true is 1/2, should she accept a bet on E at even odds or better? Yes, but only given certain conditions. This paper is about what those conditions are. In particular, we think that there is a condition that has been overlooked so far in the literature. We discovered it in response to a paper by Hitchcock (2004) in which he argues for the 1/3 answer to the Sleeping Beauty problem. Hitchcock argues that this credence follows from calculating her fair betting odds, plus the assumption that Sleeping Beauty’s credences should track her fair betting odds. We will show that this last assumption is false. Sleeping Beauty’s credences should not follow her fair betting odds due to a peculiar feature of her epistemic situation.