A Fixpoint Semantics of Event Systems with and without Fairness Assumptions

We present a fixpoint semantics of event systems. The semantics is presented in a general framework without concerns of fairness. Soundness and completeness of rules for deriving "leads-to" properties are proved in this general framework. The general framework is instantiated to minimal progress and weak fairness assumptions and similar results are obtained. We show the power of these results by deriving sufficient conditions for "leads-to" under minimal progress proving soundness of proof obligations without reasoning over state-traces

Quantum logic is undecidable

We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature $(\lor,\perp,0,1)$, where `$\perp$' is the orthogonality relation. Our main result is that already its quasi-identities are undecidable: there is no algorithm to decide whether an implication between equations and orthogonality relations implies another equation. This is a corollary of a recent result of Slofstra in combinatorial group theory. It follows upon reinterpreting that result in terms of the hypergraph approach to quantum contextuality, for which it constitutes a proof of the inverse sandwich conjecture. It can also be interpreted as stating that a certain quantum satisfiability problem is undecidable.Comment: 11 pages. v3: improved exposition. v4: minor clarification

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

UNITY and Büchi automata

UNITY is a model for concurrent specifications with a complete logic for proving progress properties of the form ``$P$ leads to $Q$''. UNITY is generalized to U-specifications by giving more freedom to specify the steps that are to be taken infinitely often. In particular, these steps can correspond to non-total relations. The generalization keeps the logic sound and complete. The paper exploits the generalization in two ways. Firstly, the logic remains sound when the specification is extended with hypotheses of the form ``$F$ leads to $G$''. As the paper shows, this can make the logic incomplete. The generalization is used to show that the logic remains complete, if the added hypotheses ``$F$ leads to $G$'' satisfy ``$F$ unless $G$''. The main result extends the applicability and completeness of UNITY logic to proofs that a given concurrent program satisfies any given formula of LTL, linear temporal logic, without the next-operator which is omitted because it is sensitive to stuttering. For this purpose, the program, written as a UNITY program, is extended with a number of boolean variables. The proof method relies on implementing the LTL formula, i.e., restricting the specification in such a way that only those runs remain that satisfy the formula. This result is a variation of the classical construction of a B\"uchi automaton for a given LTL formula that accepts precisely those runs that satisfy the formula

Completeness of the ZX-Calculus

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. It comes equipped with an equational presentation. We focus here on a very important property of the language: completeness, which roughly ensures the equational theory captures all of quantum mechanics. We first improve on the known-to-be-complete presentation for the so-called Clifford fragment of the language - a restriction that is not universal - by adding some axioms. Thanks to a system of back-and-forth translation between the ZX-Calculus and a third-party complete graphical language, we prove that the provided axiomatisation is complete for the first approximately universal fragment of the language, namely Clifford+T. We then prove that the expressive power of this presentation, though aimed at achieving completeness for the aforementioned restriction, extends beyond Clifford+T, to a class of diagrams that we call linear with Clifford+T constants. We use another version of the third-party language - and an adapted system of back-and-forth translation - to complete the language for the ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the added axioms, and finally, we provide a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler

Computational reverse mathematics and foundational analysis

Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert's program due to Simpson [1988], and predicativism in the extended form due to Feferman and Sch\"{u}tte. Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only $\omega$-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a $\Pi^1_1$ complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than $\Pi^1_1\text{-}\mathsf{CA}_0$.Comment: Submitted. 41 page

Greek and Roman Logic

In ancient philosophy, there is no discipline called “logic” in the contemporary sense of “the study of formally valid arguments.” Rather, once a subfield of philosophy comes to be called “logic,” namely in Hellenistic philosophy, the field includes (among other things) epistemology, normative epistemology, philosophy of language, the theory of truth, and what we call logic today. This entry aims to examine ancient theorizing that makes contact with the contemporary conception. Thus, we will here emphasize the theories of the “syllogism” in the Aristotelian and Stoic traditions. However, because the context in which these theories were developed and discussed were deeply epistemological in nature, we will also include references to the areas of epistemological theorizing that bear directly on theories of the syllogism, particularly concerning “demonstration.” Similarly, we will include literature that discusses the principles governing logic and the components that make up arguments, which are topics that might now fall under the headings of philosophy of logic or non-classical logic. This includes discussions of problems and paradoxes that connect to contemporary logic and which historically spurred developments of logical method. For example, there is great interest among ancient philosophers in the question of whether all statements have truth-values. Relevant themes here include future contingents, paradoxes of vagueness, and semantic paradoxes like the liar. We also include discussion of the paradoxes of the infinite for similar reasons, since solutions have introduced sophisticated tools of logical analysis and there are a range of related, modern philosophical concerns about the application of some logical principles in infinite domains. Our criterion excludes, however, many of the themes that Hellenistic philosophers consider part of logic, in particular, it excludes epistemology and metaphysical questions about truth. Ancient philosophers do not write treatises “On Logic,” where the topic would be what today counts as logic. Instead, arguments and theories that count as “logic” by our criterion are found in a wide range of texts. For the most part, our entry follows chronology, tracing ancient logic from its beginnings to Late Antiquity. However, some themes are discussed in several eras of ancient logic; ancient logicians engage closely with each other’s views. Accordingly, relevant publications address several authors and periods in conjunction. These contributions are listed in three thematic sections at the end of our entry