Counterfactual Logic and the Necessity of Mathematics

This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious program by Yli-Vakkuri and Hawthorne (2018), who seek to establish that mathematics is committed to its own necessity. I claim that their argument fails to establish this result for two reasons. First, their assumptions force our hand on a controversial debate within counterfactual logic. In particular, they license counterfactual strengthening— the inference from ‘If A were true then C would be true’ to ‘If A and B were true then C would be true’—which many reject. Second, the system they develop is provably equivalent to appending Deduction Theorem to a T modal logic. It is unsurprising that the combination of Deduction Theorem with T results in necessitation; indeed, it is precisely for this reason that many logicians reject Deduction Theorem in modal contexts. If Deduction Theorem is unacceptable for modal logic, it cannot be assumed to derive the necessity of mathematic

A Note on the Complexity of the Satisfiability Problem for Graded Modal Logics

Graded modal logic is the formal language obtained from ordinary (propositional) modal logic by endowing its modal operators with cardinality constraints. Under the familiar possible-worlds semantics, these augmented modal operators receive interpretations such as "It is true at no fewer than 15 accessible worlds that...", or "It is true at no more than 2 accessible worlds that...". We investigate the complexity of satisfiability for this language over some familiar classes of frames. This problem is more challenging than its ordinary modal logic counterpart--especially in the case of transitive frames, where graded modal logic lacks the tree-model property. We obtain tight complexity bounds for the problem of determining the satisfiability of a given graded modal logic formula over the classes of frames characterized by any combination of reflexivity, seriality, symmetry, transitivity and the Euclidean property.Comment: Full proofs for paper presented at the IEEE Conference on Logic in Computer Science, 200

Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.

There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory. In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area. Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available

Intensional Logic and Topology

This thesis is concerned with mathematical logic, in particular it is an investigation of a branch of mathematical logic called modal logic. This branch of mathematical logic extends the propositional calculus by adding two unary operators □ and 0 to the standard set of logical operators. This extension of classical logic has many interpretations; traditionally it is said to be the logic of necessity, denoted by the box operator, and possibility, denoted by the diamond operator. The notion of necessity within modal logic is ubiquitous and lends itself to a vast sea of metaphysics. For example, if X is necessarily true, denoted O X , then it is said to be true in all possible worlds. This way of understanding modalities gave imputes for a semantics that provided fodder for the first completeness proofs in modal logic. Modalities in logic have its roots in philosophy and dates back as far as Aristotle’s M etaphysics, but was brought into the limelight with the work of the philosopher mathematician Saul Kripke who in 1959, as a high school student, published the first completeness proof for a class of modal logics [Kripke]. His method used the so-called semantic-tableaux which was introduced by B eth’s The foundations of mathematics to obtain quick completeness proof for the propositional and predicate calculus. In this thesis, we are also interested in completeness for modal logics, but will use a more modern method known in the.literature as canonical model constructions . Moreover, we wish to provide a semantics for modal logics that is not the traditional possible world semantics. Our models will be topological in nature. Our goal is to provide a completeness proof for a particular modal logic called S4 which interprets the modal operators as the interior and closure operators on topological spaces. We will also prove that the logic S4 is complete with respect to the class of transitive and reflexive trees. This gives us two new completeness proof for the modal logic S4