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We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete for the basic positive distributive substructural logic
, that collection frames on multisets are sound and complete for
(the relevant logic
, without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic
. The completeness of set frames for
is, currently, an open question.
This Element is an introduction to recent work proofs and models in philosophical logic, with a focus on the semantic paradoxes the sorites paradox. It introduces and motivates different proof systems and different kinds of models for a range of logics, including classical logic, intuitionistic logic, a range of three-valued and four-valued logics, and substructural logics. It also compares and contrasts the different approaches to substructural treatments of the paradox, showing how the structural rules of contraction, cut and identity feature in paradoxical derivations. It then introduces model theoretic treatments of the paradoxes, including a simple fixed-point model construction which generates three-valued models for theories of truth, which can provide models for a range of different non-classical logics. The Element closes with a discussion of the relationship between proofs and models, arguing that both have their place in the philosophers' and logicians' toolkits.
In this paper, I motivate a cut free sequent calculus for classical logic with first order quantification, allowing for singular terms free of existential import. Along the way, I motivate a criterion for rules designed to answer Prior’s question about what distinguishes rules for logical concepts, like conjunction from apparently similar rules for putative concepts like Prior’s tonk, and I show that the rules for the quantifiers—and the existence predicate—satisfy that condition.
Australian Realists are fond of talking about truth-makers. Here are three examples from the recent literature
… suppose a is F … What is needed is something in the world which ensures that a is F, some truth-maker or ontological ground for a's being F. What can this be except the state of affairs of a's being F?
(Armstrong 1989a: 190)
If b entails Π, what makes Φ true also makes Π true (at least when Φ and Π are contingent).
(Jackson 1994: 32)
The hallowed path from language to universals has been by way of the correspondence theory of truth: the doctrine that whenever something is true, there must be something in the world which makes it true. I will call this the Truthmaker axiom. The desire to find an adequate truthmaker for every truth has been one of the sustaining forces behind traditional theories of universals … Correspondence theories of truth breed legions of recalcitrant philosophical problems. For this reason I have sometimes tried to stop believing in the Truthmaker axiom. Yet, I have never really succeeded. Without some such axiom, I find I have no adequate anchor to hold me from drifting onto the shoals of some sort of pragmatism or idealism. And this is altogether uncongenial to me; I am a congenital realist about almost everything, as long as it is compatible with some sort of naturalism or physicalism, loosely construed.
It is twelve years since the publication of “Truth-makers, entailment and necessity”, and some of my views on the topics discussed have changed in that time. More recent work on truth-making (in particular, Lewis's “Truth-making and difference-making”, this vol., Ch. 6) has convinced me that the idea that truth depends on ontology can be captured in a number of different ways. Which ways we take to be most appealing will depend on a whole host of views or commitments, not the least being those concerning what different kinds of things there are.
Elsewhere I have shown that the robust picture of truth-making in “Truthmakers, entailment and necessity” can apply in a very simple “world” in which the truth-makers are regions in which atoms are either present or not (Restall 2000). No strange “negative” or “universal” facts need to be added to that picture; regions together with their inhabitants can suffice, so worries about the queerness of “negative” or “universal” facts need not worry the friend of robust truthmaking if the metaphysics is kind enough to supply everyday objects that do the job.
However, some of my views have changed more significantly, since 1996. I am now convinced by an argument of Stephen Read, in “Truthmakers and the Disjunction Thesis” (2000) that the “or and shmor” argument in §3 of “Truthmakers, Entailment and Necessity” is too swift. In the rest of this note, I will give an account of Read's concern, and then chart four possible responses to it from the perspective of one who wishes to maintain the broad outline of the position of “Truth-makers, entailment and necessity”.
Stephen Read (2002, 2006) has recently discussed Bradwardine's theory of truth and defended it as an appropriate way to treat paradoxes such as the liar. In this paper, I discuss Read's formalisation of Bradwardine's theory of truth and provide a class of models for this theory. The models facilitate comparison of Bradwardine's theory with contemporary theories of truth.
Abstract. In this paper I introduce a sequent system for the propositional modal logic S5. Derivations of valid sequents in the system are shown to correspond to proofs in a novel natural deduction system of circuit proofs (reminiscient of proofnets in linear logic [9, 15], or multipleconclusion calculi for classical logic [22, 23, 24]).
The sequent derivations and proofnets are both simple extensions of sequents and proofnets for classical propositional logic, in which the new machinery—to take account of the modal vocabulary—is directly motivated in terms of the simple, universal Kripke semantics for S5. The sequent system is cut-free (the proof of cut-elimination is a simple generalisation of the systematic cut-elimination proof in Belnap's Display Logic [5, 21, 26]) and the circuit proofs are normalising.
In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's multiple conclusion systems for classical logic and Girard's proofnets for linear logic.