A 4-valued logic of strong conditional

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions (Section 1), we argue that a stronger account of conditional can be obtained in two steps: firstly, by reminding its historical roots inside modal logic and set-theory (Section 2); secondly, by revising the meaning of logical values, thereby getting rid of the paradoxes of material implication whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections (Section 3). Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed through a broader definition of logical consequence that encompasses both a normal relation of truth propagation and a weaker relation of falsity non-propagation from premises to conclusion (Section 3)

A Hybrid Linear Logic for Constrained Transition Systems

Linear implication can represent state transitions, but real transition systems operate under temporal, stochastic or probabilistic constraints that are not directly representable in ordinary linear logic. We propose a general modal extension of intuitionistic linear logic where logical truth is indexed by constraints and hybrid connectives combine constraint reasoning with logical reasoning. The logic has a focused cut-free sequent calculus that can be used to internalize the rules of particular constrained transition systems; we illustrate this with an adequate encoding of the synchronous stochastic pi-calculus

Students' understandings of logical implication

We report results from an analysis of responses to a written question in which highattaining students in English schools, who formed part of a longitudinal nation-wide survey on proof conceptions, were asked to assess the equivalence of two statements about elementary number theory, one a logical implication and the other its converse, to evaluate the truth of the statements and to justify their conclusions. We present an overview of responses at the end of Year 8 (age 13 years) and an analysis of the approaches taken, and follow this with an analysis of the data collected from students who answered the question again in Year 9 (age 14 years) in order to distinguish learning trajectories. From these analyses, we distinguished three strategies, empirical, focussed-empirical and focussed-deductive, that represent shifts in attention from an inductive to a deductive approach. We noted some progress from Year 8 to Year 9 in the use of the focussed strategies but this was modest at best. The most marked progress was in recognition of the logical necessity of a conclusion of an implication when the antecedent was assumed to be true. Finally we present some theoretical categories to capture different types of meanings students assign to logical implication and the rationale underpinning these meanings. The categories distinguish responses where a statement of logical implication is (or is not) interpreted as equivalent to its converse, where the antecedent and consequent are (or are not) seen as interchangeable, and where conclusions are (or are not) influenced by specific data

A Hybrid Linear Logic for Constrained Transition Systems with Applications to Molecular Biology

Linear implication can represent state transitions, but real transition systems operate under temporal, stochastic or probabilistic constraints that are not directly representable in ordinary linear logic. We propose a general modal extension of intuitionistic linear logic where logical truth is indexed by constraints and hybrid connectives combine constraint reasoning with logical reasoning. The logic has a focused cut-free sequent calculus that can be used to internalize the rules of particular constrained transition systems; we illustrate this with an adequate encoding of the synchronous stochastic pi-calculus. We also present some preliminary experiments of direct encoding of biological systems in the logic

The Epistemic Significance of Valid Inference – A Model-Theoretic Approach

The problem analysed in this paper is whether we can gain knowledge by using valid inferences, and how we can explain this process from a model-theoretic perspective. According to the paradox of inference (Cohen & Nagel 1936/1998, 173), it is logically impossible for an inference to be both valid and its conclusion to possess novelty with respect to the premises. I argue in this paper that valid inference has an epistemic significance, i.e., it can be used by an agent to enlarge his knowledge, and this significance can be accounted in model-theoretic terms. I will argue first that the paradox is based on an equivocation, namely, it arises because logical containment, i.e., logical implication, is identified with epistemological containment, i.e., the knowledge of the premises entails the knowledge of the conclusion. Second, I will argue that a truth-conditional theory of meaning has the necessary resources to explain the epistemic significance of valid inferences. I will explain this epistemic significance starting from Carnap’s semantic theory of meaning and Tarski’s notion of satisfaction. In this way I will counter (Prawitz 2012b)’s claim that a truth-conditional theory of meaning is not able to account the legitimacy of valid inferences, i.e., their epistemic significance

Logic and Intelligibility

All inquirers must have a grasp of implication and contradiction which they employ to structure their investigations. Logical ability is thus some kind of prerequisite for cognition. My dissertation scrutinizes this relationship and argues that different ways of understanding it underlie a deep debate about naturalism and the objectivity of our knowledge. Frege’s dismissal of logical aliens as mad exposes his conviction that logical ability is exhibited in our practice of demonstrative reasoning, and is a constitutive necessary condition for cognition. By denying the existence of an independent standard for objective truth that a codification of inferential principles must meet, Frege avoids logical “sociologism” (under which the validity of inferential principles is identified with their agreement with our practice). Quine objects to ascribing a “pre-logical mentality” in radical translation, but only because doing so would represent one’s interlocutor as affirming something one finds obviously false. Under his naturalism, the logician is guided by usefulness to ongoing empirical inquiry, not the search for the constitutive prerequisites of thinking. I argue that the properly-understood naturalist excises various skeptical attacks from epistemology. Davidson recovers a privileged status for logic as central in the theories of truth that are necessary to interpret another as—and also to be—a cognizer. Under his humanism, it is only through interpreting others that one can grasp the objective/subjective contrast and acquire beliefs that are properly about the world. We do not exhibit our grasp of objective truth by engaging in a practice informed by logic, but by interpreting others who are engaged with, and through, us in such a practice. Despite initial appearances, naturalism and humanism are not incompatible positions. After examining Quine’s “sectarian” and Davidson’s “ecumenical” attitude to the truth of empirically equivalent theories, I endorse ecumenism about their metaphilosophical disagreement. By renouncing a proprietary attitude to truth, this particular form of tolerance avoids the fragmentation of philosophy into distinct, yet totalizing, and hence warring, programs

Towards a Logical Framework with Intersection and Union Types

International audienceWe present an ongoing implementation of a dependent-type theory (∆-framework) based on the Edinburgh Logical Framework LF, extended with Proof-functional logical connectives such as intersection , union, and strong (or minimal relevant) implication. Proof-functional connectives take into account the shape of logical proofs, thus allowing to reflect polymorphic features of proofs in formulae. This is in contrast to classical Truth-functional connec-tives where the meaning of a compound formula is only dependent on the truth value of its subformulas. Both Logical Frameworks and proof functional logics consider proofs as first class citizens. But they do it differently namely, explicitly in the former while implicitly in the latter. Their combination opens up new possibilites of formal reasoning on proof-theoretic semantics. We provide some examples in the extended type theory and we outline a type checker. The theory of the system is under investigation. Once validated in vitro, the proof-functional type theory can be successfully plugged in existing truth-functional proof assistants

Layers of Logical Consequence

Model-theory and proof-theory are two long-standing alternative descriptions of logical consequence. Proof-theory characterises truth in terms of logical implication. Namely, according to proof-theory, the statement ‘A implies B’ is true iff there exists a proof from A to B. In contrast, model-theory characterises truth based on possible states of the world. By model-theory, ‘A implies B’ is true iff for any model m, if m satisfies A then m satisfies B. In this paper I argue that we can reconcile the views, by making an appropriate distinction between epistemic nature and metaphysical nature. Namely, I will argue that we can view logical consequence as epistemically model-theoretic and metaphysically prooftheoretic