An Argument for Minimal Logic

The problem of negative truth is the problem of how, if everything in the world is positive, we can speak truly about the world using negative propositions. A prominent solution is to explain negation in terms of a primitive notion of metaphysical incompatibility. I argue that if this account is correct, then minimal logic is the correct logic. The negation of a proposition A is characterised as the minimal incompatible of A composed of it and the logical constant ¬. A rule based account of the meanings of logical constants that appeals to the notion of incompatibility in the introduction rule for negation ensures the existence and uniqueness of the negation of every proposition. But it endows the negation operator with no more formal properties than those it has in minimal logic

On the connections between PCTL and Dynamic Programming

Probabilistic Computation Tree Logic (PCTL) is a well-known modal logic which has become a standard for expressing temporal properties of finite-state Markov chains in the context of automated model checking. In this paper, we give a definition of PCTL for noncountable-space Markov chains, and we show that there is a substantial affinity between certain of its operators and problems of Dynamic Programming. After proving some uniqueness properties of the solutions to the latter, we conclude the paper with two examples to show that some recovery strategies in practical applications, which are naturally stated as reach-avoid problems, can be actually viewed as particular cases of PCTL formulas.Comment: Submitte

Contradiction-tolerant process algebra with propositional signals

In a previous paper, an ACP-style process algebra was proposed in which propositions are used as the visible part of the state of processes and as state conditions under which processes may proceed. This process algebra, called ACPps, is built on classical propositional logic. In this paper, we present a version of ACPps built on a paraconsistent propositional logic which is essentially the same as CLuNs. There are many systems that would have to deal with self-contradictory states if no special measures were taken. For a number of these systems, it is conceivable that accepting self-contradictory states and dealing with them in a way based on a paraconsistent logic is an alternative to taking special measures. The presented version of ACPps can be suited for the description and analysis of systems that deal with self-contradictory states in a way based on the above-mentioned paraconsistent logic.Comment: 25 pages; 26 pages, occurrences of wrong symbol for bisimulation equivalence replaced; 26 pages, Proposition 1 added; 27 pages, explanation of the phrase 'in contradiction' added to section 2 and presentation of the completeness result in section 2 improved; 27 pages, uniqueness result in section 2 revised; 27 pages, last paragraph of section 8 revise

A comprehensive theory of induction and abstraction, part II

This is part II in a series of papers outlining Abstraction Theory, a theory that I propose provides a solution to the characterisation or epistemological problem of induction. Logic is built from first principles severed from language such that there is one universal logic independent of specific logical languages. A theory of (non-linguistic) meaning is developed which provides the basis for the dissolution of the `grue' problem and problems of the non-uniqueness of probabilities in inductive logics. The problem of counterfactual conditionals is generalised to a problem of truth conditions of hypotheses and this general problem is then solved by the notion of abstractions. The probability calculus is developed with examples given. In future parts of the series the full decision theory is developed and its properties explored