Read e-book online An Introduction to Modal Logic PDF

By G. E. Hughes, M. J. Cresswell

ISBN-10: 0416294707

ISBN-13: 9780416294705

Be aware: This e-book was once later changed via "A New creation to Modal common sense" (1996).

Modal good judgment could be defined in short because the common sense of necessity and danger, of 'must be' and 'may be'.

We had major goals in scripting this e-book. One used to be to provide an explanation for intimately what modal good judgment is and the way to do it; the opposite was once to provide an image of the total topic at present level of its improvement. the 1st of those goals dominates half I, and to a lesser quantity half II; the second one dominates half III. half i may be used by itself as a text-book for an introductory process guide at the simple thought and strategies of modal logic.

We have attempted to make the publication self-contained by means of together with on the acceptable issues summaries of all of the non-modal common sense we use within the exposition of the modal platforms. it might probably for this reason be tackled via anyone who had now not studied any common sense in any respect ahead of. To get the main out of it, besides the fact that, this kind of reader will be good prompt to shop for himself one other ebook on common sense besides and to benefit anything extra concerning the Propositional Calculus and the decrease Predicate Calculus than now we have been in a position to inform him right here.

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Example text

If a set of clauses is unsatisfiable, then there is guaranteed to be a resolution derivation of the empty clause from . More generally, if a set of Propositional Logic sentences is unsatisfiable, then there is guaranteed to be a resolution derivation of the empty clause from the clausal form of . As an example, consider the clauses {p, q}, {p, ¬q}, {¬p, q}, and {¬p, ¬q}. There is no truth assignment that satisfies all four of these clauses. Consequently, starting with these clauses, we should be able to derive the empty clause; and we can.

This is guaranteed by a metatheorem called the refutation theorem. 1 (Refutation Theorem) Let and only if ∪ {¬φ} is unsatisfiable. be a set of sentences. Let φ be a sentence. |= φ if Proof. Suppose that |= φ. If a truth assignment satisfies , then it must also satisfy φ. But then it cannot satisfy ¬φ. Therefore, ∪ {¬φ} is unsatisfiable. Suppose that ∪ {¬φ} is unsatisfiable. , it must satisfy φ. Therefore, |= φ. ✷ An interesting consequence of this result is that we can determine logical entailment by checking for unsatisfiability.

We begin this lesson with a discussion of simple, linear proofs. We then move on to hierarchically structured proofs, where proofs can be nested inside of other proofs. Once we have seen both linear and structured proofs, we show how they are combined in the popular Fitch proof system. We finish with definitions for soundness and completeness—the standards by which proof systems are judged. 2 LINEAR PROOFS As we saw in the introductory lesson, the essence of logical reasoning is symbolic manipulation.

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An Introduction to Modal Logic by G. E. Hughes, M. J. Cresswell

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