The Semantic Foundations of Logic Volume 1: Propositional by Richard L. Epstein

By Richard L. Epstein

This publication grew out of my confusion. If common sense is aim how can there be such a lot of logics? Is there one correct common sense, or many correct ones? Is there a few underlying harmony that connects them? what's the importance of the mathematical theorems approximately good judgment which i have discovered in the event that they haven't any connection to our daily reasoning? The solutions I suggest revolve round the belief that what one can pay consciousness to in reasoning determines which common sense is suitable. The act of abstracting from our reasoning in our ordinary language is the stepping stone from reasoned argument to good judgment. we can't take this step on my own, for we cause jointly: good judgment is reasoning which has a few goal worth. so that you can comprehend my solutions, or maybe higher, conjectures, i've got retraced my steps: from the concrete to the summary, from examples, to normal thought, to additional confirming examples, to reflections at the value of the work.

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Finally, we say that two propositions (or wffs) are semantically equivalent if each is a semantic consequence of the other: in every model they have the same truth-value. Thus in classical logic 'l(Ralph is a dog A l(George is a duck))' is semantically equivalent to 'Ralph is a dog ~ George is a duck', as you can show. Semantic equivalence is one formalization of the informal notion of two propositions meaning the same thing for all our logical purposes. Let's review what we've done. We made the assumption that the only aspects of a proposition that are of concern to logic are its truth-value and its form as built SECTION E The Logical Fonn of a Proposition 21 up from sentence connectives.

Valid propositions are those which are true due to their (propositional) form only (relative to our semantics). Valid wffs exemplify = SECTION D Validity and Semantic Consequence 19 those forms. In the following chapters when we want to stress that this notion of validity is for the classical interpretation of the connectives we 'II use the terms classical tautology or classically valid. We sometimes say that a sentence in our ordinary language, English, is valid if there is a straightforward formalization of it into semi-formal English on which we feel certain we'll all agree and which is valid.

Thus p 1 A1p2 is false in the model which assigns 'Grass is green' to p 1 and 'Every dog is a canine' to p 2 • Whenever I give an example like this I'm assuming that we take the obvious commonly agreed upon truth-values for the propositions assigned to the variables. We say that M is a model for a collection ofwffs of the formal language if every wff in the collection is true in M. p2 , 1(1p 1 A1p2), p 1 vp2 }. Some wffs are true in every model: their truth-value is independent of any particular realization.

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