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https://doi.org/10.1007/978-3-030-64811-4Mathematical Logic and Foundations; Propositional Logic; Predicate Logic; Modal Logic; First-Order Logic

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Basic Notions,here is one thing on which most definitions agree, it is that logic deals with the principles of correct reasoning. To explain what this means, we will start with some preliminary clarifications about the terms ‘reasoning’, ‘correct’, and ‘principles’.

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The Symbols of Predicate Logic, expressions. .. has a denumerable set of . ., ., .…, which represent singular terms, and a denumerable set of . ., ., .…, which represent predicates. Each predicate letter has . places, for some .. One-place predicate letters represent monadic predicates, that is, predicates that apply to single objects and express properties.

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The Symbols of Propositional Logic,This chapter introduces a propositional language called .. The alphabet of . is constituted by three categories of symbols:

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The Language L,Chapter . introduced the symbols of ., explained their meaning, and illustrated how they can be used to formalize sentences of a natural language. Now it is time to define . in a rigorous way by making fully explicit its syntax and its semantics.

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Logical Consequence in L,As anticipated in Sect. ., there are two ways to characterize a set of valid forms expressible in a language: one is semantic, the other is syntactic.

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The System ,,This chapter outlines a natural deduction system in . called .. As explained in Sect. ., a natural deduction system is constituted by a set of inference rules that are taken to be intuitively correct. Assuming our definition of validity as necessary truth preservation, this is to say that the rules of . necessarily preserve truth.