LOGIC: Lecture Notes for Philosophy, Mathematics, and Computer Science

Springer (2021)
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Abstract

This textbook is a logic manual which includes an elementary course and an advanced course. It covers more than most introductory logic textbooks, while maintaining a comfortable pace that students can follow. The technical exposition is clear, precise and follows a paced increase in complexity, allowing the reader to get comfortable with previous definitions and procedures before facing more difficult material. The book also presents an interesting overall balance between formal and philosophical discussion, making it suitable for both philosophy and more formal/science oriented students. This textbook is of great use to undergraduate philosophy students, graduate philosophy students, logic teachers, undergraduates and graduates in mathematics, computer science or related fields in which logic is required.

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ISBN(s)

3030648109   9783030648107   3030648133   9783030648114

DOI

10.1007/978-3-030-64811-4
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Chapters

Basic Notions

Logic has been defined in many ways in the course of its history, as different views have been held about its aim, scope, and subject matter. But if there 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 wil... see more

Theories and Models

This chapter presents some general results that hinge on the notion of cardinality.

Consistency, Soundness, Completeness

This chapter shows that Q is consistent, sound, and complete. The proof methods that will be employed to establish these results are the same that have been employed in Chapter 10 to prove the consistency, soundness, and completeness of L.

Gödel’s Incompleteness Theorems

In his famous article On formally undecidable propositions of Principia Mathematica and related systems I , Gödel established two results that marked a point of no return in the history of logic.

Quantification

Although propositional logic provides a formal account of a wide class of valid arguments, its explanatory power is limited. Many arguments are valid in virtue of formal properties that do not depend on the truth-functional structure of their premises and conclusion, so their validity is not explain... see more

The System L

This chapter outlines an axiomatic system called L. The language of L is L−, the fragment of L whose logical constants are ∼ and ⊃. So, L may be regarded as an axiomatic version of G−, the poor cousin of G considered in Sect. 8.5.

Formality

As anticipated in section 1.1, the validity of an argument can be explained in terms of its form. To illustrate the idea of formal explanation, consider the following argument:

The Language Lq

Now Lq will be defined in a rigorous way.

The Language L

Chapter 4 introduced the symbols of L, explained their meaning, and illustrated how they can be used to formalize sentences of a natural language. Now it is time to define L in a rigorous way by making fully explicit its syntax and its semantics.

The System Q

This chapter sets out an axiomatic system in Lq called QQ. The axioms of Q are all the formulas of Lq that instantiate the following schemas:

Validity

In order to elucidate the understanding of validity that underlies logic, it is useful to introduce some symbols that belong to the vocabulary of set theory. A setSet is a collection of things, called its elementsElement. We will write a ∈ A∈ to say that a is an element of A, and a∉A∉ to say that a ... see more

The Symbols of Predicate Logic

This chapter introduces a predicate language called LqLq. The alphabet of Lq is constituted by the following symbols: a, b, c…P, Q, R…∼, ⊃, ∀x, y, z…Let us start with the non-logical expressions. Lq has a denumerable set of individual constantsIndividual constanta, b, c…, which represent singular te... see more

First-Order Logic

So far we have focused on Lq. But there are many predicate languages, for the alphabet of Lq can be enlarged or restricted in various ways. One can add to Lq further individual constants, further predicate letters, further variables, the connectives ∧, ∨, ∃, or the symbol =.

Derivability in G

To say that a formula α is derivable from a set of formulas Γ in a system S is to say that there is a derivation of α from Γ in SDerivability.

Rudiments of Modal Logic

This last chapter aims to provide a concise presentation of modal logic, the logic of necessity and possibility. A modal language is a language that contains, in addition to the symbols of a propositional or predicate language, the modal operators and ◊, which mean respectively ‘it is necessary that... see more

The Symbols of Propositional Logic

This chapter introduces a propositional language called L.L The alphabet of L is constituted by three categories of symbols:

Logical Consequence in L

As anticipated in Sect. 3.4, there are two ways to characterize a set of valid forms expressible in a language: one is semantic, the other is syntactic.

The System G

This chapter outlines a natural deduction system in L called GG. As explained in Sect. 3.4, 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 rul... see more

Undecidability and Related Results

This chapter dwells on some facts about Q that concern decidability and related notions.

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Andrea Iacona
Università di Torino

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