Logic: The Basics

Jc Beall is Board of Trustees Distinguished Professor of Philosophy at the University of Connecticut, Storrs, USA; and Professor of Philosophy at the University of Tasmania, Hobart, Australia. Shay Allen Logan is a Postdoctoral Scholar in Logic at North Carolina State University, USA.

I BACKGROUND IDEAS

1 Consequences

1.1 Relations of support

1.2 Logical consequence: the basic recipe

1.3 Valid arguments and truth

1.4 Summary, looking ahead, and reading

2 Models, Modeled, and Modeling

2.1 Models

2.2 Models in science

2.3 Logic as modeling

2.4 A note on notation, metalanguages, etc.

2.5 Summary, looking ahead, and reading

3 Language, Form, and Logical Theories

3.1 Language and formal languages

3.2 Languages: syntax and semantics

3.3 Atoms, connectives, and molecules

3.4 Connectives and form

3.5 Validity and form

3.6 Logical theories: rivalry

3.7 Summary, looking ahead, and reading

4 Set-theoretic Tools

4.1 Sets

4.2 Ordered sets: pairs and n-tuples

4.3 Relations

4.4 Functions

4.5 Sets as tools

4.6 Summary and looking ahead

II THE BASIC CLASSICAL THEORY

5 Basic Classical Syntax and Semantics

5.1 Cases: complete and consistent

5.2 Classical ‘truth conditions’

5.3 Basic classical consequence

5.4 Motivation: precision

5.5 Formal picture

5.6 Defined connectives

5.7 Some notable valid forms

5.8 Summary and looking ahead

6 Basic Classical Tableaux

6.1 What are tableaux?

6.2 Tableaux for the Basic Classical Theory

6.3 Summary and looking ahead

7 Basic Classical Translations

7.1 Atoms, Punctuation, and Connectives

7.2 Syntax, altogether

7.3 Semantics

7.4 Consequence

7.5 Summary and Looking Ahead

III FIRST-ORDER CLASSICAL THEORY

8 Atomic Innards: Unary

8.1 Atomic innards: names and predicates

8.2 Truth and falsity conditions for atomics

8.3 Cases, domains, and interpretation functions

8.4 Classicality

8.5 A formal picture

8.6 Summary and looking ahead

9 Everything and Something

9.1 Validity involving quantifiers

9.2 Quantifiers: an informal sketch

9.3 Truth and falsity conditions

9.4 A formal picture

9.5 Summary and looking ahead.

10 First-Order Language with Any-Arity Innards

10.1 Truth and falsity conditions for atomics

10.2 Cases, domains, and interpretation functions

10.3 Classicality

10.4 A formal picture

10.5 Summary and looking ahead

11 Identity

11.1 Logical expressions, forms, sentential forms

11.2 Validity involving identity

11.3 Identity: informal sketch

11.4 Truth conditions: informal sketch

11.5 Formal picture

11.6 Summary and looking ahead

12 Tableaux for First-Order Logic with Identity

12.1 A Few Reminders

12.2 Tableaux for Polyadic First-Order Logic

12.3 Summary and looking ahead

13 First-Order Translations

13.1 Basic Classical Theory with Innards

13.2 First-Order Classical Theory

13.3 Polyadic Innards

13.4 Examples in the polyadic language

13.5 Adding Identity

13.6 Summary and Looking Ahead

IV NONCLASSICAL THEORIES

14 Alternative Logical Theories

14.1 Apparent unsettledness

14.2 Apparent overdeterminacy

14.3 Options

14.4 Cases

14.5 Truth and falsity conditions

14.6 Logical Consequence

14.7 Summary, looking ahead, and reading

15 Nonclassical Sentential Logics

15.1 Syntax

15.2 Semantics, Broadly

15.3 Defined connectives

15.4 Some notable forms

15.5 Summary and looking ahead

16 Nonclassical First-order Theories

16.1 An Informal Gloss

16.2 A formal picture

16.3 Summary and looking ahead

17 Nonclassical Tableaux

17.1 Closure Conditions

17.2 Tableaux for Nonclassical First-Order Logics

17.3 Summary and looking ahead

18 Nonclassical Translations

18.1 Syntax and Semantics

18.2 Consequence

18.3 Summary and looking ahead

V FREEDOM, NECESSITY AND BEYOND

19 Speaking Freely

19.1 Speaking of non-existent ‘things’

19.2 Existential import

19.3 Freeing our terms, expanding our domains

19.4 Truth conditions: an informal sketch

19.5 Formal picture

19.6 Summary and looking ahead

20 Possibilities

20.1 Possibility and necessity

20.2 Towards truth and falsity conditions

20.3 Cases and consequence

20.4 Formal picture

20.5 Remark on going beyond possibility

20.6 Summary and looking ahead

21 Free and Modal Tableaux

21.1 Free Tableaux

21.2 Modal Tableaux

21.3 Summary and looking ahead

22 Glimpsing Different Logical Roads

22.1 Other conditionals

22.2 Other negations

22.3 Other alethic modalities: actuality

22.4 Same connectives, different truth conditions

22.5 Another road to difference: consequence

22.6 Summary and looking behind and ahead

References