## An Introduction to Non-Classical Logic: From If to IsThis revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. |

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### Contents

3 Normal Modal Logics | 36 |

restricted class of interpretations in our case those appropriate for | 38 |

4 Nonnormal Modal Logics Strict | 64 |

5 Conditional Logics | 82 |

6 Intuitionist Logic | 103 |

7 Manyvalued Logics | 120 |

8 First Degree Entailment | 142 |

A false ie untrue Define a relational interpretation p by | 160 |

14 Constant Domain Modal Logics | 308 |

15 Variable Domain Modal Logics | 329 |

16 Necessary Identity in Modal Logic | 349 |

Proof | 363 |

17 Contingent Identity in | 367 |

a Q i | 380 |

18 Nonnormal Modal Logics | 384 |

Imatrix | 397 |

9 Logics with Gaps Gluts | 163 |

10 Relevant Logics | 188 |

C13 If a e N a E a | 199 |

In the Completeness Theorem we have to check that the | 216 |

11 Fuzzy Logics | 221 |

For an account of the variety of fuzzy logics and | 239 |

Manyvalued | 241 |

11a42 As we saw in chapter 8 FDE can be | 245 |

12 Classical Firstorder Logic | 263 |

A HHB means A H B and B H A | 271 |

13 Free Logics | 290 |

1347 It has been suggested by some that sentences in | 295 |

19 Conditional Logics | 399 |

20 Intuitionist Logic | 421 |

not be true Choose any constant c with entry number | 448 |

21 Manyvalued Logics | 456 |

D | 463 |

2169 One final example Some have argued that paradoxical sentences | 465 |

22 First Degree Entailment | 476 |

23 Logics with Gaps | 504 |

24 Relevant Logics | 535 |

25 Fuzzy Logics | 564 |

2545 Finally before we turn to identity I note that | 572 |

### Common terms and phrases

apply the rule appropriate argument Ax(a Ax(b Ax(c Ax(kd chapter classical logic complete with respect Completeness Lemma Completeness Theorem constant domain contingent identity counter-model defined Denotation Lemma example extension f shows fA(w faithful finite first-order logic follows free logic Hence holds induced interpretation induction hypothesis intuitionist logic language logical truth many-valued logic modus ponens n-place predicate necessary identity negation Negativity Constraint node non-normal worlds normal modal logics normal worlds object obtained occurs open branch premises proof propositional logic propositional parameter proved relational semantics relevant logics Routley sentence sound and complete Soundness Lemma Soundness Theorem supervaluation Suppose t-norm tableau rules tableau systems tableaux of kind takes the value tense logic true at f(i truth conditions truth value v(an valid variable domain Vx(Px x(Px