## An Introduction to Non-Classical Logic: From If to IsThis revised and considerably expanded 2nd edition, published in 2008, 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 |

4 Nonnormal Modal Logics Strict | 64 |

5 Conditional Logics | 82 |

6 Intuitionist Logic | 103 |

7 Manyvalued Logics | 120 |

8 First Degree Entailment | 142 |

Note that vwA 1 iff vwA 0 iff | 152 |

9 Logics with Gaps Gluts | 163 |

16 Necessary Identity in Modal Logic | 349 |

17 Contingent Identity in | 367 |

Ancients the Evening Star The latter for example could have | 368 |

The countermodel determined by the tableau may be depicted as | 371 |

18 Nonnormal Modal Logics | 384 |

188 History | 397 |

19 Conditional Logics | 399 |

1929 Variable domain C VC is obtained by modifying CC | 402 |

10 Relevant Logics | 188 |

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 |

Since p holds at w Op holds at wo | 250 |

12 Classical Firstorder Logic | 263 |

A HI B means A h B and B | 271 |

13 Free Logics | 290 |

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

14 Constant Domain Modal Logics | 308 |

1436 The countermodel determined by the tableau can be depicted | 313 |

15 Variable Domain Modal Logics | 329 |

20 Intuitionist Logic | 421 |

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

21 Manyvalued Logics | 456 |

e d | 463 |

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

22 First Degree Entailment | 476 |

2233 Here is another to show that VxPxVxPx D Qx | 480 |

23 Logics with Gaps | 504 |

24 Relevant Logics | 535 |

2423 Validity is defined in terms of truth preservation at | 536 |

25 Fuzzy Logics | 564 |

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

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### Common terms and phrases

apply the rule argument Ax(a Ax(b Ax(c Ax(kd Barcan Formula chapter classical logic complete with respect Completeness Lemma Completeness Theorem constant domain contingent identity counter-model deﬁned deﬁnition Denotation Lemma difﬁcult example extension fA(w faithful ﬁnite ﬁrst follows free logic Hence holds induced interpretation induction hypothesis inﬁnite intuitionist logic 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 quantiﬁers relational semantics relevant logics Routley satisﬁes 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 wi true atf(i truth conditions truth value valid variable domain Vx(A Vx(Px VxPx wiRwj x(Px