Formal Logic and Deduction Systems
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1 Formal Logic and Deduction Systems Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) Formal Logic and Deduction Systems MFES 2008/09 1 / 9
2 What is a (formal) logic? Logic is defined as the study of the principles of reasoning. One of its branches is symbolic logic, that studies formal logic. A formal logic is a language equipped with rules for deducing the truth of one sentence from that of another. A logic consists of A logical language in which (well-formed) sentences are expressed. A semantics that distinguishes the valid sentences from the refutable ones. A proof system for constructing arguments justifying valid sentences. Examples: propositional logic, first-order logic, higher-order logic, and modal logic. Maria João Frade (DI-UM) Formal Logic and Deduction Systems MFES 2008/09 2 / 9
3 What is a logical language? A logical language consists of logical symbols whose interpretations are fixed non-logical symbols whose interpretations vary These symbols are combined together to form well-formed formulas. Maria João Frade (DI-UM) Formal Logic and Deduction Systems MFES 2008/09 3 / 9
4 Logic and computer science Logic and computer science share a symbiotic relationship Logic provides language and methods for the study of theoretical computer science. Computers provide a concrete setting for the implementation of logic. Formal logic makes it possible to calculate consequences at the symbolic level, so computers can be used to automate such symbolic calculations. Moreover, logic can be used to model the situations we encounter as computer science professionals, in such a way that we can reason about them formally. Maria João Frade (DI-UM) Formal Logic and Deduction Systems MFES 2008/09 4 / 9
5 Classical logic versus intuitionistic logic The classical understanding of logic is based on the notion of truth. The truth of a statement is absolute and independent of any reasoning, understanding, or action. Statements are either true or false. ( false not true ) tertium non datur principle A A must hold no matter what the meaning of A is. Intuitionistic logic is a branch of formal logic that rejects this guiding principle. A statement A is true if we can prove it, or is false if we can show that if we have a proof of A we get a contradiction. One judgements about a statement are based on the existence of a proof or construction of that statement. To show A A one have to show A or A. If neither of these can be shown, then the putative truth of the disjunction has no justification. Maria João Frade (DI-UM) Formal Logic and Deduction Systems MFES 2008/09 5 / 9
6 Classical logic versus intuitionistic logic Much of standard mathematics can be done within the framework of intuitionistic logic, but the task is very difficult, so mathematicians use methods of classical logic (as proofs by contradiction). However the philosophy behind intuitionistic logic is appealing for a computer scientist. For an intuitionist, a mathematical object (such as the solution of an equation) does not exist unless a finite construction (algorithm) can be given for that object. Maria João Frade (DI-UM) Formal Logic and Deduction Systems MFES 2008/09 6 / 9
7 Course overview Classical Propositional Logic Classical First-Order Logic Higher-Order Logic Induction Intuitionism and the Curry-Howard Isomorphism First-Order Theories Decision Procedures for Satisfiability... The Coq proof-assistant Maria João Frade (DI-UM) Formal Logic and Deduction Systems MFES 2008/09 7 / 9
8 Bibliography A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity Shawn Hedman Oxford (2004) Michael Huth & Mark Ryan Logic in Computer Science: Modelling and Reasoning About Systems Cambridge University Press; 2nd edition (2004). Aaron R. Bradley & Zohar Manna The Calculus of Computation: Decision Procedures with Applications to Verification Springer (2007) Maria João Frade (DI-UM) Formal Logic and Deduction Systems MFES 2008/09 8 / 9
9 Bibliography Morten Heine Sørensen & Pawel Urzyczyn Lectures on the Curry-Howard Isomorphism Studies in Logic and the Foundations of Mathematics, 149. Elsevier (2004) Yves Bertot & Pierre Castéran Interactive Theorem Proving and Program Development Coq Art: The Calculus of Inductive Constructions Springer (2004) The Coq proof assistant The latest version: Coq Maria João Frade (DI-UM) Formal Logic and Deduction Systems MFES 2008/09 9 / 9
The Curry-Howard Isomorphism
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