Download Propositional Logic - Institute for Software Systems

2. Propositional Logic
Last time:
Introduction
Testing vs. Formal Specification
Examples (Petri Nets, CCS, Alloy, ...)
Today:
Propositional Logic
The following slides are based on the slides from M. Lawford
(http://www.cas.mcmaster.ca/~lawford/) and J. Bowen
(http://www.cs.ucl.ac.uk/staff/J.Bowen/GS03/ )
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What is Propositional Logic?
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Logical Connectives
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Computing with truth tables
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Example for implication
?
We expect that if n is greater than 3, then it
should be greater than 1,
so n > 3 should imply n > 1.
Therefore, the formula (n > 3) ⇒ (n > 1) should
equal true.
Substituting 4, 2, and 0 for n in this formula
explains why F ⇒ G means F implies G or,
equivalently, if F then G.
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Truth Tables
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Precedence of Logical Connectives
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Tautologies and Contradictions
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Logical (Semantic) Equivalence
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Logical (Semantic) Equivalence
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Normal Forms
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Normal Forms
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Normal Forms
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Logical Implication, Entailment
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Modus Ponens
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Checking validity (soundness)
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Example
See board
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Solution
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Contradicton
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Validity & Satisfiability
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Conjunctive Normalform
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Using CNF to Check |= Ф
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...
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Language of Propositional Calculus
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BNF and Parse Trees
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Subformulas and Subtrees
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Motivating Proofs
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Motivating Proofs
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Proof rules and Proofs
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Proof rules and Natural Deduction
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Proof rules: Implication Elimination
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Proof rules: Implication Introduction
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A First Formal Proof
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Remark
In just 6 rows not 23=8 rows of Truth Table.
Try the second and checkt that the numbers
now are 8 vs. 24=16.
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Proof rules: And Introduction
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Proof rules: And Elimination
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Proof rules: Negation Elimination/Intr.
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Proof rules: Modus Tollens
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MT
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Proof rules: Or Introduction
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Proof rules: Or Elimination
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Proof rules: Or Elimination
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...
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Overview
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Overview
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Example
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Overview
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...
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Copy
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LEM
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All Rules
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Soundness
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Completeness
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Propositional logic is decidable
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Summary
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Conclusion
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