### II.1 Introduction

#### Logic [Slide 18]

Logic is a tool for careful reasoning.

• Software and hardware development
• Formal logic can lead to defect-free products
• Heavily used in chip design (since Pentium disaster)
• Used in safety-critical software design
• Positive inﬂuence on all software development
• Foundation for programming languages (PL)
• Together with Functions (Lambda calculus) form the semantic base for all PLs
• Type checking and type inference - basis in logic
• Computation model - limits of expressiveness
• Artiﬁcial intelligence, database systems
• Grounded in formal logic

#### Propositional Logic [Slide 19]

• Proposition - atomic entity, true or false
• letters P,Q,... denote propositional variables standing for speciﬁc propositions P,Q, speciﬁc propositions - value: true or false
• letters f,f1,f2, are meta-variables denoting propositional variables or formulae f,f1,f2, value: proposition, such as P, (P Q)
• Propositional Calculus
• Scheme for calculating with logic formulae
• Three diﬀerent propositional calculi
• semantics-based reasoning - truth table
• syntax-based reasoning - inference rules
• equational reasoning - Boolean algebra

#### Logical Operators [Slide 20]

Logical operators provide ways to combine propositions for calculation. These are:

• (And)
• P Q, (P Q) R,
• (Or)
• P Q,P (Q R),
• ¬ (Not)
• ¬P, ((¬P) Q),
• (Implication)
• P Q, (P Q) Q,
• (Equivalence)
• P Q, (P Q) (Q P),

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