2.2 Statements, Connectives, and Quantifiers

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Presentation transcript:

2.2 Statements, Connectives, and Quantifiers Identify statements in logic. Represent statements symbolically using five connectives. Rewrite symbolic statements in English. Understand the difference between the universal and existential quantifiers. Write the negations of quantified statements.

In symbolic logic, we only care whether statements are true or false – not their content. A statement in logic is a declarative sentence that is either true or false. We represent statements by lowercase letters such as p, q, or r. Simple statements – express a single idea . Compound statements – contain several ideas combined together. Connectives – words used to join the ideas of compound sentences.

In logic, we connect ideas using not, and, or, if … then, and if and only if. The connectives we use in logic generally fall into five categories: negation, conjunction, disjunction, conditional, and biconditional. Negation – a statement expressing the idea that something is not true. We represent negation by the symbol ~ . Conjunction – a statement expressing the idea of and. We use the symbol ^ to represent conjunction.

Disjunction – a statement that conveys the notion of or Disjunction – a statement that conveys the notion of or. We use the symbol to represent a disjunction. Conditional – a statement that expresses the notion of if…then. We use an arrow, , to represent a conditional. Biconditional – a statement that represents the idea of if and only if. Its symbol is a double arrow, .

Quantifiers state how many objects satisfy a given property. Quantifiers – tell us “how many.” Universal quantifiers – words such as all and every that state that all objects of a certain type satisfy a given property. Existential quantifiers – words such as some, there exists, and there is at least one that state there are one or more objects that satisfy a given property.

Negating statements with quantifiers The phrase Not all are has the same meaning as At least one is not. Negate the statement: All dogs bark. At least one dog does not bark. The phrase Not some are has the same meaning as All are not. Negate the statement: Some flowers smell good. All flowers do not smell good.

Classwork/Homework Classwork – Page 90 (11 – 49 odd) Homework – Page 90 (12 – 50 even)