Statements The most important idea in logic: Validity of an argument.

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Presentation transcript:

Statements The most important idea in logic: Validity of an argument.

Statements The most important idea in logic: Validity of an argument. However, there are other important concepts that concern statements.

Statements Logical Truths (Tautologies) Contradictions Contingent Statements

Logical Truth A statement is a logic truth (tautology) iff it cannot be F.

Logical Truth A statement is a logic truth (tautology) iff it cannot be F. So its truth table has all Ts on the output column.

Logical Truth A statement is a logic truth (tautology) iff it cannot be F. So its truth table has all Ts on the output column. Samples: P>P Pv-P

Logical Truth A statement is a logic truth (tautology) iff it cannot be F. Samples: P>P Pv-P Good news: Logical Truths are always true.

Logical Truth A statement is a logic truth (tautology) iff it cannot be F. Samples: P>P Pv-P Good news: Logical Truths are always true. Bad news: Logical Truths do not carry information.

Logical Truth A statement is a logic truth (tautology) iff it cannot be F. Samples: P>P Pv-P Good news: Logical Truths are always true. Bad news: Logical Truths do not carry information. Desperate Weather Report: If it is raining then it is raining.

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. P P>P T F T * P P v -P T T F F T T *

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. P P>P T F T * P P v -P T T F F T T *

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. P P>P T F T * P P v -P T T F F T T * P>PGOAL

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. P P>P T F T * P P v -P T T F F T T * 1) PPA 2) P>P1-1 >I

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. P P>P T F T * P P v -P T T F F T T * 1) PPA 2) P>P1-1 >I Pv-PGOAL

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. P P>P T F T * P P v -P T T F F T T * 1) PPA 2) P>P1-1 >I 1) -(Pv-P)PA ?&-? Pv-P1-? -O

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. P P>P T F T * P P v -P T T F F T T * 1) PPA 2) P>P1-1 >I 1) -(Pv-P)PA 2) -P&--P 1 DM 3) Pv-P1-2 -O

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. with a tree:The tree for -A closes. P P>P T F T * P P v -P T T F F T T * 1) PPA 2) P>P1-1 >I 1) -(Pv-P)PA 2) -P&--P 1 DM 3) Pv-P1-2 -O

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. with a tree:The tree for -A closes. P P>P T F T * P P v -P T T F F T T * 1) PPA 2) P>P1-1 >I -(P>P) 1) -(Pv-P)PA 2) -P&--P 1 DM 3) Pv-P1-2 -O

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. with a tree:The tree for -A closes. P P>P T F T * P P v -P T T F F T T * 1) PPA 2) P>P1-1 >I -(P>P) P -P * 1) -(Pv-P)PA 2) -P&--P 1 DM 3) Pv-P1-2 -O

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. with a tree:The tree for -A closes. P P>P T F T * P P v -P T T F F T T * 1) PPA 2) P>P1-1 >I -(P>P) P -P * -(Pv-P) 1) -(Pv-P)PA 2) -P&--P 1 DM 3) Pv-P1-2 -O

Logical Truth To show a statement A is a logic truth... with a table: The output row for A has all Ts. with a proof: Prove A. with a tree:The tree for -A closes. P P>P T F T * P P v -P T T F F T T * 1) PPA 2) P>P1-1 >I -(P>P) P -P * -(Pv-P) -P --P * 1) -(Pv-P)PA 2) -P&--P 1 DM 3) Pv-P1-2 -O

Logical Truth To show a statement A is a logic truth (tautology)... with a table: The output row for A has all Ts. with a proof: Prove A. with a tree:The tree for -A closes. For more click here