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# 6.1 Symbols And Translation

## by Nicholas Lykins, Computer Science Research Assistant at Kentucky State University on Jun 05, 2009

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Course lecture I developed over section 6.1 of Patrick Hurley\'s "A Concise Introduction to Logic".

Course lecture I developed over section 6.1 of Patrick Hurley\'s "A Concise Introduction to Logic".

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## 6.1 Symbols And TranslationPresentation Transcript

• 6.1 Symbols and translation
• Formalizing logic
• We’re going to start diagramming statements now using logical operators (symbols).
• We’ll be using these symbols together with different variables (S, P, etc.) to mean different things.
• Different statements
• Different possible statements
• It is not the case that A. (A could mean “people are happy” or “dogs are animals”.)
• D and C. (D could mean “people are happy” and “dogs are not people”, or any kind of combination.)
• Either P or E.
• If W then F.
• B if and only if R.
• Five logical symbols
• ~ (Tilde)
• Symbolizes negation.
• Translates statements that say “not X”, or “it is not the case that X”.
• Example:
• ~A
• This means it is not the case that A.
• ~S
• There is not S.
• Logical symbols, continued
• · (dot)
• Indicates a conjunction of two things.
• And, but.
• Example:
• S · P
• There is both S and P
• There are dogs and cats.
• X · Y
• It is true that both X and Y.
• There are planets and moons.
• Logical symbols, continued
• v (wedge)
• Disjunctive, it means either one thing or the other.
• Or, unless.
• P v Q
• Either we have P or we have Q.
• Either we’re having fish or chicken for dinner.
• S v P
• There is either S or there is P.
• There are either happy students or sad students.
• Logical symbols, continued
• > (horseshoe)
• Implication (Conditional statement)
• If/then, only if, implies.
• This symbol means “If one thing, then another”.
• Example:
• P > Q
• If P is true, then Q is true.
• P entails Q.
• If there is no more liquor, then I’ll be very upset.
• S > P
• If S is true, then P is true.
• S entails P.
• Lots of friends entails a good social life.
• Formulas not to be confused
• A if B
• B > A
• A only if B
• A > B
• A if and only if B
• A = B
• Logical symbols, continued
• = (Triple bar)
• Indicates equivalence (biconditional)
• Translates the statement “if and only if”
• Example:
• A = B
• (A > B) · (B > A)
• If A then B, and if B then A.
• A entails B and B entails A.
• If there is no poverty then people will be happy. And if people are happy, then it implies there is no poverty.
• More complex formulas
• ~M v P
• Either there is not M, or there is P.
• Either we have no food, or there are happy people.
• (A v B) · (C v D)
• There is either A or B, and either C or D.
• People are either poor or wealthy, and people are either happy or sad.
• (A > B) v (C > D)
• Either A entails B or C entails D.
• Either a good economy entails lots of jobs or a poor job market entails unhappy workers.
• Complex formulas, continued
• Not both A and B.
• ~ (A · B)
• Can also means ~A v ~B (Either there is not A, or there is not B)
• Both not A and not B.
• ~A · ~B
• There is not A and there is not B.