6.1 Symbols And Translation

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

1. 6.16.1
Symbols and translationSymbols and translation

2. Formalizing logicFormalizing logic
•We’re going to start diagramming statements now using logical We’re going to start diagramming statements now using logical
operators (symbols).operators (symbols).
•We’ll be using these symbols together with different variables (S, P, We’ll be using these symbols together with different variables (S, P,
etc.) to mean different things.etc.) to mean different things.

3. Different statementsDifferent statements
•Different possible statementsDifferent possible statements
–It is not the case that A. (A could mean “people are happy” or “dogs are It is not the case that A. (A could mean “people are happy” or “dogs are
animals”.)animals”.)
–D and C. (D could mean “people are happy” and “dogs are not people”, D and C. (D could mean “people are happy” and “dogs are not people”,
or any kind of combination.)or any kind of combination.)
–Either P or E.Either P or E.
–If W then F.If W then F.
–B if and only if R.B if and only if R.

4. Five logical symbolsFive logical symbols
•~ (Tilde)~ (Tilde)
–Symbolizes negation.Symbolizes negation.
–Translates statements that say “not X”, or “it is not the case that X”.Translates statements that say “not X”, or “it is not the case that X”.
–Example:Example:
•~A~A
–This means it is not the case that A.This means it is not the case that A.
•~S~S
–There is not S.There is not S.

5. Logical symbols, continuedLogical symbols, continued
•· (dot)· (dot)
–Indicates a conjunction of two things.Indicates a conjunction of two things.
–And, but.And, but.
–Example:Example:
•S · P S · P
–There is both S and PThere is both S and P
There are dogs and cats.There are dogs and cats.
•X · YX · Y
–It is true that both X and Y.It is true that both X and Y.
There are planets and moons.There are planets and moons.

6. Logical symbols, continuedLogical symbols, continued
•v (wedge)v (wedge)
–Disjunctive, it means either one thing or the other.Disjunctive, it means either one thing or the other.
–Or, unless.Or, unless.
–P v QP v Q
•Either we have P or we have Q.Either we have P or we have Q.
–Either we’re having fish or chicken for dinner.Either we’re having fish or chicken for dinner.
–S v PS v P
•There is either S or there is P.There is either S or there is P.
–There are either happy students or sad students.There are either happy students or sad students.

7. Logical symbols, continuedLogical symbols, continued
•> > (horseshoe)(horseshoe)
–Implication (Conditional statement)Implication (Conditional statement)
–If/then, only if, implies.If/then, only if, implies.
•This symbol means “If one thing, then another”.This symbol means “If one thing, then another”.
•Example:Example:
–P P >> Q Q
–If P is true, then Q is true. If P is true, then Q is true.
–P entails Q.P entails Q.
If there is no more liquor, then I’ll be very upset.If there is no more liquor, then I’ll be very upset.
–S S > > PP
–If S is true, then P is true.If S is true, then P is true.
–S entails P.S entails P.
Lots of friends entails a good social life.Lots of friends entails a good social life.

8. Formulas not to be confusedFormulas not to be confused
•A if BA if B
–B B >> A A
•A only if BA only if B
–A A >> B B
•A if and only if BA if and only if B
–A A == B B

9. Logical symbols, continuedLogical symbols, continued
•= = (Triple bar)(Triple bar)
–Indicates equivalence (biconditional)Indicates equivalence (biconditional)
–Translates the statement “if and only if”Translates the statement “if and only if”
–Example:Example:
•AA = = BB
•(A(A > > B) · (BB) · (B > > A) A)
–If A then B, and if B then A.If A then B, and if B then A.
–A entails B and B entails A.A entails B and B entails A.
If there is no poverty then people will be happy. And if people are happy, If there is no poverty then people will be happy. And if people are happy,
then it implies there is no poverty.then it implies there is no poverty.

10. More complex formulasMore complex formulas
•~M v P~M v P
–Either there is not M, or there is P.Either there is not M, or there is P.
•Either we have no food, or there are happy people.Either we have no food, or there are happy people.
•(A v B) · (C v D)(A v B) · (C v D)
–There is either A or B, and either C or 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.People are either poor or wealthy, and people are either happy or sad.
•(A > B) v (C > (A > B) v (C > D)D)
–Either A entails B or C entails D.Either A entails B or C entails D.
•Either a good economy entails lots of jobs or a poor job market entails Either a good economy entails lots of jobs or a poor job market entails
unhappy workers.unhappy workers.

11. Complex formulas, continuedComplex formulas, continued
•Not both A and B.Not both A and B.
–~ (A · B)~ (A · B)
•Can also means ~A v ~B (Either there is not A, or there is not B)Can also means ~A v ~B (Either there is not A, or there is not B)
•Both not A and not B.Both not A and not B.
–~A · ~B~A · ~B
•There is not A and there is not B.There is not A and there is not B.