Copyright © Peter Cappello Propositional Logic

Copyright © Peter Cappello Sentence Restrictions Building more precise tools from less precise tools Precise use of natural language is difficult.Precise use of natural language is difficult. We want a sublanguage suited to precision.We want a sublanguage suited to precision. Restrict discussion to sentences that are:Restrict discussion to sentences that are: declarative either true or false but not both. Such sentences are called propositions.Such sentences are called propositions.

Copyright © Peter Cappello Examples of Propositions Which of the sentences below are propositions? “Mastercharge, dig me into a hole!” “Peter Cappello thinks this class is fascinating.” “Do I exist yet?” “This sentence is false.”

Copyright © Peter Cappello Not Operator Not ( ~ ): p is true exactly when ~p is false.Not ( ~ ): p is true exactly when ~p is false. Let p denote “This class is the greatest entertainment since Game of Thrones.”Let p denote “This class is the greatest entertainment since Game of Thrones.” ~p denotes “It is not the case that this class is the greatest entertainment since Game of Thrones.”~p denotes “It is not the case that this class is the greatest entertainment since Game of Thrones.”

Copyright © Peter Cappello Or Operator (Disjunction) Or (  ): proposition p  q is true exactly when either p is true or q is true:

Copyright © Peter Cappello And Operator (Conjunction) And (  ): proposition p  q is true exactly when p is true and q is true:

Copyright © Peter Cappello If and Only If Operator (IFF) If and only if (  ): proposition p  q is true exactly when (p  q) or (~ p  ~ q):

Copyright © Peter Cappello Exclusive-Or Exclusive-or (  ) is the negation of .

Copyright © Peter Cappello Implies Operator (If … Then) Implies (  ): proposition p  q is true exactly when p is false or q is true:Implies (  ): proposition p  q is true exactly when p is false or q is true:

Copyright © Peter Cappello If … Then... Example: “If pigs had wings they could fly.”Example: “If pigs had wings they could fly.” In English, implies normally connotes a causal relation:In English, implies normally connotes a causal relation: p implies q means that p causes q to be true. Not so with the mathematical definition!Not so with the mathematical definition! If 1  1 then Peter hates Family Guy.

Copyright © Peter Cappello Converse & Inverse The converse of p  q is q  p.The converse of p  q is q  p. The inverse of p  q is ~p  ~q.The inverse of p  q is ~p  ~q. The contrapositive of p  q is ~q  ~p.The contrapositive of p  q is ~q  ~p. If p  q then which, if any, is always true:If p  q then which, if any, is always true: Its converse? Its inverse? Its contrapositive? Use a truth table to find the answer. Describe the contrapositive of p  q in terms of the converse & inverse.Describe the contrapositive of p  q in terms of the converse & inverse. Compare the truth tables of the converse & inverse.Compare the truth tables of the converse & inverse.

Copyright © Peter Cappello p  q may be expressed as p implies q if p then q q if p q follows from p q provided p q is a consequence of p q whenever p p is a sufficient condition for q p only if q (if ~q then ~p) q is a necessary condition for p (if ~q then ~p)

Copyright © Peter Cappello Abstraction Capture the logical form of a Proposition in English Let g, h, and b be propositions:Let g, h, and b be propositions: g: Grizzly bears have been seen in the area. h: Hiking is safe on the trail. b: Berries are ripe along the trail. Translate the following sentence using g, h, and b, and logical operators:Translate the following sentence using g, h, and b, and logical operators: If berries are ripe along the trail, hiking is safe on the trail if and only if grizzly bears have not been seen in the area.

Copyright © Peter Cappello 1.If berries are ripe along the trail, hiking is safe on the trail if and only if grizzly bears have not been seen in the area. 2.If b, ( h if and only if  g ). 3. b  ( h   g ).

Copyright © Peter Cappello Truth Table of a Compound Proposition bhg  g g g g h   gh   gh   gh   g b  ( h   g ) TTT TTF TFT TFF FTT FTF FFT FFF

System Specification Systems are increasing in complexity.Systems are increasing in complexity. e.g., software, hardware, workflow, security, legal Can we know that a system works as intended?Can we know that a system works as intended? 1.Specify a set of desired system properties Each property is expressed as a compound proposition. 2.Verify that such a system is feasible. All compound propositions are simultaneously satisfiable. Z specification languageZ specification language Allow: Copyright © Peter Cappello

Knights & Knaves An island’s only inhabitants are knights (truth tellers) & knaves (liars).An island’s only inhabitants are knights (truth tellers) & knaves (liars). You are approached by 2 inhabitants, A & B.You are approached by 2 inhabitants, A & B. Determine, if possible, what A & B are, if B says nothing & A says:Determine, if possible, what A & B are, if B says nothing & A says: 1.“At least 1 of us is a knave.” 2.“I am a knave or B is a knight.” 3.“We are both knaves.” Copyright © Peter Cappello

1.“At least 1 of us is a knave.” A is a knight; B is a knave. 2.“I am a knave or B is a knight.” A & B are knights 3.“We are both knaves.” A is a knave; B is a knight. Copyright © Peter Cappello

Google Search Operators Query: “US states” “income tax rate” Beatles: “Taxman” (Query: Beatles Taxman) Let me tell you how it will be There's one for you, nineteen for me 'Cause I'm the taxman, yeah, I'm the taxman Should five per cent appear too small Be thankful I don't take it all 'Cause I'm the taxman, yeah I'm the taxman If you drive a car, I'll tax the street, If you try to sit, I'll tax your seat. If you get too cold I'll tax the heat, If you take a walk, I'll tax your feet. Don't ask me what I want it for If you don't want to pay some more 'Cause I'm the taxman, yeah, I'm the taxman Now my advice for those who die Declare the pennies on your eyes 'Cause I'm the taxman, yeah, I'm the taxman And you're working for no one but me. Copyright © Peter Cappello