Chapter 3: Basic Propositional Logic

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Chapter 3 Basic Propositional Logic

• Based on Harry Genslers book
• For CS2209A, 2009
• By Dr. Charles Ling cling_at_csd.uwo.ca

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3.1 Translation from English to Logic

• Need a (formal) language to
• deal with simple statements that may be true and
  false use capital letters (P, Q, )
• They are called propositions
• deal with if-then, and, or, not, etc.

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Well-Formed Formula (wff)

• 1. Any capital letter is a wff.
• 2. The result of prefixing any wff with is a
  wff.
• 3. The result of joining any two wffs by or
  ? or ? or and enclosing the result in
  parentheses is a wff.
• Examples and usual meaning of connectives

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Parentheses are important
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Examples of Invalid wff

• (P), (Q), ((R))
• PQ, P Q R, (P Q R)
• Logic (thus, wff) is very precise

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Some useful rules

• Rule put ( wherever you see both, either,
  or if.

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• Rule Group together parts on either side of a
  comma.

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• Rule have your capital letters stand for whole
  statements

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Exercise (also LogiCola C (EM ET))

• 1. Not both A and B.
• 2. Both A and either B or C.
• 3. Either both A and B or C.
• 4. If A, then B or C.
• 5. If A then B, or C.
• 6. If not A, then not either B or C.
• 7. If not A, then either not B or C.
• 8. Either A or B, and C.
• 9. Either A, or B and C.
• 10. If A then not both not B and not C.
• 11. If you get an error message, then the disk is
  bad or its a Macintosh disk.
• 12. If I bring my digital camera, then if my
  batteries dont die then Ill take pictures of my
  backpack trip and put the pictures on my Web
  site.

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3.2 Simple truth tables the meaning/semantics of
wff

• A truth table gives a logical diagram for a wff.
  It lists all possible truth-value combinations
  for the letters and says whether the wff is true
  or false in each case.
• Define connectives first

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Real-life or may have different meanings

• Inclusive or
• Exclusive or A or B but not both
  ((A?B)(AB))

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An interesting example

• The Conservatives have issued another apology,
  this time for comments caught on video Wednesday
  by an assistant to Transport Minister Lawrence
  Cannon.
• The exchange was caught on video and broadcast as
  the lead item Wednesday by the Aboriginal Peoples
  Television Network.
• If you behave and you're sober and there's no
  problems and if you don't do a sit down and
  whatever, I don't care, said Mr. Cannon's
  assistant Darlene Lannigan to Mr. Matchewan. She
  then added One of them showed up the other day
  and was drinking.
• Are you calling me an alcoholic? replied Mr.
  Matchewan.
• If A then B does NOT imply A but often
  misunderstood.

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3.3 Truth evaluations

• We can calculate the truth value of a wff if we
  know the truth value of its letters.
• LogiCola D (TM TH)

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3.3a Exercise

• Assume that A1 and B1 (A and B are both true)
  while X0 and Y0 (X and Y are both false).
  Calculate the truth value of each wff below.

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3.4 Unknown evaluations

• We can sometimes figure out a formulas truth
  value even if we dont know the truth value of
  some letters.
• ExerciseLogiCola D (UE, UM, UH)
• T1 (T is true), F0 (F is false), and U?

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3.5 Complex truth tables

• A formula with n distinct letters has 2n possible
  truth-value combinations

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• The truth table for (P?P) is true in all
  caseswhich makes the formula a tautology
• the law of the excluded middle, says that every
  statement is true or false. This law holds in
  propositional logic
• The truth table for (PP) is false in all
  caseswhich makes the formula a
  self-contradiction
• Otherwise, the formula is a contingent
• 3.5a ExerciseLogiCola D (FM FH)

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Logical Paradox

• Everything I say is a lie.
• Barber paradox An adult male barber shaves all
  men who do not shave themselves, and no one else.
  Can he shave himself?
• One thing is certain in this world nothing is
  certain.

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Logical Paradox

• Logical proof that God does not exist. ?
• God is supposed to be omnipotent (all-powerful). 
  
• If He is omnipotent, then He can create a rock so
  big that He can't pick it up. 
• If He cannot make a rock like this, then He is
  not omnipotent. 
• If He can make a rock so big that He can't pick
  it up, then He isn't omnipotent either. 
• Either way we demonstrated that God cannot do
  something. 
• Therefore God is not omnipotent. 
• Therefore God does not exist.  

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3.6 The truth-table test

• To prove a propositional argument (given premises
  and conclusion)
• Construct a truth table showing the truth value
  of the premises and conclusion for all possible
  cases.
• The argument is VALID if and only if no possible
  case has the premises all true and conclusion
  false (VALID if and only if for all possible
  cases that the premises are all true, the
  conclusion is also true).

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• If youre a dog, then youre an animal. Youre
  not a dog. ? Youre not an animal

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3.6a ExerciseLogiCola D (AE, AM, AH)
Short-cut table do it faster
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3.6a Exercise (selected)

• 3. If television is always right, then Anacin is
  better than Bayer. If television is always right,
  then Anacin isnt better than Bayer. ? Television
  isnt always right.     Use T and B.
• 4. If it rains and your tent leaks, then your
  down sleeping bag will get wet. Your tent wont
  leak. ? Your down sleeping bag wont get
  wet.     Use R, L, and W.
• 7. If ethics depends on Gods will, then
  something is good because God desires it.
  Something isnt good because God desires it.
  (Instead, God desires something because its
  already good.) ? Ethics doesnt depend on Gods
  will.     Use D and B this argument is from
  Platos Euthyphro.
• 9. Ill go to Paris during spring break if and
  only if Ill win the lottery. I wont win the
  lottery. ? I wont go to Paris during spring
  break.  Use P and W.

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• If an argument passes the truth-table test, it
  means that the premises entails the conclusion
  (in semantics).
• The truth-table test can get tedious for long
  arguments. Arguments with 6 letters need 64
  linesand ones with 10 letters need 1024 lines
• Can we do it based only on syntax? Yes, see
  3.10- , Chapter 4,

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3.7 The truth-assignment test

• Take a propositional argument. Set each premise
  to 1 and the conclusion to 0. The argument is
  VALID if and only if no consistent way of
  assigning 1 and 0 to the letters will make this
  workso we cant make the premises all true and
  conclusion false.
• You REFUTE the argument if you can find such an
  assignment

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• ExerciseLogiCola E (S) E (E)

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• (Read Chapter 3.8 by yourselves)
• (Read Chapter 3.9 by yourselves)
• 3.8a ExerciseLogiCola C (HM HT)
• 3.9a ExerciseLogiCola E (F I)

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3.8a ExerciseLogiCola C (HM HT)
Multiple Choice
Translation
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3.9a ExerciseLogiCola E (F I)
Finding Conclusion
Valid or not?
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Some extra topics

• See extra slides posted
• Adequate set of connectives (set2)
• Normal forms (set3)

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3.10/3.11 S-rules, I-rules

• Inference rules, which state that certain
  formulas can be derived with validity from
  certain other formulas, mechanically
• Deduce, formally deducible
• Will be building blocks for formal proofs
• Also check mechanically a proof
• They reflect common forms of reasoning
• What we hope to have (see later)
• Everything that is deduced is valid. Sound
• Everything that is valid can be deduced. Complete

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3.10a ExerciseLogiCola F (SE SH)
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All S-rules
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3.11a ExerciseLogiCola F (IE IH)
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3.12 Combining S- and I-rules

• 3.12a ExerciseLogiCola F (CE CH)

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3.13 Extended inferences
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• Exercise

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Rules you can use
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3.14 Logic gates and computers