Formal Foundations I: All About Logic and Stuff

1
Formal Foundations IAll About Logic and Stuff

• Helpful Reading
• Rosen 1.1-1.3

2
Why study logic?

• In order to make assertions about, well,
  anything, we need to be precise
• The system of typographic logic we will study at
  the beginning of this course aims to give
  structure to the way we will discuss the
  characteristics of computer science
• Logic is itself directly applicable to computer
  circuitry (logic gates, bit representations)

3
The Fundamental Unit of LogicThe Proposition

• Notion of truth and falsity from our everyday
  conversation
• A proposition is a statement that is either true
  or false (not both, and not neither) at some
  given time
• I am a bottle of wine is a proposition
• I hate wine is a proposition
• Drink this wine is NOT a proposition

4
The Binary Connectives

• More interesting statements can be created
  through the combination of two propositions using
  binary connectives
• A binary connective is a method for determining
  the aggregate truth value of two propositions --
  the result of connecting two propositions is
  another proposition
• Common binary connectives are AND, OR, and XOR,
  and they get their meanings from English

5
Meanings of the Connectives

• AND (Ù)
• If p is a proposition, and q is a proposition,
  then p Ù q is true exactly when both p and q are
  true (I hate wine AND I hate beer)
• OR (Ú)
• p Ú q is true if either p is true or q is true
  (She can visit on Sunday OR Tuesday) -- both
  can be true
• XOR (Å)
• p Å q is true if either p is true or q is true
  but NOT both (You can have soup OR salad)
• English is a terribly ambiguous language! (when
  it comes to logic)

6
Implications

• Another common form of binary connective is the
  implication ()
• Stated as if p, then q written pq
• Alternately p implies q. q if p.
• If it is raining, then we wont go to the
  beach.
• Implications are one-way!
• Nothing says we will go to the beach if its not
  raining, but we definitely wont go if it is

7
Negations

• If p is a proposition, then Øp is its negation
  (opposite)
• p I like wine., Øp I dont like wine.
• p Bobs new car is red., Øp Bobs new car
  is not red.
• Øp is true exactly when p is false, and vice
  versa
• Negation is a unary operator on propositions -
  precedence over binary

8
A Propositional Calculus

• Represent absolute truth with T, absolute falsity
  with F
• Assign letters to various propositions
• Binary connectives are used to join the
  propositions
• GOAL Particular assignments of T and F to each
  proposition consistently yield the overall truth
  value of the proposition as a whole

9
Truth Tables
10
Logical Equivalence

• Two compound propositions over some set of
  component propositions are logically equivalent
  if they have the same truth value for each
  possible assignment to component propositions
• This also means that they are logically
  equivalent if they imply each other, i.e. if
  (pq) Ù (qp) is always true
• Leads to some well-known reductions

11
Common Equivalences

• Used to reduce complex propositions
• De Morgan 1 Ø(p Ù q) Û Øp Ú Øq
• De Morgan 2 Ø(p Ú q) Û Øp Ù Øq
• Implication pq Û Øp Ú q
• Double negation Ø(Øp) Û p
• Distributive 1 p Ù (q Ú r) Û (p Ù q) Ú (p Ù r)
• Distributive 2 p Ú (q Ù r) Û (p Ú q) Ù (p Ú r)
• Replace parts of complex propositions with
  logically equivalent ones

12
Propositions vs. Predicates

• In computer science, we are often concerned with
  statements whose truth or falsity depends on
  unknown values
• x gt 5 is an interesting statement that is NOT a
  proposition (what is x?)
• Algorithm A is faster than Algorithm B is
  another such statement (what is A? B?)
• A statement whose truth is dependent on unknowns
  is a predicate

13
The Predicate Calculus

• Predicates become propositions when their unknown
  values are specified from some universe of
  discourse, then propositional calculus rules
  apply
• Denote a predicate P over unknown x by P(x) --
  note capital P
• If P(x) x gt 2 and Q(x) x lt 4, and the
  universe of discourse is the integers, then
• P(3) Ù Q(2) Û T
• P(3) Ù Q(4) Û F

14
Quantifiers

• Often we are interested in proving that a
  predicate either holds for the whole universe of
  discourse, or for at least one element in the
  universe of discourse
• Universal Quantifier ("), for all
• P(x) is true for all x from the universe of
  discourse, written "x P(x)
• Existential Quantifier (), exists
• P(x) is true for at least one x from the
  universe of discourse, written x P(x)
• Quantifying a predicate makes a proposition

15
Caveats for Quantifiers

• Caveat 1 Negation is tricky
• Ø("x P(x)) Û x Ø P(x)
• Ø(x P(x)) Û "x Ø P(x)
• Caveat 2 Existence does not equal specification
• (x P(x)) Ù (x Q(x)) DOES NOT imply that x
  (P(x) Ù Q(x)) -- it may not be the same x that
  satisfies P(x) and Q(x).

16
Logic and Computers

• A bit is either ON or OFF (1 or 0), and a
  proposition is either true or false -- a truth
  value can be represented as a bit
• Bit strings are sequences of 0s and 1s
• Binary connectives can be applied to bit strings
  by applying the connective to each bit in
  sequence
• 01100101 Ù 11010101 01000101
• 10010101 Ú 01101010 11111111
• 10010110 Å 11111111 01101001

17
What Next?

• The notion of proof
• Generally, proofs aim to show beyond a doubt that
  for some p and q, pq
• Inductive proofs aim to show the truth of a
  proposition "n P(n) where n is from the universe
  of discourse of some infinite subset of all the
  integers (e.g. n gt 1)
• Also, techniques for proving x P(x)
• We can use the underlying rules of the
  propositional calculus to simplify life