CS1502 Formal Methods in Computer Science

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CS1502 Formal Methods in Computer Science

• Lecture Notes 2
• Introduction to Logic
• Part 2

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What is an argument?

• A series of statements in which one (called the
  conclusion) is meant to follow from the others
  (called the premises).
• Not talking about the type of argument

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Fitch-style Argument

• P1 P2 ... Pn Q

premises
conclusion
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Valid Argument

• A valid argument is one that guarantees the truth
  of its conclusion on the assumption that the
  premises are true.
• A valid argument ensures the conclusion is true
  provided the premises are true.
• Often written premises conclusion

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Valid Argument - Example

• Large(b) v Cube(b) ?Cube(b) Large(b)

?
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Sound Argument

• If an argument is valid and its premises are
  true, then the argument is said to be sound.

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Sound Argument
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Argument is not sound
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ExamplesWhich are valid? Sound?

• (worked out in lecture)
• All men are mortal. Socrates is a man. So,
  Socrates is Mortal.
• Bill is a man. After all, Bill is mortal and all
  men are mortal.
• All women are taller than all men. Ralph is a
  woman and Bill is a man. Therefore, Ralph is
  taller than Bill.

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ExamplesWhich are valid? Sound?

• Since this class meets Tuesday after 1245pm, it
  is January.
• Tom Hanks is a good actor. After all, all rich
  actors are good actors, and Tom Hanks is a rich
  actor.

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Methods of Proof

• Formal We will use a Fitch-style proof employed
  in the software.
• Informal This style of proof, used by
  mathematicians, is just as rigorous. It consists
  of sentences describing the situation at hand,
  the inferences being made, and the justification
  of each inference.
• Difference? The amount of explicit detail.

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What constitutes a proof?

• Proof that P1,P2,,Pn Q is
• a step-by-step demonstration showing that Q must
  be true in any circumstances in which all of
  P1,P2,,Pn are true.

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Fitch-style Proof
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Proof Rules

• Proof rules are used to construct proofs (both
  formal and informal)
• That is, each step but the premises has to be
  justified by a proof rule
• As we introduce more pieces of FOL, we will
  introduce more proof rules
• Well start now with proof rules involving
  identity

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Rules

• EliminationIf b c and P(b) then P(c).
• Introductiona a
• Symmetry of IdentityIf a b then b a.
• Transitivity of IdentityIf a b and b c then
  a c

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Elimination

• P(n) n m P(m)

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Introduction

• n n

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Symmetry of Identity

• a b

1)2)3)
a a
Introduction
b a
Elimination 1, 2
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Example Formal Proof

• Larger(a,b) c b

1)2)3)4) 5)6)
Smaller(b,a)
Ana Con 1
Introduction
c c
b c
Elim 2, 4
Smaller(c,a)
Elim 5, 3
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Explicit Proof of Ana Con step
This is a look ahead we havent seen these
proof rules before.
Note As this is displaying, the line is in the
wrong place. It should be between lines 2 and 3.
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Example Informal Proof
Prove If a is smaller than b and c is identical
to b then c is larger than a.
Since we are given that a is smaller than b, it
follows that b must be larger than a. Moreover,
since c is identical to b, it follows that c must
be larger than a.
QED