Download - Truth, deduction, computation; lecture 3
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Truth, Deduction, ComputationLecture 3The Logic of Atomic Sentences
Vlad PatryshevSCU2013
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Introducing Arguments...
Premise1, premise2… conclusion!Or: conclusion - because premise1,...
E.g.● All men are mortal; Superman is a man, hence
Superman is mortal● Pavlova is a man: after all, Pavlova is mortal,
and all men are mortal
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Introducing Arguments...
Premise1, premise2… conclusion!Or: conclusion - because: premise1,...
E.g.● All men are mortal; Superman is a man,
hence Superman is mortal● Pavlova is a man: after all, Pavlova is mortal,
and all men are mortal
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Arguments
● Valid arguments (true, assuming premises are true)
● Sound arguments (valid, and premises are true)
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Fitch Notation (LPL version)
All cactuses have needlesPrickly pear is a cactus
Prickly pear has needles
Fitch Bar Conclusion
Premises
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What is a Proof?
Definition. Proof is a step-by-step demonstration that a conclusion follows from premises.
Counterexample:
I ride my bicycle every dayThe probability of an accident is very low
I will never have an accident
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Good Example of a Proof
1. Cube(c)2. c=b
3. Cube(b) = Elim: 1,2
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Elimination Rule
Aka the Indiscernibility of IdenticalsAka Substitution Principle (weaker than Liskov’s)
Aka Identity Elimination
If P(a) and a = b, then P(b).E.g.
x2 - 1 = (x+1)*(x-1)x2 > x2 - 1
x2 > (x+1)*(x-1)
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Introduction Rule
Aka Reflexivity of Identity
P1P2…Pn
x = x
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Symmetry of Identity
If a = b then b = a
a = ba = a
b = a
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Transitivity of Identity
If a = b and b = c then a = c
a = bb = c
a = c
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Other relationships may be transitive
If a < b and b < c then a < c
a < bb < cc < da < d
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F-notation (specific to LPL book)
(Has nothing to do with System F)We include in intermediate conclusions
P1P2…Pn
S1S2…SmS
For example:
1. a = b
2. a = a = Intro3. b = a = Elim: 2, 1
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Introduction Rule in Fitch
P1P2…Pn
x = x
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Introduction Rule (= Intro) in F
= Intro
x = x
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Elimination Rule in F
= Elim
P(n)…n = m…P(m)
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Reiteration Rule in F
= Reit
P………P
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“Bidirectionality of Between” in F
Between(a,b,c)………Between(a,c,b)
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Now, How Does It Work?
From premises SameSize(x, x) and x = y, prove SameSize(y, x).
1. SameSize(x, x)2. x = y…?. SameSize(y, x)
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Now, How Does It Work? (take 2)
From premises SameSize(x, x) and x = y, prove SameSize(y, x).
1. SameSize(x, x)2. x = y…?. y = x?. SameSize(y, x) = Elim: 1, ?
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Now, How Does It Work? (take 3)
From premises SameSize(x, x) and x = y, prove SameSize(y, x).
1. SameSize(x, x)2. x = y…3. y = y = Intro4. y = x = Elim: 3, 25. SameSize(y, x) = Elim: 1, 4
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Analytical Consequence in Fitch
This is something like a rule, but is based on “common sense” and external knowledge. E.g.
Cube(a)SameShape(a, b)
Cube(b) =Ana Con (“because we know what Cube means”)
Can be used to prove anything as long as we believe in our rules. It’s okay.
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Proving Nonconsequence
E.g.Are all binary operations associative? Addition is, multiplication is, even with matrices or complex number
1. op(a, b) = x2. op(b, c) = y
?. op(a, y) = op(x, c)
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Proving Nonconsequence
E.g.Are all binary operations associative? Addition is, multiplication is, even with matrices or complex number
1. op(a, b) = x2. op(b, c) = y
?. op(a, y) = op(x, c)
No!!!
Take binary trees. Take terms (from Chapter 1)
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Proving Nonconsequence
Given premises P1,...,Pn, and conclusion Q.
Q does not follow from P1,...,Pn if we can provide a counterexample.
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References
What Fitch actually is: http://en.wikipedia.org/wiki/Fitch-style_calculus
Fitch Online: http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-fitch.html
LPL software online (in Java Applets) http://softoption.us/content/node/339