Truth, deduction, computation; lecture 3
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Transcript of Truth, deduction, computation; lecture 3
Truth, Deduction, ComputationLecture 3The Logic of Atomic Sentences
Vlad PatryshevSCU2013
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
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
Arguments
● Valid arguments (true, assuming premises are true)
● Sound arguments (valid, and premises are true)
Fitch Notation (LPL version)
All cactuses have needlesPrickly pear is a cactus
Prickly pear has needles
Fitch Bar Conclusion
Premises
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
Good Example of a Proof
1. Cube(c)2. c=b
3. Cube(b) = Elim: 1,2
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)
Introduction Rule
Aka Reflexivity of Identity
P1P2…Pn
x = x
Symmetry of Identity
If a = b then b = a
a = ba = a
b = a
Transitivity of Identity
If a = b and b = c then a = c
a = bb = c
a = c
Other relationships may be transitive
If a < b and b < c then a < c
a < bb < cc < da < d
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
Introduction Rule in Fitch
P1P2…Pn
x = x
Introduction Rule (= Intro) in F
= Intro
x = x
Elimination Rule in F
= Elim
P(n)…n = m…P(m)
Reiteration Rule in F
= Reit
P………P
“Bidirectionality of Between” in F
Between(a,b,c)………Between(a,c,b)
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)
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, ?
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
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.
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)
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)
Proving Nonconsequence
Given premises P1,...,Pn, and conclusion Q.
Q does not follow from P1,...,Pn if we can provide a counterexample.
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