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