May 4, 2018

Sets

  • There are things: \(x,y,z\)
  • Things have attributes
  • The group of all things \(x\) with an attribute \(\mathcal A\) is the set \(A\) \[A=\{x: x\text{ has property }\mathcal A \}\]
  • For each attribute \(\mathcal A\) there is a set \(A\) of all things \(x\) with that attribute

Summary of Logic

Predicates \(P,Q,R\) can be either true (\(T\)) or false (\(F\)). For example:

  • \(P\) says that a thing has an attribute \[P:\ x\in A\]
  • \(Q\) says that there exist something with an attribute \[Q:\ \exists x, x\in A\qquad\text{that is}\qquad A\not=\emptyset\]
  • \(R\) declares that all \(A\) things have an attibute \(B\) \[R:\ \forall x, x\in A \Rightarrow x\in B \qquad\text{that is}\qquad A\subset B\]

Summary of Logic

  • We can combine predicates using AND, OR, NOT, IMPLIES
  • We write \(P \wedge Q\), \(P \vee Q\), \(\neg P\), \(P\Rightarrow Q\)
  • We can know the truth value of any predicate following some rules
  • if \(P\) and \(Q\) have always the same truth value, we write \(P\Leftrightarrow Q\)

Some rules

  • AND rule: \(\forall P, (P\wedge F)\Leftrightarrow F, (P\wedge T)\Leftrightarrow P\)
  • OR rule: \(\forall P, (P\vee F)\Leftrightarrow P, (P\vee T)\Leftrightarrow T\)
  • NOT rule: \(\neg T =F, \neg F=T\)
  • IMPLICATION: \((P\Rightarrow Q)\Leftrightarrow (\neg P\vee Q) \Leftrightarrow (P\wedge Q\Leftrightarrow P)\)

This is called Boolean Algebra

  • Double negation: \(\neg (\neg P) \Leftrightarrow P\)
  • Commutativity: \((P \wedge Q) \Leftrightarrow (Q \wedge P)\)
  • Distributivity: \((P \wedge Q)\vee R \Leftrightarrow (P\vee R) \wedge (Q\vee R)\)

De Morgan’s laws

  • \(\neg (P \wedge Q) \Leftrightarrow (\neg P) \vee (\neg Q)\)
  • \(\neg (P \vee Q) \Leftrightarrow (\neg P) \wedge (\neg Q)\)

We can check any logical argument

An arguments \(A\) is a logical sentence like \[A: P_1\wedge P_2\wedge\ldots\wedge P_n\Rightarrow Q\] The predicates \(P_i\) are called premises, and \(Q\) is the conclusion

The argument \(A\) is correct if it is true for all feasible \(P_i\)

The argument \(A\) is sound if the premises \(P_i\) can be true in reality

Some arguments

  • All men are mortal, Socrates is a man \(\Rightarrow\) Socrates is mortal \[(A\subset B) \wedge (x\in A) \Rightarrow (x\in B)\]
  • Modus ponens: \[(A\Rightarrow B)\wedge (A) \Rightarrow (B)\]
  • Modus tollens: \[(A\Rightarrow B)\wedge (\neg B) \Rightarrow (\neg A)\]

Extending Logic

When we don’t know if it is true or not

Now we will accept that in many cases we don’t know the truth value of some predicates

For example:

  • “It will rain tomorrow”
  • “I will win the lottery next month”
  • “I will finish my thesis”

How much we think it is true?

Plausibility

literally, we can clap for it

How much do someone believes/thinks about the truth value of a predicate

credible, reasonable, believable, likely, feasible, tenable, possible, conceivable, convincing, persuasive, cogent, sound, rational, logical, thinkable.

ANTONYM
unlikely.

Plausibility always depends on context

Predicate \(A\): ”it will snow tomorrow” can be true or false

The truth value depends on the context.

Context is a predicate \(Z\) that is true to our knowledge

We write the plausibility of \(A\) given \(Z\) as

\[(A\vert Z)\]

Wishlist for extended logic

according to Jaynes

  1. Plausibility should be a real number
  2. Qualitative Correspondence with common sense
  3. Consistency

1. Plausibility should be a real number

  • Greater plausibility corresponds to a greater number
  • If \(A\) is more plausible than \(B\) given the context \(Z\), then \[(A\vert Z)>(B\vert Z)\]
  • If the context \(Z_1\) changes to \(Z_2\) and makes \(A\) more plausible, then \[(A\vert Z_2)>(A\vert Z_1)\]

2. Qualitative agreement with common sense

  • This should be an extension of formal logic

  • If \(Z_1\) changes to \(Z_2\) and \(A\) becomes more plausible, then \(\neg A\) is less plausible

    \[(A\vert Z_2)>(A\vert Z_1)\Rightarrow(\neg A\vert Z_2)<(\neg A\vert Z_1)\]

  • If in the same case \(B\) plausibility does not change, then \(A\wedge B\) cannot be less plausible: \[(A\vert Z_2)>(A\vert Z_1)\quad\wedge\quad (B\vert A\wedge Z_2)=(B\vert A\wedge Z_1)\] \[\Rightarrow(A\wedge B\vert Z_2)\geq(A\wedge B\vert Z_1)\]

3. Consistency

  • If we can reason in several ways, every way must give the same result
  • All evidence must be considered
  • Do not arbitrarily ignore any information
  • Equivalent states of knowledge are represented by equivalent plausibility.