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: $(A\Rightarrow B)\wedge (\neg B) \Rightarrow (\neg A)$

## 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.