1. Why logic

Why logic?

### What is logic?
**In philosophy:** Logic was formed to "formalise correct argumentation and deduction".
**In mathematics:** Rules of logic are used to distinguish between correct and incorrect mathematical arguments (otherwise known as proofs).

Propositions

Proposition

A proposition is a **declarative sentence** of which we can meaningfully ask whether it is true $(T)$ or false $(F)$.

Sample syntax:

• Propositional / boolean random variables: $\text{Cavity = true}\mapsto \text{cavity}$ $\text{Cavity = false}\mapsto \neg \text{cavity}$
• Discrete random variables (finite or infinite) Given $\text{Weather}$ is one of $\{sunny,rain,cloudy,snow\}$ "$\text{Weather}=rain$" is a proposition
• Continuous random variables (bounded or unbounded) $\text{Temp}=21.6$; $\text{Temp}<22.0$; $\text{Temp}\ge 5.0\text{ AND }\text{Temp}\le 23.0$

Link to original

• Notations for true: $T,1,+$.
• Notations for false: $F,\perp ,0,-$.

For example:

• Today is Monday.
• $2+2=4$
• $2+2=5$

Not declarative:

• Hi, how are you?
• $x+y=4$

Propositional Logic

Propositional logic is the area of logic which deals with propositions, it is sometimes referred to as boolean logic.

Link to original

• We use propositional variables, denoted as lower case letters ($p,q,r$ or $p_1,p_2,..$) to denote propositions whose structure we don’t want to analyse further.
• We build compound expressions called formulas, denoting compound propositions. They are built from logical connectives and parentheses.

We can define a formula as a compound expression made up of propositions, logical connectives and parentheses.