# Boolean Logic

## Examples

Boolean logic (a.k.a. Propositional logic) is restrictive in what it can express.

It is useful because

if(a < b || (a >= b && c == d)) ...

can be simplified to

if(a < b || c == d)

## Syntax of Boolean Logic

 $$\top$$ and $$\bot$$ Denote true and false $$P$$ and $$Q$$ Denote propositions $$P\wedge Q$$ Denoting conjunction $$P\vee Q$$ Denoting disjunction $$\neg P$$ Denoting negation

Other people define different syntax

## Alternative Syntax

 $$1$$ and $$0$$ Denote true and false $$P$$ and $$Q$$ Denote propositions $$P.Q$$ Denoting conjunction $$P + Q$$ Denoting disjunction $$\overline{P}$$ Denoting negation

let's call these terms.

## Forming Sentences

All terms are sentences.

If $$\alpha$$ and $$\beta$$ are sentences then $$(\alpha \wedge \beta)$$ is a sentence.

Similarly $$(\alpha \vee \beta)$$ is a sentence.

## Semantics of Boolean Logic

A sentence in Boolean logic evaluates to either true or false.

# Misunderstandings

Mostly stem from applying linguistic interpretation of a sentence rather than an logical interpretation.

## 'and'

Aidan bought a motorbike and went to work.

The English sentence has issues

• temporal implicature
• causality

Which is why we formalise in the first place.

## 'or'

• Exclusive or versus inclusive or.

Logical or is inclusive. Exclusive or more precisely translates as

## implication

Some people add implication to their set of Boolean connectives.

$$P\rightarrow Q$$ read as "if P then Q"

This shorthand for $$(\neg P \vee Q)$$, it causes significant confusion.

# Conclusion

• Boolean Logic is a useful system for demonstrating computational thinking.
• A visual approach to Boolean Logic is useful.
• The visual approach extends to First-order logic (and Second-order).
• Learners make consistent types of mistakes -- knowing these allow us to target interventions.

## Thank you

• I'm always happy to answer questions over email.