# Propositions in Natural Language

## 21.01.2021

### Contents/Index

1. Intro
@2. Propositions in Natural Language
3. Syntax
4. Semantics
5. Natural Deduction
6. Equivalences

Declarative sentences (or propositions) are sentences of the kind that are either $true$ or $false$. Some examples are

1. I have a cat.
2. Adding 3 to 8 results in 11.
3. I have a cat, and I have a dog.
4. If I have a cat, then I do not have a dog.

These can be evaluated. "I have a cat" is absurd if I do not have one. The second example is a statement within arithmetic. Here it is true. The third and fourth examples are compound. The third depends on two sub statements, namely "I have a cat" and "I have a dog". We call these atomic expressions. Here we have a logical operator in form of "and", we call it conjunction and the set of operators for connectives. There are 4 in total. In the fourth example we again have a connective, her an implication which in language takes the form "if ..., then". Again we have two sub statements, "I have a cat" and "I do not have a dog". The latter takes on the form of a negation, it is the negated version of "I have a dog". Let's present the list of the four connectives:

• Conjunction: Primarily has the form of "... and ...". Though it can have the form of for example "but" as well.
• Disjunction: Has the form of "... or ...".
• Implication: Has the form of "If ..., then ...".
• Negation: Has the form of "not ...".

As you can see I have placed ... as placeholders for each of the connectives. The goal of formal logic is to abstract away all the dogs, cats and so on. Instead of these details we are interested in reasoning about the connectives.