# Propositional Logic – A Primer

## 1. Introduction

Citation Information

Allan, Leslie 2023. Propositional Logic – A Primer, URL = <https://www.rationalrealm.com/philosophy/logic/propositional-logic-primer.html>.

Contents

Propositional logic is a type of formal logic that deals with the logical relationships between propositions. Propositions are statements that are either true or false. The purpose of propositional logic is to provide a formal system for representing and analyzing these logical relationships. Propositional logic uses logical operators, such as 'and', 'or' and 'not' to combine propositions and create more complex statements. Propositional logic is sometimes called 'sentential logic' or 'propositional calculus'.

The main function of propositional logic is to provide a way to reason logically and systematically about the truth or falsity of propositions, and to create proofs or arguments that are logically valid. It is a powerful tool for identifying the logical structure of arguments and evaluating the validity of conclusions based on premises.

As such, propositional logic is an important tool in philosophy and mathematics for analyzing arguments and proofs. It has many other practical applications, such as in computer science for designing logical circuits and in natural language processing for analyzing the logical structure of sentences. Overall, propositional logic is a fundamental tool for reasoning and problem-solving in a wide range of fields.

What is an 'argument' in propositional logic? An argument consists of one or more propositions, each listed as a premise in the argument. It also consists of another proposition labeled the 'conclusion'. An argument is considered valid if it is impossible for all of the premises to be true and the conclusion false. Here is a simple argument:

Premise 1:    If it is raining, it is wet outside.

Premise 2:    It is raining.

Conclusion:  It is wet outside.

In this primer, I will introduce you to all three methods.

First, what are the components of propositional logic? Propositional logic consists of a system of syntactical symbols and their arrangement, semantics that give meaning to the symbols and rules of inference that determine the validity of the conclusions of arguments.

The basic syntactical elements of propositional logic are:

Propositions:
Propositions are statements represented by letters, such as 'P' and 'Q'.
Logical Operators:
Logical operators are used to connect propositions and create more complex statements. The most common operators in propositional logic are:
Negation:
The negation operator (¬) is used to express the opposite of a proposition. For example, if 'P' is the proposition 'It is raining', then '¬P' is the proposition 'It is not raining'.
Conjunction:
The conjunction operator (∧) is used to express that two propositions are true at the same time. For example, if 'P' is the proposition 'It is raining' and 'Q' is the proposition 'It is cold', then 'P ∧ Q' is the proposition 'It is raining and cold'.
Disjunction:
The disjunction operator (∨) is used to express that at least one of two propositions is true. For example, if 'P' is the proposition 'It is raining' and 'Q' is the proposition 'It is sunny', then 'P ∨ Q' is the proposition 'It is raining or sunny'.
Implication:
The implication operator (→) is used to express that if one proposition is true, then another proposition is also true. For example, if 'P' is the proposition 'It is raining' and 'Q' is the proposition 'The streets are wet', then 'P → Q' is the proposition 'If it is raining, then the streets are wet'.
Biconditional:
The biconditional operator (↔) is used to express that two propositions are true if and only if each other is true. For example, if 'P' is the proposition 'You are a cat' and 'Q' is the proposition 'You have fur', then 'P ↔ Q' is the proposition 'You are a cat if and only if you have fur'.

The semantics of propositional logic is based on truth values of propositions. In this system, we stipulate that propositions can only be either true or false and we use logical operators to combine them to create more complex propositions.

Truth tables are often used to determine the truth value of a complex compound proposition. A truth table lists all possible combinations of truth values for the propositions that make up a compound proposition. The final column of the truth table displays the resulting truth value of the compound proposition as a function of the truth values of the component propositions.

Lastly, propositional logic includes a set of rules of inference that are used to determine whether the conclusion of an argument is valid based on the propositions listed in its premises. Some of the most common rules of inference in propositional logic include:

Modus Ponens:
If 'If P then Q' is true and 'P' is true, then 'Q' is true.
Modus Tollens:
If 'If P then Q' is true and 'Q' is false, then 'P' is false.
Disjunctive Syllogism:
If 'P' or 'Q' is true and the negation of 'P' is true, then 'Q' is true.
Conjunction:
If 'P' is true and 'Q' is true, then 'P and Q' is true.

Now that we've covered the basics, in the next sections, we'll delve more into the inner workings of propositional logic.