Symbols and Connectives

The simple language of logic

Let’s start by creating a simple language of modern logic. We know that a useful language has to, minimally, have a syntax and a way to resolve the meaning of syntactically correct sentences. This lesson will detail the syntax and show how to resolve the truth values (meaning) of correct propositional logic sentences through truth tablesA truth table is a mathematical table used to determine if a compound statement is true or false..

Fundamental unit

In this logic, the most basic element is a complete sentence. However, these sentences are limited to propositions.

Propositions

Each proposition is represented as a symbol, typically using alphabets such as $a,b,c,...,p,q,r,..., x,y,z$. If there’s a need to use more than $26$ propositions, they can have names like $a123$— letters appended by digits.

Connectives

To add complexity to sentence structures, there’s an allowance for using logical operators. Modern propositional logic includes the following connectives:

• $¬$ (negation)

• $∨$ (disjunction, which represents the logical OR)

• $∧$ (conjunction, which represents the logical AND)

• $→$(implication, which represents the logical IF-THEN)

• $↔$(biconditional, which represents the logical IF AND ONLY IF)

Syntactic complexity

Introducing connectives brings syntactic complexity to this modern logic, allowing us to create more logical statements.

Semantic meaning

To determine the truth or falsehood of statements, we can automate the process using truth tables, which specify the truth value of a compound statement based on the truth values of its constituent propositions and connectives.

Truth tables for each connective

In this lesson, we’ll reduce everything to symbols and their truth resolution.

The logical AND operation

The $∧$ symbol is also called conjunction, representing the logical AND operation between two symbols (propositions) at a time. If we use the AND operation between two sentences, the resulting compound sentence will only be true when both symbols are true.