Syntax of Propositional Logic

In the previous post of this thread, we introduced propositional logic and its purpose: to provide a formal system for analyzing and evaluating statements using logical structures. Now, we turn to the syntax of propositional logic, which defines the fundamental building blocks of this system.

Propositions and Atomic Statements

At the heart of propositional logic are propositions, which are statements that are either true or false. These propositions serve as the basic units of reasoning, forming the foundation upon which logical structures are built. The need for propositions arises because natural language can be ambiguous, making it difficult to determine the validity of arguments. By representing statements as precise logical symbols, we eliminate ambiguity and ensure rigorous reasoning.

Atomic statements are the simplest propositions that cannot be broken down further. These statements capture fundamental mathematical facts or real-world assertions. In mathematics, statements such as “5 is a prime number” or “A function is continuous at x = 2” are examples of atomic statements. In everyday language, sentences like “The sky is blue” or “It is raining” serve as atomic statements.

By introducing atomic statements, we create a standardized way to express truth values and establish logical relationships between different facts, allowing us to construct more complex reasoning systems.

Logical Connectives

While atomic statements provide the basic building blocks, more complex reasoning requires combining them. This is where logical connectives come into play. Logical connectives allow us to form compound statements from atomic ones, preserving precise meaning and facilitating logical deductions.

The primary logical connectives are:

1. Negation (NOT, \(\neg\)): Negation reverses the truth value of a proposition. If a statement is true, its negation is false, and vice versa.
   • Example: If \(P\) represents “It is raining,” then \(\neg P\) means “It is not raining.”
2. Conjunction (AND, \(\land\)): The conjunction of two propositions is true only if both propositions are true.
   • Example: \(P \land Q\) means “It is raining AND it is cold.”
3. Disjunction (OR, \(\lor\)): The disjunction of two propositions is true if at least one of them is true.
   • Example: \(P \lor Q\) means “It is raining OR it is cold.”
4. Implication (IMPLIES, \(\rightarrow\)): Implication expresses a logical consequence. If the first proposition (antecedent) is true, then the second (consequent) must also be true. This is often misunderstood because an implication is still considered true when the antecedent is false, regardless of the consequent.
   • Example: \(P \rightarrow Q\) means “If it is raining, then the ground is wet.” Even if it is not raining, the implication remains valid as long as there is no contradiction.
   • A common confusion arises because people often think of implication as causation, but in formal logic, it represents a conditional relationship rather than a cause-effect mechanism.
5. Biconditional (IF AND ONLY IF, \(\leftrightarrow\)): A biconditional statement is true when both propositions have the same truth value.
   • Example: \(P \leftrightarrow Q\) means “It is raining if and only if the ground is wet.” This means that if it is raining, the ground must be wet, and conversely, if the ground is wet, it must be raining.

Well-Formed Formulas (WFFs)

A well-formed formula (WFF) is a syntactically correct expression in propositional logic. The rules for forming WFFs include:

• Every atomic proposition (e.g., \(P, Q\)) is a WFF.
• If \(\varphi\) is a WFF, then \(\neg \varphi\) is also a WFF.
• If \(\varphi\) and \(\psi\) are WFFs, then \(\varphi \land \psi\), \(\varphi \lor \psi\), \(\varphi \rightarrow \psi\), and \(\varphi \leftrightarrow \psi\) are WFFs.
• Parentheses are used to clarify structure and avoid ambiguity (e.g., \((P \lor Q) \land R\)).

Conventions and Precedence Rules

To simplify expressions, we often omit unnecessary parentheses based on operator precedence. The order of precedence for logical operators is as follows:

1. Negation (\(\neg\)) has the highest precedence.
2. Conjunction (\(\land\)) comes next, meaning \(P \land Q\) is evaluated before disjunction.
3. Disjunction (\(\lor\)) follows, evaluated after conjunction.
4. Implication (\(\rightarrow\)) has a lower precedence, meaning it is evaluated later.
5. Biconditional (\(\leftrightarrow\)) has the lowest precedence.

For example, \(\neg P \lor Q \land R\) is interpreted as \((\neg P) \lor (Q \land R)\) unless explicitly parenthesized otherwise. Similarly, \(P \lor Q \land R \rightarrow S\) is evaluated as \(P \lor (Q \land R) \rightarrow S\) unless parentheses dictate otherwise.

Understanding these precedence rules helps avoid ambiguity when writing logical expressions.

Next Steps

Now that we understand the syntax of propositional logic, the next step is to explore truth tables and logical equivalence, which provide a systematic way to evaluate and compare logical expressions.