Symbolic and Deductive Logic – CUET PG Philosophy – Notes

Symbolic Logic

The Need for a Formal Language

Natural languages like English or Hindi are rich and nuanced, but they are also filled with ambiguity, vagueness, and emotional overtones. These features,while useful in poetry or everyday conversation, are problematic for logic, which demands precision and clarity. Symbolic logic was developed to overcome these limitations. It is a formal, artificial language that uses specific symbols with uniquely defined meanings to represent logical propositions and their relationships. By translating arguments from natural language into this symbolic system, we can analyze their structure and determine their validity with mathematical rigor, free from the distracting complexities of ordinary speech.

The primary advantage of symbolic logic is its ability to reveal the underlying logical form of an argument. For instance, the two arguments below seem different in natural language:

• If it is raining, then the ground is wet. It is raining. Therefore, the ground is wet.
• If the company’s profits increase, its stock price will rise. The company’s profits increased. Therefore, its stock price will rise.

Propositional Logic (Sentential Calculus)

The foundational level of symbolic logic is called Propositional Logic or Sentential Calculus. It deals with propositions (or statements) as whole, unanalyzed units and the ways they can be connected to form more complex propositions.

Propositions: The Building Blocks

A proposition is a declarative sentence that can be definitively classified as either true or false. It cannot be both, and it cannot be neither. This property of being true or false is known as its truth value.

• “New Delhi is the capital of India.” (This is a true proposition.)
• “The moon is made of cheese.” (This is a false proposition.)
• “What is your name?” (This is a question, not a proposition.)
• “Close the door!” (This is a command, not a proposition.)

Propositions are categorized into two types:

1. Simple Proposition: A proposition that does not contain any other proposition as a component part. For example, “The sky is blue.” We use capital letters (e.g., A, B, C, P, Q, R) to represent simple propositions. These are called propositional variables.
2. Compound Proposition: A proposition formed by connecting two or more simple propositions using logical operators. For example, “The sky is blue and the grass is green.”

Logical Connectives: The Glue of Logic

Logical connectives (or operators) are symbols used to combine simple propositions into compound ones. The truth value of a compound proposition is determined entirely by the truth values of its simple components and the rules of the connectives used. These are called truth-functional connectives.

• Negation (~,): This is a unary operator, meaning it applies to a single proposition. It reverses the truth value of the proposition it modifies. In English, it corresponds to “not” or “it is not the case that”. If P is true, then~Pis false. If P is false, then~Pis true.
• Conjunction (., Λ): This is a binary operator that connects two propositions, called conjuncts. It corresponds to “and”, “but”, “also”, “while”. A conjunction P.Q is true if and only if both P and Q are true. If either conjunct is false, the entire conjunction is false.
• Disjunction (V): This operator connects two propositions, called disjuncts, and corresponds to “or”. Logic uses the inclusive sense of “or”, meaning “one, or the other, or both”. A disjunction P ∨ Q is false if and only if both P and Q are false. In all other cases, it is true.
• Conditional (⊃, →): This is also known as material implication. It corresponds to “if…then…” statements. In a conditional statement PQ, P is called the antecedent and Q is the consequent. A conditional statement is false if and only if the antecedent is true and the consequent is false. This is the only case where the implication fails. This rule can seem counter-intuitive (e.g., a false antecedent implying a true consequent results in a true statement), but it is essential for logical consistency.
• Biconditional (=, →): This is also known as material equivalence_ It corresponds to “if and only if” (often abbreviated as “iff”). A biconditional statement P = Q is true if and only if P and Q have the same truth value (both are true or both are false).

Logical Flow: Natural Language Statement → Identify Simple Propositions → Identify Connectives → Translate into Symbolic Form.