Unit 1: Symbols, Symbolization, and Truth-Functions

Special Symbols: Variables and Constants
Symbolization
Five Basic Truth-Functions

Special Symbols: Variables and Constants

In modern logic, we use special symbols to remove the ambiguity of ordinary language and reveal the underlying logical structure of an argument. The basic units are statements (or propositions), which are sentences that can be either True or False.

1. Statement Variables

Symbol: Lowercase letters, typically p, q, r, s...
Function: They are placeholders for any simple statement. They are "variable" because 'p' could stand for "It is raining" or "It is Tuesday."
Example: The logical form "If p then q" can represent "If it rains (p) then the ground is wet (q)" or "If you study (p) then you will pass (q)."

2. Statement Constants

Symbol: Uppercase letters, typically A, B, C...
Function: They stand for a specific simple statement. They are "constant" because 'A' will always mean "Assam is a state" throughout the argument.
Example: To symbolize "Assam is a state and Bihar is a state," we would write "A • B". Here, A and B are constants.

Exam Tip: For most logic problems, you will use statement variables (p, q, r) to analyze the *form* of an argument. You use constants when translating a specific argument from English into symbolic form.

Symbolization

Symbolization is the process of translating arguments from ordinary language into the symbols of modern logic. This helps us to clearly see the argument's form and test its validity.

Steps to Symbolize:

Identify the simple statements and assign them uppercase letters (constants).
Identify the logical connectives (like "and," "or," "if...then," "not").
Combine the constants using the symbols for the connectives.

Example:

Argument: "If it is raining (R), then the ground is wet (W). It is raining (R). Therefore, the ground is wet (W)."

Simple Statements:
- R = "It is raining."
- W = "The ground is wet."
Connectives: "If...then...", "Therefore"
Symbolized Form:
R ⊃ W
R
/ ∴ W

Five Basic Truth-Functions

A truth-function (or connective) is a logical operator that builds a complex statement from simple ones. The truth value (True/False) of the complex statement is completely determined by (is a "function" of) the truth values of the simple statements.

Function	Symbol	Name	Meaning	Truth-Table
Negation	~ (tilde)	"Not p"	Reverses the truth value of the statement it precedes.	p \| ~p T \| F F \| T
Conjunction	• (dot)	"p and q"	True only if both conjuncts (p, q) are true.	p \| q \| p • q T \| T \| T T \| F \| F F \| T \| F F \| F \| F
Disjunction	v (wedge)	"p or q"	This is the inclusive "or". It is true if at least one disjunct is true. It is false only if both are false.	p \| q \| p v q T \| T \| T T \| F \| T F \| T \| T F \| F \| F
Implication	⊃ (horseshoe)	"If p then q"	Called "Material Implication." It is false only if the antecedent (p) is true and the consequent (q) is false.	p \| q \| p ⊃ q T \| T \| T T \| F \| F F \| T \| T F \| F \| T
Equivalence	≡ (triple bar)	"p if and only if q"	Called "Material Equivalence." It is true only if both p and q have the same truth value.	p \| q \| p ≡ q T \| T \| T T \| F \| F F \| T \| F F \| F \| T

Key Points to Memorize:

A Conjunction (•) is true only when both are true.
A Disjunction (v) is false only when both are false.
An Implication (⊃) is false only when T ⊃ F. (This is the most important rule to remember!)
An Equivalence (≡) is true only when both sides are the same (T ≡ T or F ≡ F).