Propositions in Logic

Propositions in Logic

• Propositions are declarative sentences that assert facts and are either true or false, not both.
• Examples of propositions:
  • "Washington, D.C., is the capital of the United States." (True)
  • "Toronto is the capital of Canada." (False)
  • "1 + 1 = 2." (True)
  • "2 + 2 = 3." (False)
• Non-propositional sentences include:
  • "What time is it?" (Not declarative)
  • "Read this carefully." (Not declarative)
  • "x + 1 = 2" and "x + y = z" (Not true or false without assigned values)
• Propositional variables (p, q, r, s, etc.) are used to represent propositions in logical expressions.

Truth Values

• A true proposition is denoted by T.
• A false proposition is denoted by F.

Aristotle and the Foundations of Logic

• Aristotle (384-322 B.C.E.) was a significant figure in developing propositional logic.
• He was born in Stagira, Greece, and his father was the physician to the King of Macedonia.
• Orphaned at a young age, Aristotle was educated in various subjects, including poetry and Greek rhetoric.
• At 17, he attended Plato's Academy in Athens, where he studied and eventually lectured for around 20 years.
• After Plato's death in 347 B.C.E., Aristotle did not succeed him as the head of the Academy.

Truth Tables Overview

• Truth tables are visual representations to determine the truth values of compound propositions formed from basic propositions.
• Each row in a truth table corresponds to a distinct combination of truth values for its propositions.

Key Logical Connectives

• Conjunction (p ∧ q): True only if both p and q are true.
• Disjunction (p ∨ q): True if at least one of p or q is true.
• Exclusive Or (p ⊕ q): True if p and q have different truth values; false if they are the same.
• Conditional Statement (p → q): True unless p is true and q is false.

Structure of Truth Tables

• Truth tables contain columns for each proposition and the resulting compound statement.
• The total number of rows in a truth table depends on the number of propositions involved, specifically 2^n, where n is the number of propositions.
• For two propositions, there are four possible combinations of truth values (TT, TF, FT, FF).

Compound Propositions

• Compound propositions are created using logical connectives to combine simpler propositions.
• Important for analyzing conditions under which statements are true or false.

Importance of Truth Tables

• Essential in logic, mathematics, and computer science for evaluating logical expressions and determining their validity.
• Helps in understanding and constructing logical arguments and reasoning processes.