/* MINDFIRE BLOG */

“A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so.”
— Aristotle, Prior Analytics

Believe it or not, Aristotle was the first Artificial Intelligence researcher! Artificial Intelligence research is thus at least 2,300 years old.

He invented the first form of formal logic to figure out what it meant to be rational. Fast-forward more than two millennia, and George Boole carried that torch into the 19th century. Boole’s brilliant insight transformed Aristotle’s syllogisms into an algebraic playground—what we now call Boolean logic—and he even dubbed it “The Laws of Thought.” From there, the journey continued into what we call propositional logic, a nimble framework that treats whole statements as building blocks and applies clear, truth-functional rules to combine them.

How Aristotle’s Logic Became Propositional Logic

Aristotle imagined that by stating “All humans are mortal” and “Socrates is human,” the conclusion “Socrates is mortal” must leap out without any extra magic. He invented rules so ironclad that doubting his conclusion meant doubting one of the premises. In his day, that felt revolutionary—reason was no longer just instinctive; it could be pinned down.

Then came George Boole in the 1800s with a spark of genius: “What if we treat truth like numbers?” In Boole’s world, a proposition became a symbol—1 for true, 0 for false—and operations like AND, OR, and NOT behaved like algebraic operators. Suddenly, logical relationships could be written as equations and manipulated just like polynomials. Overnight, logic stepped out of philosophy textbooks and got ready for the age of machines.

By the late 19th century, thinkers realized they could bundle any full sentence—“It is raining,” “The ground is wet,” or “If it is raining, then the ground is wet”—into a single unit called an atomic proposition. Instead of playing with categories, you simply ask, “Is this statement true or false?” Propositional logic weaves those atomic pieces together with truth-functional rules, marrying Aristotle’s insistence on necessary conclusions with Boole’s algebraic flair. The result is a lean, mean framework that can express everything from simple “if-then” statements to nested chains of “and,” “or,” and “not.”

How Propositional Logic Works

• Atomic Propositions
  Each basic statement is represented by P, Q, or R, and each symbol can be True (T) or False (F). For instance, let P be “It is raining” and Q be “I carry an umbrella.” When you see P, you don’t break it down—you just know it’s T or F.

• Logical Connectives

  • NOT (¬): If P is true, then ¬P (“not P”) is false; if P is false, then ¬P is true.
  • AND (∧): P ∧ Q is true only if both P and Q are true.
  • OR (∨): P ∨ Q is true if at least one of P or Q is true.
  • IMPLIES (→): P → Q is false exactly when P is true and Q is false; otherwise it’s true.
  • EQUIVALENT (↔): P ↔ Q is true only when P and Q share the same truth value (both true or both false).

• Truth Tables
  Here’s a version of the truth table for P ∧ Q:

Here is what all the Truth Tables look like so far:

Recall that that P ∧ Q should only be True if both P and Q are both True. And that is what we find via the truth table.

By consulting these tables (in HTML form), you know exactly whether any compound statement is true or false under a given assignment of truth values.

• Compound Formulas and Evaluation
  You can build complex formulas like (P ∨ ¬Q) → (R ∧ S). To test it, assign truth values to P, Q, R, and S, then use the table to compute ¬Q, P ∨ ¬Q, R ∧ S, and finally check whether (P ∨ ¬Q) implies (R ∧ S).

• Inference Rules
  From a set of premises—each written as a propositional formula—you use rules that guarantee any conclusion is true if the premises are true. For example, Modus Ponens says: if P is true and P → Q is true, then Q must be true. Modus Tollens says: if P → Q is true and ¬Q is true, then ¬P must be true. These rules let you build chains of rock-solid reasoning, ensuring each step follows without fail.

Through these simple yet powerful mechanisms, propositional logic becomes a toolkit for checking whether conclusions actually follow from premises or whether a set of statements can ever be true together.

Short Examples for Each Rule

• Atomic Proposition
  Let P be “My phone is charging.” At any moment, either P is True (T) or False (F).

• NOT (¬)
  If P = “My phone is charging” is T, then ¬P = “My phone is not charging” is F.

• AND (∧)
  Let P = “I have bread” and Q = “I have butter.” Then P ∧ Q = “I have bread and I have butter,” which is T only if both are T.

• OR (∨)
  Let P = “It is sunny” and Q = “It is warm.” Then P ∨ Q = “It is sunny or it is warm,” which is T if at least one is T.

• IMPLIES (→)
  Let P = “I press the button” and Q = “The light turns on.” Then P → Q is F only if I press the button (P = T) and the light does not turn on (Q = F); otherwise it’s T.

• EQUIVALENT (↔)
  Let P = “Switch is up” and Q = “Light is on.” Then P ↔ Q is T when both are T (switch up, light on) or both are F (switch down, light off).

• Modus Ponens
  If P = “I am hungry” is T and P → Q = “If I am hungry, I eat” is T, then Q = “I eat” must be T.

• Modus Tollens
  If P → Q = “If it rains, the ground gets wet” is T and ¬Q = “The ground is not wet” is T, then ¬P = “It is not raining” must be T.

Key Logical Laws: Patterns that Always Hold

Beyond inference rules, propositional logic includes deep structural laws that always hold true—no matter what the actual truth values are. These help us transform and simplify logical formulas in predictable ways:

• Idempotence
  Repeating a statement doesn't change its meaning.

  • P ∧ P ≡ P
  • P ∨ P ≡ P

• Commutativity
  The order of terms doesn’t matter for AND or OR.

  • P ∧ Q ≡ Q ∧ P
  • P ∨ Q ≡ Q ∨ P

• Associativity
  Grouping doesn’t affect the result with AND or OR.

  • (P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R)
  • (P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R)

• Distributivity
  AND distributes over OR and vice versa.

  • P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
  • P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)

• De Morgan’s Laws (Duality)
  Negation flips the logic and the connectives.

  • ¬(P ∧ Q) ≡ ¬P ∨ ¬Q
  • ¬(P ∨ Q) ≡ ¬P ∧ ¬Q

These identities are the algebra of logic. They allow formulas to be rewritten for simplification, for proof, or for input into solvers—like algebra lets us balance and rearrange equations. In essence, they are the “grammar rules” that preserve truth while changing form.

Conclusion: Where We Go from Here

Don’t let the simplicity of propositional logic fool you—it’s the foundation of every modern reasoning and computational system. By boiling reasoning down to atomic propositions and truth-driven rules, it shows us exactly how to make conclusions inevitable once assumptions are in place. The adventure that began with Aristotle’s first syllogisms and continued through Boole’s algebraic innovations never ends; it just keeps branching into richer and more exciting territories.

Propositional logic is just the foundation. Once you cross into first-order logic, you gain the power to talk about objects, properties, and quantifiers—so you can precisely state “Every human is mortal” or “There exists a bird that cannot fly.” Step further into probability theory and Bayesian inference, and you layer uncertainty onto propositions, turning “It is raining” into a probability that you update as new evidence rolls in. Meanwhile, automated theorem provers and SAT solvers lean on propositional logic’s clarity to tackle massive formulas, powering everything from hardware checks to cryptographic puzzles. In the same vein, I wrote a DPLL algorithm and logic parser to let someone quickly check if a set of logical statements entails a conclusion. These are the sorts of things you can do once you've formalized logic in this say.

At its heart, propositional logic is the bridge between Aristotle’s bold quest to formalize rational thought and today’s sprawling universe of AI techniques that learn, decide, and act on rules and evidence. Grasp it, and you’re ready to dive into the systems that shape our digital world.