Understanding Logic Simply: Making "Propositional Logic" Your Weapon at the Very Least

Understanding Logic Simply: Making "Propositional Logic" Your Weapon at the Very Least Introduction: Why Logic Now? "Be logical." It's a phrase we hear often, and frankly, it sounds a bit stiff. However, the word "Logic" comes from the Greek word Logos, which has a broad meaning encompassing words, reasoning, and laws. In modern society, where social media is flooded with fake news, dubious claims parading as "facts," and people trying to assert dominance with their own versions of "justice," logic becomes your strongest defense. The famous Japanese columnist Natsuhiko Yamamoto once declared, "Everyone embellishes when they speak." Indeed, everyone decorates their reasoning to justify themselves. To see through these "decorations," avoid fruitless arguments, and protect yourself from being deceived, let's quickly equip ourselves with university-level "Propositional Logic" as an "Intellectual Aegis" (shield). Just knowing this will change the way you see the world a little. Chapter 1: What are the "Basic Rules" of Logic? In a nutshell, logic is "a game of deriving true conclusions from true premises." Crucially, logic is not responsible for whether the premises are actually true (factually correct). Logic guarantees the correctness of the process (validity): "If we assume the premises are true, and we follow the rules, the conclusion must absolutely be true." 1. Propositions and Symbols First, let's simplify language by replacing sentences with symbols. Propositions (sentences) are represented by alphabet letters ($P, Q, A$, etc.). Example: $P$ = "I am a human." 2. Negation (NOT) Symbol: $\neg P$ (or $-P$) Meaning: "Not $P$" 3. Conjunction (AND) Symbol: $P \land Q$ (or $P \cap Q$) Meaning: "$P$ and $Q$" Feature: The whole is true only when both are true. 4. Disjunction (OR) Symbol: $P \lor Q$ (or $P \cup Q$) Meaning: "$P$ or $Q$" Feature: The whole is true if at least one is true. 5. Conditional (IF...THEN) Symbol: $P \to Q$ Meaning: "If $P$, then $Q$" This is the main character of logic. Chapter 2: Just Know This! Powerful Argument Patterns 1. Modus Ponens (Affirming the Antecedent) "The Absolute Basics" Premise 1: If $P$, then $Q$. ($P \to Q$) Premise 2: $P$ is true. Conclusion: Therefore, $Q$ is true. [The Memorable "Tortoise" Episode] There is a legend that the ancient Greek tragedian Aeschylus died when an eagle dropped a tortoise on his head. Let's turn this into a logical formula. Premise 1: If a tortoise falls on my head, I am unlucky. Premise 2: A tortoise fell on my head! Conclusion: I am unlucky. If the premises are true (however tragic), the conclusion is necessarily true. 2. Modus Tollens (Denying the Consequent) "The Logic of Contraposition" Premise 1: If $P$, then $Q$. ($P \to Q$) Premise 2: Not $Q$. ($\neg Q$) Conclusion: Therefore, not $P$. ($\neg P$) Premise 1: If a tortoise falls on my head, I will be in pain. Premise 2: I am not in pain. Conclusion: A tortoise has not fallen on me. (Thank goodness!) Chapter 3: Advanced Rules for Using Logical "Magic" This part gets a bit puzzle-like, but understanding this will dramatically increase your logical power. 1. Doraemon's "What-If Phone Booth": Assumption and Conditional Proof (CP) In the world of logic, you are allowed to "assume for the moment" things you don't yet know to be true (Rule of Assumption). If you reach a conclusion within that hypothetical world, you can bring it back to reality (Conditional Proof). [Example: If the wind blows, the bucket maker profits] (A famous Japanese proverb implying a butterfly effect). To derive "$A \to C$" from "$A \to B$" and "$B \to C$": Assumption (A): Let's assume "The wind blows." Inference: Since the wind blows, dust rises. (From $A \to B$) Inference: Since dust rises, the bucket maker profits. (From $B \to C$) Summary (CP): When we assumed "The wind blows," the result was "The bucket maker profits." Conclusion: Therefore, we can say, "If the wind blows, the bucket maker profits." Thus, Assumption (spreading the cloth) and CP (folding the cloth) are used as a set. 2. The Same Result Either Way: Disjunction Elimination ($\lor E$) This is powerful in situations of "It's either $A$ or $B$." [Example: The Dessert Argument] Premise: I will eat either "Cake" or "Ice Cream." Case 1 (If Cake): Cake is sweet. Therefore, I eat something sweet. Case 2 (If Ice Cream): Ice Cream is sweet. Therefore, I eat something sweet. Conclusion: Either way, I eat something sweet. This is "Disjunction Elimination." The name is formidable, but it's simply the work of "case analysis" to confirm that you reach the same goal regardless of the path. 3. Breaking the Alibi: Reductio ad Absurdum (RAA) The strongest method of proof. It is a technique to "dare to go along with the opponent's claim, point out that it leads to a contradiction, and refute it." [Example: Breaking an Alibi in a Mystery] We want to prove "He is not the culprit." Dare to Assume: "For the sake of argument, let's assume he IS the culprit." Proceed with reasoning: If he is the culprit, he must have been at the crime scene at the time of the crime. Find the contradiction: However, at that time, he was in Okinawa (there is security camera footage). Table Flip: It is impossible to have been "at the scene" AND "not at the scene"! Conclusion: The contradiction arose because the initial assumption was wrong. Therefore, he is NOT the culprit. Conclusion: Update Your Thinking OS The rules of "Propositional Logic" introduced here are actually the very mechanism by which computers work (logic circuits) and the foundation of programming languages. In a sense, it is the "machine code" that runs modern society. Just as the generation that learned BASIC or machine code understood the "backside" of computers, knowing the rules of logic leads to understanding the "backside (OS)" of human thought and debate. You don't need to write these out strictly in everyday life. However, just realizing "Ah, I can use Reductio ad Absurdum here" or "This is correct because it's a contrapositive" turns your intelligence into a powerful defense wall (Aegis) against fraud and sophistry. We spend one-third of our lives sleeping, but to improve the quality of thinking during our waking hours, why not love logic—as a "gentleman's/gentlewoman's cultivation"—just a little bit more?