Understanding Logic in a Friendly Way — At Least Propositional Logic…

Understanding Logic in a Friendly Way — At Least Propositional Logic… 0. Hook: Ever Been Told to “Think Logically”? Online, at work, even in casual conversation, people casually say things like: “That’s not logical.” “I’m just making a logical argument.” “Let’s stick to facts and logic.” But if you ask: “Okay, but what exactly do you mean by ‘logic’?” it suddenly becomes much harder to explain. Fake news vs. facts Discussions of “justice” and its deconstruction or relativization People trying to win status by claiming “I’m being logical and you’re not” In these kinds of situations, even just a basic, college-level grasp of propositional logic is enough to: keep your bearings, spot weird reasoning, and sometimes protect yourself from being dragged into nonsense. This article aims to be gentle and approachable, and to: show what propositional logic is trying to do, and explain some of the classic rules (MP, MT, A, CP, ∧ / ∨ rules, and reductio) with everyday-style examples. 1. So What Does Logic Actually Study? Very roughly, logic is about the question: Given true premises, how can we get to true conclusions by valid rules? Premises: one or more statements (propositions) Conclusion: another statement Rules: inference rules connecting the two The key points: Whether the premises are really true is not logic’s job. In science, experiments decide that. In history, source criticism decides that. In politics, it’s about facts, values, and power. Logic’s job is to study the form of reasoning: “If the premises are true, then by this rule the conclusion must be true.” So we say: A piece of reasoning is sound when all its premises are true, its inferences follow the rules, and its conclusion is true. Logic is mainly concerned with the middle part: Are the inferences valid given the form of the argument? 2. The Main Characters in Propositional Logic In propositional logic, we look at sentences that are either true or false. For example: “I am a human being.” “It is raining today.” “A turtle falls on my head.” All of these can be treated as propositions. We usually abbreviate them with capital letters: A, P, Q, R, … Then we introduce some basic logical symbols to connect propositions: Negation: ¬P … “not P” Conjunction: P ∧ Q … “P and Q” Disjunction: P ∨ Q … “P or Q” Conditional: P → Q … “if P, then Q” Once you’re comfortable with these four, you’ve already understood about 80% of elementary propositional logic. 3. Classic Rule 1: Modus Ponens (MP) 3.1 The Shape Modus Ponens (MP) 𝑃 ,    𝑃 → 𝑄    ⊢    𝑄 P,P→Q⊢Q In words: Premise 1: P Premise 2: If P then Q (P→Q) Conclusion: Q 3.2 A Deliberately Weird Example Let’s choose silly propositions on purpose: P: “A turtle falls on my head.” Q: “I am a human being.” Then MP says: A turtle falls on my head. If a turtle falls on my head, then I am a human being. Therefore I am a human being. The situation is bizarre, but the reasoning is perfectly valid. If someone says: “That’s not logical!” what they really mean is “that’s not realistic” or “that’s ridiculous as a story”, not that the logical form is wrong. MP is the basic rule: If “if P then Q” is true, and P is true, then Q must be true. It’s one of the fundamental “moves” in deductive reasoning. 4. Classic Rule 2: Modus Tollens (MT) Modus Tollens (MT) ¬ 𝑄 ,    𝑃 → 𝑄    ⊢    ¬ 𝑃 ¬Q,P→Q⊢¬P Premise 1: Not Q (¬Q) Premise 2: If P then Q (P→Q) Conclusion: Not P (¬P) Using the same silly setup: P: “A turtle falls on my head.” Q: “I am a human being.” Then: I am not a human being (¬Q). If a turtle falls on my head, I am a human being (P→Q). Therefore A turtle does not fall on my head (¬P). This is basically the rule of the contrapositive: If “P→Q” is true, and Q is false, then P must also be false. 5. Assumptions (A) and Conditional Proof (CP) — Using a “What If” Box to Build “If … Then …” 5.1 Rule of Assumptions (A) The Rule of Assumptions says: You’re allowed to temporarily assume any proposition as a premise in order to see what follows from it. It’s like saying, “Let’s suppose P is true and see where that leads.” Think of it as opening a little “what if” bubble or a “what-if box.” 5.2 Conditional Proof (CP) Conditional Proof (CP) is the rule that lets you close that “what-if box” and turn it into a real conditional: If, assuming P, you can derive Q, then you may conclude “P→Q” and discharge (close) that assumption. A and CP are almost always used together: A: open the “what if P?” world CP: close that world, and record the result as “P→Q” 5.3 Example: “If the Wind Blows, the Cooper Profits” Let’s take the classic chain: P: The wind blows. Q: Dust swirls. R: The cooper (barrel maker) makes money. Premises: If the wind blows, dust swirls. (P→Q) If dust swirls, the cooper profits. (Q→R) We want to prove: If the wind blows, the cooper profits. (P→R) Proof sketch: P→Q (premise) Q→R (premise) Assume P. (A) From P and P→Q, get Q. (MP) From Q and Q→R, get R. (MP) We assumed P and derived R, so we conclude P→R. (CP) Conceptually: A opens the “P is true” world, we walk through to see that R follows, CP lets us return to the real world and say “If P, then R.” 5.4 Another Example: (P ∧ Q) → (Q ∧ P) This says: If P and Q are both true, then Q and P are both true. Obvious to intuition, but let’s write it logically: Assume P ∧ Q. (A) From P ∧ Q, get P. (∧-elimination) From P ∧ Q, get Q. (∧-elimination) From Q and P, form Q ∧ P. (∧-introduction) Thus, from the assumption P ∧ Q we derived Q ∧ P, so (P ∧ Q) → (Q ∧ P). (CP) Again, A opens the assumption, CP closes it as a conditional. 6. “And” and “Or”: The Rules for ∧ and ∨ 6.1 Conjunction “and” (∧) P ∧ Q … “P and Q” Conjunction Introduction (∧I) 𝑃 ,    𝑄    ⊢    𝑃 ∧ 𝑄 P,Q⊢P∧Q If P is true and Q is true, then P ∧ Q is true. Conjunction Elimination (∧E) 𝑃 ∧ 𝑄    ⊢    𝑃 and 𝑃 ∧ 𝑄    ⊢    𝑄 P∧Q⊢PandP∧Q⊢Q If P ∧ Q is true, then each of P and Q is true. Intuitively very straightforward: if “P and Q” is true, both parts are true; if both parts are true, “P and Q” is true. 6.2 Disjunction “or” (∨) P ∨ Q … “P or Q” Disjunction Introduction (∨I) 𝑃    ⊢    𝑃 ∨ 𝑄 P⊢P∨Q If P is true, then “P or Q” is true, regardless of Q. Example: “I am drinking coffee.” therefore “I am drinking coffee or tea.” You’ve broadened the statement in a way that stays true. 6.3 Disjunction Elimination (∨E): Case Analysis This one looks more complicated, but the idea is simple: case analysis. Formally: If P ∨ Q is true, from P you can get R, and from Q you can also get R, then you may conclude R. In symbols: 𝑃 ∨ 𝑄 ,    [ 𝑃 ] ⋮ 𝑅 ,    [ 𝑄 ] ⋮ 𝑅    ⊢    𝑅 P∨Q,[P]⋮R,[Q]⋮R⊢R Example: Rain or Snow, the Road Is Wet P: It is raining. Q: It is snowing. R: The road is wet. Premises: P ∨ Q … “It’s either raining or snowing.” P → R … “If it’s raining, the road is wet.” Q → R … “If it’s snowing, the road is wet.” Reasoning: Case 1: Suppose P (rain). Then from P→R, you get R. Case 2: Suppose Q (snow). Then from Q→R, you get R. Since either P or Q holds, and either way R follows, you can conclude: R: The road is wet. Intuition: Think of a Y-shaped path. You start at “P or Q,” the path splits into P and Q, but whichever branch you take, you meet again at R. That “meeting point” is what ∨E formalizes. 7. Reductio ad Absurdum (RAA): The Art of Riding Your Opponent’s Assumption into a Wall Reductio ad absurdum (RAA, “proof by contradiction”) has the following pattern: To prove P, temporarily assume ¬P, derive a contradiction, then reject ¬P and conclude P. It’s the formal version of: “Okay, let’s grant your claim for a moment and see what happens. …See? It leads to nonsense. So your claim must be wrong.” 7.1 A Mystery-Drama Example: Proving an Alibi Let: P: “He is the culprit.” Q: “He was at the crime scene at the time.” Premises: If he is the culprit, he must have been at the scene. (P→Q) In fact, he was not at the scene at that time (¬Q). We want to prove: He is not the culprit. (¬P) Using RAA: P→Q (premise) ¬Q (premise) Assume P. (“Suppose he is the culprit.”) From P and P→Q, get Q. But we already have ¬Q, so Q ∧ ¬Q → contradiction (⊥). Since assuming P led to a contradiction, we conclude ¬P. (RAA) The structure is: Step into the “what if you’re right” world. Show that world contains a logical bug (contradiction). So that world must be rejected, and its initial assumption thrown away. 7.2 The Simplest Logical Example: ¬¬P ⊢ P We want to prove: From ¬¬P, conclude P. In words: Premise: “It’s not the case that ‘P is not true’.” (¬¬P) Conclusion: “P is true.” Proof via RAA: ¬¬P (premise) Assume ¬P. (A) From ¬¬P and ¬P, you get a contradiction (⊥). Therefore, reject ¬P and conclude P. (RAA) So the rule “double negation elimination” ¬ ¬ 𝑃 ⊢ 𝑃 ¬¬P⊢P can itself be justified using reductio. 8. At This Point, You’ve Basically Met the Main Moves of Propositional Logic Let’s list what we’ve seen: Basic logical symbols Negation ¬ Conjunction ∧ Disjunction ∨ Conditional → Key inference rules Modus Ponens (MP) Modus Tollens (MT) ∧-Introduction / ∧-Elimination ∨-Introduction / ∨-Elimination (case analysis) Rule of Assumptions (A) Conditional Proof (CP) Reductio ad absurdum (RAA) If you understand how and why these rules work, you already have a solid grasp of “logic as a tool” for everyday reasoning and reading. Of course, logic goes much further: Predicate logic (“for all x…”, “there exists some x…”), completeness and consistency theorems, intuitionistic logic, many-valued logics, connections to computer science, category theory, and so on. But all of that can come later, if your curiosity pulls you in. 9. Why Is It Worth Knowing At Least Propositional Logic? Even this limited toolkit is useful in many ways: You’re less likely to fall for bad arguments. You can detect when someone’s “logic” is just rhetoric. You notice when the conclusion doesn’t actually follow from the premises. You can partially “mechanize” your own fact-checking. Logic doesn’t tell you whether the premises are true, but it can tell you: “If the premises are true, does this conclusion really follow?” You can proofread your own arguments. When writing or speaking, you can step back and ask: “Did I just sneak in a new premise?” “Am I changing the meaning of a key term halfway through?” That internal “referee” makes your reasoning more robust. It’s a shield against pure manipulation. In an age of clever rhetoric, marketing, and “weaponized narratives,” even basic logical awareness can serve as a kind of cognitive firewall. 10. Logic Isn’t Everything, But Not Knowing It Is a Handicap Humans are not made of logic alone. Emotions Trust Shared “atmosphere” or “vibes” Culture, history, stories All of these play a huge role in real life, and sometimes are more important than strict deduction. Still, it’s very helpful if you can say: “At least at the level of propositional logic, I can check whether this reasoning holds together.” As: part of a general education, a foundation for reading philosophy or contemporary critical theory, an intellectual survival skill in the age of fake news and viral outrage, the motto “at least propositional logic” is not a bad guiding line.