Logic Study Guide

📖 Core Concepts Logic – the systematic study of correct reasoning; examines how premises support a conclusion. Deductive reasoning – guarantees the conclusion if premises are true (validity). Ampliative (non‑deductive) reasoning – inductive, abductive, conductive; adds new information beyond premises. Formal vs. informal logic – formal uses abstract symbols & strict syntax; informal deals with natural‑language arguments and context. Validity – an argument’s form ensures that no possible interpretation makes all premises true and the conclusion false. Soundness – a valid argument whose premises are in fact true. Logical truth – a proposition true in every possible interpretation (depends only on logical vocabulary). Classical logic principles – law of excluded middle ($p\lor\neg p$), double‑negation elimination ($\neg\neg p\Rightarrow p$), principle of explosion ($p\land\neg p\Rightarrow q$), bivalence (exactly one of true/false). 📌 Must Remember Modus Ponens: $p$, $p\rightarrow q$ ⟹ $q$. Law of Excluded Middle: $p\lor\neg p$ is always true (classical). Double‑negation: $\neg\neg p \equiv p$ (classical only). Explosion: $p\land\neg p \Rightarrow q$ (classical). Soundness: every provable statement is semantically true. Completeness: every semantically true statement is provable. Universal Instantiation: $\forall x\,P(x) \Rightarrow P(c)$. Existential Generalization: $P(c) \Rightarrow \exists x\,P(x)$. Modal relationship: $\Box p \Rightarrow \Diamond p$ and $\Box p \equiv \neg\Diamond\neg p$. 🔄 Key Processes Assessing Deductive Validity Translate argument into a formal language. Identify form (e.g., modus ponens, hypothetical syllogism). Apply truth‑table or proof‑system rules; if every row with true premises yields a true conclusion → valid. Constructing a Formal Proof List premises. Repeatedly apply definitory inference rules (e.g., conjunction introduction, universal instantiation). Use strategic rules to choose steps that move toward the target conclusion efficiently. Evaluating an Inductive Argument Gather observed instances. Determine statistical strength (sample size, representativeness). Assess whether the generalization is warranted. Modal Reasoning Translate statements with “possible/necessary” into $\Diamond$ / $\Box$. Apply modal inference rules (e.g., from $\Box p$ infer $\Diamond p$). 🔍 Key Comparisons Deductive vs. Inductive → certainty vs. probability; premises guarantee conclusion vs. provide support. Formal fallacy vs. Informal fallacy → error in logical form vs. error in content/context. Classical logic vs. Intuitionistic logic → accepts LEM & double‑negation vs. rejects both; truth = proof vs. truth = bivalence. Paraconsistent vs. Classical → allows contradictions without explosion vs. explosion makes any statement derivable. Propositional vs. First‑order logic → deals with whole statements only vs. can analyze internal structure with predicates & quantifiers. ⚠️ Common Misunderstandings “If premises are true, conclusion must be true” – true only for valid (deductive) arguments; non‑deductive arguments can be false even with true premises. Equating a logical truth with a contingent truth – logical truths hold in every interpretation; contingent truths depend on the world. Assuming “not p” implies “p is false” – in intuitionistic logic $\neg p$ means “a proof of $p$ leads to contradiction,” not simply “p is false.” Believing “modus tollens” works in all logics – valid in classical and most non‑paraconsistent systems, but may fail where explosion is denied. 🧠 Mental Models / Intuition Validity as a “shape” – think of an argument’s form as a geometric skeleton; swapping out the concrete content (labels) never changes the shape’s validity. Truth tables as “lookup grids” – each row is a possible world; if any world makes all premises true and conclusion false, the argument is invalid. Modal worlds – imagine multiple possible worlds; $\Box p$ means “p is true in all worlds,” $\Diamond p$ means “p is true in some world.” 🚩 Exceptions & Edge Cases Intuitionistic logic – no LEM, no double‑negation elimination; constructivist proofs required. Multi‑valued logics – third value (e.g., “indeterminate”) breaks bivalence; truth tables must include extra rows. Paraconsistent logic – $p\land\neg p$ does not explode; careful to track which inference rules still hold. Denial of the antecedent – a classic formal fallacy: from $p\rightarrow q$ and $\neg p$, infer $\neg q$ (invalid). 📍 When to Use Which Propositional logic – when the argument’s content can be treated as whole, indivisible units (e.g., circuit design). First‑order logic – when you need to reason about properties of objects or quantify (“all,” “some”). Modal logic – when statements involve necessity, possibility, obligation, time, or knowledge. Intuitionistic logic – in constructive mathematics or computer‑science settings where existence must be witnessed. Paraconsistent logic – when dealing with inconsistent databases or theories where explosion is undesirable. 👀 Patterns to Recognize Repeated premise → conclusion chain – look for a “bridge” where one conclusion becomes the next premise. Quantifier scope errors – e.g., confusing $\forall x\,\exists y$ with $\exists y\,\forall x$. Hidden logical connectives in natural language (“unless” = “if not … then …”). Fallacy templates – denying the antecedent, affirming the consequent, equivocation, straw‑man. 🗂️ Exam Traps Choosing the wrong logic level – treating a predicate‑involving argument as merely propositional leads to loss of necessary quantifier rules. Assuming LEM in intuitionistic contexts – a question may explicitly state a constructive setting; answer requiring $p\lor\neg p$ will be wrong. Misreading “if…then” – in formal logic it is material implication ($p\rightarrow q$), not causal implication; avoid treating it as “because.” Over‑applying explosion – in a paraconsistent‑logic question, deriving arbitrary $q$ from a contradiction is a trap. Confusing “valid” with “sound” – an argument can be valid with false premises; only “sound” guarantees true conclusion. --- Keep this sheet handy; it condenses the most exam‑relevant ideas from the outline into bite‑size, review‑ready bullets.