formal-logic

Formal Logic

Logic is the study of valid reasoning. It provides the scaffolding on which all philosophical argument rests — without it, philosophy is rhetoric. This skill covers formal systems (propositional, predicate, syllogistic, modal) and informal reasoning (fallacy detection, argument mapping), with worked examples drawn from the philosophical tradition.

Agent affinity: aristotle (logic and epistemology, Opus)

Concept IDs: philo-argument-structure, philo-deductive-reasoning, philo-logical-fallacies

The Logic Landscape at a Glance

| # | Domain | Core question | Key tools | |---|---|---|---| | 1 | Propositional logic | Is the compound statement true? | Connectives, truth tables, tautologies | | 2 | Predicate logic | Does the claim hold for all/some objects? | Quantifiers, domains, free/bound variables | | 3 | Syllogistic logic | What follows from categorical premises? | Figures, moods, Venn diagrams | | 4 | Natural deduction | Can we derive the conclusion step by step? | Introduction/elimination rules | | 5 | Modal logic | Is it necessarily or possibly true? | Box/diamond, possible worlds, accessibility | | 6 | Informal fallacies | Where does the reasoning go wrong? | Fallacy taxonomy, argument reconstruction | | 7 | Argument mapping | What is the logical structure of this passage? | Premise identification, diagram construction |

1 — Propositional Logic

Core idea. Propositional logic studies the truth-functional relationships between whole statements (propositions) connected by logical operators.

The Five Connectives

| Symbol | Name | English | True when | |---|---|---|---| | neg | Negation | "not P" | P is false | | /\ | Conjunction | "P and Q" | Both P and Q are true | | \/ | Disjunction | "P or Q" | At least one is true | | -> | Conditional | "if P then Q" | P is false, or Q is true | | <-> | Biconditional | "P if and only if Q" | Both have the same truth value |

Truth Tables

A truth table exhaustively lists all possible truth-value assignments for a compound proposition.

Worked example. Show that P -> Q is logically equivalent to neg-P \/ Q.

| P | Q | P -> Q | neg-P | neg-P \/ Q | |---|---|--------|-------|-----------| | T | T | T | F | T | | T | F | F | F | F | | F | T | T | T | T | | F | F | T | T | T |

The columns for P -> Q and neg-P \/ Q are identical. The equivalence holds. This is the material conditional — a fact that surprises students, since "if P then Q" in ordinary language carries pragmatic implications (relevance, causation) that the material conditional does not.

Tautologies, Contradictions, Contingencies

• Tautology: True under every assignment. Example: P \/ neg-P (law of excluded middle).
• Contradiction: False under every assignment. Example: P /\ neg-P.
• Contingency: True under some assignments, false under others. Example: P -> Q.

Key tautologies for philosophy:

• Modus ponens: ((P -> Q) /\ P) -> Q
• Modus tollens: ((P -> Q) /\ neg-Q) -> neg-P
• Hypothetical syllogism: ((P -> Q) /\ (Q -> R)) -> (P -> R)
• De Morgan's laws: neg(P /\ Q) <-> (neg-P \/ neg-Q) and neg(P \/ Q) <-> (neg-P /\ neg-Q)
• Contraposition: (P -> Q) <-> (neg-Q -> neg-P)
• Double negation: neg-neg-P <-> P (classical logic; fails in intuitionistic logic)

2 — Predicate Logic

Core idea. Predicate logic extends propositional logic by analyzing the internal structure of propositions — subjects, predicates, and quantifiers.

Quantifiers

• Universal (forall x): "For every x in the domain, P(x) holds."
• Existential (exists x): "There is at least one x in the domain such that P(x) holds."

Validity in Predicate Logic

Worked example. Analyze: "All philosophers are mortal. Socrates is a philosopher. Therefore Socrates is mortal."

Let P(x) = "x is a philosopher," M(x) = "x is mortal," s = Socrates.

1. forall x (P(x) -> M(x)) [Premise]
2. P(s) [Premise]
3. P(s) -> M(s) [Universal instantiation from 1]
4. M(s) [Modus ponens from 2, 3]

The argument is valid. The conclusion follows necessarily from the premises. Whether it is sound depends on whether the premises are true — a separate question.

Validity vs. Soundness

This distinction is among the most important in philosophy:

• Valid: The conclusion follows logically from the premises (the form is truth-preserving).
• Sound: The argument is valid AND all premises are true.
• Invalid arguments can have true conclusions. "All cats are mammals. All mammals are cats. Therefore all cats are mammals." The conclusion is true, but the argument is invalid (the second premise is false, and the form is flawed).

Worked example. A valid but unsound argument:

1. All birds can fly. [False premise — penguins, ostriches]
2. Penguins are birds. [True]
3. Therefore penguins can fly. [False conclusion]

The argument is valid (the conclusion follows from the premises) but unsound (premise 1 is false).

3 — Syllogistic Logic

Core idea. Aristotle (384-322 BCE) systematized reasoning about categorical propositions — statements asserting that all, no, or some members of one category belong to another.

The Four Categorical Forms

| Form | Name | Example | |------|------|---------| | A: All S are P | Universal affirmative | All humans are mortal | | E: No S are P | Universal negative | No reptiles are mammals | | I: Some S are P | Particular affirmative | Some philosophers are scientists | | O: Some S are not P | Particular negative | Some animals are not domesticated |

Figures and Moods

A categorical syllogism has two premises and a conclusion, each in one of the four forms, with exactly three terms (major, minor, middle). The four figures are determined by the position of the middle term:

• Figure 1: Middle is subject of major, predicate of minor (Barbara, Celarent, Darii, Ferio)
• Figure 2: Middle is predicate of both (Cesare, Camestres, Festino, Baroco)
• Figure 3: Middle is subject of both (Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison)
• Figure 4: Middle is predicate of major, subject of minor (Bramantip, Camenes, Dimaris, Fesapo, Fresison)

Worked example — Barbara (AAA-1):

1. All mammals are animals. [A: All M are P]
2. All dogs are mammals. [A: All S are M]
3. Therefore all dogs are animals. [A: All S are P]

The medieval mnemonic names encode the mood: "Barbara" = A-A-A in Figure 1. There are exactly 24 valid syllogistic forms (15 without existential import assumptions).

4 — Natural Deduction

Core idea. Natural deduction formalizes reasoning as a sequence of rule applications, each introducing or eliminating a logical connective. Developed independently by Gentzen (1934) and Jaskowski (1934).

Introduction and Elimination Rules

| Connective | Introduction | Elimination | |------------|-------------|-------------| | /\ | From A and B, infer A /\ B | From A /\ B, infer A (or B) | | \/ | From A, infer A \/ B | From A \/ B, if A leads to C and B leads to C, infer C | | -> | Assume A, derive B; discharge assumption, infer A -> B | From A -> B and A, infer B (modus ponens) | | neg | Assume A, derive contradiction; infer neg-A | From neg-neg-A, infer A (classical) | | forall | If P(a) for arbitrary a, infer forall x P(x) | From forall x P(x), infer P(t) for any term t | | exists | From P(t), infer exists x P(x) | From exists x P(x), if P(a) leads to C (a fresh), infer C |

Worked example. Prove: from (P -> Q) and (Q -> R), derive (P -> R).

1. P -> Q [Premise]
2. Q -> R [Premise]
3. | P [Assumption for conditional proof]
4. | Q [Modus ponens: 1, 3]
5. | R [Modus ponens: 2, 4]
6. P -> R [Conditional introduction: 3-5, discharge assumption P]

This is the hypothetical syllogism derived as a theorem rather than taken as an axiom.

5 — Modal Logic

Core idea. Modal logic extends classical logi