Logical Arguments

Types of Arguments in Logic

Deductive Arguments

Modus Ponens

Definition: If the premise is true, then the conclusion must be true.

Structure: P → Q, P ∴ Q

Example: If it's raining, the ground is wet. It's raining. Therefore, the ground is wet.

Modus Tollens

Definition: If the consequent is false, then the antecedent must be false.

Structure: P → Q, ¬Q ∴ ¬P

Example: If it's sunny, it's not raining. It is raining. Therefore, it's not sunny.

Hypothetical Syllogism

Definition: If A implies B, and B implies C, then A implies C.

Structure: P → Q, Q → R ∴ P → R

Example: If I study, I'll pass the exam. If I pass the exam, I'll graduate. Therefore, if I study, I'll graduate.

Disjunctive Syllogism

Definition: If one of two options is false, the other must be true.

Structure: P ∨ Q, ¬P ∴ Q

Example: Either it's raining or it's snowing. It's not raining. Therefore, it's snowing.

Categorical Syllogisms

Barbara (AAA-1)

Definition: All members of a group have a property, and a subject is a member of that group.

Structure: All M are P, All S are M ∴ All S are P

Example: All mammals are warm-blooded. All dogs are mammals. Therefore, all dogs are warm-blooded.

Celarent (EAE-1)

Definition: No members of a group have a property, and a subject is a member of that group.

Structure: No M are P, All S are M ∴ No S are P

Example: No reptiles are warm-blooded. All snakes are reptiles. Therefore, no snakes are warm-blooded.

Propositional Logic

Conjunction Introduction

Definition: If two statements are true individually, their conjunction is true.

Structure: P, Q ∴ P ∧ Q

Example: It's sunny. It's warm. Therefore, it's sunny and warm.

Disjunction Introduction

Definition: If a statement is true, its disjunction with any other statement is true.

Structure: P ∴ P ∨ Q

Example: It's raining. Therefore, it's raining or snowing.

Predicate Logic

Universal Instantiation

Definition: If a property applies to all members of a set, it applies to any specific member.

Structure: ∀x P(x) ∴ P(a)

Example: All humans are mortal. Therefore, Socrates is mortal.

Existential Generalization

Definition: If a property applies to a specific member of a set, then there exists a member with that property.

Structure: P(a) ∴ ∃x P(x)

Example: Socrates is wise. Therefore, there exists someone who is wise.

Inductive Arguments

Enumerative Induction

Definition: If a property is observed in many instances, it might apply to all instances.

Structure: P(a₁), P(a₂), ..., P(aₙ) ∴ ∀x P(x)

Example: Swan 1 is white, Swan 2 is white, ..., Swan 100 is white. Therefore, all swans are white.

Statistical Syllogism

Definition: Most members of a group have a property, and a subject is a member of that group.

Structure: x% of F are G, a is F ∴ a is G (with x% probability)

Example: 90% of students pass this course. John is a student in this course. Therefore, John will probably pass this course (with 90% probability).

Abductive Reasoning

Inference to the Best Explanation

Definition: If a hypothesis best explains an observation, it is probably true.

Structure: P is observed, Q would best explain P ∴ Q is probably true

Example: The grass is wet. The best explanation for wet grass is that it rained. Therefore, it probably rained.

Modal Logic

Necessity Elimination

Definition: If something is necessarily true, it is true.

Structure: □P ∴ P

Example: It is necessarily true that 2+2=4. Therefore, 2+2=4.

Possibility Introduction

Definition: If something is true, it is possible.

Structure: P ∴ ◇P

Example: It is raining. Therefore, it is possible that it is raining.

Analogical Arguments

Basic Form of Analogical Argument

Definition: If two things are similar in some respects, they are probably similar in other respects.

Structure: A is similar to B in respects R₁, R₂, ..., Rₙ, A has property P ∴ B probably has property P

Example: Earth and Mars are similar in having a solid surface, orbiting the sun, and having an atmosphere. Earth has life. Therefore, Mars probably has (or had) life.

Causal Arguments

Simple Causal Argument

Definition: If two events are correlated, one precedes the other, and there's no better explanation, the first event likely causes the second.

Structure: A is correlated with B, A precedes B, There is no better explanation for B than A causing B ∴ A causes B

Example: Regular exercise is correlated with lower blood pressure. Exercise precedes the reduction in blood pressure. There is no better explanation for lower blood pressure than regular exercise causing it. Therefore, regular exercise causes lower blood pressure.