Logic Cheat Sheet

The core ideas of Logic distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Propositional Logic

The branch of logic that deals with propositions (statements that are either true or false) and the logical connectives that combine them, such as conjunction, disjunction, negation, and implication. It provides the simplest formal system for analyzing the logical structure of arguments.

Predicate Logic

An extension of propositional logic that introduces variables, predicates, and quantifiers (universal and existential) to express properties of objects and relationships between them. It is far more expressive than propositional logic and forms the basis of most mathematical reasoning.

Validity and Soundness

An argument is valid if its conclusion necessarily follows from its premises, regardless of whether the premises are actually true. An argument is sound if it is both valid and all of its premises are in fact true. This distinction is central to evaluating arguments rigorously.

Logical Fallacy

An error in reasoning that renders an argument logically invalid or seriously weakened. Formal fallacies violate the structural rules of valid inference, while informal fallacies involve errors in content, context, or relevance that undermine persuasive force.

Deductive Reasoning

A form of reasoning in which the conclusion is guaranteed to be true if the premises are true. Deductive arguments aim for certainty and are evaluated by their validity, meaning the truth of the premises necessitates the truth of the conclusion.

Inductive Reasoning

A form of reasoning in which the premises provide probabilistic support for the conclusion but do not guarantee it. Inductive arguments are evaluated by their strength, that is, how likely the conclusion is given the evidence provided.

Truth Table

A mathematical table used to determine the truth value of a compound proposition for every possible combination of truth values of its component propositions. Truth tables provide a mechanical method for checking validity in propositional logic.

Modus Ponens

A fundamental rule of inference stating that if a conditional statement ('if P then Q') is true and its antecedent (P) is also true, then its consequent (Q) must be true. It is one of the most basic and widely used valid argument forms.

Quantifiers

Symbols in predicate logic that specify the quantity of subjects to which a predicate applies. The universal quantifier (for all) asserts that a property holds for every element in a domain, while the existential quantifier (there exists) asserts that at least one element satisfies the property.

Logical Equivalence

Two statements are logically equivalent if they have the same truth value in every possible interpretation. Logical equivalences such as De Morgan's laws and double negation are used to simplify and transform logical expressions without changing their meaning.

Key Terms at a Glance

Antecedent:The 'if' part of a conditional statement 'If P then Q.' P is the antecedent, and its truth or falsehood determines conditions under which the conditional may be evaluated.

Consequent:The 'then' part of a conditional statement 'If P then Q.' Q is the consequent, representing the result or outcome that is asserted to follow when the antecedent holds.

Conjunction:A logical connective (AND) that produces a compound statement which is true only when both component statements are true. Symbolized as P AND Q or P & Q.

Disjunction:A logical connective (OR) that produces a compound statement which is true when at least one component statement is true. In logic, this is inclusive OR, meaning it is also true when both components are true.

Negation:A logical operation that reverses the truth value of a proposition. If P is true, NOT P is false, and vice versa. It is the simplest logical connective, taking only one operand.

Conditional:A compound statement of the form 'If P then Q' (also called material implication). It is false only when P is true and Q is false; in all other cases it is considered true.

Biconditional:A compound statement of the form 'P if and only if Q' that is true when both components share the same truth value (both true or both false) and false otherwise.

Tautology:A compound proposition that evaluates to true under every possible assignment of truth values to its component propositions. Tautologies represent logical truths that hold universally.

Contradiction:A compound proposition that evaluates to false under every possible assignment of truth values to its component propositions. No interpretation can make a contradiction true.

Contingency:A compound proposition that is neither a tautology nor a contradiction, meaning it is true under some truth-value assignments and false under others. Most everyday statements are contingencies.

Syllogism:A form of deductive argument with two premises and a conclusion, originally formalized by Aristotle. Categorical syllogisms involve statements about classes of objects using terms like 'all,' 'some,' and 'no.'

Predicate:A statement template containing one or more variables that becomes a proposition when specific values are substituted for the variables. Predicates express properties or relationships of objects.

Quantifier:A logical symbol that specifies the extent to which a predicate applies to elements in a domain. The two primary quantifiers are the universal quantifier (for all) and the existential quantifier (there exists).

Domain of Discourse:The set of all objects or entities over which variables in a logical expression can range. The domain must be specified or understood in order to determine the truth of quantified statements.

Inference Rule:A logical rule that justifies the derivation of a conclusion from premises. Standard inference rules include modus ponens, modus tollens, hypothetical syllogism, and disjunctive syllogism.

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