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Predicate Logic·Advanced·5 lessons·~320 min
How internal sentence structure changes formal reasoning
What you'll learn
Lessons
Motivates the move from propositional to predicate logic by showing arguments that propositional logic cannot capture and introduces the basic vocabulary of predicates, constants, and variables.
Introduces the universal and existential quantifiers, teaches students to translate simple quantified English claims into first-order form, and establishes the relationship between quantifiers and the connectives that typically accompany them.
Tackles the hardest translation challenges in predicate logic: multiple quantifiers, nested scope, mixed universal and existential claims, and the subtleties of relational predicates.
Introduces the four quantifier inference rules (universal instantiation, existential instantiation, universal generalization, existential generalization) and explains the restrictions that each rule imposes.
An integrative lesson that asks students to translate ordinary-language arguments involving quantifiers and predicates, construct short proofs using quantifier rules, and diagnose scope errors.
How to study
Each lesson opens with a guided walkthrough — read it before the activity.
Look at why each step follows, not just what the answer is.
Know which rule applies and what would make the response weak before you start.
Optional context for the unit. Each lesson surfaces the concepts and rules it uses — these are here when you want the bigger picture.
An expression that attributes a property to an object or a relation among several objects; a one-place predicate applies to a single object, while a relational predicate applies to two or more.
A symbol that names a specific object in the domain of discourse, typically written with lowercase letters like a, b, c.
A symbol like x, y, or z that does not name a specific individual but can range over the domain when bound by a quantifier.
The set of objects that the variables of a predicate-logic formula are taken to range over in a given interpretation.
An operator that binds a variable and states how much of the domain it ranges over; the universal quantifier ∀ means 'for all,' and the existential quantifier ∃ means 'for some' or 'there exists.'
The scope of a quantifier is the portion of a formula it governs; a variable occurrence is bound if it falls within the scope of a quantifier using the same variable, and free otherwise.
The quantifier ∀x, read 'for all x,' which claims that the formula it binds holds for every object in the domain.
The quantifier ∃x, read 'there exists x such that,' which claims that the formula it binds holds for at least one object in the domain.
A proof move that goes from a quantified claim to a claim about a specific individual; universal instantiation picks any individual, while existential instantiation introduces a fresh name for a witness.
A proof move that goes from a claim about an individual to a quantified claim; universal generalization is allowed only when the individual was arbitrary, and existential generalization is always allowed when a specific instance has been found.