What Are Not Represented by Using Propositional Logic?
Propositional logic represents whole statements as Proposition symbols and reasons using logical connectives like Negation and Conjunction. However, it does not include a mechanism for talking about objects (things) or relations (between things), because it has no variables that range over a domain and no predicate symbols that take arguments (objects) as inputs—capabilities provided by First-Order Logic instead of propositional logic.
A key intuition: propositional logic can express that “P is true”, but it cannot express “for every object x, property R holds of x” or “object a stands in relation S to object b”—those require quantifiers and predicates.
Answer to the multiple-choice question: (iii) Both objects and relations.
Propositional Logic vs First-Order Logic (Predicates, Objects, Relations)
Key learning terms
- Propositional symbol
- Truth assignment
- Quantifier
- Predicate
- Argument
Why propositional logic can’t represent objects
Propositional logic has only atomic sentence symbols (propositions). It lacks the syntax/semantics to interpret “object terms” (like , , ) as members of a domain of discourse. That’s why it cannot express statements of the form:
- “Object is a student” (needs predicates + variables)
- “There exists an object with property P” (needs existential quantification)
In contrast, first-order logic introduces variables and quantifiers over a domain, enabling statements about objects and their properties.
Why propositional logic can’t represent relations
Relations require two or more entities connected by a relation predicate, such as:
- “ is taller than ”
- “ is related to by ”
Propositional logic cannot do this because it has no predicate application structure (no “relation predicate to arguments” form like ). At best, it can only represent a flat atomic proposition such as —but that loses the ability to specify which objects are related and to express general rules about relations across many objects.
Quick test
If your intended meaning needs “for every …”, “there exists …”, or “ holds”, then propositional logic is the wrong tool; use first-order predicate logic.
Common confusion
You might encode relations indirectly by creating a proposition per pair (e.g., Tall(A,B)), but that still does not truly represent relations generically—it's only a pre-enumerated set of propositions, and you can't quantify over objects.
Deciding which choice fits (i)-(iv)
- 1Step 1
Objects require variables or a domain; propositional logic has neither, so (i) is not representable.
- 2Step 2
Relations require predicate structure with arguments (like ); propositional logic has only flat proposition symbols, so (ii) is not representable.
- 3Step 3
Since both objects (i) and relations (ii) require predicate/quantifier machinery absent from propositional logic, the correct combined option is (iii).
- 4Step 4
Choose (iii) Both objects and relations.
Logic Expressiveness (high level)
Propositions only
Propositional levelAtomic truth-bearers; reasoning with connectives. No objects/relations/quantifiers."
Predicates over a domain
First-order levelVariables + quantifiers + predicates let you represent objects and relations."
More expressive frameworks
Higher levels (contextual)Sometimes adds additional structure (e.g., functions, equality, higher-order quantification)."
Knowledge Check
Propositional logic can directly represent statements like “P is true,” but it cannot represent statements that require objects as variables. What is the correct option?