📖 Glossary#
Glossary of Logical Terms#
Proposition#
A proposition is a declarative statement that is either true or false, but not both. It makes an assertion about reality and can be evaluated for truth. The linguistic form (English, math symbols, etc.) doesn’t matter; what matters is whether the content has a definite truth value.
Examples:
“The Earth orbits the Sun” is a proposition because it can be evaluated as true.
“The Moon is made of cheese.” (False)
“2 + 2 = 4” (True)
Compound Proposition#
A compound proposition is a logical statement formed by combining two or more simple propositions using logical connectives such as and (∧), or (∨), not (¬), if…then (→), and if and only if (↔). It expresses a more complex relationship between truth values.
Examples:
“The Earth orbits the Sun and 2 + 2 = 4” — combines two true propositions using ∧.
“If it rains, then the ground gets wet” — implication using →.
“Either the Moon is made of cheese or pigs can fly” — disjunction using ∨.
Simple (Non-Compound) Proposition#
A simple proposition is a single declarative statement that does not contain any logical connectives. It stands alone and can be evaluated as either true or false.
Examples:
“The Earth orbits the Sun” — true
“The Moon is made of cheese” — false
“2 + 2 = 4” — true
Propositional Function#
A propositional function is a statement containing one or more variables that becomes a proposition once specific values are substituted for those variables. Until the variables are assigned, the statement is neither true nor false.
Examples:
\(P(x):\; x > 5\) — not a proposition until \(x\) is given a value.
If \(x = 7\), then \(P(7): 7 > 5\) is true.
If \(x = 3\), then \(P(3): 3 > 5\) is false.
\(Q(x,y):\; x + y = 10\) — becomes a proposition once values of \(x\) and \(y\) are specified.
Predicate#
A predicate is essentially the property or relation expressed in a propositional function. In logic, it acts as a function that returns a truth value when applied to one or more subjects. Predicates are the building blocks of propositional functions. In practice, propositional functions and predicates often look the same (P(x), Q(x,y)), but predicate emphasizes the meaning (the property/relationship), while propositional function emphasizes the form (a function that turns into a proposition when variables are instantiated).
Examples:
In “Socrates is mortal”, “is mortal” is the predicate and “Socrates” is the subject. In predicate logic: Mortal(x)
IsTall(x): “x is tall”
Loves(x, y): “x loves y”
Prime(n): “n is a prime number”
Premise#
In mathematics, a premise is often called a hypothesis. In a conditional statement (“If p, then q”), the antecedent (the “if” part) is a specific type of premise. Thus, every antecedent is a premise, but not every premise is an antecedent.
Examples:
“All mammals are warm-blooded.”
“If it rains, the ground gets wet.”
“Jane is taller than Tom.”
Conclusion#
A conclusion is the proposition that is supported or proven by the premises in an argument. In a conditional statement (“If p, then q”), the consequence (the “then” part) corresponds to the conclusion of that conditional. Thus, every consequence is a conclusion, but not every conclusion is a consequence.
Examples:
“Therefore, the Earth is round.” (based on previous evidence/premises)
“Hence, she must be at home.”
“It follows that all bachelors are unmarried.”
Argument#
An argument is a group of propositions in which some (premises) are intended to support another (conclusion). It’s the fundamental structure of reasoning.
Example:
Premise: All birds have feathers.
Premise: A sparrow is a bird.
Conclusion: Therefore, a sparrow has feathers.
Example:
Premise: If you study, you will pass.
Premise: You studied.
Conclusion: Therefore, you will pass.
Inference (Reasoning)#
Inference, or reasoning, is the process of deriving logical conclusions from premises or known facts.
Examples:
From “All dogs bark” and “Rex is a dog,” we infer “Rex barks.”
From “If the light is red, stop” and “The light is red,” infer “Stop.”
From “All metals conduct electricity” and “Copper is a metal,” infer “Copper conducts electricity.”
Validity (Valid)#
An argument is valid if the conclusion necessarily follows from the premises. Validity is about form, not the actual truth of the statements.
Examples: 1.
Premise: All cats are animals.
Premise: Felix is a cat.
Conclusion: Felix is an animal. ✅ Valid
Premise: All unicorns are reptiles.
Premise: Bob is a unicorn.
Conclusion: Bob is a reptile. ✅ Valid (though premises are false)
Premise: Some dogs are brown.
Premise: Rex is brown.
Conclusion: Rex is a dog. ❌ Invalid (conclusion doesn’t logically follow)
Converse#
The converse of a conditional proposition \(p \to q\) is the statement \(q \to p\). It is formed by swapping the hypothesis and the conclusion. The converse is not logically equivalent to the original statement in general.
Examples:
Original: “If it rains, then the ground gets wet.” (\(p \to q\))
Converse: “If the ground gets wet, then it rains.” (\(q \to p\)) — This is not always true (sprinklers, for example).
Contrapositive#
The contrapositive of a conditional proposition \(p \to q\) is the statement \(\sim q \to \sim p\). It is formed by swapping and negating both the hypothesis and the conclusion. The contrapositive is logically equivalent to the original statement.
Examples:
Original: “If it rains, then the ground gets wet.” (\(p \to q\))
Contrapositive: “If the ground does not get wet, then it did not rain.” (\(\sim q \to \sim p\)) — Always true if the original is true.
Inverse#
The inverse of a conditional proposition \(p \to q\) is the statement \(\sim p \to \sim q\). It is formed by negating both the hypothesis and the conclusion without swapping them. The inverse is not logically equivalent to the original statement in general.
Examples:
Original: “If it rains, then the ground gets wet.” (\(p \to q\))
Inverse: “If it does not rain, then the ground does not get wet.” (\(\sim p \to \sim q\)) — Not always true (sprinklers again).