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👁️‍🗨️formal logic i review

Quantified statements

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

Quantified statements are logical expressions that specify the quantity of instances in a domain that satisfy a given property. These statements are often expressed using quantifiers, such as 'for all' (universal quantifier) and 'there exists' (existential quantifier), which allow us to make generalizations or assertions about objects in a certain set.

5 Must Know Facts For Your Next Test

  1. Quantified statements can express broad truths or existence claims about elements in a particular domain, making them powerful tools in logical reasoning.
  2. The universal quantifier '∀x P(x)' states that the property P holds for every element x in the domain.
  3. The existential quantifier '∃x P(x)' asserts that there exists at least one element x in the domain for which the property P is true.
  4. Using quantified statements, one can derive logical conclusions through rules such as Universal Generalization and Existential Instantiation.
  5. Quantified statements can be combined with logical connectives like conjunction, disjunction, and negation to form more complex expressions.

Review Questions

  • How do quantified statements enhance logical reasoning compared to non-quantified statements?
    • Quantified statements enhance logical reasoning by allowing us to express general truths and existence claims about entire sets rather than individual instances. For example, while a non-quantified statement might say 'Some birds can fly,' a quantified statement can clarify this by saying 'For all x, if x is a bird, then x can fly.' This ability to generalize or specify conditions makes quantified statements crucial for constructing logical arguments and proofs.
  • Discuss how Universal Generalization can be applied to a quantified statement and provide an example.
    • Universal Generalization is a rule that allows us to conclude that if a property holds for an arbitrary object, it holds for all objects in the domain. For instance, if we have shown that 'a specific dog barks' is true and that 'barking' applies generally to all dogs, we can then assert the universal statement 'For all x, if x is a dog, then x barks.' This logical step demonstrates how we can move from specific cases to general principles using quantified statements.
  • Evaluate the importance of Existential Instantiation in logical proofs involving quantified statements.
    • Existential Instantiation is vital in logical proofs because it allows us to take a statement that claims the existence of an object satisfying a property and create a specific instance from it. For example, if we know 'There exists an x such that P(x) is true,' we can instantiate this by saying 'Let c be such that P(c) holds.' This step transforms abstract existence into concrete terms, enabling us to derive further conclusions or build upon existing knowledge within logical arguments effectively. It helps bridge the gap between general claims and practical application in proofs.