Module 07 : Direct Proofs

Logic of Quantified Statements

Expressions in logic that assert that a certain condition holds for all or some members of a set.
Various logical operations can be applied to quantified statements, examining their implications and relationships through equivalencies.

Universal Instantiation: If a property is true for everything in a set, it is true for any specific entity from that set.
- Example: "All men are mortal; Socrates is a man, thus Socrates is mortal."
Universal Modus Ponens: Combination of universal instantiation with modus ponens.
- Formal Version: ∀x, if P(x) then Q(x). P(a) [for some particular a] implies Q(a).
- Example: If an integer is even, then its square is even; k is an even integer, so k² is even.

To prove properties involving specific values based on their universal properties. For example, triangles and the Pythagorean theorem or even integers and their sums.
Application of Universal Modus Ponens: Presents structured arguments leading to codified conclusions.

Based on universal instantiation and modus tollens.
Formal version involves negating the conclusion, leading to the conclusion that the hypothesis must also be false.
- Example: If all humans are mortal and Zeus is not mortal, then Zeus cannot be human.

An argument form is valid if the truth of its premises ensures the truth of its conclusion.
Sound Argument: An argument form is valid and its premises are true.
Knowledge of logical principles aids in evaluating the structure and validity of complex arguments.

Diagrams can help visualize logical relationships between premises and conclusions.
Valid Argument Example: Using a diagram, showing that all human beings are mortal, followed by proving that a specific individual is not mortal, thus concluding they are not human.

Converse Error: Assuming that if Q holds for a, then P holds for a if P(x) implies Q(x).
Inverse Error: Assuming that the failure of P for a fails Q for a.
Examples illustrate common missteps in logic related to erroneous assumptions about relationships.
- Importance of correctly identifying valid forms and understanding their contrapositives.

Proof by Counterexample: Disproves universal statements by finding a specific case where the hypothesis holds but the conclusion fails.
Existential Statements: Argues for the existence of specific cases meeting defined criteria.
The practice of mathematical reasoning cultivates precision and aids comprehensive understanding.

Structure proofs using specific statements and definitions to establish truth in universal contexts.
Ensure clarity in exposition for reproducibility by peers.
Practical examples include proving sums of integers based on their characteristics defined earlier.
Theorem Example: The sum of two even integers is even, justified through substitution and algebraic identities.