Mathematical Logic and Formality

Logic and Formality

• Logic is the science of correct reasoning.
• Reasoning is the process of drawing inferences or conclusions from known or assumed facts.
• Mathematical logic is a framework that defines, proves theorems, lemmas, conjectures, corollaries, propositions, and proof methods precisely.

Formality

• An expression is considered completely formal when it's context-independent and precise (Heylighen & Dewaele, "Formality of Language").
• Definitions are formal statements of a word or phrase’s meaning, capable of standing alone.
  • Examples include “right triangle” and “carabao.”

Theorem

• A theorem is a statement demonstrably true using accepted mathematical operations and arguments.
• Theorems embody general principles within a larger theory.
• The process of proving a theorem is called a proof.
  • Example: Pythagorean Theorem

Proposition

• A proposition is a declarative statement that's either true or false, but not both.
• Propositions are precise and concise, forming a crucial part of formality.

Corollary

• A corollary is a proposition that readily follows from a proven proposition with little or no proof required.
• Example: A corollary to the statement that opposite sides of a triangle are congruent is that an equilateral triangle is also equiangular.

Lemma

• A lemma is a short theorem used in proving a larger theorem. It’s akin to a "helping theorem."

Conjecture

• A conjecture is a proposition consistent with known data but not yet verified or proven false.
• It’s also called an educated guess or hypothesis.
• A conjecture can only be disproven by finding a counterexample.

Two Basic Types of Human Reasoning

• Deductive reasoning: moving from general premises to specific and certain conclusions.
• Inductive reasoning: moving from specific instances to general but potentially uncertain conclusions.

Problem Solving

• A problem is a statement needing a solution, often involving mathematical operations or geometric constructions.
• Problem-solving is an ongoing process of using known information to discover unknown solutions.
• Problem-solving involves seeking information, generating new knowledge, and making decisions.

Mathematical Reasoning

• Mathematical reasoning involves analyzing situations, constructing logical arguments, justifying/creating hypotheses, and developing conceptual foundations and connections for information processing.

Inductive Reasoning

• Inductive reasoning is a process of reaching a general conclusion by examining specific examples.

Use Inductive Reasoning to Predict a Number

• Example sequence: 3, 6, 9, 12, 15, ... ⇒ Next number is 18.
• Example sequence: 1, 3, 6, 10, 15, ... ⇒ Next number is 21.

Use Inductive Reasoning to Make a Conjecture

• A procedure: Multiply a number by 8, add 6, divide by 2, subtract 3.
• Conjecture: The result is 4 times the original.

Use Inductive Reasoning to Solve an Application

• Example concerning pendulum periods: if the length of a pendulum is quadrupled, its period is doubled.

Counterexamples

• A statement is only true if it holds in all cases.
  • A counterexample is a specific case in which a statement is false, proving it is not universally true.

Examples of Counterexamples

• Example: "Every number multiple of 10 is divisible by 4."

  • Counterexample: 110 (110 / 4 = 27.5)

• Verifying False Statements with Counterexamples:

  • Statement: |x| > 0 for all numbers x ⇒ x = 0, |0| = 0, so false.
  • Statement: x² > x for all numbers x ⇒ x = 1, 1² =1, so false.
  • Statement: √x²= x for all numbers x ⇒ x = -3, √(-3)²=3, so false.

Deductive Reasoning

• Deductive reasoning involves concluding something based on general rules, assumptions, or procedures.
• Example: If a number is divisible by 2, it's even. 12 is divisible by 2, therefore, 12 is even.
• Geometry often utilizes deductive reasoning to justify statements and solve problems.