Discrete Mathematics - Rules of Inference

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Questions and Answers

What is an argument?

  • A conclusion
  • A sequence of statements that end with a conclusion (correct)
  • An incorrect reasoning
  • None of the above

What does it mean for a conclusion to be valid?

A conclusion that must follow the truth of the preceding statements.

What are premises in an argument?

Proceeding statements.

What are fallacies?

<p>Incorrect reasoning.</p>
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What are rules of inference?

<p>Simple argument forms.</p>
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What is the Law of Detachment?

<p>A tautology also known as Modus Ponens mode that affirms.</p>
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Match the logical forms with their descriptions:

<p>Modus Ponens = (p ∧ (p → q)) → q Modus Tollens = (¬q ∧ (p → q)) → ¬p Hypothetical Syllogism = ((p → q) ∧ (q → r)) → (p → r) Disjunctive Syllogism = ((p ∨ q) ∧ ¬p) → q Addition = p → (p ∨ q) Simplification = (p ∧ q) → p Conjunction = ((p) ∧ (q)) → (p ∧ q) Resolution = ((p ∨ q) ∧ (¬p ∨ r)) → (q ∨ r)</p>
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Flashcards

What is an argument?

A sequence of statements that end with a conclusion.

What is a valid conclusion?

A conclusion that inevitably follows if the preceding statements are true.

Premises in arguments?

Statements that precede and support the conclusion.

What are fallacies?

Errors in reasoning that invalidate an argument.

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Rules of inference?

Basic valid argument forms used to deduce conclusions from premises.

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Law of Detachment?

If P, then Q. P is true, therefore Q is true.

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Study Notes

Argument Fundamentals

  • An argument consists of a sequence of statements culminating in a conclusion.
  • Validity indicates that a conclusion necessarily follows from its preceding statements.

Key Components of Arguments

  • Premises are the preceding statements that support the conclusion.
  • Fallacies represent incorrect reasoning that undermines an argument's validity.

Rules of Inference

  • Rules of inference are basic argument structures that allow for the derivation of valid conclusions from given premises.

Specific Rules of Inference

  • Law of Detachment (Modus Ponens): If "p" is true and "p → q" is true, then "q" is necessarily true.

  • Example structure: If "p" and "p → q", then conclude "q".

  • Modus Tollens: If "¬q" (not q) is true and "p → q" is true, then "¬p" (not p) must also be true.

  • Example structure: If "¬q" and "p → q", then conclude "¬p".

  • Hypothetical Syllogism: If "p → q" and "q → r" are both true, then "p → r" is also true.

  • Example structure: If "p → q" and "q → r", then conclude "p → r".

  • Disjunctive Syllogism: From "p ∨ q" (either p or q) and "¬p" (not p), we can conclude "q".

  • Example structure: If "p ∨ q" and "¬p", then conclude "q".

  • Addition: If "p" is true, we can conclude "p ∨ q" (p or q).

  • Example structure: If "p", then conclude "p ∨ q".

  • Simplification: If "p ∧ q" (both p and q) is true, we can conclude "p".

  • Example structure: If "p ∧ q", then conclude "p".

  • Conjunction: If both "p" and "q" are true, we can form "p ∧ q".

  • Example structure: If "p" and "q", then conclude "p ∧ q".

  • Resolution: From "p ∨ q" and "¬p ∨ r", we can conclude "q ∨ r".

  • Example structure: If "p ∨ q" and "¬p ∨ r", then conclude "q ∨ r".

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