Validity of Arguments: Informal and Formal Proofs

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Questions and Answers

What is the primary goal of the formal deductive system introduced in the course?

To make informal arguments more persuasive

To establish opinions based on previous research

To derive a conclusion from a set of premises using well-defined transformations (correct)

To derive conclusions from assumptions without using transformations

In informal proofs, which of the following is typically allowed?

Using accepted results like axioms and theorems (correct)

Borrowing hypothetical conclusions as premises

Employing personal beliefs to support arguments

Skipping critical steps in reasoning

What defines a direct proof in mathematical reasoning?

It applies valid inference steps to show a conclusion follows from assumptions (correct)

It infers conclusions through contrapositive reasoning

It automatically assumes the conclusion without premises

It requires verifying each step by independent means

Which statement is an example of an axiom in a reasoning system for geometry?

Two points lie on one and only one line (B) Signup and view all the answers

What inference property is characteristic of the statement 'If n is an odd integer, then n^2 is odd'?

It can only be false if the antecedent is true and the consequent is false (C) Signup and view all the answers

What is one of the interim objectives of informal proofs?

To relate reasoning patterns in proofs with known tautologies (D) Signup and view all the answers

What distinguishes a formal proof from an informal proof?

Formal proofs derive conclusions from premises using defined transformations (D) Signup and view all the answers

What is the significance of the Deduction Theorem in proofs?

It establishes a formal relationship between premises and conclusions (B) Signup and view all the answers

What does the symbol ⊢ represent in formal proofs?

The conclusion of a proof (A) Signup and view all the answers

Which statement best describes modus ponens?

It affirms the antecedent to derive a consequence. (C) Signup and view all the answers

In an inference rule notation, what do A1, A2, ..., An represent?

The conditions needed to apply the rule (A) Signup and view all the answers

Which of the following is a self-justified premise in a proof?

When it is labeled as 'given' or 'assumption' (A) Signup and view all the answers

What does the notation W1, W2, ..., Wn signify in a proof?

The sequence of statements and their justifications (B) Signup and view all the answers

What must be included when a line in a proof derives from other formulas?

The line numbers of the dependent formulas (B) Signup and view all the answers

How is the 'justification' for a premise in a proof indicated?

Using the words 'given', 'assumption', or 'data' (D) Signup and view all the answers

Considering the proof sequence example, what is the role of the line that follows '1.P'?

To serve as a foundation for all other conclusions (A) Signup and view all the answers

What is the primary goal of a natural deduction proof system?

To derive the conclusion from some of its premises (A) Signup and view all the answers

Which operation involves removing a connective from a complex formula to generate a new conclusion?

Elimination rule (C) Signup and view all the answers

In a proof using the (∧E) rule, what must be present before applying the rule?

A line with the formula A ∧ B (D) Signup and view all the answers

What is a consequence of A1, ..., An ̸|= B?

It indicates A1, ..., An does not imply B (D) Signup and view all the answers

What does the (→E) rule enable you to do?

Deduce a conclusion from a conditional and its antecedent (D) Signup and view all the answers

What does it mean if A1, ..., An ̸⊢ B?

There is no proof found for B from A1, ..., An (B) Signup and view all the answers

Which statement is true regarding the application of the (∧E) rule?

It needs A ∧ B as a premise in the proof (C) Signup and view all the answers

Which of these is the correct sequence of applying rules to prove C from the premises A ∧ B and A → C ∧ D?

<ol> <li>A ∧ B, 2. A, 3. C ∧ D, 4. C (B)</li> </ol> Signup and view all the answers

What is a primary characteristic of a formal proof?

It includes a sequence of justified steps. (D) Signup and view all the answers

What symbol is used to denote a consequence relation between a set of formulas and a formula?

⊢ (B) Signup and view all the answers

What is the implication of using truth-tables to evaluate validity?

They can involve checking an exponential number of rows. (D) Signup and view all the answers

In a forward reasoning approach, what characteristic is emphasized?

Obtaining conclusions through a sequence of manipulations. (A) Signup and view all the answers

Which of the following best describes an informal proof?

A simplified explanation often missing key justifications. (B) Signup and view all the answers

Which two premises were identified as key to ascertaining the conclusion in the motivating example?

P and P→Q (A) Signup and view all the answers

What type of argument structure does a formal proof utilize?

A finite sequence involving axioms and inference rules. (B) Signup and view all the answers

Why is it often more useful to visualize how conclusions follow from premises?

It can simplify complex logical evaluations. (C) Signup and view all the answers

What does the deduction theorem state about the proof of B using A?

There is a proof of B that does not depend on A itself. (C) Signup and view all the answers

In the example provided, which implication is being demonstrated?

If n is an odd integer, then n2 is odd. (B) Signup and view all the answers

Which step in the example directly employs the property of odd integers?

Step 2 (D) Signup and view all the answers

What must be true about the assumptions in a proof according to the notes about proofs?

Only premises are allowed as assumptions in a proof. (C) Signup and view all the answers

Which step in the example represents the conclusion being derived from the prior assumptions?

Step 5 (D) Signup and view all the answers

What does the expression $S ∪ {A} ⊢ B$ signify in the deduction theorem?

B can be concluded using premise A and any subset of S. (D) Signup and view all the answers

What follows from applying modus ponens to lines 1 and 2?

Line 3 is accepted as true. (C) Signup and view all the answers

What implication is established if N1 is assumed true?

N1 directly leads to the proof of N2 being odd. (D) Signup and view all the answers

What type of logical fallacy is represented by the argument "If P then M, M is true, therefore P is true"?

Fallacy of affirming the conclusion (C) Signup and view all the answers

Which statement correctly describes the argument structure of "If P then M, P is false, therefore M is false"?

It commits the fallacy of denying the hypothesis. (B) Signup and view all the answers

What does the fallacy of affirming the conclusion imply about an implication's antecedent when the consequent is true?

It does not guarantee the truth of the antecedent. (B) Signup and view all the answers

Why is the formula $((P→M) ∧ ¬P)→¬M$ considered a fallacy?

The premises do not guarantee the conclusion. (B) Signup and view all the answers

Which statement correctly identifies a flaw in the argument: "If you do every problem in this book (P), then you will learn discrete mathematics (M). You did not do every problem. Therefore, you did not learn discrete mathematics."?

The argument does not consider other ways to learn. (A) Signup and view all the answers

In logical reasoning, what does denying the antecedent of an implication indicate?

The implication remains true. (D) Signup and view all the answers

What is a counter-model for the argument that 'you did every problem in this book (P) therefore you learned discrete mathematics (M)'?

P is false and M is true. (A) Signup and view all the answers

Which of the following illustrates the fallacy of denying the hypothesis accurately?

If A, then B; A is false, therefore we cannot conclude about B. (D) Signup and view all the answers

What characterizes the fallacy of affirming the conclusion?

The truth of the conclusion guarantees the truth of the antecedent. (B) Signup and view all the answers

What is the implication of the statement 'If Odinaldo was born in Britain, then he is a British citizen' followed by 'Odinaldo is a British citizen'?

Odinaldo could have gained citizenship through other means. (D) Signup and view all the answers

Flashcards

Formal Proof

A proof that derives a conclusion from a set of premises using a sequence of valid transformations and relies on a system that fully agrees with the semantical notion of logical consequence.

Informal Proof

A proof that may skip steps, use axioms, theorems, or borrow established results. It doesn't require a fully defined system and can be more intuitive.

Direct Proof

A proof that directly applies valid inference steps to show how the conclusion follows from the assumptions.