Propositional Logic

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Questions and Answers

Given a domain of discourse comprising exclusively meromorphic functions on the Riemann sphere, the proposition $\forall f \exists g : f \circ g = z$, where $f \circ g$ denotes functional composition and $z$ is the identity function, constitutes a veridical assertion.

False (B)

Within the framework of Zermelo-Fraenkel set theory with the axiom of choice (ZFC), the assertion that every non-empty set admits a choice function is logically independent of the remaining axioms of ZF.

False (B)

The validity of an argument exhibiting the formal structure of modus ponens is contingent upon the semantic interpretation of the propositional variables involved.

False (B)

In the context of intuitionistic logic, the proposition $P \lor \neg P$ (the law of excluded middle) is universally affirmed for all propositions $P$.

<p>False (B)</p> Signup and view all the answers

Affirming the consequent, wherein one deduces the antecedent from the consequent and an implicative premise, constitutes a fallacious argument form exclusively within classical bivalent logic; multi-valued logics preclude such fallacies.

<p>False (B)</p> Signup and view all the answers

Within a first-order theory encompassing Peano Arithmetic (PA), Gödel's incompleteness theorems demonstrably preclude the formal derivation of every arithmetical truth.

<p>True (A)</p> Signup and view all the answers

In a Venn diagram depicting sets A, B, and C within a universal set U, the shaded region representing $(A \cap B) \cup (C - A)$ is necessarily disjoint from the region representing $(A \cap C) - B$.

<p>False (B)</p> Signup and view all the answers

The application of De Morgan's Laws to the quantified statement $\neg \forall x (P(x) \rightarrow Q(x))$ invariably yields the logically equivalent statement $\exists x (P(x) \land \neg Q(x))$, irrespective of the domain of discourse.

<p>True (A)</p> Signup and view all the answers

Under the assumption of the continuum hypothesis, there exists a surjection $f: \mathbb{N} \rightarrow \mathbb{R}$, where $\mathbb{N}$ denotes the set of natural numbers and $\mathbb{R}$ represents the set of real numbers.

<p>False (B)</p> Signup and view all the answers

An argument exhibiting the structure of an appeal to authority is rendered inherently fallacious, irrespective of the cited authority's expertise, relevance, and potential biases concerning the argument's conclusion.

<p>False (B)</p> Signup and view all the answers

Flashcards

Proposition

A declarative sentence that is either true or false, but not both.

Logical Connective

Connects propositions to form compound propositions.

Negation (¬)

“Not P”. True if P is false, and false if P is true.

Conjunction (∧)

“P and Q”. True only if both P and Q are true; otherwise, false.

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Disjunction (∨)

“P or Q”. True if either P or Q (or both) are true; false only if both are false.

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Implication (→)

“If P, then Q”. False only if P is true and Q is false; otherwise, true.

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Biconditional (↔)

“P if and only if Q”. True if P and Q have the same truth value; false otherwise.

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Tautology

Always true, regardless of the truth values of its variables.

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Contradiction

Always false, regardless of the truth values of its variables.

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Predicate

A statement that contains variables and may be true or false depending on the values of those variables.

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Study Notes

  • Logic is the study of reasoning and argumentation
  • Used to determine good and bad arguments

Propositional Logic

  • Studies statements and their logical relationships
  • A proposition is a declarative sentence that can be either true or false, but not both
  • Propositional variables represent propositions (e.g., P, Q, R)
  • Logical connectives combine propositions to form compound propositions

Common Logical Connectives

  • Negation (¬): "not P" - true if P is false, and false if P is true
  • Conjunction (∧): "P and Q" - true if both P and Q are true; otherwise, false
  • Disjunction (∨): "P or Q" - true if either P or Q (or both) are true; false only if both are false
  • Implication (→): "if P, then Q" or "P implies Q" - false only if P is true and Q is false; otherwise, true; P is the hypothesis/antecedent, Q is the conclusion/consequent
  • Biconditional (↔): "P if and only if Q" or "P is equivalent to Q" - true if P and Q have the same truth value (both true or both false); false otherwise

Truth Tables

  • Used to define the meaning of logical connectives by specifying the truth value of a compound proposition for all possible truth values of its constituent propositions
  • Negation:
    • P | ¬P
    • T | F
    • F | T
  • Conjunction:
    • P | Q | P ∧ Q
    • T | T | T
    • T | F | F
    • F | T | F
    • F | F | F
  • Disjunction:
    • P | Q | P ∨ Q
    • T | T | T
    • T | F | T
    • F | T | T
    • F | F | F
  • Implication:
    • P | Q | P → Q
    • T | T | T
    • T | F | F
    • F | T | T
    • F | F | T
  • Biconditional:
    • P | Q | P ↔ Q
    • T | T | T
    • T | F | F
    • F | T | F
    • F | F | T

Tautology, Contradiction, and Contingency

  • Tautology: A compound proposition that is always true, regardless of the truth values of its variables
  • Contradiction: A compound proposition that is always false, regardless of the truth values of its variables
  • Contingency: A compound proposition that is neither a tautology nor a contradiction

Logical Equivalence

  • Two propositions are logically equivalent if they have the same truth value in all possible cases
  • Can be shown using truth tables or logical equivalences (e.g., De Morgan's Laws)
  • De Morgan's Laws:
    • ¬(P ∧ Q) ≡ ¬P ∨ ¬Q
    • ¬(P ∨ Q) ≡ ¬P ∧ ¬Q

Predicate Logic

  • Extends propositional logic to include predicates, variables, quantifiers, and objects
  • Predicate: A statement that contains variables and may be true or false depending on the values of those variables, it defines a property or relationship
  • Variable: A symbol representing an object or entity
  • Quantifier: Expresses the extent to which a predicate is true over a range of elements

Types of Quantifiers

  • Universal Quantifier (∀): "for all" - ∀x P(x) means P(x) is true for every x in the domain
  • Existential Quantifier (∃): "there exists" - ∃x P(x) means P(x) is true for at least one x in the domain

Negating Quantified Statements

  • ¬(∀x P(x)) ≡ ∃x ¬P(x) (Not all x have property P is equivalent to: There exists an x that does not have property P)
  • ¬(∃x P(x)) ≡ ∀x ¬P(x) (It is not the case that there exists an x with property P is equivalent to: All x do not have property P)

Logical Fallacies

  • Errors in reasoning that invalidate an argument
  • Formal Fallacy: A defect in the structure of an argument which makes the argument invalid
  • Informal Fallacy: A defect in the content of an argument which makes the argument weak

Common Logical Fallacies

  • Ad Hominem: Attacking the person making the argument instead of the argument itself
  • Appeal to Authority: Arguing that a statement is true because an authority figure said it, without providing other evidence
  • Appeal to Ignorance: Arguing that a statement is true because it has not been proven false, or vice versa
  • Bandwagon Fallacy: Arguing that something is true because it is popular
  • Straw Man: Misrepresenting an opponent's argument to make it easier to attack
  • False Dilemma: Presenting only two options when more exist
  • Hasty Generalization: Drawing a conclusion based on insufficient evidence
  • Post Hoc Ergo Propter Hoc: Assuming that because one event preceded another, the first event caused the second

Deductive Reasoning

  • Starts with general statements (premises) and reaches a specific conclusion
  • If the premises are true and the argument is valid, the conclusion must be true
  • Example:
    • Premise 1: All men are mortal.
    • Premise 2: Socrates is a man.
    • Conclusion: Therefore, Socrates is mortal.

Inductive Reasoning

  • Starts with specific observations and arrives at a general conclusion
  • The conclusion is likely but not guaranteed to be true, even if the premises are true
  • Example:
    • Observation: Every swan I have seen is white.
    • Conclusion: Therefore, all swans are white. (This is false, as black swans exist)
  • Inductive reasoning is used to form hypotheses and theories

Venn Diagrams

  • Diagram that utilizes overlapping circles to represent sets and their relationships
  • The universal set is typically represented by a rectangle enclosing all sets under consideration.
  • Each circle represents a set, and the overlapping areas represent the intersection of sets

Shading in Venn Diagrams

  • Shading is used to represent set operations visually
  • A ∪ B (A union B): Shade everything in set A and set B
  • A ∩ B (A intersection B): Shade the region where set A and set B overlap
  • A' (Complement of A): Shade everything outside of set A within the universal set
  • A - B (A minus B): Shade the region of A that does not overlap with B
  • Shading can be used to determine set identities and simplify set expressions

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