Logic Course: Propositional and Modal Logic
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Questions and Answers

Which of the following models demonstrates the invalidity of the formula K 2φ Ñ 3φ?

  • A model where 2φ is true but 3φ is false. (correct)
  • A model where both 2φ and 3φ are true.
  • A model where both 2φ and 3φ are false.
  • A model where 2φ is false while 3φ is true.

    • In classical propositional logic, which statement is true regarding the premises and conclusion relationship?

  • The conclusion is always a semantic consequence of its premises. (correct)
  • No false premises can lead to a true conclusion.
  • Some premises can be false under certain interpretations.
  • All interpretations lead to the same set of false premises.

    • Which of these is a characteristic of Kleene's three-valued logic?

  • It exclusively uses true and false values.
  • It concludes that every argument is valid.
  • It incorporates an additional value representing indeterminacy. (correct)
  • It is not useful for semantic arguments.

    • What does the abbreviation Cφ represent?

    <p>3φ ^ 3„φ</p> Signup and view all the answers

    In Priorean tense logic, what is the conclusion of the formula φ Ñ HFφ?

    <p>That if φ is true now, it will be true at some future moment.</p> Signup and view all the answers

    Which of the following is a valid formula in the logic of paradox (LP)?

    <p>φ ^ ¬φ</p> Signup and view all the answers

    What is the primary concern when establishing validities in modal logics?

    <p>Considering non-standard interpretations and constraints.</p> Signup and view all the answers

    What does the formula D 2φ Ñ 3φ imply about the relationship between necessity and possibility?

    <p>Necessarily 2φ guarantees that φ is possible.</p> Signup and view all the answers

    What condition must be satisfied for $VI(pCφ, wq)$ to equal 1 in terms of $VI(pφ, uq)$?

    <p>The conditions of $u$ must imply $w$.</p> Signup and view all the answers

    What should students who have not studied Elements of Deductive Logic do regarding the ‹’ questions?

    <p>Omit them on their first pass</p> Signup and view all the answers

    Which statement fails when replacing S5 with S4?

    <p>$S5 Cφ Ñ „CCφ$</p> Signup and view all the answers

    If $O1$ expresses the same modality as $O2$ in S, what can be concluded about their concatenated strings $OO1$ and $OO2$?

    <p>$O1$ and $O2$ can be different but must produce the same modality.</p> Signup and view all the answers

    Which of the following expressions does VI pφ Ñ ψq equal to when it is true?

    <p>VI pφq = 0 or VI pψq = 1</p> Signup and view all the answers

    In propositional logic, what does the expression VI p„φq = 1 indicate?

    <p>φ is false</p> Signup and view all the answers

    How many modalities are expressed by strings in S5?

    <p>Exactly two</p> Signup and view all the answers

    What implication does the interpretation I ` have on any sentence φ with no negation?

    <p>$VI`(φ) = 1$</p> Signup and view all the answers

    What is the truth condition for the ‘Peirce arrow’ Ó in propositional logic?

    <p>VI pφ Ó ψq = 1 iff VI pφq = 0 and VI pψq = 0</p> Signup and view all the answers

    Which of the following represents a semantic consequence in propositional logic?

    <p>(PL pφ Ñ pψ Ñ χqq Ñ ppφ Ñ ψq Ñ pφ Ñ χqq</p> Signup and view all the answers

    What can be said about the occurrence of parentheses in an MPL-sentence?

    <p>It is twice the number of occurrences of Ñ.</p> Signup and view all the answers

    When using Sider’s presentation of the semantics of propositional logic, which condition holds for conjunction?

    <p>VI pφ ^ ψq = 1 iff VI pφq = 1 and VI pψq = 1</p> Signup and view all the answers

    Which of the following is a true statement about the axiomatic proofs required for PL?

    <p>Each proof must confirm the structure of logical implications.</p> Signup and view all the answers

    Which statement represents an interpretation that includes negation?

    <p>A sentence with at least one occurrence of negation.</p> Signup and view all the answers

    If φ contains at most one occurrence of any sentence letter, what does PL φ represent?

    <p>A proposition that ensures unique implication</p> Signup and view all the answers

    What is the relationship between the truth conditions provided and Halbach’s presentation in The Logic Manual?

    <p>Halbach presents a more complex version</p> Signup and view all the answers

    Which of the following binary generalized quantifiers can be symbolized in L“?

    <p>Most α: φψ</p> Signup and view all the answers

    What condition must be met for the quantifier 'Finitely many α: φψ' to be true?

    <p>|φM,g,α X ψM,g,α | is finite</p> Signup and view all the answers

    Which of the following identifies a valid schema in SQML?

    <p>Dα2φ Ñ 2Dαφ</p> Signup and view all the answers

    In the context of semantic equivalence, what is required for two sentences?

    <p>Both sentences must be true in the same models.</p> Signup and view all the answers

    Which of the following statements about second-order logic is true?

    <p>Second-order logic can capture more generalized quantifiers than first-order logic.</p> Signup and view all the answers

    What does the symbol '2' typically represent in the schemas listed in SQML?

    <p>A necessity operator</p> Signup and view all the answers

    Which of the following helps justify whether a generalized quantifier can be symbolized in L“?

    <p>The truth values assigned in models</p> Signup and view all the answers

    What is a condition for a quantifier to be considered as capturing a certain meaning in logic?

    <p>It must remain true across all interpretations.</p> Signup and view all the answers

    What does the selection function f ensure in relation to the validity of the expression VM pφ € ψ, wq?

    <p>It provides a specific world where φ is false.</p> Signup and view all the answers

    Which of the following represents a semantic consequence for the material conditional?

    <p>ψ (SC φ Ñ ψ</p> Signup and view all the answers

    What is a non-consequence for Stalnaker’s conditional?

    <p>ψ *SC φ € ψ</p> Signup and view all the answers

    In the formalization of the argument regarding the coin flip, what condition is applied to conclude the argument?

    <p>It could have landed either heads or tails.</p> Signup and view all the answers

    Which logical system is argued as the correct logic for metaphysical necessity according to Salmon?

    <p>S5 is the correct logic.</p> Signup and view all the answers

    What does the countermodel in part (b) intends to demonstrate regarding the formal argument?

    <p>It indicates that the conclusion does not follow from the premises.</p> Signup and view all the answers

    Which of the following conditions fails to preserve truth in the English counterfactual conditional?

    <p>φ € ψ *SC pφ ^ χq</p> Signup and view all the answers

    What does the expression pφ ∧ ψq denote in the context of semantic non-consequences for Stalnaker’s conditional?

    <p>It indicates a false condition for the implication.</p> Signup and view all the answers

    Study Notes

    • Students who have not read Elements of Deductive Logic (EDL) should skip the questions marked ‹'
    • Students who have read EDL should attempt the questions marked ‹', but may skip the questions marked :
    • The : and ‹ markings will become less common as the term progresses, as the material will bridge the gap between EDL and the current material.

    Propositional Logic

    • Students should read the provided chapters from "Vagueness" by Tim Williamson and "Sorites Paradoxes and the Semantics of Vagueness" by Michael Tye to prepare for later questions in the course.

    Variations of PL

    • The "Peirce arrow" (Ó) is a connective where VI pφ Ó ψq “ 1 iff VI pφq “ 0 and VI pψq “ 0
    • The "Peirce arrow" (Ó) is also known as "nor"

    Semantics for MPL

    • K, D, T, B, S4, and S5 are modal logic systems
    • The semantics for these systems should be explored through informal semantic arguments

    Tense Logic

    • "Cφ" is an abbreviation for 3φ ^ 3„φ
    • "Cφ" is called "C"
    • The semantics of "C" are determined by the value of VI pφ, uq ### Induction on Complexity
    • The number of occurrences of parentheses in an MPL-sentence is twice the number of occurrences of Ñ

    Axiomatic Proofs in PL

    • PL P Ñ pP Ñ P q is a valid axiomatic proof
    • PL P Ñ P is a valid axiomatic proof
    • PL p„P Ñ P q Ñ P is a valid axiomatic proof

    The Logic of Metaphysical Necessity - Task A

    • Students should read the chapters from "The Logic of What Might Have Been" by Nathan Salmon and sections 3.1–3.3 from "Modal Logic as Metaphysics" by Tim Williamson

    Task B

    • Students should consider if second-order logic can be seen as "logic"

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    Description

    This quiz covers key concepts from Propositional Logic and Semantics, including the Peirce arrow and various modal logic systems. Students are encouraged to reference 'Elements of Deductive Logic' and provided readings for a comprehensive understanding. Prepare for questions related to tense logic and the nuances of vagueness.

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