Exam 2: Symbolic Logic Assessment Guidelines and Overview

Exam 2 Due Oct 2 at 11:59pm Points 40 Questions 17 Available until Oct 2 at 11:59pm Time Limit 120 Minutes Instructions This quiz was locked Oct 2 at 11:59pm. Attempt History Attempt Time Score LATEST Attempt 1 99 minutes 21 out of 40 Score for this quiz: 21 out of 40 Submitted Oct 2 at 11:35pm This attempt took 99 minutes. Please read the following instructions before you begin: Exams evaluate your comprehensive of the material. Exam 2 covers the material and activities in Lesson 4 . • You should not take this assessment on a mobile device. • Once you begin, you must complete the assessment in one sitting. The clock will start when you begin and will continue to run even if you go back to another page or log out of CANVAS. Once time has expired, the assessment will be automatically submitted, even if you have not answered all the questions. • You will only be able to see your assessment responses immediately after you complete the assessment, but you may not receive a final grade for some exams require until your instructor has graded the relevant portions. • If there is a file-upload associated with this assessment, you must upload it as a PDF. • If you are asked to input logical notation, do not paste symbols unless directed. While taking any assessment in this course, you should not: (i) ask other students for assistance or (ii) solicit the help of a tutor. 0 / 2 pts Question 1 Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 1 of 15 10/13/22, 1:57 PM

What decomposition rule should you use on the following wff: ou Answered ou Answered orrect Answer orrect Answer 2 / 2 pts Question 2 What decomposition rule should you use on the following wff: Correct! Correct! 2 / 2 pts Question 3 What decomposition rule should you use on the following wff: Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 2 of 15 10/13/22, 1:57 PM

Correct! Correct! 0 / 2 pts Question 4 What decomposition rule should you use on the following wff: orrect Answer orrect Answer ou Answered ou Answered 2 / 2 pts Question 5 What decomposition rule should you use on the following wff: Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 3 of 15 10/13/22, 1:57 PM

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Correct! Correct! 0 / 2 pts Question 6 Suppose we were to test the following argument to determine whether or not it is deductively valid (a case of semantic entailment): . The first step of using the truth-tree test is to setup the tree into what is called the stack''. Which one of the following formulas would NOT be in the initial stack: ou Answered ou Answered orrect Answer orrect Answer Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 4 of 15 10/13/22, 1:57 PM

When using the truth-tree method to test an argument for deductive validity, the tree is formed by stacking the premises and the literal negation of the conclusion. The non-negated version of the conclusion should NOT appear in the stack. The literal negation of the conclusion R is ¬ R . This means that ¬ R should appear in the stack. Since each of the other options here is one of the premises of the argument, each of them should be in the stack as well. A truth tree is a means of checking whether there is some way of assigning a truth value to each of the atomic propositions that occurs in the stack (in this case, P , Q , S , M , and R ) such that all of the propositions in the stack turn out to be true. By including the negation of the conclusion, we are using the truth tree to test whether it is possible to assign these values in such a way that the premises and the negation of the conclusion are true. In other words, we are using it to test whether it is possible for the premises to be true and the conclusion false. For more on how to set up the truth tree, see PL Truth Trees - Part 1 ( https://youtu.be/YdTTdArmUKA ) . For more on how to use the truth-tree method to determine if a PL argument is deductively valid, see PL Truth Trees - Validity ( https://youtu.be/l3bqzTixAls ) and see Agler Symbolic Logic (1st ed.), pp.148-150. For more on literal negation, see PL Syntax Part 3 (Literal negation) ( https://youtu.be /8QUaFqIkwDQ?t=2m4s ) and Agler Symbolic Logic (1st ed.), pp.42-43. 0 / 2 pts Question 7 Suppose we were to test the following argument to determine whether or not it is deductively valid (a case of semantic entailment): . The first step of using the truth- tree test is to setup the tree into what is called the stack''. Which one of the following formulas would NOT be in the initial stack: Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 5 of 15 10/13/22, 1:57 PM

orrect Answer orrect Answer ou Answered ou Answered When using the truth-tree method to test an argument for deductive validity, the tree is formed by stacking the premises and the literal negation of the conclusion. The literal negation of the conclusion ¬ P ∨ ¬ R is ¬ ( ¬ P ∨ ¬ R ) and not P ∨ R . To form the literal negation of a formula, negate the entire formula. A truth tree is a means of checking whether there is some way of assigning a truth value to each of the atomic propositions that occurs in the stack (in this case, just P ) such that all of the propositions in the stack turn out to be true. By including the negation of the conclusion, we are using the truth tree to test whether it is possible to assign these values in such a way that the premises and the negation of the conclusion are true. In other words, we are using it to test whether it is possible for the premises to be true and the conclusion false. Since this is a zero-premise argument, the test amounts to a test to see if it is possible for the conclusion to be false. In other words, it amounts to a test of whether or not the conclusion is a tautology. For more on how to set up the truth tree, see PL Truth Trees - Part 1 ( https://youtu.be/YdTTdArmUKA ) . For more on how to use the truth-tree method to determine if a PL argument is deductively valid, see PL Truth Trees - Validity ( https://youtu.be /l3bqzTixAls ) and see Agler Symbolic Logic (1st ed.), pp.148-150. For more on literal negation, see PL Syntax Part 3 (Literal negation) ( https://youtu.be/8QUaFqIkwDQ?t=2m4s) and Agler Symbolic Logic (1st ed.), pp.42-43. Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 6 of 15 10/13/22, 1:57 PM

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

2 / 2 pts Question 8 Answer 1: Answer 2: Answer 3: (To answer the following question, type upper case T or F): Assign an interpretation to T , F , and F assuming the following literal wffs were recovered from a completed open branch of a truth tree: T Correct! Correct! F Correct! Correct! F Correct! Correct! 2 / 2 pts Question 9 The following argument is invalid: . Which assignment of truth values to the propositional letters (or their literal negations) shows the argument to be invalid? Use the truth-tree procedure (using either Proof Tools or pen/pencil and paper) to help you determine the answer. In other words, use the truth-tree method to pick out a valuation that shows that the argument is deductively invalid. Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 7 of 15 10/13/22, 1:57 PM

. Correct! Correct! 2 / 2 pts Question 10 The following argument is invalid: . Which assignment of truth values to the propositional letters (or their literal negations) shows the argument to be invalid? Use the truth-tree procedure (using either Proof Tools or pen/pencil and paper) to help you determine the answer. In other words, use the truth-tree method to pick out a valuation that shows that the argument is deductively invalid. . Correct! Correct! 0 / 2 pts Question 11 Use the truth-tree decision procedure (relying on Proof Tools or pen/pencil and paper) to determine whether is deductively valid. invalid. ou Answered ou Answered Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 8 of 15 10/13/22, 1:57 PM

valid. orrect Answer orrect Answer A truth-tree test reveals a tree with all closed branches. What this means is that there is no way of assigning truth values (valuations) to the propositional letters so that all of the propositions in the stack are true. In testing for validity, we test a stack of propositions consisting of the premises R ∧ S , R ∧ ¬ T , ( R ∨ S ) → ¬ S and the negation of the conclusion, i.e. ¬ ( ( R ∨ T ) ↔ ( S ∨ ¬ T ) ) . As there is no way to assign truth values to the propositional letters in this set of propositions so that they are all true, it is not possible for the premises and the negation of the conclusion to be true. In other words, it is not possible for the premises to be true and the conclusion false. Thus, the argument is valid. POINTS: 2 2 / 2 pts Question 12 Use the truth-tree decision procedure (relying on Proof Tools or pen/pencil and paper) to determine whether is deductively valid. invalid. Correct! Correct! Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 9 of 15 10/13/22, 1:57 PM

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

0 / 2 pts Question 13 Use the truth-tree decision procedure (relying on Proof Tools or pen/pencil and paper) to determine whether is deductively valid. valid. ou Answered ou Answered invalid. orrect Answer orrect Answer Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 10 of 15 10/13/22, 1:57 PM

A truth-tree test for the argument shows this argument to be invalid. Decomposing the stack that consists of the premises and the negation of the conclusion yields a tree with at least one completed open branch. The branch where v ( Q ) = T and v ( ¬ P ) = T shows us that it is possible for all the premises to be true while the conclusion is false. Translated into English P → Q , Q ⊢ P is "If P then Q , Q , therefore P ." This argument is deductively invalid as it is possible for the conclusion " P " to be false while the premises "If P then Q " and " Q " are true. Think about the following as an argument that provides a counter-example: If it is snowing, then it is cold, it is cold, therefore it is snowing. Of course, even if it snows only when it is cold, it is still possible to have a very cold day without snow. Combining the general rule that 'If it is snowing, then it is cold' with the fact that 'it is cold', does not allow us to conclude legitimately that it is snowing. It might be snowing under these conditions, but, then again, it also might not be snowing under these conditions. This invalid argument form is often confused with the valid argument form P → Q , P ⊢ Q . The invalid form is called "affirming the consequent", and is referred to as a formal fallacy. The valid argument form is called "affirming the antecedent" (or sometimes modus ponens, from the Latin for "the way of affirming"). "Affirming the antecedent" or modus ponens is a trusted rule for use in deductive logic. To see why, use a truth-tree to test it for validity, then compare your results to the test for the invalid form "affirming the consequent". 2 / 2 pts Question 14 Under what condition does the truth-tree test show an argument to be valid? if and only if determines a closed tree. Correct! Correct! Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 11 of 15 10/13/22, 1:57 PM

2 / 2 pts Question 15 Under what condition does the truth-tree test show a set (collection) of propositions to be consistent? under the condition that determines a completed open tree. Correct! Correct! 2 / 5 pts Question 16 This question asks you a conceptual question concerning two semantic properties of propositional logic. First, assume that are all complex well-formed formulas (wffs). Next, assume that . That is, assume that semantically entails . Given this assumption, is a tautology? Explain your reasoning. Note 1: are propositional variables (metavariables) for Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 12 of 15 10/13/22, 1:57 PM

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Your Answer: propositional logic wffs. Note 2: You should try to reason from the assumption that to whether is a tautology or not a tautology. You should not deny since this is being assumed. If then proved tautology. This means that would never have be false and be true. It would only be the case if and are true but . However due to our original the previous statement is false. Yes, if , then is a tautology. Proof: If , then there is no interpretation where are true and is false. If there is no interpretation where are true and is false, then is a tautology since is false if and only if and are true and is false (but this is ruled out by ). And so, if is not false under any permissible interpretation, is true under every interpretation. For more on truth trees and validity, see Propositional Logic Truth Trees, Part 8 (Validity) - ( https://youtu.be/l3bqzTixAls) 1 / 5 pts Question 17 This question involves two parts. 1. Translate the argument provided in this prompt into formal logic and then use the truth-tree decision procedure (relying on Proof Tools or pencil/pen and paper) to determine whether the argument is deductively valid or invalid (entailment / non-entailment). 2. If the argument is invalid (a case of non-entailment), determine an assignment of truth values (interpretation) to the propositional Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 13 of 15 10/13/22, 1:57 PM

Your Answer: letters that would show the argument to be invalid (non-entailment). Here is the argument: John is happy if and only if John is both rich and in love. If John is in love, then he is rich. John is rich. Therefore, John is happy. H=John is happy R=John is rich L=John is in love (H (R L)) L (R H) Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 14 of 15 10/13/22, 1:57 PM

H ↔ ( R ∧ L ) , L → R , R ⊢ H . Invalid. Shown to be invalid by: v ( R ) = T , v ( H ) = F , v ( L ) = F . For details on translation , see Propositional Logic: Translation, P1 (Atomic and Negated Wffs) ( https://youtu.be/J2byFuZEGFI) Propositional Logic; Translation, P2 (Conjunctions) ( https://youtu.be/YWSYX1azWfk) Propositional Logic; Translation, P3 (Disjunctions) ( https://youtu.be/XnuFcahdYSI) Propositional Logic; Translation, P4 (Conditionals) ( https://youtu.be/YHRGKQ13UiU) Propositional Logic: Translation, P5 (Biconditionals) ( https://youtu.be/BvHHDJKi6WY) Propositional Logic: Translation, P6 (More Complex Translations) ( https://youtu.be/wJqzWCF4KmY) For details on determining if an argument is valid or invalid (entailment / non-entailment) and for how to recover an interpretation from a completed open branch, see Propositional Logic Truth Trees, Part 8 (Validity) ( https://youtu.be/l3bqzTixAls ) Propositional Logic: Truth Trees, Part 4 (Recovering an Interpretation) ( https://youtu.be/_106Le_WBKs) Quiz Score: 21 out of 40 Firefox https://psu.instructure.com/courses/2204563/quizzes/4533223?headless=1 15 of 15 10/13/22, 1:57 PM

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Exam 2 [22FA] PHIL 12, Sec 1 Symbolic Logic (Fisher)

Related Documents