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Q1: PHIL 37 Symbolic Logic Summer I 2023 Logic Problem Set #5 Read the following arguments. If there is an informal fallacy in the argument, identify it. If there is no fallacy, then state "no fallacy." 1. During years in which there is more pasta exported from Italy, there also tends to be fewer traffic crashes in the midwestern United States. Therefore pasta can reduce traffic injuries. 2. Jorge said that he has an issue with an increase on the sales tax for sugary drinks, so he must be a conservative Republican. Everyone either approves of more taxes or they are conservative Republicans. 3. You cannot trust the lectures from Professor Musa. She does not dress like a proper academic. 4. Lying is always wrong because it is always wrong deliberately to deceive someone into believing something that is not true. 5. It is true that a high-fiber diet is healthy for the general population because all dietitians agree on this fact. 6. People who want to build bike lanes think that it will solve the environmental crisis. It will not solve the crisis because there are other sorts of harmful emissions apart from cars. 7. Classical moral philosophy teaches that we should seek the good. My uncle thinks that milkshakes from McBurgerPrince are good. Therefore we should all seek milkshakes from McBurgerPrince. 8. Taylor Swift endorsed Jenna Liesalot in the electoral race for house representative, so you should vote for Jenna! 9. The more time spent on social media is correlated with a lower grade point average. Therefore social media can have a negative causal impact on your grades. 10. Nadia said that America is a free country. But she's wrong about that. It is in fact very expensive to live in America.See Answer
Q2:Using the provided keys, symbolize the following English sentences into PL sentences. On your assignment, include your name and student number. Be sure that your assignment is legible, and that you show your work to receive full marks. You may submit either handwritten work (jpg, PNG, etc.), or (preferably) typed up in Word and submitted as a .docx or a PDF. If you're typing it up, it is highly recommended that you do your work by hand first. You can find a tutorial on how to input connective symbols in Word at the top of the main Moodle page, right below the syllabus. Hint: we did not discuss this in class, but a sentence that has a reference like "a positive integer" is quantified with a universal even though the indefinite article ("a") might suggest otherwise, because it is referencing positive integers in a general sense. Part A UD The set (Brandon University faculty} Tx x is a tenured professor Sx x is a sessional instructor Ax x is a member of the Arts Faculty Ex x is a member of the Education Faculty Cxy x co-teaches with y 1. Not all sessional instructors are members of the Education Faculty. 2. No sessional instructors are members of both the Arts and the Education Faculties. 3. All Arts Faculty members are either tenured professors or sessional instructors. 4. Some Arts Faculty members don't co-teach with any Education Faculty members. 5. Some sessional instructors in the Arts Faculty co-teach with each other. →→→Part B is on the next page! The set (all positive integers) Part B UD Px x is prime Ox x is odd Ex x is even Gxy x is greater than y Lxy x is less than y Dxy x is evenly divisible by y Txyz Sxyz x is the product of y and z x is the sum of y and z a 1 b 2 6. For each prime, there is a greater non-prime. 7. A positive integer is even if it is evenly divisible by 1, 2, and itself. 8. There is no even prime positive integer that is not evenly divisible by 1, 2, and itself. 9. The product of a prime and an even positive integer is not prime. 10. Every sum of any two positive integers is even if and only if those integers are both even or both odd.See Answer
Q3:Question 6: Using the indirect method, determine if the following statements are consistent or inconsistent. Keep in mind that you may have more rows than you need in this table. (B&A) (A C) [(A & B) C] (B & C) (A & B) Are these statements consistent? Type YES or NO here.See Answer
Q4:Question 1: Are these statements in the table below consistent? Solve the truth table below and determine consistency or inconsistency. A v (B C) (A Are these statements consistent? & B) с Type YES or NO here. B & O сSee Answer
Q5:5. Statement to evaluate: "If Mary doesn't sing, then Bob doesn't sing; and if Mary sings, then Don doesn't dance." Overall statement:See Answer
Q6:3. Statement to evaluate: "If Bob sings, or Mary sings, then Don dances." Overall statement:See Answer
Q7:1. Statement to evaluate: "Mary sings and Bob sings." Overall statement:See Answer
Q8: ment3 - Sp24 PC X e/content/6441397/viewContent/65399714/View Et For each argument listed on the next page, please type the statement in symbolic form using: for negation, & for conjuction, v for disjunction, an arrow -> for conditional, and a double arrow <-> for biconditional. Example: Tu POL221-Class3-Spring 2024.mp4 x | + If Bill and Marci clean the kitchen, then Jenny will vacuum the house. Statement: 1. Mary sings and Don dances. 2. Mary sings and Bob sings. 3. If Mary sings, Carlita dances. 4. It is not the case that if Mary sings, then Don dances. Download 5. If Bob sings, or Mary sings, then Carlita dances. 6. If Bob sings or Mary sings, then Don dances. Print hp a (B & M) -> J Symbolic form: P u/d21/le/content/6441397/viewContent/65399714/View 6: 7. If Don dances or Mary sings, then Carlita dances and Bob doesn't sing. 8. If Don dances, then Carlita dances; and if Bob sings, then Mary sings. 9. If Carlita dances, then Mary sings; and Carlita dances. 10. If Mary doesn't sing, then Bob doesn't sing; and if Mary sings, then Don doesn't dance. 11. If Bob doesn't sing, then Carlita dances, and if Mary doesn't sing, then Don dances. 12. Carlita dances if and only if Bob doesn't sing Download 13. Mary sings and Bob sings if and only if Carlita dances and Don dances. 14. Carlita dances, and she dances if and only if Bob sings or Don dances. 15. Carlita dances and Mary sings, if and only if Bob doesn't sing or Don doesn't dance. 16. Carlita dances and Mary sings, if and only if Bob doesn't sing and Bob doesn't sing O Print oces a ate.edu/d21/le/content/6441397/viewContent/65399714/View 100 Download 16. Carlita dances and Mary sings, if and only if Bob doesn't sing; and Bob doesn't sing. 17. If either Mary or Bob sing, then Carlita dances; but Don dances if and only if Carlita dances and Bob doesn't sing. 18. If, if Carlita dances then Don dances, then Don dances if and only if Mary doesn't sing. 19. It is not the case that Mary sings and Bob sings if and only if Carlita dances and Don, dances; however, Mary sings and Don dances if and only if Bob sings and Carlita dances. 20. If it is not the case that if Carlita dances, then Don dances, then it is not the case that either Mary sings and Bob sings or that Carlita doesn't dance and Don doesn't dance. Print g WSee Answer
Q9:Problem 3 Do Exercise 71 on page 46 in [EML.Chapter_4.pdf]. 2 For LEAN_4 exercises in this assignment, your solutions should be restricted to tactics and lemmas which are introduced up to the end of [MoP, Section 6.4]. We will take off points from solutions that use tactics and lemmas introduced in [MoP, Section 6.5] and later sections or nowhere used in [MoP]. import Mathlib.Data.Real. Basic import Library. Theory.Parity import Library. Theory. Comparison import Library. Theory. Prime import Library. Tactic. Induction import Library. Tactic. ModCases import Library. Tactic. Extra import Library. Tactic. Numbers import Library. Tactic. Addarith import Library. Tactic. Use import Mathlib. Tactic.GCongr import Library. Tactic. Cancel 0 In your solutions for Problem 4 and Problem 5 as LEAN_4 scripts, you may want to include the following imports from the LEAN_4 library (possibly not all of them, but you will have to experiment to find out if all are needed):See Answer
Q10:Problem 1 Let Σ = {R, f, g, co, c₁} be a first-order signature, where R is a binary relation symbol, f is a unary function symbol, g is a binary function symbol, and ₁ and ₁ are constant symbols. In this exercise you are asked to revisit parts of the proof of Theorem 58 relative to a particular E-structure A. More precisely, you have to show below that, given an arbitrary WFF ZOL (Σ, Ø): A = 4 ⇒ H = 4, where H is the Herbrand E-structure induced by A. We define A (A, RA, JA, GA, C4, C₁4) where A = Nand: RA = fA g-A = {(m, n) | m, n € N with m<n} {(m, m + 2) | m €N} {(m, n, m+n) | m, n {N} (R is the binary less-than relation on N) (fA is the two-successor function on N) (g4 is the addition function on N) and c = 0 and ₁ = 1. More succintly, we can therefore write A (N,<, fA, +, 0, 1) where we use < and + in infix position. There are four parts: 1. We want you to write an inductive definition of Terms(E, 0) in a particular way. The set Terms(E, ) is the smallest such that: Terms(Σ, Ø) 2 {co, c₁} U {f(t)|te Terms(Σ, Ø)} U ... You have to complete the definition. 2. Define the Herbrand E-structure H induced by A: H(Terms(Σ, Ø), R², f, g, co, c₁) As in the proof of Theorem 58, you only need to define the relation RH. 3. Let t₁,12 € Terms(E, Ø) be arbitrary terms. Define an algorithm that decides whether HRH (t₁, 1₂) or HR (t1, t2) depending on the syntactic structure of t1 and 12. Hint 1: An initial part of your algorithm transforms an arbitrary t Terms(Σ, Ø) into a term u of the form f(P) co or fe such that t = u, for some p>0, where we write f(P) to denote that f is applied p times. Hint 2: The rest of your algorithm decides whether HR (₁, 2) or H = R¹ (U₁, 1₂) where each of u₁ and u₂ is of the form fp) co or f()c₁. H by structural induction on the 4. Prove, for every WFF (E, Ø), that A 201 definition of p.See Answer
Q11:Problem 3 LCS, page 163: Exercise 2.4.6. For LEANĻĀ exercises in this assignment, your solutions should be restricted to tactics and lemmas which are introduced up to the end of [MoP, Section 6.1). We will take off points from solutions that use tactics and lemmas introduced in [MoP, Section 6.2] and later sections or nowhere used in [MoP]. In your solutions for Problem 4 and Problem 5 as LEAN 4 scripts, you may want to include the following imports from the LEAN-4 library (possibly not all of them, but you will have to experiment to find out if all are needed): import Mathlib.Data.Real Basic isport Library Theory Parity isport Library. Tactic. Induction isport Library. Tactic ModCases isport Library. Tactic Extra isport Library. Tactic Numbers isport Library. Tactic Addarith isport Library. Tactic. UseSee Answer
Q12:Problem 1 The divisibility predicate, the perfect square predicate, and the prime predicate, can be defined as follows: m|n perfectSq(n) prime(n) def def def true false true false true false if m is a divisor of n, if m is not a divisor of n. if there is m € N such that n = m², otherwise. if n is a prime number > 2, otherwise. There are two parts: (a) Show that perfect Sq: N→ {true, false) is first-order definable in the structure (N, I, +,0). Hint: z = y²iff x + y = lcm(y, y + 1). (b) Show that prime : N→ {true, false) is first-order definable in the structure (N, ], +,0).See Answer
Q13:Exercises 2.4 def * 1. Consider the formula of VrVy Q(g(x, y), g(y, y), z), where Q and g have arity 3 and 2, respectively. Find two models M and M' with respective environments I and I' such that MF, but M' .See Answer
Q14:In this assignment, you will use what you learn in Logic module to: Create a compound statement that contains at least 2 connectives (negation, conjunction, disjunction, conditional, biconditional). Express the compound statement in the symbolic form. Construct a truth table for the compound statement that you created.See Answer
Q15:36 The next proof makes explicit reference to notions in topology. As indicated in the preceding alternative proof, a truth assignment can be denoted by a path in the full binary tree Tall, now viewed as an w-sequence in the product space (T,F). (We write for the first infinite ordinal, which is the set of natural numbers listed in their standard order.) We view {T,F) as the underlying space of a product topology ({T,F)", O), thus making every truth assignment a "point" in that topology. O is a family of open sets that define the topology, which are in this case subsets of points in {T, F)~ and satisfy the usual requirements of a topology: • The empty set and the full space {T,F) are in O. • O is closed under arbitrary unions, • O is closed under finite intersections. We can define a subset UC {T,F}" to be open iff there is a finite set of indices IC such that: U = II { Ai | i €w and A; C {T,F}} where A. = {T,F) for every i Ew-I. In words, in the infinite product U = Ao × A₁ × --- × A; x--, for all but finitely many indices i it is the case that A; = {T,F). A set U C {T,F)" is closed iff it is the complement of an open APPENDIX F. ALTERNATIVE COMPACTNESS PROOFS set. In the case of the product topology, every open subset UC (T,F) is also closed, and thus called clopen. Let A be a subset of (T,F). An open covering of A is a family of open sets {U; | i I} CO such that A CU{U; i 1}. And A is said compact if every open covering of A has a finite subcovering; i.e., there exists a finite subfamily U₁₁, UU₁ of (U; | i I} such that A (U₁, UU, U...UU.). By Tychonoff's Theorem in topology, the product topology ({T,F), O) is compact, which means that every open covering of a subset of points AC {T,F) has a finite subcovering. Alternative Proof II of Theorem 2 (Compactness for Propositional Logic). Again here, we only need to consider the non-trivial implication "+": If I is finitely satisfiable, then I is satisfiable. For every propositional wffy, let A, C {T,F}" be the collection of all points/truth assignments that satisfy p. The set A, is a closed (and open) subset of {T,F}" in the topology ({T,F)",0), which follows from the fact that only mentions finitely many propositional variables. It is easy to check that, for every finite subset A of I, if A is satisfiable, then {A | A} is not empty. Hence, the family of closed sets {Apyr} satisfies the finite intersection property. Moreover, the product topology ({T.F),O) is compact, as noted above. Hence, {A, ET) is not empty, which is the desired conclusion. 0/nExercise 147. This exercise is couched in a language which is a little more familiar to computer scientists. Given a set IWFF₁(P), the binary tree T(I) induced by I' is defined in the 35 proof above. In T(I), every finite path is a finite sequence, and every infinite path is an infinite sequence, where all the entries are in {T,F). The set of all such finite sequences is denoted {T,F)", and the set of all such infinite sequences is denoted {T,F). The root node of T(I) is represented by the empty sequence e, every leaf node is connected to the root node by a finite path, and every infinite path is one that can be traversed without ever reaching a leaf node. An arbitrary binary tree can therefore be represented by a subset of {T, F}*U{T,F}", satisfying two conditions: • U is prefix-closed, i.e., for every (possibly infinite) path ₁ € {T,F)* U{T,F) and every finite path #₂ € {T,F)", if ₁ EU and ₂ is a prefix of ₁, then #₂ EU. • For every finite path = {T,F)", it holds that T U iff F EU, i.e., every non-leaf node has two immediate successors. Given an arbitrary IWFF (P), let T(T) be the subset of infinite paths in T(I): T(T) = T(r)n(T.F)". By the preceding proof, T. (T) ‡ Ø iff the set I is satisfiable. Let To, F₁, and I2, be defined as follows: To = {-P₁ ^P₂}, T₁ = {P₁ → (P₂A---A-PE) | A>2}, del 1₂ = {~-P₁ → (P²2 A---A-¬-Pk) |>2}. There are two parts in this exercise: 1. Define the sets of infinite paths T. (To), T.(₁UF₂), and T(Tour₁ UF₂) as subsets of {T,F). In your answers, use the notational conventions of regular languages and regular w-languages that we have used earlier in the presentation of this exercise. 2. For each of the three sets defined in part 1, explain how they indicate whether the corre- sponding set of wff's is satisfiable or not. You will find it useful to consult Figure F.1. 0See Answer
Q16:Exercise 144. Let R be a binary relation symbol and f a unary function symbol. def 1. Show that the sentence of Vr R(x, f(x)) → Vry R(x, y) is valid. Do it in two different ways: (a) proof-theoretically, p, using natural deduction, and (b) semantically, = 4. def 2. Show that the sentence Valy R(x,y) → Vr R(x, f(x)) is not valid. Note that is just the converse implication of . Hint: Try a semantic approach, i.e., show. You need to define a structure A so that the left-hand side of "" in is true in A but the right-hand side of "" is false in A. 3. Conclude that Vrly R(x, y) and Vr R(x, f(x)) are not equivalent first-order wff's. Remark: Despite the conclusion in part 3, Proposition 142 asserts Vray R(x, y) and Vr R(x, f(x)) are equisatisfiable, i.e., if there is a model for one, then there is a model for the other, and vice- versa. 42 Review the definition of o[a] in Section B.3.See Answer
Q17:Exercise 126 (Hilbert-styles vs. Natural Deduction). Read Exercise 125, without necessarily solving it, before you attempt this one. The preceding exercise proposes a strictly proof-theoretic approach to showing that a Hilbert-style proof system and a natural-deduction proof system have equal deductive power. In this exercise we consider an alternative semantic approach, which invokes Soundness and Completeness for both proof systems. Specifically, each of the two sys- tems as here presented is sound and complete relative to the standard (classical) semantics of propositional logic based on Boolean algebras. There are three parts in this exercise, the first two of which are just plans in outline to prove the equivalence of the two systems: 1. If I, then I by Soundness of the Hilbert system. By Completeness of the natural deduction system, it follows that I END 4. 2. If I END 4, then I by Soundness of the natural deduction system. By Completeness of the Hilbert system, it follows that I ₁9. Provide the details of the two preceding parts, given only in outline here, pointing out missing prerequisites (e.g., we do not prove Soundness and Completeness for the Hilbert system in these notes) and propose ways of filling the gaps and how to prove them. (We do not add subscripts "H" and "ND" to "", because the two systems are sound and complete relative to the same semantics.) 3. Compare and discuss the pros and the cons of the proof-theoretic approach in Exercise 125 and the approach in this Exercise 126 which makes a detour through semantics. 0See Answer
Q18:16 Exercise 27 (Queens Problem). The n-Queens Problem is the problem of placing n queens on an nxn chessboard so that no two queens can attack each other. A solution of the problem when n-6 is shown on the left of Figure 1.2. In this exercise we specify the requirements of a solution for the n-Queens Problem as a propositional wif , with one such wff for every n 4. (There are no solutions for n-2 and 3.) For convenience, we use a set of doubly-indexed propositional variables, instead of P, where the indices range over the positive integers: Q={LJE(L2}}- The desired wif, in this exercise is in WFF(Q). We set the variable to truth value trae (resp. false) if there is (resp. there is not) a queen placed in position (i, j) of the board, where we take the first index i (resp. the second index j) to range over the vertical axis downward (resp. CHAPTER 1. PROPOSITIONAL LOGIC the horizontal axis rightward); that is, i is a row number and j is a column mmber. There are four parts in this exercise: 1. Write the wife and justify how it accomplishes its task. Hint: Write as a conjunction (a) (b) (e) (d) Aqide Age, where is satisfied if there is exactly one queen in each row, is satisfied if there is exactly one queen in each column, is satisfied if there is at most one queen in each diagonal, is satisfied if there is at most one queen in each antidiagonal. Further Hist: Given any two distinct positions (is, ji) and (ia, ja) along a diagonal, it is always the case that is −j 12-ja. And if the two positions are along an antidiagonal, then it is always the case that i, +₁=₂+/ 2. Imagine now an infinite chessboard, which occupies the entire south-east quadrant of the Cartesian plane. The coordinates along the vertical and horizontal axes are, respectively, i (increasing downward) and j (increasing rightward), both ranging over the positive integers (1,2,...). In an attempt to repeat the argument in Example 20 and Exercise 21, someone once defined the set of wis {n>4), and wrote the following (in outline here): The set I is finitely satisfiable and, therefore, satisfiable by Compactness. Hence, there exists a solution of the Infinite Queens Problem, which satisfies conditions {(a), (b), (c), (d)) for all 4. 3. Your task is to define an infinite set that: What is wrong with the preceding argument? The answer is subtle and you need to be careful. (₁|k> 1} of distinct propositional weff's such (a) For every > 1 there is n1 such that satisfaction of wff 8, implies satisfaction of wif, defined in part 1 of this exercise, ie, satisfaction of 6, defines a solution of the x-Queens Problem. (b) For all >k> 1, if satisfaction of welf's By and define solutions of the n'-Queens Problem and Queens Problem, respectively, then n²>n. (e) Every finite subset ofis satisfiable. Hint: For part (a), make 8 define a particular n-Queens Problem, ie, is satisfied by exactly one truth assigment of the variables occurring in 6. For (b) and (c), read and understand the subsection entitled "A second solution of the infinite Queens Problem" in Appendix G. 4. Let be the infinite set of weff's defined in the preceding part. Use Compactness for PL to give a rigorous argument that the Infinite Queens Problem has indeed a solution. 0See Answer
Q19:Exercise 8. We can restrict the logical connectives to {, V, A}. The set of propositional vari- ables is again P = {po,P1, P2,...}, which is countably infinite. Suppose we extend this syntax with two new connectives, denoted W and M, each taking as a single argument a countably infinite set of previously defined wff's. The resulting syntax is one version of Infinitary PL. If I is the countably infinite set {91, 92, 93,...), then WI can be informally understood as: Wr = 41 V 4₂ V 43 V..., and similarly for Mr. Keep in mind that "41 V 42 V 43 V..." is not a legal formal expression; we use Was a unary connective, applied to a single argument which must be a set, and similarly for . There are three parts in this exercise: 1. Define the syntax of Infinitary PL, preferably in an extended BNF (Backus-Naur Form). Try to be as precise as you can, paying special attention to the presence of ellipses "..." in the definition - or can you think of a formulation that avoids any mention of ellipses? 2. Define the semantics of Infinitary PL, by structural induction on the syntax in Part 1, starting from an assignment of truth values to every member of P (for the base case of the induction). 3. Show that Theorem 2 does not hold, and therefore nor does Corollary 7, for Infinitary PL. Hint: Define a countably infinite set I of wff's such that every finite To CI is satisfiable, but I is not. Further Hint: Include wff = W{-Po: P₁, P2,...} in your proposed r. Remark 9. The infinitary propositional logic (Infinitary PL) extends finitary propositional logic, which is just what is commonly called propositional logic (our PL here). The distinction between infinitary and finitary extends to other formal logics and their proof systems besides PL. You can take the phrase "finitary proof system" to qualify a system that generates new finite expressions (e.g., the sequents of PL in natural-deduction style or the wff's of propositional logic in Hilbert- or Frege-style) from previously generated ones by means of finitely many rules that require each finitely many premises or antecedents - without using any notion of infinite sequence or any notion of infinite set.9See Answer
Q20:Exercises 1.5 1. Show that a formula is valid iff T = 0, where T is an abbreviation for an instance p V -p of LEM. 2. Which of these formulas are semantically equivalent to p → (q V r)? (a) qv (-p Vr) (b) q^-r-p (c) p^rq * (d)¬q^r-p. 3. An adequate set of connectives for propositional logic is a set such that for every formula of propositional logic there is an equivalent formula with only connectives from that set. For example, the set {-, V} is adequate for propositional logic, because any occurrence of A and → can be removed by using the equivalences → ¬V and o^= -(-V¬). (a) Show that {¬, ^}, {¬,→} and {→,1} are adequate sets of connectives for propositional logic. (In the latter case, we are treating as a nullary con- nective.) (b) Show that, if C C {¬, ^, V, →, 1} is adequate for propositional logic, then ¬EC or LEC. (Hint: suppose C contains neither nor and consider the truth value of a formula o, formed by using only the connectives in C, for a valuation in which every atom is assigned T.) (c) Is {,} adequate? Prove your answer.See Answer

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