Problem 1 There are two parts: [Lecture Slides 10, page 18]:

2. Let r be the proposition "It is raining," s the proposition "The sun is shining," and w the proposition "It is windy." Express each of the following compound propositions as an English sentence. (a) w V (r^8) (b) r→ (-8 ^ w) (c)¬8 → (w Vr)

+ A ALEKS-Ahkia Holloway - HW 3.1 X G Write the statement in symbols. L x s.com/alekscgi/x/Isl.exe/1o_u-IgNslkr7j8P3jH-li0BWOJBInLRZL18Trqeg6PUZ9BEvQEFqlFxsjsXHHKJvMnqEfhfiubz6nu28sVwqAJLE60... Q : ch HW 3.1-3.3 Question 10 of 25 (1 point) | Question Attempt: 3 of Unlimited Check Write the statement in symbols. Let p= "Sara is a political science major" and let q = "Jane is a quantum physics major". Jane is a quantum physics major, or Sara is not a political science major. The statement, in symbols, is written as % 9 15 5 10 f6 i 48 6 & 7 AD OVO 0-0 58 f8 8 $ 0 0 0 fg hp ( 9 ((( f10 16 ▷II Ⓒ2023 McGraw Hill LLC. All Rights Reserved. O E ✓ 18 f11 Save For Later ✓ 19 Terms of Use f12 Ahkia ins Espand 8 prt sc © E₂ Submit Assignment K Privacy Center Accessibility 91°F Windy delete backsp

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. 0

3. Construct a complete truth table for the following SL sentence, determine whether or not it is truth- functionally true, truth-functionally false, or truth-functionally indeterminate, and briefly state why. (/10) (X = 0) & (X = ~0) 4. Construct a complete truth table for the following SL sentence, determine whether or not it is truth- functionally true, truth-functionally false, or truth-functionally indeterminate, and briefly state why. (/10) ~D> [(DVF) >F] 5. Construct a complete truth table for the following SL sentence, determine whether or not it is truth- functionally true, truth-functionally false, or truth-functionally indeterminate, and briefly state why. (/10) ([(HI) & (I>J)] & H) & ~J

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.

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.

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. 0

Construct a complete truth table for the following SL sentence, determine whether or not it is truth- functionally true, truth-functionally false, or truth-functionally indeterminate, and briefly state why.

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' .

Exercise 4. In the definition of the nested sequence of A's in the preceding proof, we did not write: Ait1 [A₁U {4₁} A; U{-₁} if A, U {₁} is finitely satisfiable, if A, U{-₁} is finitely satisfiable. Explain why. Hint: Exhibit a set I of wff's and a single wffy such that both IU {p} and TU{-} are satisfiable.