Linguistics Answer Note #15

Question 1 prove using truth table p or [p and (p or q)] biconditional p Answer To prove this statement using a truth table, we first need to understand what the proposition p or [p and (p or q)] biconditional p means. A biconditional statement is a logical statement that is true if both of its components are either true or false. In this case, the statement is true if both p and [p and (p or q)] are either true or false. Using a truth table, we can represent all possible combinations of truth values for p and q, and then determine if the statement is true or false for each combination. The truth table for this statement would look like this: | p | q | p or [p and (p or q)] | p biconditional p | |---|---|-------------------------|--------------------| |T|T|T|T| |T|F|T|T| |F|T|F|T| |F|F|F|T| Looking at the truth table, we can see that the statement p or [p and (p or q)] biconditional p is true for all possible combinations of truth values for p and q. This is because both p and [p and (p or q)] have the same truth value for each combination. To further understand why this statement is true, let's break it down into two separate propositions: p or [p and (p or q)] and p biconditional p. The first proposition, p or [p and (p or q)], can be simplified by using the distributive property: p or [p and (p or q)] = p. Similarly, the second proposition, p biconditional p, is always true since p is logically equivalent to itself. In essence, this means that the statement p or [p and (p or q)] is equivalent to p, which explains why the truth table shows that the biconditional statement is always true. Question 2 All clowns are funny. Some mascots aren’t funny. Which of the following is not consistent with the given statements? A Some mascots are not clowns. B Some clowns are not mascots. C No mascot is a clown. D Some mascots are clowns too. Answer Question: "All clowns are funny. Some mascots aren’t funny. Which of the following is not consistent with the given statements? A Some mascots are not clowns. B Some clowns are not mascots. C No mascot is a clown. D Some mascots are clowns too." Response: The correct answer is C - No mascot is a clown. This statement is not consistent with the given statements because the first two statements indicate that there are some mascots that are not funny, while the third statement suggests that there are no mascots which are clowns. This is a contradictory statement as it implies that all mascots are not funny, but there could be some mascots who are clowns, which could potentially be funny. Question 3 Show that p → (q → s) is tautologically implied by p → (q → r) and q → (r → s) using CP rule. Answer In order to prove that p → (q → s) is tautologically implied by p → (q → r) and q → (r → s) using CP rule, we will use the rules of propositional logic to construct a logical argument. Let's first break down the statement p → (q → s). This can be rewritten as p → (q implies s). In other words, if p is true, then q implies s. This means that if q is true, s must also be true. To prove this statement using CP rule, we will use two premises: p → (q → r) and q → (r → s). We will also use the principle of conditional proof, which allows us to assume the antecedent of a conditional statement in order to prove its consequent. Step 1: Assume p is true. Since we are using CP rule, we can assume the antecedent of the first premise, p. This means that for this argument, we can assume that p is true. Step 2: Apply the first premise. Using the first premise p → (q → r), we can substitute p with our assumption and rewrite the premise as p is true, then q implies r. Step 3: Apply the second premise. Using the second premise q → (r → s), we can substitute q with our assumption (since we've established that q implies r) and rewrite the premise as q is true, then r implies s. Step 4: Apply the principle of conditional proof. Since we've assumed p and q to be true, we can now use the principle of conditional proof to prove our desired statement p → (q → s). This means that we can assume the antecedent of this statement, q, in order to prove its consequent, s. Step 5: Combine the two premises. By combining the two premises, we now have r implies s, which is what we need in order to prove our desired statement. Step 6: Apply modus ponens. Using modus ponens (if p implies q and p is true, then q must also be true), we can conclude that s is true. Step 7: Combine all steps. Combining all the steps, we have shown that assuming p and q to be true, then s must also be true. In other words, p → (q → s) is tautologically implied by p → (q → r) and q → (r → s) using CP rule. Question 4 Choice Type Question The question below consists of a question and two statements numbered I and ll given below it. You have to decide whether the data provided in the statements are sufficient to answer the questions. Read both statements and give an answer. What is the code of "called"? 1. In XYZ language," meeting had been called" is written as "sx vt bv mp" and " they called every time" is written as " bv it faj!". Il. In XYZ language," you called every hour" is written as "ca bv jl ma". • if the data in statements I and Il together are sufficient to answer the question. if the data in statements I and Il together are not sufficient to answer the question. • if the data in statement I alone are sufficient to answer the question, while the data in statement Il alone are not sufficient to answer the question. if the data in statement Il alone are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question. •if the data in either statement I alone or in statement Il alone are sufficient to answer the question. Answer After reviewing both statements I and II, it can be determined that the data provided is sufficient to answer the question. In statement I, it is stated that in XYZ language, "meeting had been called" is written as "sx vt bv mp" and "they called every time" is written as "bv it faj!". In statement II, it is stated that in XYZ language, "you called every hour" is written as "ca bv jl ma". Based on the information provided in both statements, it can be concluded that the word "called" in XYZ language is represented by "bv" in both instances. Therefore, the code for "called" in XYZ language is "bv". Question 5 Multi Choice Type Question The question below consists of a question and two statements numbered I and Il given below it. You have to decide whether the data provided in the statements are sufficient to answer the questions. Read both the statements and give answer: In a certain code, nop all ed' means 'They like flowers'. Which code word means 'flowers'? I. 'id nim nop' means 'They are innocent. Il. 'gob ots all' means 'We like roses'. • If the data given in both statements | and Il together are not sufficient to answer the question If the data in statement I alone are sufficient to answer the question, while the data in statement Il alone are not sufficient to answer the question • If the data in both statements I and Il together are necessary to answer the question data in statement Il alone are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question Answer In this question, we are given a code and we have to determine which code word represents 'flowers'. Let us analyze the two statements provided. Statement I: 'id nim nop' means 'They are innocent'. This statement does not provide any information related to flowers. Therefore, it cannot help us in determining the code word for 'flowers'. Statement II: 'gob ots all' means 'We like roses'. Similar to statement I, this statement also does not provide any information about the word 'flowers'. It only mentions 'roses', which is not relevant to our question. Therefore, neither of the statements alone are sufficient to answer the question. However, when we combine the information from both statements, we can infer that the code word for 'flowers' could be 'nop' as it appears in both statements and it is associated with liking or being fond of something. Hence, we can conclude that the data in both statements I and II together are necessary to answer the question.