Which of the following statements concerning symbolic logic are true and false?
1. A truth-functional connective forms a compound sentence from other sentences such that the truth value of the compound is entirely determined by the truth values of the component sentences.
2. A pair of two-value propositions can be related by 16 possible truth functions.
3. To deny a particular negative categorical proposition (an O-type sentence) one must affirm a universal negative proposition (an E-type sentence).
4. Assume existential import AND that "No hedgehogs can read" is false. Which of the following propositions are also true according to Aristotle's square of opposition: A. all hedgehogs can read. B. some hedgehogs cannot read. C. some hedgehogs can read. D. not all hedgehogs can read. E. None of the above are true.
5. Consider the corresponding conditional formed from the premises and conclusion of an argument. Such a conditional is: A. a tautology, if the original argument is valid. B. a tautology, only if the original argument is valid. C. a contradiction, if the original argument is invalid. D. a contingent statement, if the original argument is invalid. E. none of the above.
6. Suppose p is a contradiction, q is a tautology, and r is a contingent sentence. Which of the following are true? A. r, therefore q is an invalid argument. B. p, therefore r is a valid argument. C. q unless r is a tautology. D. r, there fore p is an invalid argument. E. p, therefore q is an valid argument.
7. Symbolized "E will happen if, but only if, there is no G happens".
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As we are limited in space, below are a few ideas and sources to help get you started.
With respect to your first question, a truth-functional connective does not necessarily have to be a compound sentence. What we call a conjunction is a type of truth-functional connective, but connectives do not always have to take the form of conjunctions. What we call conjunctions in logic "corresponds to the English expression 'and'" ("Truth Functional Connectives," p. 34). In English grammar, we can use and to join two sentences together, called a compound sentence. In logic, conjunctions are compound sentences. The following is one example: "It is raining and it is sleeting" (p. 35). However, other types of connectives exist, including one-place, two-place, and three-place connectives. An example of a two-place connective would be, "I will pass only if I study" (p. 31). This sentence actually contains both an independent and a dependent clause, making it, in terms of English grammar, a complex sentence ("Sentences: Simple, Compound, and Complex"). However, what is true is that a truth-functional connective can only be truth-functional as a compound STATEMENT, but a compound statement is not necessarily the same as a conjunction, or compound sentence (p. 34) The following is an example of a compound statement: "If it is not true that all swans are white, AND the president believes that all swans are white, THEN the president is fallible" (p. 32). As you can see, while this is a compound statement, it is actually a complex sentence rather than a compound sentence. Therefore, the first answer should inevitably be false.
With respect to your second question, you first want to note that the term propositions can also mean the same thing as statements. It is true that a truth function will yield a number of truth statements, but it is not true that a truth statement will yield a number of truth functions. A two-valued truth function, more technically called a conjunction truth function, can yeild 4 possible truth statements; therefore, two pairs of conjunctive truth functions will yeild a total of 16 possible truth statements. But, the reverse is not true: A truth statement, or two-valued proposition, cannot yeild a truth function. According to Hardegree in Symbolic Logic, one conjunction truth function can be broken down into the following series of 4 truth statements:
(1) T & T = T
(2) T & F = F
(3) F & T = F
(4) F & F = F (p. 36)
Therefore, the second answer should also inevitably be false since the question inverses the usage of terms.
With respect to your third question, you first want to note that an O-type negative categorical proposition states, "Some S are not P" (Fallacy Files, "Glossary"). The following is one example of an O-type categorical proposition: "Some marsupials are not koalas" ("Glossary"). If we were to deny the truth of the above statement, we would be denying marsupials exist that are koalas; hence we would also be confirming a universal negative proposition, or an E-type categorical proposition. The E-type categorical proposition takes the form "No S are P" ("Glossary"). The following would be an example of an E-type proposition: No marsupials are koalas. The following is another example of an E-type categorical proposition: "No monkeys are marsupials" ("Glossary"). Looking again at our first E-type proposition example as well as our O-type proposition, if we are to deny that some marsupials exist as koalas, then we would indeed be saying, or confirming, that no marsupials are koalas; therefore, your third answer should inevitably be true.
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