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Converse of Implication

Clock
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Posted 03/31/08 - 05:26 PM: Subject: Converse of Implication	#1
I apologise if this is a really simple/newby question; we just started on some propositional logic at uni and I'm having trouble getting my head around implication. Now, there was an example in the course textbook (Discrete Mathematics and its Applications, 5th Ed) that runs like this: "How can this English sentence be translated into a logical expression? 'You can access the Internet from campus only if you are a computer science major or you are not a freshman'". Assuming proposition a = "You can access the Internet from campus", proposition c = "you are a computing science major" and proposition f = "you are a freshman", the result I came to was: (c OR ¬f) → a However, the book gave the answer: a → (c OR ¬f) This confuses me since the book also states the converse of an implication does not have the same truth value of the implication, and as far as I can see, both answers are correct. Can someone help me out? Edited by Clock on 03/31/08 - 06:59 PM

Timothy
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Posted 03/31/08 - 05:54 PM:	#2
The "only if" states that "you can access the internet from campus" is true only when c OR ¬f. This is best expressed by the conditional where a is the antecedent, for if a is true, then c OR ¬f must be true for the conditional to be true. Your answer allows the case where a is true, yet c OR ¬f is false; check the truth-table for it.
""Physics investigates the essential nature of the world, and biology describes a local bump. Psychology, human psychology, describes a bump on the bump." W.V.O. Quine

Clock
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Posted 03/31/08 - 06:53 PM:	#3
Argh, sorry, I mixed around two of the propositions and added a not to one that shouldn't be there. That said -- if a is the antecedent, that must mean (according to the truth table) that there are examples where a is false (No access to the internet on campus), and (c OR ¬f) is true (You are a CompSci major or you're not a Freshman), doesn't it? Does that actually make sense?

Brian Bosse
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Posted 03/31/08 - 08:53 PM:	#4
Hello Clock, You can access the Internet from campus only if you are a computer science major or you are not a freshman. Consider this statement: If you have water, then you have hydrogen. Consider this statement: You have water only if you have hydrogen. Consider this statement: If you have hydrogen, then you have water. The third statement is false. The reason for this is that hydrogen is a necessary but not a sufficient condition for water. That is to say, the antecedent can be true when the consequent is false. Now, apply this to the problem at hand. What is the sufficient condition for being able to access the Internet from campus? The sentence does not tell us. What it does tell us is that [being a computer science major or not being a freshman] is a necessary condition to being able to access the Internet from campus. In other words... If you can access the Internet from campus, then you are a computer science major or you are not a freshman. Now, you want to say that the following is true as well... If you are a computer science major or you are not a freshman, then you can access the Internet from campus. The problem with this is that it assumes [being a computer science major or a not a freshman] is a sufficient condition to access the Internet from campus. But you don't know this! You ask... That said -- if a is the antecedent, that must mean that there are examples where a is false, and you are a CompSci major or you're not a Freshman, doesn't it? Does that actually make sense? Yes, it makes perfect sense. For instance, what if you are not at campus, but you are a senior? Can you still access the Internet from campus? Surely not! What if the Internet is down due to electrical issues, but you are a computer science major? Can you still access the Internet from campus? No! What if you have to have a password that no one told you about, but you are both a Junior and a computer science major? Can you accesss the internet from on Campus? Doubtful. Why? Because [being not a freshman or being a computer science major] is not a sufficient condition for being able to access the Internet from campus. It is only a necessary condition. However, in all of these cases I just mentioned it still is true that... If you can access the Internet from campus, then you are a computer science major or you are not a freshman. Here is a good translation tip: (A only if B) <--> (If A, then B) Your interpretation of the sentence reads more into the sentence than is warranted. This is why logic is such a great subject. It really helps in critical thinking. Sincerely, Brian Edited by Brian Bosse on 03/31/08 - 09:04 PM

7
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Posted 03/31/08 - 10:54 PM:	#5
Clock wrote: (c OR ¬f) → a However, the book gave the answer: a → (c OR ¬f) This confuses me since the book also states the converse of an implication does not have the same truth value of the implication The two might have the same truth value. For example, have c = T and a = T. Both conditionals are true, then. It is impossible for both to be false, though. I am sure that what the book said is that the truth table for a conditional is not equivalent to the truth table for its converse, which is indeed so. Edited by 7 on 03/31/08 - 10:58 PM

Timothy
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Posted 04/01/08 - 04:21 AM:	#6
clock wrote: Argh, sorry, I mixed around two of the propositions and added a not to one that shouldn't be there. I thought it was a tricky formulation of the book itself. Oh well. clock wrote: That said -- if a is the antecedent, that must mean (according to the truth table) that there are examples where a is false (No access to the internet on campus), and (c OR ¬f) is true (You are a CompSci major or you're not a Freshman), doesn't it? Does that actually make sense? It does make sense. The conditional in that case is true simply because the antecedent is false. This is so because conditional statements only commit themselves to the case where the antecedent is true. To state a conditional a -> b means that b has to be true whenever a is true. Since it does not say what happens when a is false, we assume the conditional as trivially true. This is why the only line in the truth table of the conditional where it is false is when the antecedent is true and the consequent false. This line shows that b fails to be true when a is true; hence, b is not true whenever a is true, thus the conditional a -> b is false (it's like breaking a contract). The original sentence, which we assume is true, with the "only if" states that a is true only when c OR ¬f is true. This means that the truth of a guarantees the truth of c OR ¬f, for a cannot be true and c OR ¬f false. This is committing to the conditional that if a is true, then c OR ¬f has to be true. Whether it actually is the case or not is beyond the analysis of logic. We just focus on expressing in logical terms what was being expressed in the english sentence. clock wrote: This confuses me since the book also states the converse of an implication does not have the same truth value of the implication... Amore accurate reading would be that the converse of an implication has not necessarily the same truth-value. For example, take a -> b on the one hand and b -> a on the other. Assume that b is false and a is true. a -> b would be false on that assumption, while b -> a would be true under the same assumption. This is not always the case thou. if you take two tautologies a and b and connect them in a conditional a -> b, this will be always true, as well as the converse; they would be equivalent conditionals.
""Physics investigates the essential nature of the world, and biology describes a local bump. Psychology, human psychology, describes a bump on the bump." W.V.O. Quine

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Posted 04/01/08 - 08:03 AM:	#7
I don't like the question. If I hear the statement 'so-and-so can access the internet from campus,' I don't conclude that he must be able to do it right this moment. That would be like thinking that someone who goes to a job interview and says 'I can operate the following machines...' is a liar because he doesn't have access to the machines at that very moment.

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