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Some basic logic questions

Some basic logic questions

DavidKent
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Posted 03/24/08 - 02:07 PM: Subject: Some basic logic questions	#1
Hello, I'm a first year in philosophy and as part of that I do a logic course. I apologise for the elementary nature of these questions, but I couldn't find the answers on the net. Any light you can shed on these problems which have been bugging me would be greatly appreciated. Apparently, this is a valid argument: P1 All vegetarians are healthy, P2 Babs is a vegetarian, p3 Bab has cancer c Bab is healthy. The leacturer said that the definition of a valid argument is one where IF all the premises are true, then it is IMPOSSIBLE for the conclusion to be false. Taking this definition as correct, clearly p1 + p2 equal the conclusion. Or, in other words, IF P1 and P2 are true, C can not be false. However, assuming that P3 does, we know from experience, mean that C can't be true, how does this fit in? How is it we can just ignor it? My next question is to do with the arrow connective (if, then). The truth table for it is: 0. A -> B 1. T T T 2. T F F 3. F T T 4. F T F Line 3: Why is 'if A then B' true when A is false and B is true? Surely it shouldn't be considered at 'true', but rather 'irrelevant'. If the presence of A means B is definetly there, why does it mean that the 'If, then' part is true? To me, is hasn't been either proven true or proven false... Like wise, in 'if and only if' [ <- -> ], why is 'if and only if' true when both A and B are false? Again, surely it should be labeld as 'inconclusive'? Why is this not so? Thanks a lot in advance for any help you are able to give me, David

Goaswerfraiejen
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Posted 03/24/08 - 04:11 PM:	#2
DavidKent wrote: Apparently, this is a valid argument: P1 All vegetarians are healthy, P2 Babs is a vegetarian, p3 Bab has cancer c Bab is healthy. The leacturer said that the definition of a valid argument is one where IF all the premises are true, then it is IMPOSSIBLE for the conclusion to be false. The argument is in fact invalid, so long as cancer is understood to constitute something unhealthy--and assuming that by "Babs" and "Bab" you mean the same person. "Validity" basically just means "law of non-contradiction". In this case, the conclusion contradicts one of the premises, so it should be invalid. However, assuming that P3 does, we know from experience, mean that C can't be true, how does this fit in? How is it we can just ignor it? Logic doesn't care about real-world analogues; you could say "Either George Bush is president of the U.S.A. or aliens will eat my sister before she is born," and that's a valid argument (so long as one or other part of the disjunction is true). Real-world states of affairs can be helpful for understanding what's going on in a logical symbolisation (for example, my last example could be symbolised "B v A"), but often it's just confusing. Just because something is true in real life doesn't mean it's true in the context of a logical proposition. In the case that you listed above, however, the argument is invalid, so I'd chalk it up to a blackboard mistake. They happen. My next question is to do with the arrow connective (if, then). The truth table for it is: 0. A -> B 1. T T T 2. T F F 3. F T T 4. F T F Line 3: Why is 'if A then B' true when A is false and B is true? Surely it shouldn't be considered at 'true', but rather 'irrelevant'. If the presence of A means B is definetly there, why does it mean that the 'If, then' part is true? To me, is hasn't been either proven true or proven false... Remember the definition of validity? In line 3 of that truth table, it is impossible to make the the premises true and the conclusion false, therefore it must be valid. Truth tables do not test the truth of a proposition, but rather, the conditions under which it is true (and valid). There is an error in line 4, however: it should read FTT. With the material conditional, false premises will always yield a valid conclusion, since it is impossible to make those premises true and the conclusion false. Tricky, eh? Like wise, in 'if and only if' [ <- -> ], why is 'if and only if' true when both A and B are false? Again, surely it should be labeld as 'inconclusive'? Why is this not so? Ordinary language is confusing you. IFF is, in fact, a biconditional, which means that it can only be valid when both conditions are identical. Does that help?

Timothy
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Posted 03/24/08 - 06:40 PM:	#3
DavidKent, There's a specialised forum for logic, both technical and philosophical (or philosophical issues of) DK wrote: Hello, I'm a first year in philosophy and as part of that I do a logic course. I apologise for the elementary nature of these questions, but I couldn't find the answers on the net. Any light you can shed on these problems which have been bugging me would be greatly appreciated. Apparently, this is a valid argument: P1 All vegetarians are healthy, P2 Babs is a vegetarian, p3 Bab has cancer c Bab is healthy. The leacturer said that the definition of a valid argument is one where IF all the premises are true, then it is IMPOSSIBLE for the conclusion to be false. Taking this definition as correct, clearly p1 + p2 equal the conclusion. Or, in other words, IF P1 and P2 are true, C can not be false. However, assuming that P3 does, we know from experience, mean that C can't be true, how does this fit in? How is it we can just ignor it? As long as there is not a explicit premise stating that "All persons with cancer are unhealthy", the argument is valid. Formalize the premises as abstract propositions. You would have as premises "All A's are B's", "C is an A" and "C is a D" (where D stands for having cancer). Check for validity, and you will see that it is valid. Add the premise "All Ds are not Bs" and it will turn out invalid. DK wrote: My next question is to do with the arrow connective (if, then). The truth table for it is: 0. A -> B 1. T T T 2. T F F 3. F T T 4. F T F Line 3: Why is 'if A then B' true when A is false and B is true? Surely it shouldn't be considered at 'true', but rather 'irrelevant'. If the presence of A means B is definetly there, why does it mean that the 'If, then' part is true? To me, is hasn't been either proven true or proven false... Material conditional does not reflect a causal relation between A and B. It just states that if a proposition A is true, then a proposition B is also true. It commits itself just to the case where A is true. Hence, when A is false, we cannot say that the conditional is false, for it hasn't said anything about what happens when A is false. It only does when A is true. One could, perhaps, see line 3 as a "charitable" truth assignment. DK wrote: Like wise, in 'if and only if' [ <- -> ], why is 'if and only if' true when both A and B are false? Again, surely it should be labeld as 'inconclusive'? Why is this not so? Let's remember that a biconditional are two conditionals joined by a conjuntion like so: (A -> B) & (B -> A) =df. A <-> B Now, if A is false, then left part is true. If the conjunction is true, then that means that B has also to be false, for is B is not false, the right part would be false, and the conjunction false. A same reasoning applies when we assume that B is false. This shows that A and B must have the same truth-value; that is, A is true only if B is true, and A is false only if B is false. Both A and B false yield the conjunction true; hence the biconditional is true when both A and B are false. Cheers
_____________________ "I think I'll stop here." -- Andrew Wiles immediately after presenting the proof of Fermat's Last Theorem, Cambridge, 23 June 1993

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