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 2 added 142 characters in body edited May 5 '14 at 19:36 Chris Sunami 21.8k11 gold badge3232 silver badges6767 bronze badges Using mathematical logic, we often find that statements that seem quite different on the surface are equivalent, which is to say, they have the same truth value under all distinguishable conditions. Or, to put it another way, if statement A is true in all and only the cases that statement B is true, they are equivalent. One simple and basic case of two statements that are equivalent, but that may not look equivalent is "P implies Q" P IMPLIES Q and "Not-P or Q" NOT P OR Q. Both Both are true in all and only the cases where Q is true or P is false. This is because the way "implies"IMPLIES is defined (in logics such as the one you are studying), is as meaning that if P is true, Q must also be true (if P is false, Q can be either true or false), and the way OR is defined is as meaning that at least one of the two sides of the OR is true (but both can be). In some ways, we might say that the whole purpose of mathematical logic is to go beyond the dictates of intuition in cases such as these. Using mathematical logic, we often find that statements that seem quite different on the surface are equivalent, which is to say, they have the same truth value under all distinguishable conditions. Or, to put it another way, if statement A is true in all and only the cases that statement B is true, they are equivalent. One simple and basic case of two statements that are equivalent, but that may not look equivalent is "P implies Q" and "Not-P or Q". Both are true in all and only the cases where Q is true or P is false. This is because the way "implies" is defined (in logics such as the one you are studying), is as meaning that if P is true, Q must also be true (if P is false, Q can be either true or false). In some ways, we might say that the whole purpose of mathematical logic is to go beyond the dictates of intuition in cases such as these. Using mathematical logic, we often find that statements that seem quite different on the surface are equivalent, which is to say, they have the same truth value under all distinguishable conditions. Or, to put it another way, if statement A is true in all and only the cases that statement B is true, they are equivalent. One simple and basic case of two statements that are equivalent, but that may not look equivalent is P IMPLIES Q and NOT P OR Q. Both are true in all and only the cases where Q is true or P is false. This is because the way IMPLIES is defined (in logics such as the one you are studying), is as meaning that if P is true, Q must also be true (if P is false, Q can be either true or false), and the way OR is defined is as meaning that at least one of the two sides of the OR is true (but both can be). In some ways, we might say that the whole purpose of mathematical logic is to go beyond the dictates of intuition in cases such as these. 1 answered May 5 '14 at 19:28 Chris Sunami 21.8k11 gold badge3232 silver badges6767 bronze badges Using mathematical logic, we often find that statements that seem quite different on the surface are equivalent, which is to say, they have the same truth value under all distinguishable conditions. Or, to put it another way, if statement A is true in all and only the cases that statement B is true, they are equivalent. One simple and basic case of two statements that are equivalent, but that may not look equivalent is "P implies Q" and "Not-P or Q". Both are true in all and only the cases where Q is true or P is false. This is because the way "implies" is defined (in logics such as the one you are studying), is as meaning that if P is true, Q must also be true (if P is false, Q can be either true or false). In some ways, we might say that the whole purpose of mathematical logic is to go beyond the dictates of intuition in cases such as these.