Asked 3 years, 7 months ago. Active 3 years, 7 months ago. Viewed 3k times. Converse : If the home team wins, then it is raining. The converse allows that. The statement does not. What if it's sunny and the home team wins. The converse doesn't allow that but the statement does. Why is it not equivalent to "if I am a math. Depending on the surrounding context, the sentence may be interpreted as "the home team wins if and only if it is raining", in which case the converse is equivalent. A If it is raining, then the home team wins.
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B If the home team wins, then it is raining. The only way this can be false is for the home team to win while it is not raining. Brian M. Scott Brian M. Scott k 40 40 gold badges silver badges bronze badges. Brian Tung Brian Tung The statement does not say: Only if it is raining will the home team win. If I have more than 5 coins, I have more than 10 coins. You wouldn't regard the above as equivalent, would you?
Such people would provide counter-examples to the following: If you will pass the test, then you have studied. Abhijit Abhijit 11 1 1 bronze badge. Sign up or log in Sign up using Google. Construct a truth table for each of the expressions you determined in Part 4. Are the expressions logically equivalent?
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Assume that Statement 1 and Statement 2 are false. What do you observe? We now define two important conditional statements that are associated with a given conditional statement. Which statement in the list of conditional statements in Part 1 is the converse of Statement 1a? Which is the contrapositive of Statement 1a? The following theorem gives two important logical equivalencies. The first equivalency in Theorem 2.
Table 2. It is possible to develop and state several different logical equivalencies at this time. However, we will restrict ourselves to what are considered to be some of the most important ones. Since many mathematical statements are written in the form of conditional statements, logical equivalencies related to conditional statements are quite important. The negation of a conditional statement can be written in the form of a conjunction. So what does it mean to say that the conditional statement. So the negation of this can be written as.
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This conditional statement is false since its hypothesis is true and its conclusion is false. Consequently, its negation must be true. Its negation is not a conditional statement. So, the negation can be written as follows:. We have seen that it often possible to use a truth table to establish a logical equivalency. However, it is also possible to prove a logical equivalency using a sequence of previously established logical equivalencies.
For example,. When proving theorems in mathematics, it is often important to be able to decide if two expressions are logically equivalent. Sometimes when we are attempting to prove a theorem, we may be unsuccessful in developing a proof for the original statement of the theorem.
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However, in some cases, it is possible to prove an equivalent statement. One can prove as he of course knows by exactly the same argument that all "valid" sentences are logically equivalent. All logic texts prove this. It is, for example, general law 12 on p. The argument goes through so long as "valid" means "true in every interpretation" and "equivalent" means "have the same truth-value in every interpretation". In particular, it doesn't matter what is meant by an interpretation at least as long as the interpretations are classical.
First, when Rapaport says that "two [interpreted] sentences And "Snow is white" and "Obama was born in Hawaii" are not, of course, truth-functionally equivalent. The sole atomic constituent of each sentence is that sentence itself. Constituency is so defined as to make it a reflexive relation, trivially. And an interpretation can perfectly well assign these two sentences different truth-values. So this is just a case of P not being equivalent to Q.
Second, as Rapaport notes, there is some controversy whether the basic notion here should apply to sentences or formulas or to propositions. Note that in this example classical logic is assumed. Some non-classical logics do not deem 1 and 2 logically equivalent. Logical equivalence is different from material equivalence. The statements are logically equivalent if, in every model, they have the same truth value.
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