We know how to write the conditional, but what does it mean? As before, we will take the meaning to be given by the truth conditions-that is, a description of when the sentence is either true or false. The second sentence (the one after the arrow, which in this example is “ Q”) is called the “consequent”. The first constituent sentence (the one before the arrow, which in this example is “ P”) is called the “antecedent”. It is also sometimes called a “material conditional”. This kind of sentence is called a “conditional”. There are several ways to do this, but the most familiar (although not the most elegant) is to use parentheses. In that case, we need a way to identify that this is a single sentence when it is combined with other sentences. We might want to combine this complex sentence with other sentences. One last thing needs to be observed, however. The most commonly used such symbol is “→”. It will be useful, however, to replace the English phrase “if…then…” by a single symbol in our language. Then, the whole expression could be represented by writing We could thus represent this sentence by lettingīe represented in our logical language by The sentence, “If Lincoln wins the election, then Lincoln will be President” contains two atomic sentences, “Lincoln wins the election” and “Lincoln will be President”. Thus, it would be useful if our logical language was able to express these kinds of sentences in a way that made these elements explicit. To make these relations explicit, we will have to understand what “if…then…” and “not” mean. And the second sentence above will, one supposes, have an interesting relationship to the sentence, “The Earth is the center of the universe”. For example, the first sentence tells us something about the relationship between the atomic sentences “Lincoln wins the election” and “Lincoln will be President”. We could treat these like atomic sentences, but then we would lose a great deal of important information. The Earth is not the center of the universe. If Lincoln wins the election, then Lincoln will be President. “If…then….” and “It is not the case that….” 2.1 The ConditionalĪs we noted in chapter 1, there are sentences of a natural language, like English, that are not atomic sentences. Then the law of syllogism tells us that if we turn of the water (p) then we don't get wet (r) must be true.2. If the water stops pouring (q) then we don't get wet any more (r). If we turn of the water (p), then the water will stop pouring (q). The law of syllogism tells us that if p → q and q → r then p → r is also true. This is called the law of detachment and is noted: This means that if p is true then q will also be true. If we call the first part p and the second part q then we know that p results in q. If we turn of the water in the shower, then the water will stop pouring. The most common patterns of reasoning are detachment and syllogism. If the conditional is true then the contrapositive is true.Ī pattern of reaoning is a true assumption if it always lead to a true conclusion. The contrapositive does always have the same truth value as the conditional. We could also negate a converse statement, this is called a contrapositive statement: if a population do not consist of 50% women then the population do not consist of 50% men. The inverse always has the same truth value as the converse. The inverse is not true juest because the conditional is true. If we negate both the hypothesis and the conclusion we get a inverse statement: if a population do not consist of 50% men then the population do not consist of 50% women. A conditional and its converse do not mean the same thing If both statements are true or if both statements are false then the converse is true. If we exchange the position of the hypothesis and the conclusion we get a converse statement: if a population consists of 50% women then 50% of the population must be men. Our conditional statement is: if a population consists of 50% men then 50% of the population must be women. If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional. The example above would be false if it said "if you get good grades then you will not get into a good college". Hypotheses followed by a conclusion is called an If-then statement or a conditional statement.Ī conditional statement is false if hypothesis is true and the conclusion is false. The part after the "if": you get good grades - is called a hypotheses and the part after the "then" - you will get into a good college - is called a conclusion. If you get good grades then you will get into a good college. We will explain this by using an example. If we instead use facts, rules and definitions then it's called deductive reasoning. When we previously discussed inductive reasoning we based our reasoning on examples and on data from earlier events.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |