Consider the statement:. It is not clear what the logical form of this statement is. The King and Queen are visiting dignitaries. Visiting dignitaries is always boring. Therefore, the King and Queen are doing something boring. Because of the difficulty in identifying the logical form of an argument, and the potential deviation of logical form from grammatical form in ordinary language, contemporary logicians typically make use of artificial logical languages in which logical form and grammatical form coincide.
The use of an artificially constructed language makes it easier to specify a set of rules that determine whether or not a given argument is valid or invalid. In short, a deductive argument must be evaluated in two ways. First, one must ask if the premises provide support for the conclusion by examing the form of the argument. If they do, then the argument is valid.
Then, one must ask whether the premises are true or false in actuality. Only if an argument passes both these tests is it sound. However, if an argument does not pass these tests, its conclusion may still be true, despite that no support for its truth is given by the argument.
Note: there are other, related, uses of these words that are found within more advanced mathematical logic. Moreover, an axiomatic logical calculus in its entirety is said to be sound if and only if all theorems derivable from the axioms of the logical calculus are semantically valid in the sense just described.
The author of this article is anonymous. The IEP is actively seeking an author who will write a replacement article. Validity and Soundness A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
The following argument is valid, because it is impossible for the premises to be true and the conclusion nevertheless to be false: Elizabeth owns either a Honda or a Saturn. Consider, then an argument such as the following: All toasters are items made of gold. However, the following argument is both valid and sound: In some states, no felons are eligible voters, that is, eligible to vote.
For example, consider these two arguments: All tigers are mammals. Now consider: All basketballs are round. The Earth is round. Therefore, the Earth is a basketball. Consider, for example, the following arguments: My table is circular. Therefore, it is not square shaped. Juan is a bachelor. Therefore, he is not married. These arguments, at least on the surface, have the form: x is F; Therefore, x is not G.
Take for example the two statements: 1 Tony is a ferocious tiger. Consider the statement: 3 The King and Queen are visiting dignitaries. Only crows are black. So, John is a crow. It is saying essentially that all black things are crows. Not true of course, but not relevant to judging the reasoning. The first premise is false, but this is not relevant to judging the reasoning. IF these premises are true, we are locked into the conclusion.
If we gave these premises to a computer, it would give us back the conclusion. It would simply judge the implications of the alleged information it is given. See the credit score example in the video for Chapter 1. Which one of the following is true about valid arguments? They always have true premises. If the premises are true, then the conclusion will be true. If we find out that a premise is false, then we change our mind about whether an argument is valid.
In 6 when we see that the first premise is false, we know the argument is now invalid. Valid arguments always have true conclusions, even if some of the premises are false.
Saying "Only crows are black" means that "If anything is black, it is a crow. All Internet spies for the Chinese government are Chinese. Wen Ho Lee is Chinese. The first premise is not saying that every Chinese person in the world is an Internet spy for the Chinese government. Question : What if we discover that the conclusion is true? Is the argument now valid?
The reasoning is still invalid, but we can get lucky and still have true conclusions. Important -- Invalid arguments can have as a matter of luck true premises and a true conclusion, BUT the key difference is that the premises do not guarantee the conclusion as they do with valid arguments.
All Democrats always tell the truth. President Obama is a democrat. So, President Obama always tells the truth. IF the premises are true, we are locked into the conclusion.
We go by what the premises are saying. Question: What if the conclusion of the Obama example is false? What do we know about the premises? IF an argument is valid, but if we know the conclusion is false, then we know at least one premise is false. As it turns out, reasoning is exemplified by arguments, in other words, when one creates an argument one is participating in the process of reasoning.
Suppose we have an argument. Then by definition, we have a set of premises and a main conclusion. Let P be the conjunction of all of the premises meaning we just take each individual premise, connect them each with the word "and" and form a larger proposition which we are calling P , and let C be the conclusion. We will call the truth functional relationship between P and C the argument's inference. Definition : The truth functional relationship between the conjunction of an argument's premises and the argument's conclusion is called the argument's inference.
In Section 1 we explored three distinct and types of relations between statements, namely implication, relevance and independence.
Since an argument is something created by humans for now at least then we can ignore any cases where all of the premises of an argument are actually independent of the conclusion. Since an argument's inference is the relationship between the argument's premises and its conclusion, we will classify arguments by whether the argument's inference is that of implication or relevance.
Emily points out that the pattern suggests that the sum of 1 odd number is 1 2 and the sum of the first 2 consecutive odd number is 2 2 and the sum of the first 3 consecutive odd numbers is 3 2 and so forth. Emily then tells you that she has a proof showing that the sum of the first n odd numbers is n 2 for any whole number n.
She then gives you that proof, which consists of a set of premises that lead to the conclusion that the sum of the first n odd numbers is n 2. What does Emily mean when she says she has a proof? In logic and mathematics and systems which use formal reasoning to say one has a proof of some proposition C is to say that if the premises one presents in the proof are true, then the conclusion C of the proof must be true.
In other words, to prove a claim C is to offer reasons premises such that if those premises are true then the conclusion C must be true. To recast the notion of proof in terms of inference and relations between statements we introduce a key term used in logic which is more common than the term "proof". Definition : Let P be the conjunction of each premise for a given argument, and let C be the argument's conclusion.
If P implies C , then we say the argument is valid. Equivalently, if an argument has the property that it is impossible to have all true premises and a false conclusion, then the argument is said to be valid.
In other words a valid argument is an argument whose inference between the conjunction of its premises and the conclusion is one of implication. When the inference of an argument is one of implication, then the inference is said to have the validity property and the argument is said to be valid.
Note that an argument's inference is valid or not.
0コメント