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Example 1. One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example: If someone lives in San Diego, then they live in California. Joe lives in California. Therefore, Joe lives in San Diego. There are many places to live in California other than San Diego.
However, the logical validity of an argument is a function of its internal consistency, not the truth value of its premises. For example, consider this syllogism, which involves a false premise: If the streets are wet, it has rained recently. (premise) The streets are wet. (premise) Therefore it has rained recently. (conclusion)
Affirmative conclusion from a negative premise (illicit negative) – a categorical syllogism has a positive conclusion, but at least one negative premise. [11] Fallacy of exclusive premises – a categorical syllogism that is invalid because both of its premises are negative. [11]
The argument itself could have true premises, but still have a false conclusion. [3] Thus, a formal fallacy is a fallacy in which deduction goes wrong, and is no longer a logical process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.
An argument is valid if and only if it is the case that whenever the premises are all true, the conclusion must also be true. If there exists a logical interpretation where the premises are all true but the conclusion is false, the argument is invalid. Key to evaluating the quality of an argument is determining if it is valid and sound.
Deductive reasoning is the mental process of drawing deductive inferences. Deductively valid inferences are the most reliable form of inference: it is impossible for their conclusion to be false if all the premises are true. [34] [35] This means that the truth of the premises ensures the truth of the conclusion.
To show that this form is invalid, we demonstrate how it can lead from true premises to a false conclusion. All apples are fruit. (True) All bananas are fruit. (True) Therefore, all bananas are apples. (False) A valid argument with a false premise may lead to a false conclusion, (this and the following examples do not follow the Greek syllogism):
The example's first premise is false – there are people who eat carrots who are not quarterbacks – but the conclusion would necessarily be true, if the premises were true. In other words, it is impossible for the premises to be true and the conclusion false. Therefore, the argument is “valid”, but not “sound”.