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4.2: Valid patterns of inference

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    If someone says to us, Either Joe is crazy or he is lying, and he is not crazy, and we believe the speaker to be truthful and well-informed, we will naturally conclude that Joe is lying. This is an example of inference: knowing that one fact or set of facts is true gives us an adequate basis for concluding that some other fact is also true.

    Logic is the science of inference. One important goal of logic is to provide a systematic account for the kinds of reasoning or inference that we intuitively know to be correct, like the example mentioned in the previous paragraph. In thinking about such examples it is helpful to lay out each of the premises (the facts which form the basis for the inference) and the conclusion (the fact which is inferred) as shown in (1). For longer and more complex chains of inference, the same format can be used to lay out each step in the reasoning and thereby provide a proof that the conclusion is true.

    (1) Premise 1: Either Joe is crazy or he is lying.
    Premise 2: Joe is not crazy.

    Conclusion: Therefore, Joe is lying.

    As we will see, the kind of inference illustrated in (1) does not depend on the meanings of the “content words” (nouns, verbs, adjectives, etc.) but only on the meaning of the logical words, in this case or and not. Propositional logic, the topic of §4.3, deals with patterns of this type. Some other kinds of reasoning that we intuitively recognize as being correct are illustrated in (2):

    (2) a. Premise 1: All men are mortal.
    Premise 2: Socrates is a man.
    Conclusion: Therefore, Socrates is mortal.

    b. Premise 1: Arthur is a lawyer.
    Premise 2: Arthur is honest.
    Conclusion: Therefore, some (= at least one) lawyer is honest.

    The kinds of inference illustrated in (2) are clearly valid, and have been studied and discussed for over 2000 years. But these patterns cannot be explained using propositional logic alone. Once again, these inferences do not depend on the meanings of the “content words” (mortal, lawyer, honest, etc.). In these examples the inferences follow from the meaning of the quantifiers all and some. Predicate logic, the topic of §4.4, provides a way of dealing with such cases.

    Now consider the inference in (3):

    (3) Premise: John killed the wasp.
    Conclusion: Therefore, the wasp died.

    This inference is not determined by the meanings of logical words or quantifiers, but only by the meanings of the verbs kill and die. Neither propositional logic nor predicate logic actually addresses this kind of inference. Logic deals with general patterns or forms of reasoning, rather that the meanings of individual words. However, predicate logic provides a notation for representing the meanings of the content words within each proposition, and thus gives us a way of expressing lexical entailments (e.g., kill entails die; see Chapter 6).

    It is important to remember that a valid form of inference does not (by itself) guarantee a true conclusion. For example, the inferences in (4) both make use of a valid pattern discussed in §4.3.2, which is called Modus Tollens ‘method of rejecting/denying’:

    (4) a. Premise 1: If dolphins are fish, they are cold-blooded.
    Premise 2: Dolphins are not cold-blooded.
    Conclusion: Dolphins are not fish.

    b. Premise 1: If salmon are fish, they are cold-blooded.
    Premise 2: Salmon are not cold-blooded.
    Conclusion: Salmon are not fish.

    Even though both of these examples employ the same logic, the results are different: (4a) leads to a true conclusion while (4b) leads to a false conclusion. Obviously this difference is closely related to the premises which are used in each case: (4b) starts from a false premise, namely Salmon are not cold-blooded. Valid reasoning guarantees a true conclusion if the premises are true, but if one or more of the premises is false there is no guarantee.

    Example (4b) shows that a false conclusion does not necessarily mean that the reasoning is invalid. Conversely, a true conclusion does not necessarily mean that the reasoning is valid. The examples in (5) both make use of an invalid form of reasoning called ‘denying the antecedent.’ This is in fact a common fallacy, i.e., an invalid pattern of inference which people nevertheless often try to use to support an argument. Now, the conclusion in (5a) is true, but the truth of this statement (Crocodiles are not warm-blooded) does not show that the reasoning is valid. It is simply a coincidence that in our world, crocodiles happen to be coldblooded. It is easy to imagine a slightly different sort of world which is much like our own except that crocodiles and other reptiles are warm-blooded. In that context, the same reasoning would lead to a false conclusion. This shows that the conclusion is not a necessary truth in all contexts for which the premises are true.

    (5) a. Premise 1: If crocodiles are mammals, they are warm-blooded.
    Premise 1: Crocodiles are not mammals.
    Conclusion: Crocodiles are not warm-blooded.

    b. Premise 1: If bats are birds, then they have wings.
    Premise 1: Bats are not birds.
    Conclusion: Bats do not have wings.

    Another way of showing that this pattern of inference is invalid is to change the content words while preserving the same logical structure, as illustrated in (5b). In this example the conclusion is false even though both premises are true, showing that the logical structure of the inference is invalid.

    We have said that one important goal of logic is to provide a systematic account for the kinds of reasoning or inference that we intuitively know to be correct. In addition, logic can help us move beyond our intuitions in at least two important ways. First, it provides a way of analyzing very complex arguments, for which our intuitions do not give reliable judgements. Second, our intuitive reasoning may sometimes be based on patterns of inference which are not in fact valid. Logic provides an objective method for distinguishing valid from invalid patterns of inference, and a way of proving which patterns belong to each of these types. We now procede to survey the basic notation and concepts used in the two primary branches of logic, beginning with propositional logic.


    This page titled 4.2: Valid patterns of inference is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Paul Kroeger (Language Library Press) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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