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## Meaning ConditionsThe notion of a self-contradictory argument as introduced in the previous section is based on “the way in which a proposition is used in an argument.” In this section we aim at making it more precise what we mean by this, and we will outline how the notion of a self-contradictory argument will be formally expressed in the succeeding sections. Given an argument and a proposition of this argument we shall in the following consider Let us consider an example. Let because In the previous section, “a self-contradictory argument” was explained to be an argument in which there is a proposition which is used in two or more ways such that not all of the ways of using the proposition are compatible. In this section “the meaning forced on a proposition, by the steps of the argument” expresses precisely the way in which the proposition is used in the argument. Hence, we can explain what “a self-contradictory argument” is by saying that it is an argument such that the steps of the argument force several meanings on one of the propositions of the argument and that not all of these meanings are compatible. Yet another way to put this is to say that an argument is self-contradictory if and only if the steps of the argument force an Now we change to how to formally express “a self-contradictory argument.” Let us by we shall write to define consisting of an ⊃E inference, minor premise and the conclusion, respectively, of
Reasoning in the same way as in the previous example, we know that since
Hence the meaning condition for the major premise
The meaning conditions, as we shall give them, are related to the Observe that an elimination rule is, in a sense, the inverse of the corresponding introduction rule: by an application of an elimination rule one essentially only restores what had already been established if the major premise of the application was inferred by an application of an introduction rule. We may say that, for a given deduction, the constraint expressed by the meaning conditions is an attempt to make the inversion principle global, in the deduction. But this attempt is successful if and only if the deduction is not self-contradictory, since otherwise there is no assignment of formal meanings to the formulas in the deduction such that this assignment satisfies the meaning conditions. The Curry-Howard interpretation may resemble what designates meanings in the meaning conditions. However, the similarity is only superficial. In general, it is not the case that the assignment of Curry-Howard interpretations to the formula occurrences in a deduction satisfies the meaning conditions. Since the Curry-Howard interpretation is just a representation of an argument, there are always Curry-Howard interpretations of the formula occurrences in a deduction, but there need not be an assignment of formal meanings to the formulas in the deduction such that this assignment satisfies the meaning conditions. |

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