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Meaning Conditions

The 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 the meaning forced on the proposition, by the steps of the argument. 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.

Let us consider an example. Let D be the following argument: The wind is blowing

because it's snowing and the wind is blowing. Let A be the proposition it's snowing and the wind is blowing and let B be the proposition the wind is blowing. Thus D consists of one step and A and B are the premise and the conclusion, respectively, of this step. If we forget about which propositions A and B represent we still know something about them by remembering what kind of step the inference of D is. That is, knowing only that the inference of D is of the kind that informally corresponds to one of the &E inference schemata in natural deduction for naïve set theory N (see Appendix below), we know that since A is the premise of the step, A is A1 and A2 for some propositions A1 and A2. Moreover, if A is A1 and A2 then B is A2. The meaning forced on the propositions A and B by the inference of D is this knowledge about A and B given by the knowledge about what kind of step the inference of D is. Hence the meaning of the meaning forced on the proposition, by the steps of the argument depends on what is considered to be known, when knowing only what kind of steps the steps of the argument are.

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 ambiguous meaning on one of the propositions of the argument. Note that, as is clear from the example above, the meaning forced on a proposition by an argument is not an interpretation of the proposition but a constraint on how it may be interpreted.

Now we change to how to formally express “a self-contradictory argument.” Let us by the meaning of a proposition mean an interpretation of the proposition. For instance, the wind is blowing is the meaning of the proposition B in the example above. Let A be a formula occurrence in a deduction in some formal system. To denote that A has a certain meaning, m say, we decorate A with m. More precisely,

we shall write m : A to denote that A has the meaning m. We use these decorations

to define meaning conditions. Meaning conditions are formal representations of the constraints given by the meaning forced on a proposition by an argument. For every formal system considered in this article we shall do the following. We shall define what the set of formal meanings is for decorating the formulas in deductions in the formal system and we shall give the meaning conditions associated with the formal system. Thus, through the meaning conditions we formally define what is informally described by “the way in which a proposition is used in an argument.” By an assignment of meanings to the formulas in a deduction we mean a decoration of all of the formulas in the deduction. That a meaning is assigned to a formula means that the formula has been decorated with the meaning. The meaning conditions are given as constraints on the decorations, by formal meanings, of the formulas in the deductions. As an example let us consider, in the formal system N, a deduction

consisting of an ⊃E inference, α say. Let X , Y and Z be the major premise, the

minor premise and the conclusion, respectively, of α. Let mx , m y and mz denote some meanings assigned to X , Y and Z , respectively. We decorate the formulas in the deduction as follows.

m x : X m y : Y α

mz : Z

Reasoning in the same way as in the previous example, we know that since X is the major premise of an ⊃E inference, X must be X1 ⊃ X2 for some propositions X1 and X2. We express this constraint by requiring the meaning mx to be mn for some meanings m and n, where thus ⇒ means “implies that.” Moreover we require m y to be m and mz to be n. Thus, mx may not be it's snowing and the wind is blowing. However m y may be it's snowing and the wind is blowing and mz may be the wind is blowing. In this case mx must be it's snowing and the wind is blowing implies that the wind is blowing. We express meaning conditions given for any ⊃E inference in any deduction in the formal system N by the schema

D E

m n : A m : B E

n : C

Hence the meaning condition for the major premise A of an ⊃E inference is that A must have the meaning mn for some meanings m and n. Moreover, the meanings of the major premise, the minor premise and the conclusion respectively must have the relation to each other expressed by the schema. The notion of a self-contradictory deduction in a formal system is defined as follows.

Definition 1 Assume that F is a formal system. Assume that the set of formal meanings for decorating the formulas in the deductions in F are defined, and assume that the meaning conditions associated with the formal system are given in some way. Then a deduction D in F is self-contradictory if there is no assignment of formal meanings to the formulas in D such that this assignment satisfies the meaning conditions.

The meaning conditions, as we shall give them, are related to the inversion principle of Prawitz. In Prawitz (1965) [6] we can read the following.

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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