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## Self-contradictory Reasoning in N−∃=Under the assumption that meaning conditions formally express the way in which a proposition is used, as outlined in Sect. 2, it is a bit more complicated to define the meaning conditions associated with a formal system with quantifiers than it is to define the meaning conditions associated with a quantifier-free formal system. In this section we shall study the notion of self-contradictory deductions in the formal system = and the inference schemata corresponding to these symbols from
Let A naïve way to give the meaning conditions associated with
With meaning conditions given this way, we require that there is a one to one correspondence between the meaning of the premise and the conclusion both for ∀I inferences and for ∀E inferences. This condition is however too strong, if we consider ∀ &E inference. As an example, consider the following deduction.
)] &E
&E
∈E
∈E
⊃E ⊥ I ¬ This deduction is non-self-contradictory independently of which formulas system
∈E
∈E
⊥ I ⊃ ¬∀ This deduction is self-contradictory if the meaning conditions are given as above. We suggest the following definition of meaning conditions associated with the formal system
n/x ] : A[t /x ] ∀We have the restriction on the meaning variable, designated
Remember that the aim is to define the meaning conditions so that the meaning conditions express a constraint given by the meaning forced on a proposition given by an argument, in the sense of Sect. 2. Remember also that the meaning forced on a proposition given by an argument is arbitrary so far as what is considered to be known is arbitrary, when knowing only what kind of steps the steps of the argument are. We do not claim that the meaning conditions given are the only possible. The given meaning conditions express constraints which we judge as accurate. We have chosen the constraint defined by the meaning conditions to be no more restrictive than what is necessary to prove Theorem 2. There are however reasons to consider further restrictions on the meaning conditions. Consider the deduction
Assume that In the following we shall not assume this last restriction to be added. Of course, if Theorem 2 holds without this restriction added to the restrictions of the meaning conditions, then this theorem also is true with this restriction added. All meaning condition schemata except the ⊥E meaning condition schema define a relation between the meanings assigned to the premises and the conclusion of the inference. We can interpret this as follows: use of the ⊥E inference schema says that nothing more is known about how the premise of an ⊥E inference is derived other than that it is the premise of an ⊥E inference. Instead of having ⊥ primitively given in ⊥E inference schema as a derived schema, derived as follows, where
Then if we also take
The proof of Theorem 2 is similar to the proof of Theorem 1. To prove Theorem 1 we define a function ∗ from the set of non-self-contradictory deductions in ∗ from the set of non-self-contradictory deductions in in language of the same for are the following.
∀
B/ X ] ∀We have the restriction on deductions in rules for deductions in to one correspondence, ∗ say, between the set of variables of propositional variables of variable
to formulas in deductions in the formal system is defined as follows.
The function ∗ is extended to a function from the set of sets of meanings to be assigned to formulas in deductions in the formal system
set of non-self-contradictory deductions in in Sect. 4. I
I ∀I ≡ ∗
∀
n/x ] : A[t /x ] ∀ ≡ ∀Xm∗
The definition of
From Girard (1971) [4] it is known that deductions in |

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