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## The Liar ParadoxIn this section we shall study the liar paradox as an example of a self-contradictory argument. The liar paradox is the following. Let This argument is very similar to Russell's paradox. Below we present the formal system
¬
[
B ⊃I D
¬
The liar paradox is formally represented by the following deduction
⊥ ⎪ ¬ The set of formal meanings to be assigned to formulas in deductions in the formal system
[
Now assume that there is an assignment of formal meanings to the formulas in the deduction Assume that
above we conclude that the meaning of the premise
⎬⎪ ⊃ ? : ¬ The condition given for the ⊃I inference schema requires both of the formulas cancelled at the ⊃I inference in ## Self-contradictory Reasoning in N−∀∃=Let
In this section and the two succeeding ones we shall use the terminology of Ekman (1994) [2, Sect. 3.1], see Appendix below. Hence, by “normalizable” in Theorem 1 we mean normalizable as defined in Ekman (1994) [2, Sect. 3.1], see Appendix below. As in the formal system Assume that is no normal proof of ⊥ in contradictory, if by paradox we mean a proof of ⊥. In Ekman (1994) [2, Sect. 2.1] it is shown that there is no normal proof of the formula defined by Hence, every proof of ∈ Sect. 2.1] also a proof, named Crabbe's counterexample (see Crabbé (1974) [1]), of the formula counterexample is a self-contradictory proof. It is also argued in Ekman (1994) [2, Sect. 2.1] that Crabbe's counterexample expresses a correct argument in the formula The variables of the language of
the formal system
[
n : A ⊃ B ⊃ m ⇒ n : A ⊃ B m : A
∧ n : A & &E1B
∧ n : A & &E2B
n : A ∨ B ∨ n : + B mn : A ∨ B ∨ m 1 + m 2 : A 1 ∨ A 2 n : C n : C
Let
⎪ ⎪⎪ ∈E⎬ ⎪ ⎪ ⎭ In this case
If [ ⎪ ⎪
G ⎬
In this case
⎪
For all other cases of the kind of reduction that takes
Let the formal system in the formal system
We extend ∗ to a function from the set of sets of meanings to be assigned to formulas in deductions in the formal system We extend ∗ once more, to a function from the set of non-self-contradictory deductions in consisting of the open assumption ⎛ ⎝
Observe that there is no open assumption of the form ⊥⊃⊥ in ⊃I inference, in the deduction to the right above. ⎛ ⎝
⊃E For all other cases of the end inference of a deduction ⎛ [ ⎜ ⎟ ≡ ⎝
Since |

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