Abstract
We argue that formal logical systems are four-valued, these four values being determined by the four deductive outcomes: A without ~A, ~A without A, neither A nor ~A, and both A and ~A. We further argue that such systems ought to be three-valued, as any contradiction, A and ~A, should be removed by reconceptualisation of the concepts captured by the system. We follow by considering suitable conditions for the removal of the third value, neither A nor ~A, yielding a classically valued system. We then consider what values are appropriate for the meta-theory, arguing that it should be three-valued, but reducible to the two classical values upon the decidability of the object system.
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Acknowledgments
I would like to thank the referee for a number of incisive comments on this paper. I would also like to thank the members of the University of Melbourne Logic Seminar for their interesting and pertinent discussion of this paper and, in particular, for pointing out a glaring error of reference. I would also like to thank Thomas Ferguson for encouraging me to continue with this paper, even after I felt that I could not complete it in a reasonable time.
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Appendix
Appendix
The required axiomatization of MCQ is set out below:
MC.
Primitives: ~, &, ∨, → .
Axioms.
-
1.
A → A.
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2.
A&B → A.
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3.
A&B → B.
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4.
(A → B)&(A → C) → .A → B&C.
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5.
A → A∨B.
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6.
B → A∨B.
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7.
(A → C)&(B → C) → .A∨B → C.
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8.
~~A → A.
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9.
A → ~B → .B → ~A.
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10.
(A → B)&(B → C) → .A → C.
Rules.
-
1.
A, A → B ⇒ B.
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2.
A, B ⇒ A&B.
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3.
A → B, C → D ⇒ B → C → .A → D.
Metarule.
-
1.
If A, B ⇒ C then D∨A, D∨B ⇒ D∨C.
We now add the quantifiers to yield MCQ. As in earlier presentations, we separate free and bound variables to simplify the conditions on the axioms.
MCQ.
Primitives: ∀, ∃,
a, b, c, … (free variables)
x, y, z, … (bound variables)
-
Axioms.
-
1.
∀xA → Aa/x.
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2.
∀x(A → B) → .A → ∀xB.
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3.
Aa/x → ∃xA.
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4.
∀x(A → B) → . ∃xA → B.
Rule.
-
1.
Aa/x ⇒ ∀xA, where a does not occur in A.
Metarule.
-
1.
If A, B ⇒ C then A, ∃xB ⇒ ∃xC,
where QR1 does not generalize on any free variable in A or in B. The same applies to the premises A and B of the metarule MR1 of MC.
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Brady, R.T. (2019). The Number of Logical Values. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_3
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