Abstract
We start from the algebraic method of theorem-proving based on the translation of logic formulas into polynomials over finite fields, and adapt the case of first-order formulas by employing certain rings equipped with infinitary operations. This paper defines the notion of M-ring, a kind of polynomial ring that can be naturally associated to each first-order structure and each first-order theory, by means of generators and relations. The notion of M-ring allows us to operate with some kind of infinitary version of Boolean sums and products, in this way expressing algebraically first-order logic with a new gist. We then show how this polynomial representation of first-order sentences can be seen as a legitimate algebraic semantics for first-order logic, an alternative to cylindric and polyadic algebras and closer to the primordial forms of algebraization of logic. We suggest how the method and its generalization could be lifted successfully to n-valued logics and to other non-classical logics helping to reconcile some lost ties between algebra and logic.
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- 1.
In this paper “ring” always means commutative ring with unity.
- 2.
Distinct, obviously, from Boolean algebras.
- 3.
The operator
clearly induces a finitary closure operator \( \overline{( \ )} : Parts(F[X]) \rightarrow Parts(F[X])\), in particular, 
iff exists \(S' \subseteq _{fin} S\) such that 
. - 4.
The symbol
denotes “reduction by means of polynomial rules”; in order to ease reading, however, we shall use the symbol \(\approx \) everywhere when there is no danger of misunderstanding. - 5.
- 6.
In more details, \(v^F\) is recursively defined by: \(v^F(P_i) := v(P_i),\ v^F(\alpha \wedge \beta ) := v^F(\alpha ) \wedge v^F(\beta ),\ v^F(\alpha \vee \beta ) := v^F(\alpha ) \vee v^F(\beta ),\ v^F(\alpha \rightarrow \beta ) := v^F(\alpha ) \rightarrow v^F(\beta ),\ v^F(\lnot \alpha ) := \lnot v^F(\alpha )\).
- 7.
By definition, \(\prod S =1\) whenever \(S = \emptyset \).
- 8.
In a Boolean ring, \(ab =1\) iff \(a =b=1\). Indeed: if \(ab =1\), then \( 1= ab = a^2b =a(ab) = a.1 =a\).
- 9.
However, it must be noted that a heterodox proposal to algebraize paraconsistent logics is proposed in [6].
- 10.
I.e., if \((p,p') \in C\) and \((q,q') \in C\), then: \((-p,-p') \in C\), \((p + q, p' + q') \in C\), \((p . q, p'.q') \in C\).
- 11.
Note that the gluing \(S_{i,a} : |F(M)| \rightarrow |F(M)|\) preserves rank.
- 12.
Remember that \(rank(S_{i,a}(r) = rank(r)\).
- 13.
Remember the identifications in Exercise 27.
- 14.
This rule is equivalent to \(r \rightarrow r \approx 1\), since \((r\rightarrow r) = r.r+r+1\) and R(M) is, by construction, a ring of characteristic 2, i.e. the “index rule” \(r+r = 0\) is already true in R(M).
- 15.
I.e., for each M-homomorphism \(\mathscr {M}\)-compatible \(H : R(M) \rightarrow \mathbb {Z}_2\), the left and the right side of the rule have the same image under H.
- 16.
By the distributive law in \(R(\mathscr {M})\) and \({r + r = 0} \).
- 17.
By the correct rule \(r .(1+r)\approx 0\).
- 18.
In fact, \(j^{\star }\) constitutes a contravariant from the category of \(L'\)-structures and \(L'\)-homomorphisms into the category of L-structures and L-homomorphisms: if \(h : \mathscr {M}' \rightarrow \mathscr {N}'\) is a \(L'\)-homomorphism, then the same map \(h : j^{\star }(\mathscr {M}') \rightarrow j^{\star }(\mathscr {N}')\) is an L-homomorphism.
- 19.
I.e., the class of pairs \(\mathscr {M}= (M, [[-]])\), where M is set, \([[-]]\) is a map \((\phi (x_1, \ldots , x_k) \in Form(L)) \ \mapsto \ ([[\phi (\bar{x})]] : M^k \longrightarrow B_T)\), satisfying the usual (but conditional, since \(B_T\) may not be complete) compatibility requirements of Boolean valued models and, moreover, if \(\phi (x_1, \ldots , x_k) \in T\), then \([[\phi (\bar{x})]] = 1_{B_{T}}\).
clearly induces a finitary closure operator 
iff exists 
.
denotes “reduction by means of polynomial rules”; in order to ease reading, however, we shall use the symbol References
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Carnielli, W., Luiz Mariano, H., Matulovic, M. (2018). Reconciling First-Order Logic to Algebra. In: Carnielli, W., Malinowski, J. (eds) Contradictions, from Consistency to Inconsistency. Trends in Logic, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-98797-2_13
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