Franco Montagna (1948–2015)

1 The backstage

Two powerful streams in the first half of last century impeded the development of Mathematical Logic (ML) in Italy after Peano and his school. From one side, Italian culture was dominated by the local version of Idealistic Philosophy¹ (Gentile was Minister for Education under Mussolini) with its emphasis on the superiority of philosophy over science. From the other side, the reaction of mathematicians, in particular, was to establish their point by stressing the applicability of Mathematics (Vito Volterra was the head of the National Research Council in the same period). ML was smashed in between the two damnations: “Mathematica sunt, non leguntur–Philosophica sunt, non leguntur”. For instance, the influential mathematician and philosopher Federigo Enriques (1871–1946)² expressed dramatic judgments about the developments of ML and of Set Theory well into the thirties. For him, while at its origins in the hands of Bolzano, Dedekind or Cantor the mathematical investigation of the Foundations was a decisive step for Mathematics, the abstraction in the current ML of his time was deemed as “transcendent” or “metaphysical”; for instance, Enriques never accepted the axiom of choice or any unbridled usage of actual infinity; he apparently did not take seriously Symbolic Logic: on that he possibly agreed with Benedetto Croce, in considering it as a child’s game. There was no teaching of ML courses in the Universities and preciously few papers in ML by Italian mathematicians in that subject. All this happened while Hilbert, Gödel, Gentzen, Herbrand, Turing, Tarski, Von Neumann, Kleene, Mostowsky and their peers were establishing broad and deep roots for ML. In the sixties some young smart Philosophers and Mathematicians in Florence and elsewhere, with the important stimulus by Ludovico Geymonat,³ revived the interest in ML. Among them,⁴ Roberto Magari was considered the brightest (he got a chair at Ferrara University in 1969, albeit in Algebra, not in ML). In the fresh atmosphere brought by the 1968 student movement more than half dozen students went there from all over Italy to have him as an advisor for their master thesis in 1970–1973, when Magari was engaged primarily in Universal Algebra, and Franco Montagna was one of them.

2 Personality and academical frame

Franco Montagna was born in Broni, near Pavia, on September 27, 1948 and died in Siena, February 18, 2015. His wife Tonina, the mother of their two children, is beloved by all colleagues and coauthors of Franco: he considered her as his life’s perfect companion. She painstakingly and merrily typewrote for years umpteen papers for Franco, before Latex took care of that.⁵ Franco was a naturally polite, apparently shy but easygoing fellow. He loved some good soccer game and became fond of true music. When first meeting him, people felt a maternal urge to protect him: a mistake if any, as everybody had to admit after not long. The man had a steel determination, an absolute quest for precision and consistency, and an overcapacity for research work, wrapped inside the appearance of a motherless child.

His master thesis at the University of Pavia in 1972 was on “Operators on classes of Algebras” under the joint direction of Cesare Tonti (Pavia)⁶ and Magari (Ferrara). When Magari moved to the University of Siena, he followed him in 1973, with a grant from the National Research Council. He was a lecturer in Siena from 1976; in 1983 he became associate professor, and in a short time full professor. Magari had established in Siena an Advanced School in Mathematical Logic (the first such in Italy; the law which instituted PhD programs at Italian Universities arrived only in 1983). He was the Chairman of the Advanced School (1987–1990). When finally the PhD program in ML was opened in Siena, he was the Coordinator from 1994 to 2002 and again from 2007 to 2011. Then the Department of Mathematics was forced, by shortsighted politics and by the purse-narrowing economical policy, to merge with—in fact to become a residual appendix of—the local department of Engineering. He was spared from seeing the forthcoming disappearance (from 2015 to 2016) of the graduate courses in Pure Mathematics and of the PhD program in ML at Siena University. Because of his unassuming behavior, tameness of character, dedication to research, rather than to academic struggles, he could not have done much against such a dreadful destiny. Since 1987, he was the Coordinator of four successive National Research Projects in ML, each covering several years. He supervised nine PhD students, and was a mentor of several European post-doc fellows and researchers.

Among the generation of scholars who revived ML in Italy, he was one of the most renown. He published more than 130 scientific papers mostly in leading journals, and the number of coauthors he worked with is impressive: 69 of them. A good portion of those were young people which found in his guidance and stimulus a gentle but powerful force. Montagna had acquired a wide-ranging mathematical culture, became soon an active member of the international net of researches in pure and applied logic, always followed the recent advancements, and was able to contribute substantially to several chapters of ML: Modal Logics, Many-valued Logics and Algebraic Logic, as well as to Probability Theory and Theoretical Computer Science.

3 The mathematics

We will summarize next some chapters of his main scientific contributions in three periods of his activity: Provability Logic until the end of the 1990; Many-valued Logics and their connections with Probability Theory afterward, and finally a hint at further themes.

3.1 Logic

After a pair of papers (Montagna 1974a, b) deriving from his master’s thesis, he began a thorough investigation of Diagonalizable Algebras (which were introduced by Magari and now are called Magari Algebras) and Provability Logic.

A Magari algebra is a Boolean algebra with a further unary operation $\tau$ satisfying: $\tau (1)=1$, $\tau (x\wedge y)=\tau (x) \wedge \tau (y)$, $\tau (\tau (x) \rightarrow x)=\tau (x)$. These identities algebrize the Hilbert–Bernays’ and Löb’s conditions of the “provability predicate” $Theor_T(x)$ invented by Gödel to obtain his milestone results on incompleteness for suitably strong consistent first-order theories T. Namely, such a theory T proves a formula $\alpha$ iff in a suitable basic theory S for arithmetic, the formula $Theor_T(\alpha ^*)$ is provable, $\alpha ^*$ being the numeral of the Gödel number of $\alpha .$ Provability Logic deals with the same ideas in a propositional modal logic where the modality has the corresponding properties. As a formal modal logic, this existed already under the name K4W, but the arithmetical viewpoint came out in the same years, and it was investigated by de Jong, Smorinsky, Boolos and others, and is also known as $\mathbf {GL}$. ”Diagonalizable” made reference to the fixed point property which algebrizes Gödel Diagonalization Lemma: for any term t(x) in which the variable x appears only within the scope of some $\tau ,$ there is a (unique) term f such that $t(f)=f$ holds in the variety of Magari algebras. This was proved by Claudio Bernardi (1975, 1976), and then rediscovered by many other people, notably in an effective form by Giovanni Sambin (1976) (both of them where in Siena at that time). (As a matter of fact, a previous paper by Magari had introduced a different notion, namely the “Diagonalized Algebras” in which, besides the above identities, the fixed point property was assumed in the definition; these of course disappeared, after the proofs of the fixed point property.)

If ${{\mathrm{PA}}}$ denotes first-order Peano Arithmetic, its Lindenbaum algebra $\mathcal {L}_{{{\mathrm{PA}}}}$ is a Magari algebra with $\tau$ being the algebraization of the provability predicate $Theor_{{{\mathrm{PA}}}}(x)$ of ${{\mathrm{PA}}}$ in itself. Montagna (1975) shows that for every natural number n there are identities which hold in the free Magari algebra on n generators but fail to hold in the whole variety. The main result in this framework was Robert Solovay’s Completeness theorem (1976): $\mathcal {L}_{{{\mathrm{PA}}}}$ generates the whole variety of Magari algebras, which means that Magari’s identities capture exactly the algebraizable properties of $Theor_{{{\mathrm{PA}}}}(x)$. Montagna (1979a) got an improved version of such arithmetical completeness showing that the free Magari algebra on $\aleph _0$ generators is a subalgebra of $\mathcal {L}_{{{\mathrm{PA}}}}$ and more generally that it embeds into $\mathcal {L}_T$, whenever T is an recursively enumerable extension of ${{\mathrm{PA}}}$ (or of Elementary Arithmetic), provided in $\mathcal {L}_T$, $\tau ^n(0)\ne 1$ for every $n>0$.

After that, the equational theory of Magari Algebras is decidable and coincides with the identities of $\mathcal {L}_{{{\mathrm{PA}}}}.$ What about the first-order theory of Magari Algebras? Montagna (1980a, b) proved directly that it is undecidable; this can be also easily obtained from general results in universal algebra.⁷ What about the first-order theory of $\mathcal {L}_{{{\mathrm{PA}}}}?$ A number of people around the world addressed this problem: Shavrukov in 1994 proved that it is undecidable.

Montagna (1984b) addresses to the predicative (i.e., first order) logic of provability, showing that many properties of $\mathbf {GL}$ do not transfer to the predicative version $\mathbf {QGL}$. In particular: (1) $\mathbf {QGL}$ shares no fixed point property; (2) it is not complete with regard to any classes of Kripke frames; (3) it is not arithmetically complete: it does not contain the predicative logic of provability of ${{\mathrm{PA}}},$ which in turn implies: (4) it is different from the predicative logic of set theory ZF. Montagna (1978) produced an algebraization of the non-standard Feferman’s provability predicate. Feferman invented it to show that the arithmetical representation of a provability predicate (for ${{\mathrm{PA}}},$ say) is quite sensitive to minimal variations, specifically it can be intensionally wrong, while still correctly numeralwise representing the set of provable sentences of the theory. With a particular Feferman’s predicate, for instance, the corresponding Consistency sentence results provable (of course, such Feferman’s predicate does not satisfy all of Hilbert–Bernays Conditions). Montagna introduced $\rho$-algebras, which are Boolean algebras with a suitable further unary operation $\rho$ algebraizing Feferman’s predicate. He also introduced $\rho ,\tau$-algebras, in which both operators are present, and gave equational axioms for them. There are remarkable differences between $\rho ,\tau$-algebras and Magari algebras already regarding the existence and uniqueness of fixed points. This paper also had a lively impact: for instance, Shavukrov worked with the corresponding (propositional) bimodal logic and, with the addition of two further axioms, he showed that this logic is decidable and arithmetically complete.

Some work by Montagna (specifically Jongh and Montagna 1987) was among the precursors of so-called interpretability logic: we have binary modalities like $\alpha \vartriangleright _T \beta$ (where T is an arithmetical theory sufficiently powerful) to express that there is a relative interpretation of $T+ \beta$ into $T+\alpha$. In Di Paola and Montagna (1991); Hájek and Montagna (1992) he proved that the interpretability logic of ${{\mathrm{PA}}}$ is complete with regard to $\vartriangleright _T$-conservativity of $\Pi ^0_1$-sentences (meaning that every $\Pi ^0_1$-sentence which is provable in $T+\beta$ is provable in $T+\alpha$).

3.2 Uncertainty

From the 1990 on, Montagna was engaged with fuzzy logics and their algebrization, inspired by his long-standing interest in probability theory, under Magari’s influence. In the influential book Metamathematics of Fuzzy Logic by Petr Háyek, a friend and co-author of Montagna, the so-called Basic Logic is introduced, as the set of all formulas validated by all continuous t-norms. The corresponding algebraic semantics is the variety of BL-algebras: bounded residuated integral, commutative, prelinear and divisible lattices. Montagna’s first remarkable contributions were in Esteva et al. (2004): (1) every totally ordered BL-algebra is an ordinal sum of a family of Wajsberg hoops, the first of which is a Wajsberg algebra and (2) the variety of BL-algebras is generated by a single algebra, which is the ordinal sum of $\aleph _0$ copies of the standard MV-algebra on the real interval [0, 1].

In Montagna et al. (2003) he deals with the logic MTL: he introduced a method (known as the Jenei–Montagna method) to prove that every totally ordered MTL-algebra is embeddable into a standard MTL-algebra and hence that MTL is complete w.r.t. standard MTL-algebras.

Other contributions to many-valued logics are in Baaz et al. (2001), Montagna and Sebastiani (2001); Marchioni and Montagna (2008), Esteva et al. (2002), Bova and Montagna (2008), Hájek and Montagna (2008) and Montagna (2011c), in particular several generalizations of BL-algebras and a detailed study of interpolation and Beth property.

3.3 Foundations of probability

Montagna also devoted himself to the foundation of subjective probability in de Finetti’s approach, but with multi-valued events (rather then just two-valued events). Two outstanding technical contributions were: the theory of internal states in MV-algebras and the notion of stable coherence for assignments of conditional probability. According to some results by Daniele Mundici, states in an MV-algebra (the algebra of events) can be considered as a generalization of the classical states in a Boolean algebra. A state on an MV-algebra is a normalized additive mapping from the algebra to the real interval [0, 1]. But now a state can be internalized, thus becoming a map from the MV-algebra into itself. Franco Montagna, after introducing internal states in 2009, obtained characterization theorems (Ciabattoni and Montagna 2013) and further generalizations which led to the algebraic treatment of fuzzy probabilities (Fedel et al. 2013a; Hosni and Montagna 2014), conditional probabilities (Fedel et al. 2013b) and of their logic foundations.

As it happens, full maturity inspires some mathematicians to take up the core conceptual aspects of their discipline, and Montagna too turned to the foundation of conditional probability on events in an MV-algebra. His inspiration was de Finetti’s Coherence Theorem.⁸ Franco’s idea is as follows. Assume $\beta$ is an assignment in [0, 1] on some finitely many conditional events $a_1\mid b_1, \ldots , a_n\mid b_n$ and on their conditioning events $b_1,\ldots , b_n$ wit $a_i, b_i$ elements of a given MV-algebra. The question is whether there exists a state s on the MV-algebra such that for all $i=1,\ldots , n$, $\beta (a_i\mid b_i)=s(a_i\cdot b_i)/s(b_i)$ and $\beta (b_i)=s(b_i)$ (WLOG we can assume that $a_i$ and $b_i$ are real-valued functions, and $a_i\cdot b_i$ is their pointwise product). For that question to be meaningful, one should have $s(b_i)>0$ for all i,  which is not always the case. Thus $\beta$ is called stable coherent if: (1) it is de Finetti-coherent, (2) there exists an hyperreal-valued variant $\beta '$ which is still coherent and moreover (3) for every conditioning event $b_i$, $\beta '(b_i)>0$ (which might be positive infinitesimal), and (4) $\beta$ is infinitely close to $\beta '$, namely for every conditional event $a_i\mid b_i$, the distances $|\beta (a_i\mid b_i)-\beta '(a_i\mid b_i)|$ and $|\beta (b_i)- \beta '(b_i)|$ are both infinitesimal. Montagna proves that $\beta$ is stably coherent iff there is an hyperreal-valued state s such that for all $i=1,\ldots , n$, both $|\beta (a_i\mid b_i)- s(a_i\cdot b_i)/s(b_i)|$ and $|\beta (b_i)-s(b_i)|$ are infinitesimal.

3.4 $\ldots$ and beyond

In Computability Theory (formerly Recursion Theory), Franco dealt with completeness and universality for the preorder corresponding to provable implication in ${{\mathrm{PA}}}$ (Montagna and Sorbi 1985) and for the corresponding equivalence relation (Montagna 1982), giving in Montagna (1982) an interesting characterization in terms of recursion theory. Toward the end of the 1990s he dealt with Learning Theory (probabilistic paradigms). He also worked on the speed-up of formal proofs (see Carbone and Montagna 1989, 1990; Montagna 1992; Hájek et al. 1993; Fontani et al. 1993), in particular on speeding-up the length of proofs by means of modal rules. Let’s quote a sample result from Montagna (1992): let T be a $\Sigma _1$-valid extension of the fragment $I\Delta _0+\Omega _1$ of first-order Arithmetic; a modal rule A / B is called T-consistent if it does not add any new theorem, namely from $T\vdash A^*$ it follows $T\vdash B^*$ for every interpretation $^*$, where $(\square C)^*:= Theor_T(\ulcorner C^*\urcorner )$), and moreover, T proves $A^*$ for at least one interpretation. For such a T-consistent modal rule, he shows that the speed-up is either polynomial or else super-exponential.