Sergio Galvan | Università Cattolica del Sacro Cuore (Catholic University of the Sacred Heart) (original) (raw)

Papers by Sergio Galvan

Research paper thumbnail of Normal deductions

Oxford University Press eBooks, Aug 17, 2021

The concept of a normal deduction is introduced as a way to capture the notion of a “direct” proo... more The concept of a normal deduction is introduced as a way to capture the notion of a “direct” proof, i.e., one that makes no detours. The latter happens, e.g., when a formula is introduced first by a rule of introduction (say, starting from A and B to introduce the conjunction A ∧ B) and is followed immediately by an elimination of the same connective (yielding, for instance, A). A normalization theorem proves that if one has a proof with detours then there is a proof without detours. Proving such more advanced results in proof theory requires more complex proof methods than simple induction. A first step to more complex proofs is the use of double induction. A first application of double induction is the proof of normalization for a fragment of minimal logic containing conjunction, implication, negation, and the universal quantifier. The result is extended to full intuitionistic logic, followed by a discussion of the structure of normal deductions. Normal deductions have the sub-formula property, and so cannot be proofs of contradictions. This shows that natural deduction is consistent. Normalization can also be used to show that certain formulas cannot be proved, e.g., that the principle of excluded middle cannot be proved in intuitionistic logic. Finally, a full proof of normalization for full classical logic is given.

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Research paper thumbnail of Axiomatic calculi

Oxford University Press eBooks, Aug 17, 2021

The first results in proof theory were obtained using axiomatic systems of logical deduction. The... more The first results in proof theory were obtained using axiomatic systems of logical deduction. The chapter develops propositional and predicate logic in such an axiomatic system. It shows how reasoning under assumption can be regained in an axiomatic system by proving the so-called “deduction theorem” propositional logic (and later, predicate logic as well). The deduction theorem provides the opportunity to introduce one of the main tools needed for proving properties of formal systems, namely, proof by induction on the natural numbers. The differences between minimal, intuitionistic, and classical systems of logic according to the properties of negation characterizing each of the systems are outlined. Independence of theorems such as the law of excluded middle are here given model-theoretically (in later chapters, these are also presented proof-theoretically). After presenting classical predicate logic, the systems of classical and intuitionistic arithmetic are presented. Gentzen’s first substantial result, the so-called Gödel-Gentzen translation, is discussed in the final section where it is proved that if intuitionistic arithmetic is consistent, so is classical arithmetic.

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Research paper thumbnail of The cut-elimination theorem

Oxford University Press eBooks, Aug 17, 2021

All the rules of the sequent calculus have the property that all the formulas that are present in... more All the rules of the sequent calculus have the property that all the formulas that are present in the premises also occur in the conclusion. There is only one exception, the cut rule. In this chapter, it is shown using double induction that every theorem provable in Gentzen’s sequent calculi using the cut rule can also be proved without. One first proves that the mix rule is equivalent in proof-theoretic strength to the cut rule. Then one proves by induction on two complexity measures, the degree and the rank of a mix, that every application of mix can be eliminated. Several results follow from the cut-elimination theorem, including the so-called “mid-sequent” theorem and Herbrand’s theorem. In addition, one can prove the consistency of minimal, intuitionistic, and classical logic.

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Research paper thumbnail of The consistency of arithmetic

Oxford University Press eBooks, Aug 17, 2021

This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of... more This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of interest are not purely logical ones, but rather formalized versions of mathematical theories, and in particular the first-order version of classical arithmetic built on top of the sequent calculus. Classical arithmetic goes beyond pure logic in that it contains a number of specific axioms for, among other symbols, 0 and the successor function. In particular, it contains the rule of induction, which is the essential rule characterizing the natural numbers. Proving a cut-elimination theorem for this system is hopeless, but something analogous to the cut-elimination theorem can be obtained. Indeed, one can show that every proof of a sequent containing only atomic formulas can be transformed into a proof that only applies the cut rule to atomic formulas. Such proofs, which do not make use of the induction rule and which only concern sequents consisting of atomic formulas, are called simple. It is shown that simple proofs cannot be proofs of the empty sequent, i.e., of a contradiction. The process of transforming the original proof into a simple proof is quite involved and requires the successive elimination, among other things, of “complex” cuts and applications of the rules of induction. The chapter describes in some detail how this transformation works, working through a number of illustrative examples. However, the transformation on its own does not guarantee that the process will eventually terminate in a simple proof.

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Research paper thumbnail of Ordinal notations and induction

Oxford University Press eBooks, Aug 17, 2021

In order to prove that the simplification process for arithmetic eventually reaches a simple proo... more In order to prove that the simplification process for arithmetic eventually reaches a simple proof, it is necessary to measure the complexity of proofs in a more sophisticated way than for the cut-elimination theorem. There, a pair of numbers suffices, and the proof proceeds by double induction on this measure. This chapter develops the system of ordinal notations up to ε0 which serve as this more sophisticated measure for proofs in arithmetic. Ordinal notations are presented as purely combinatorial system of symbols, so that from the outset there is no doubt about the constructive legitimacy of the associated principles of reasoning. The main properties of this notation system are presented, and it is shown that ordinal notations are well-ordered according to its associated less-than relation. The basics of the theory of set-theoretic ordinals is developed in the second half of the chapter, so that the reader can compare the infinitary, set-theoretic development of ordinals up to ε0 to the system of finitary ordinal notations. Finally, Paris-Kirby Hydra game and Goodstein sequences are presented as applications of induction up to ε0.

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Research paper thumbnail of Exercises on axiomatic derivations

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of Natural deduction

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of Sequent calculus

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of Outline of the cut-elimination theorem

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of The Greek alphabet

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of The consistency of arithmetic, continued

Oxford University Press eBooks, Aug 17, 2021

By assigning ordinal notations to proofs in classical arithmetic it is possible to show that each... more By assigning ordinal notations to proofs in classical arithmetic it is possible to show that each step in the simplification process making up the consistency proof, the complexity of proofs, as measured by the associated ordinal notation, successively decreases. Since the system of ordinal notations is well-ordered, it is not possible to have an infinite decreasing sequence. The reduction process described in Chapter 7 therefore must bring down the ordinal notations assigned to the successive transformed proofs, and must ultimately end with a proof whose ordinal notation is less than ω, i.e. a “simple” proof. Since “simple” proofs cannot be proofs of a contradiction, this establishes the consistency of classical arithmetic.

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Research paper thumbnail of The sequent calculus

An Introduction to Proof Theory

In addition to natural deduction, Gentzen developed a different calculus, called the sequent calc... more In addition to natural deduction, Gentzen developed a different calculus, called the sequent calculus. A sequent is a configuration presenting an arrow symbol (⇒) flanked on the left and on the right by finite sequences of formulas, possibly empty. The sequent calculus is developed, with examples of how to prove statements in the calculus, and a few results about transforming proofs through variable replacements are proved. Proofs in the intuitionistic sequent calculus can be translated into natural deductions, and vice versa (this system is obtained by restricting sequents to those that have at most one formula on the right hand side of the arrow).

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Research paper thumbnail of An Introduction to Proof Theory

Oxford University Press eBooks, Aug 17, 2021

Proof theory is a central area of mathematical logic of special interest to philosophy. It has it... more Proof theory is a central area of mathematical logic of special interest to philosophy. It has its roots in the foundational debate of the 1920s, in particular, in Hilbert’s program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, “finitary” means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a “simple” proof. This is Hilbert’s central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail.

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Research paper thumbnail of Essentialism and Forms of Necessity

With the death of E.J. Lowe, the analytical philosophy lost one of the most influential thinkers ... more With the death of E.J. Lowe, the analytical philosophy lost one of the most influential thinkers of the last thirty-five years. In particular, the most innovative part of Lowe's perspective is the four-category ontology. This framework is the subject-matter of the analyses carried out in Corradini-Galvan's essay

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Research paper thumbnail of Il problema della conoscenza matematica

"Il problema della conoscenza matematica" in Scritti per Luigi Lombardi Vallauri, Vol. 1, Wolters Kluwer CEDAM, 2016

L'articolo difende una concezione aprioristica del sapere matematico che fa appello a una forma d... more L'articolo difende una concezione aprioristica del sapere matematico che fa appello a una forma di intuizione intellettiva delle entità matematiche.

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Research paper thumbnail of Reality of Phenomenal Becoming

In the current debate on the nature of time one can identify two competing positions: the eternal... more In the current debate on the nature of time one can identify two competing positions: the eternalist (B-series) and the Aristotelian (A-series) position. Which is the adequate one? I argue for the reality of becoming (A-series) on the basis of the experience, undeniable even for an eternalist, of the change of appearance in consciousness. I begin with formal characterizations of the A-theory and the B-theory on the general phenomenon of becoming. Th en, I analyse the notion of change of appearing to consciousness. It turns out that it is not possible to provide an account of it in B-theoretical terms. For at least one state of aff airs p of phenomenal nature it is certain that the reality of becoming is given: A(before, p) 27 \uacA(now, p)

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Research paper thumbnail of Il ruolo e la dimensione della ricerca: l'ambito epistemologico

Repor

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Research paper thumbnail of Teoria classica dei possibili. Possibilismo classico e suo fondamento attualistico

The current debate on possible objects entails several problems. We focused on the issue of possi... more The current debate on possible objects entails several problems. We focused on the issue of possible objects reality and sided for the possibilist position. As a matter of fact, we believe this position to be the best suited to classical metaphysical tradition. The classical possibilism we defend herein, though, has a realist foundation that is essentially different from Lewis\u2019 concretist possibilism. It construes possible objects as possible non-existing objects of an existing producing power. Consequently, they are nothing vis-\ue0-vis the modality of actual being of their own, though they are existing with regard to the modality of being of the producing power. The actualist requirement prescribed by the Frege-Quinean criterion of the quantification domain is thus fulfilled, therefore possible objects must not necessarily be actual objects. Hence the sense of coincidence between classical possibilism and actualist possibilism

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Research paper thumbnail of 11. Logic of Existence, Ontological Frames, Leibniz’s and Gödel’s Ontological Proofs

Ontological Proofs Today

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Research paper thumbnail of Realismo morale, bene, volere divino

Rivista di Filosofia Neo-Scolastica CXV (2023),1, pp. 189-200 ISSN: 00356247, 18277926, , 2023

This article discusses the theses argued in A. Aguti’s volume, Morale e Religione. Per una vision... more This article discusses the theses argued in A. Aguti’s volume, Morale e Religione. Per una visione teistica, Morcelliana, Brescia 2021, in which the issue of the theistic foundation of ethics is addressed. The article argues that theism has a convincing motivational value vis-à-vis the acceptance of ethics, while it does not have as much value from the point of view of its justification. The basis for non-naturalistic moral realism is a robust theory of the good.

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Research paper thumbnail of Normal deductions

Oxford University Press eBooks, Aug 17, 2021

The concept of a normal deduction is introduced as a way to capture the notion of a “direct” proo... more The concept of a normal deduction is introduced as a way to capture the notion of a “direct” proof, i.e., one that makes no detours. The latter happens, e.g., when a formula is introduced first by a rule of introduction (say, starting from A and B to introduce the conjunction A ∧ B) and is followed immediately by an elimination of the same connective (yielding, for instance, A). A normalization theorem proves that if one has a proof with detours then there is a proof without detours. Proving such more advanced results in proof theory requires more complex proof methods than simple induction. A first step to more complex proofs is the use of double induction. A first application of double induction is the proof of normalization for a fragment of minimal logic containing conjunction, implication, negation, and the universal quantifier. The result is extended to full intuitionistic logic, followed by a discussion of the structure of normal deductions. Normal deductions have the sub-formula property, and so cannot be proofs of contradictions. This shows that natural deduction is consistent. Normalization can also be used to show that certain formulas cannot be proved, e.g., that the principle of excluded middle cannot be proved in intuitionistic logic. Finally, a full proof of normalization for full classical logic is given.

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Research paper thumbnail of Axiomatic calculi

Oxford University Press eBooks, Aug 17, 2021

The first results in proof theory were obtained using axiomatic systems of logical deduction. The... more The first results in proof theory were obtained using axiomatic systems of logical deduction. The chapter develops propositional and predicate logic in such an axiomatic system. It shows how reasoning under assumption can be regained in an axiomatic system by proving the so-called “deduction theorem” propositional logic (and later, predicate logic as well). The deduction theorem provides the opportunity to introduce one of the main tools needed for proving properties of formal systems, namely, proof by induction on the natural numbers. The differences between minimal, intuitionistic, and classical systems of logic according to the properties of negation characterizing each of the systems are outlined. Independence of theorems such as the law of excluded middle are here given model-theoretically (in later chapters, these are also presented proof-theoretically). After presenting classical predicate logic, the systems of classical and intuitionistic arithmetic are presented. Gentzen’s first substantial result, the so-called Gödel-Gentzen translation, is discussed in the final section where it is proved that if intuitionistic arithmetic is consistent, so is classical arithmetic.

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Research paper thumbnail of The cut-elimination theorem

Oxford University Press eBooks, Aug 17, 2021

All the rules of the sequent calculus have the property that all the formulas that are present in... more All the rules of the sequent calculus have the property that all the formulas that are present in the premises also occur in the conclusion. There is only one exception, the cut rule. In this chapter, it is shown using double induction that every theorem provable in Gentzen’s sequent calculi using the cut rule can also be proved without. One first proves that the mix rule is equivalent in proof-theoretic strength to the cut rule. Then one proves by induction on two complexity measures, the degree and the rank of a mix, that every application of mix can be eliminated. Several results follow from the cut-elimination theorem, including the so-called “mid-sequent” theorem and Herbrand’s theorem. In addition, one can prove the consistency of minimal, intuitionistic, and classical logic.

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Research paper thumbnail of The consistency of arithmetic

Oxford University Press eBooks, Aug 17, 2021

This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of... more This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of interest are not purely logical ones, but rather formalized versions of mathematical theories, and in particular the first-order version of classical arithmetic built on top of the sequent calculus. Classical arithmetic goes beyond pure logic in that it contains a number of specific axioms for, among other symbols, 0 and the successor function. In particular, it contains the rule of induction, which is the essential rule characterizing the natural numbers. Proving a cut-elimination theorem for this system is hopeless, but something analogous to the cut-elimination theorem can be obtained. Indeed, one can show that every proof of a sequent containing only atomic formulas can be transformed into a proof that only applies the cut rule to atomic formulas. Such proofs, which do not make use of the induction rule and which only concern sequents consisting of atomic formulas, are called simple. It is shown that simple proofs cannot be proofs of the empty sequent, i.e., of a contradiction. The process of transforming the original proof into a simple proof is quite involved and requires the successive elimination, among other things, of “complex” cuts and applications of the rules of induction. The chapter describes in some detail how this transformation works, working through a number of illustrative examples. However, the transformation on its own does not guarantee that the process will eventually terminate in a simple proof.

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Research paper thumbnail of Ordinal notations and induction

Oxford University Press eBooks, Aug 17, 2021

In order to prove that the simplification process for arithmetic eventually reaches a simple proo... more In order to prove that the simplification process for arithmetic eventually reaches a simple proof, it is necessary to measure the complexity of proofs in a more sophisticated way than for the cut-elimination theorem. There, a pair of numbers suffices, and the proof proceeds by double induction on this measure. This chapter develops the system of ordinal notations up to ε0 which serve as this more sophisticated measure for proofs in arithmetic. Ordinal notations are presented as purely combinatorial system of symbols, so that from the outset there is no doubt about the constructive legitimacy of the associated principles of reasoning. The main properties of this notation system are presented, and it is shown that ordinal notations are well-ordered according to its associated less-than relation. The basics of the theory of set-theoretic ordinals is developed in the second half of the chapter, so that the reader can compare the infinitary, set-theoretic development of ordinals up to ε0 to the system of finitary ordinal notations. Finally, Paris-Kirby Hydra game and Goodstein sequences are presented as applications of induction up to ε0.

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Research paper thumbnail of Exercises on axiomatic derivations

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of Natural deduction

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of Sequent calculus

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of Outline of the cut-elimination theorem

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of The Greek alphabet

Oxford University Press eBooks, Aug 17, 2021

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Research paper thumbnail of The consistency of arithmetic, continued

Oxford University Press eBooks, Aug 17, 2021

By assigning ordinal notations to proofs in classical arithmetic it is possible to show that each... more By assigning ordinal notations to proofs in classical arithmetic it is possible to show that each step in the simplification process making up the consistency proof, the complexity of proofs, as measured by the associated ordinal notation, successively decreases. Since the system of ordinal notations is well-ordered, it is not possible to have an infinite decreasing sequence. The reduction process described in Chapter 7 therefore must bring down the ordinal notations assigned to the successive transformed proofs, and must ultimately end with a proof whose ordinal notation is less than ω, i.e. a “simple” proof. Since “simple” proofs cannot be proofs of a contradiction, this establishes the consistency of classical arithmetic.

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Research paper thumbnail of The sequent calculus

An Introduction to Proof Theory

In addition to natural deduction, Gentzen developed a different calculus, called the sequent calc... more In addition to natural deduction, Gentzen developed a different calculus, called the sequent calculus. A sequent is a configuration presenting an arrow symbol (⇒) flanked on the left and on the right by finite sequences of formulas, possibly empty. The sequent calculus is developed, with examples of how to prove statements in the calculus, and a few results about transforming proofs through variable replacements are proved. Proofs in the intuitionistic sequent calculus can be translated into natural deductions, and vice versa (this system is obtained by restricting sequents to those that have at most one formula on the right hand side of the arrow).

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Research paper thumbnail of An Introduction to Proof Theory

Oxford University Press eBooks, Aug 17, 2021

Proof theory is a central area of mathematical logic of special interest to philosophy. It has it... more Proof theory is a central area of mathematical logic of special interest to philosophy. It has its roots in the foundational debate of the 1920s, in particular, in Hilbert’s program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, “finitary” means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a “simple” proof. This is Hilbert’s central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail.

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Research paper thumbnail of Essentialism and Forms of Necessity

With the death of E.J. Lowe, the analytical philosophy lost one of the most influential thinkers ... more With the death of E.J. Lowe, the analytical philosophy lost one of the most influential thinkers of the last thirty-five years. In particular, the most innovative part of Lowe's perspective is the four-category ontology. This framework is the subject-matter of the analyses carried out in Corradini-Galvan's essay

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Research paper thumbnail of Il problema della conoscenza matematica

"Il problema della conoscenza matematica" in Scritti per Luigi Lombardi Vallauri, Vol. 1, Wolters Kluwer CEDAM, 2016

L'articolo difende una concezione aprioristica del sapere matematico che fa appello a una forma d... more L'articolo difende una concezione aprioristica del sapere matematico che fa appello a una forma di intuizione intellettiva delle entità matematiche.

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Research paper thumbnail of Reality of Phenomenal Becoming

In the current debate on the nature of time one can identify two competing positions: the eternal... more In the current debate on the nature of time one can identify two competing positions: the eternalist (B-series) and the Aristotelian (A-series) position. Which is the adequate one? I argue for the reality of becoming (A-series) on the basis of the experience, undeniable even for an eternalist, of the change of appearance in consciousness. I begin with formal characterizations of the A-theory and the B-theory on the general phenomenon of becoming. Th en, I analyse the notion of change of appearing to consciousness. It turns out that it is not possible to provide an account of it in B-theoretical terms. For at least one state of aff airs p of phenomenal nature it is certain that the reality of becoming is given: A(before, p) 27 \uacA(now, p)

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Research paper thumbnail of Il ruolo e la dimensione della ricerca: l'ambito epistemologico

Repor

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Research paper thumbnail of Teoria classica dei possibili. Possibilismo classico e suo fondamento attualistico

The current debate on possible objects entails several problems. We focused on the issue of possi... more The current debate on possible objects entails several problems. We focused on the issue of possible objects reality and sided for the possibilist position. As a matter of fact, we believe this position to be the best suited to classical metaphysical tradition. The classical possibilism we defend herein, though, has a realist foundation that is essentially different from Lewis\u2019 concretist possibilism. It construes possible objects as possible non-existing objects of an existing producing power. Consequently, they are nothing vis-\ue0-vis the modality of actual being of their own, though they are existing with regard to the modality of being of the producing power. The actualist requirement prescribed by the Frege-Quinean criterion of the quantification domain is thus fulfilled, therefore possible objects must not necessarily be actual objects. Hence the sense of coincidence between classical possibilism and actualist possibilism

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Research paper thumbnail of 11. Logic of Existence, Ontological Frames, Leibniz’s and Gödel’s Ontological Proofs

Ontological Proofs Today

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Research paper thumbnail of Realismo morale, bene, volere divino

Rivista di Filosofia Neo-Scolastica CXV (2023),1, pp. 189-200 ISSN: 00356247, 18277926, , 2023

This article discusses the theses argued in A. Aguti’s volume, Morale e Religione. Per una vision... more This article discusses the theses argued in A. Aguti’s volume, Morale e Religione. Per una visione teistica, Morcelliana, Brescia 2021, in which the issue of the theistic foundation of ethics is addressed. The article argues that theism has a convincing motivational value vis-à-vis the acceptance of ethics, while it does not have as much value from the point of view of its justification. The basis for non-naturalistic moral realism is a robust theory of the good.

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Research paper thumbnail of Introduzione ai Sistemi dell'Aritmetica da Q a PA

Sergio Galvan, 2021

L’aritmetica è il cuore della matematica. Essa è la disciplina che studia la struttura dei numeri... more L’aritmetica è il cuore della matematica. Essa è la disciplina che studia la struttura dei numeri naturali. L’aritmetica non è tuttavia costituita da una unica teoria. Le leggi dei numeri naturali si possono inquadrare in una gerarchia di teorie che partendo dal sistema elementare Q arrivano fino ai sistemi molto potenti dell’aritmetica degli ordini superiori. Burgess traccia nelle prime pagine di Burgess (2005) l’intero arco di tali sistemi e ne fornisce uno schema esauriente alla fine del lavoro. Nel presente volume sono trattati solo alcuni sottosistemi dell’aritmetica del primo ordine, a partire dal sistema di Robinson Q fino al sistema dell’aritmetica di Peano PA. La trattazione di questi sistemi si articola nella presentazione del sistema (assiomi o regole specifiche) e nella derivazione all’interno di ciascuno di essi dei teoremi relativi. L’aritmetica di Peano risulta così spalmata sulla serie dei sottosistemi di PA: Q, Iop, IΔ0, IΣ1, di cui ciascuno è estensione del sistema precedente secondo l’ordine della serie, e di alcune estensioni ricorsive primitive di IΔ0. La trattazione dell’aritmetica all’interno di questi sistemi avviene in parte attraverso l’uso del μ-operatore. I sistemi IΔ0 e IΣ1 costruiti secondo il metodo del μ-operatore (vale a dire IΔ0μ e IΣ1μ) presentano un vantaggio pratico insostituibile nella pratica della derivazione. IΣ1 (in particolare nella versione con μ-operatore) è, tra tutti, il sistema più importante. Infatti IΣ1 è equivalente al sistema dell’aritmetica ricorsiva primitiva (con quantificatori), nel senso che tutte le funzioni ricorsive primitive – tipiche di PRA – sono definibili in IΣ1 e che le funzioni definibili in IΣ1 coincidono esattamente con le funzioni ricorsive primitive. Nel presente lavoro viene dimostrata la prima parte di tale equivalenza. Dalla seconda segue la conservatività di IΣ1 su PRA (e, conseguentemente, sul sistema PR di Skolem dell’aritmetica ricorsiva primitiva senza quantificatori) rispetto alle formule di complessità Π2. I teoremi di IΣ1 esprimibili nel linguaggio di PR sono perciò teoremi della aritmetica ricorsiva primitiva. Tali argomenti sono svolti nelle prime 5 sezioni. Nella sezione 6 sono derivati alcuni risultati necessari per derivare le tre condizioni di derivabilità dei teoremi di incompletezza. Queste sono derivate nella sezione successiva. La dimostrazione è svolta all’interno di opportune estensioni ricorsive primitive di IΔ0. Nella sezione 8 è sviluppata la teoria formale della verità per PA ed è derivato il teorema di Tarski. Il presente volume è il risultato di una rielaborazione profonda del volume Galvan (2007). Sono innanzitutto aggiunte le sezioni 6-8. Inoltre il testo è corredato di 6 appendici che approfondiscono i rapporti tra le singole teorie aritmetiche. Tra queste appendici le prime tre mettono a fuoco il tema della interpretabilità delle teorie aritmetiche in Q, attraverso la individuazione di specifici cuts nella struttura dei numeri naturali. Altre approfondiscono il significato dei vari frammenti dell’aritmetica in ordine alla formalizzazione delle diverse posizioni fondazionali: ultrafinitismo, finitismo e infinitismo. La derivazione di molti teoremi è ripresa, con modificazioni, correzioni e integrazioni, da Galvan (1983), ove, tuttavia, essi sono presentati secondo una diversa classificazione. Lo spirito è però lo stesso. Come si dice nell’introduzione di quel libro “il senso di molti risultati logici sta nella loro dimostrazione, per cui senza l’effettiva esecuzione di quest’ultima essi non sono perfettamente acquisibili”. All’inizio vengono fornite le regole fondamentali del calcolo usato per la formalizzazione dell’aritmetica. Una trattazione più ampia del calcolo si trova in Galvan (1992), pp. 17-62.

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Research paper thumbnail of An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs

Oxford University Press, 2021

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with... more An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details worked out and many examples and exercises. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines Gentzen’s consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal proof theory are developed from scratch. The proof methods needed, especially proof by induction, are introduced in stages throughout the text.

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Research paper thumbnail of ONTOLOGIA DEL POSSIBILE

Sergio Galvan, 2009

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Research paper thumbnail of Logica

Galvan S. Logica. p. 7-304, BRESCIA:La Scuola, 2012

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Research paper thumbnail of Logiche intensionali. Sistemi proposizionali di logica modale, deontica, epistemica.

GALVAN S (1991). Logiche intensionali. Milano:Franco Angeli, ISBN: 88-204-6642-2, 1991

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Research paper thumbnail of GALVAN S. (2006). Einführung in die Unvollständigkeitstheoreme

GALVAN (2006). Einführung in die Unvollständigkeitstheoreme. p. 1-170, PADERBORN:mentis, ISBN: 3-89785-290-X , 2006

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Research paper thumbnail of L' Hauptsatz di Gentzen

GALVAN (2015), Educatt, Milano, 2015

Nel volume viene esposta analiticamente la dimostrazione del teorema di eliminazione del Cut (Hau... more Nel volume viene esposta analiticamente la dimostrazione del teorema di eliminazione del Cut (Hauptsatz) dimostrato da G. Gentzen nelle sue “Untersuchungen über das logische Schliessen”. La dimostrazione originale di Gentzen è una dimostrazione fondata su una doppia induzione, sul rango delle derivazioni e sul grado della formula soggetta al taglio. In questo testo la dimostrazione è una dimostrazione a doppia induzione, ma sulla lunghezza della derivazione e sul grado della formula soggetta al taglio. Il metodo seguito dall’autore, è ripreso con miglioramenti nel rigore formale e nella completezza analitica dei passaggi da L. Heindorf, Elementare Beweistheorie, Wissenschaftsverlag, Mannheim 1994, pp. 103-140. Inoltre, ci sono due differenze. In primo luogo, mentre Heindorf dimostra il teorema per il calcolo intuizionistico LJ, qui è dimostrato per il il calcolo classico LK. In secondo luogo, il calcolo usato nel testo contiene solo la regola strutturale di Weakening e questo consente di semplificare la dimostrazione del teorema di eliminazione del taglio, evitando di dover far ricorso alla regola di fusione equivalente a quella del taglio.
Il volume si articola in tre parti: la prima comprende una esposizione preliminare del calcolo dei sequenti LK. Nella seconda si enuncia e si dimostra l’Hauptsatz. Nella terza sono illustrate alcune importanti conseguenze dell’Hauptsatz, alcune di carattere formale, altre di significato più filosofico. La seconda parte è arricchita di una appendice tecnica, nella quale si studiano le ragioni dell’uso della regola del Mix al posto di quella del Cut.

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Research paper thumbnail of Introduzione ai teoremi di incompletezza

GALVAN (1992). F. Angeli, Milano, 1992

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Research paper thumbnail of Non contraddizione e terzo escluso. Le regole della negazione nella logica classica, intuizionistica e minimale

GALVAN (1997). F. Angeli, Milano, 1997

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