Computational Logic

Software engineering courses offer one of many opportunities for providing students with a significant experience in declarative programming. This report discusses some results from taking advantage of this opportunity in a two-semester sequence of software engineering courses for students in their final year of baccalaureate studies in computer science. The sequence is based on functional programming using ACL2, a purely

In this paper, we analyse the structure of evolving moral dilemmas with an eye of regard for the increasing importance of the role of artificial intelligence in such context. Starting with the analysis of the famous trolley problem experiment as formulated by Philippa Foot, we consider subsequent variants of this moral dilemma conceived throughout the years, culminating with formulations of the trolley problem concerning artificial intelligence, in which self-driving vehicles will have to make life or death decisions autonomously. In doing so, we investigate the basis for the construction of dilemmatic questions both for humans and machines by considering the problem from a philosophical, social and neuroscientific perspective. After considering and analysing the trolley problem in utilitarian and deontological terms, we follow Rittel and Webber’s footsteps, by highlighting the fallacies of the deontological and utilitarian traditional ‘one-right-answer’ approach, where a solution i...

OR consultants aspire to provide their clients with good reasoning. OR methods are designed to support the process of developing good reasoning. Logic examines what counts as good reasoning. The aim of the paper is to investigate the foundations of OR in terms of the relationship between logic and the activities and methods of OR inquiry. The approach taken is: (i) to identify the types of activities that OR practitioners typically engage in during an OR project; (ii) to introduce some of the main areas of logical reasoning, particularly those relevant to OR practice such as deductive, inductive, modal, and informal logic; and (iii) to bring these two lines of thinking together in order to examine the relationship between logic and OR practice and methods. The key finding is that although the core activities and methods of gathering data, analysing it and developing models are logically well founded, the management of uncertainty in the line of argument could benefit from the insights and methods provided by logic. From this analysis, some practical suggestions are made to improve the process of OR, the development of the line of argument, and the honing of advice given to clients.

(Dalla prefazione:) Questo lavoro si propone di contribuire a chiarire alcuni aspetti dei rapporti che sussistono fra le teorie filosofiche del significato di impostazione logico-modellistica e l'intelligenza artificiale (IA) di orientamento logico. In anni recenti alcuni filosofi del linguaggio hanno tentato di individuare una risposta a problemi difficilmente affrontabili all'interno del paradigma logico modellistico facendo riferimento alle ricerche svolte nell'ambito della scienza cognitiva. In particolare, grande attenzione è stata dedicata ai formalismi e alle tecniche sviluppate in intelligenza artificiale.
Contemporaneamente, tuttavia, l'IA, spinta dalla necessità di un maggiore rigore teorico e formale, ha adottato in maniera sempre più massiccia strumenti di tipo logico, e della semantica model teoretica in particolare. Quindi, questi due settori di ricerca si sono sviluppati lungo linee in parte parallele, seguendo però direzioni opposte; questo ha comportato certamente una auspicabile sinergia fra i due settori, ma ha anche posto problemi che vale la pena mettere in luce: talvolta sembra che entrambe le discipline cerchino nel campo dell'altra una soluzione ai problemi che la affliggono, dando origine in questo modo a una sorta di circolo vizioso. Lo scopo di questo lavoro consiste appunto nell'indagine e nel chiarimento di alcuni aspetti dei rapporti che sussistono fra questi due ambiti disciplinari. Tale obiettivo verrà affrontato analizzando i modelli sviluppati in IA in relazione a due settori critici per le teorie filosofiche del significato di impostazione logico-formale, vale a dire la rappresentazione del significato dei simboli primitivi extra logici (o, in altri termini, la rappresentazione del significato lessicale), e i contesti di atteggiamento proposizionale, in
particolare i contesti epistemici.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ must always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. MSC: 11-XX (Number Theory)

Three classic distinctions specify that truths can be necessary versus contingent, analytic versus synthetic, and a priori versus a posteriori. The philosopher reading this article knows very well both how useful and ordinary such distinctions are in our conceptual work and that they have been subject to many and detailed debates, especially the last two. In the following pages, I do not wish to discuss how far they may be tenable. I shall assume that, if they are reasonable and non problematic in some ordinary cases, then they can be used in order to understand what kind of knowledge the maker's knowledge is. By this I mean the sort of knowledge that Alice enjoys when she holds the information (true content) that Bob's coffee is sweetened because she just put two spoons of sugar in it herself. The maker's knowledge tradition is quite important but it is not mainstream in modern and analytic epistemology and lacks grounding in terms of exactly what sort of knowledge one is talking about. My suggestion is that this grounding can be provided by a minimalist approach, based on an information-theoretical analysis. In the article, I argue that (a) we need to decouple a fourth distinction, namely informative versus uninformative, from the previous three and, in particular, from its implicit association with analytic versus synthetic and a priori versus a posteriori; (b) such a decoupling facilitates, and is facilitated by, moving from a monoagent to a multiagent approach: the distinctions qualify a proposition, a message, or some information not just in themselves but relationally, with respect to an informational agent; (c) the decoupling and the multiagent approach enable a re-mapping of currently available positions in epistemology (Classic, Innatist, Kant's and Kripke's) on these four dichotomies; (d) within such a re-mapping, two positions, capturing the nature of a witness' knowledge and of a maker's knowledge, can best be described as contingent, synthetic, a posteriori, and uninformative and as
contingent, synthetic, weakly a priori (ab anteriori), and uninformative respectively. In the conclusion, I indicate why the analysis of the maker’s knowledge has important consequences in all those cases in which the poietic (constructive) intervention on a system determines the truth of the model of that system.

This book constitutes the proceedings of the 12th International Workshop on Computational Logic in Multi-Agent Systems, CLIMA XII, held in Barcelona, Spain, in July 2011. The 22 papers presented were carefully reviewed and selected from 43 submissions. The purpose of the CLIMA workshops is to provide a forum for discussing techniques, based on computational logic, for representing, programming and reasoning about agents and multi-agent systems in a formal way. This volume features five thematic special sessions: ...

I introduce, defend, and demonstrate, a new paradigm for artificial discovery in science, called ‘Computational Axiomatic Science’ (CAS). This doctoral dissertation has two parts. In the first part, I discuss the philosophical roots of the paradigm, and its method of representation of scientific theories. I then explicate its bearing on standard issues in the philosophy of science, such as scientific explanation, prediction, and experimentation, and defend it against objections. In the second part, I demonstrate CAS for artificial (proof) verification and discovery in the sciences of physics, chemistry, and biology. I also demonstrate its usefulness in the sphere of educational technology. Finally, I discuss the relationship, in the context of CAS, between science and metaphysics.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]

The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the Prior Analytics motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the analysis of concepts, a term that presupposes—through the root ἀνά + λύω —a mereological background.

In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parry’s original system AI was later expanded to the system PAI. The hallmark of Parry’s systems—and of what may be thought of as containment logics or Parry systems in general—is a strong relevance property called the ‘Proscriptive Principle’ (PP) described by Parry as the thesis that:

No formula with analytic implication as main relation holds universally if it has a free variable occurring in the consequent but not the antecedent.

This type of proscription is on its face justified, as the presence of a novel parameter in the consequent corresponds to the introduction of new subject matter. The plausibility of the thesis that the content of a statement is related to its subject matter thus appears also to support the validity of the formal principle.

Primarily due to the perception that Parry’s formal systems were intended to accurately model Kant’s notion of an analytic judgment, Parry’s deductive systems—and the suitability of the Proscriptive Principle in general—were met with severe criticism. While Anderson and Belnap argued that Parry’s criterion failed to account for a number of prima facie analytic judgments, others—such as Sylvan and Brady—argued that the utility of the criterion was impeded by its reliance on a ‘syntactical’ device.

But these arguments are restricted to Parry’s work qua exegesis of Kant and fail to take into account the breadth of applications in which the Proscriptive Principle emerges. It is the goal of the present work to explore themes related to deductive systems satisfying one form of the Proscriptive Principle or other, with a special emphasis placed on the rehabilitation of their study to some degree. The structure of the dissertation is as follows:

In Chapter 2, we identify and develop the relationship between Parry-type deductive systems and the field of ‘logics of nonsense.’ Of particular importance is Dmitri Bochvar’s ‘internal’ nonsense logic Σ0, and we observe that two ⊢-Parry subsystems of Σ0 (Harry Deutsch’s Sfde and Frederick Johnson’s RC) can be considered to be the products of particular ‘strategies’ of eliminating problematic inferences from Bochvar’s system.

The material of Chapter 3 considers Kit Fine’s program of state space semantics in the context of Parry logics. Recently, Fine—who had already provided the first intuitive semantics for Parry’s PAI—has offered a formal model of truthmaking (and falsemaking) that provides one of the first natural semantics for Richard B. Angell’s logic of analytic containment AC, itself a ⊢-Parry system. After discussing the relationship between state space semantics and nonsense, we observe that Fabrice Correia’s weaker framework—introduced as a semantics for a containment logic weaker than AC—tacitly endorses an implausible feature of allowing hypernonsensical statements. By modelling Correia’s containment logic within the stronger setting of Fine’s semantics, we are able to retain Correia’s intuitions about factual equivalence without such a commitment. As a further application, we observe that Fine’s setting can resolve some ambiguities in Greg Restall’s own truthmaker semantics.

In Chapter 4, we consider interpretations of disjunction that accord with the characteristic failure of Addition in which the evaluation of a disjunction A ∨ B requires not only the truth of one disjunct, but also that both disjuncts satisfy some further property. In the setting of computation, such an analysis requires the existence of some procedure tasked with ensuring the satisfaction of this property by both disjuncts. This observation leads to a computational analysis of the relationship between Parry logics and logics of nonsense in which the semantic category of ‘nonsense’ is associated with catastrophic faults in computer programs. In this spirit, we examine semantics for several ⊢-Parry logics in terms of the successful execution of certain types of programs and the consequences of extending this analysis to dynamic logic and constructive logic.

Chapter 5 considers these faults in the particular case in which Nuel Belnap’s ‘artificial reasoner’ is unable to retrieve the value assigned to a variable. This leads not only to a natural interpretation of Graham Priest’s semantics for the ⊢-Parry system S⋆fde but also a novel, many-valued semantics for Angell’s AC, completeness of which is proven by establishing a correspondence with Correia’s semantics for AC. These many-valued semantics have the additional benefit of allowing us to apply the material in Chapter 2 to the case of AC to define intensional extensions of AC in the spirit of Parry’s PAI.

One particular instance of the type of disjunction central to Chapter 4 is Melvin Fitting’s cut-down disjunction. Chapter 6 examines cut-down operations in more detail and provides bilattice and trilattice semantics for the ⊢-Parry systems Sfde and AC in the style of Ofer Arieli and Arnon Avron’s logical bilattices. The elegant connection between these systems and logical multilattices supports the fundamentality and naturalness of these logics and, additionally, allows us to extend epistemic interpretation of bilattices in the tradition of artificial intelligence to these systems.

Finally, the correspondence between the present many-valued semantics for AC and those of Correia is revisited in Chapter 7. The technique that plays an essential role in Chapter 4 is used to characterize a wide class of first-degree calculi intermediate between AC and classical logic in Correia’s setting. This correspondence allows the correction of an incorrect characterization of classical logic given by Correia and leads to the question of how to characterize hybrid systems extending Angell’s AC∗. Finally, we consider whether this correspondence aids in providing an interpretation to Correia’s first semantics for AC.

Logic provides the mathematical basis for hardware design and software development. In fact, digital circuits and computer programs are logic formulas expressed in a formal language. Accordingly, educated computer scientists should have experience in reasoning about the formulas that their digital circuits and programs represent. An exemplary way to get this experience is to use computational logic in support of such reasoning. This paper searches the typical undergraduate curriculum in computer science for opportunities to include material on computational logic in the context of hardware and software design and implementation. It explains how computational logic has been included as an element of two courses required in most computer science programs. It discusses some successes and a few missteps that the author has experienced over the past nine years in developing this material and using it in the classroom, and it suggests opportunities for similar efforts in other courses.

The reflection theorem has been proved using Isabelle/ZF. This theorem cannot be expressed in ZF, and its proof requires reasoning at the meta-level. There is a particularly elegant proof that reduces the meta-level reasoning to a single induction over formulas. Each case of the induction has been proved with Isabelle/ZF, whose built-in tools can prove specific instances of the reflection theorem upon demand.

This paper is the author's personal side project and is of no affiliation to any institution or courses. The paper is a rookie attempt to deriving mathematical expressions of the well-known puzzle, the bridge and torch problem. The problem is about finding the shortest time needed for n people to cross a bridge at night. The catch is the bridge can only support at most two persons at a time, and the two persons will cross the bridge at the speed of the slower one. Moreover, a torch is needed for each and every crossing, and the torch has a finite time of burning. Thus, the puzzle is a problem to determine the possible shortest time from the choice of the two persons per crossing against the finite burning time of the torch. The result of this paper gives into fruition of a set of three equations--the vector form describing the mechanics of the logic solution, even-valued n function and odd-valued n function.

We analyze both pre- and post-verbal clitics in Italian in the context of infinitival phrases and declarative sentences, using a new form of categorial grammar in which one assigns to each morpheme a type, an element of the free "pregroup" generated by a partially ordered set of basic types.

Let us suppose that you are a research linguist, tormented by some doubts and questions about the state of your profession, and not constrained by having to repeat a catechism of "known truths" to Linguistics 101 students, and not worried about employment tenure. How would you actually go about tackling "the central problem of linguistics", namely how we acquire and maintain knowledge of the probability of systemic relationships in a language?

Interactive theorem provers require too much effort from their users. We have been developing a system in which Isabelle users obtain automatic support from automatic theorem provers (ATPs) such as Vampire and SPASS. An ATP is invoked at suitable points in the interactive session, and any proof found is given to the user in a window displaying an Isar proof script. There are numerous differences between Isabelle (polymorphic higher-order logic with type classes, natural deduction rule format) and classical ATPs (first-order, untyped, and clause form). Many of these differences have been bridged, and a working prototype that uses background processes already provides much of the desired functionality.

To date, a number of researchers are seeking for and/or designing novel molecules which function as arithmetic molecular engines. Biomolecules such as deoxyribonucleic acid (DNA) and proteins are examples of promising candidate molecules. In the present article, we showed our view that DNA-based molecules could be used as a novel class of platforms for discrete mathematical operations or tools for natural computation. Here, we report on a novel molecular logic circuit combining exclusive disjunction (XOR) gate and conjunction (AND) gate implemented on a single DNA molecule performing arithmetic operations with simple binary numbers through polymerase chain reactions (PCR); which was inspired by previously developed protein-based computing model allowing simple polynomial algebra over fields through algebraic representation of cyclic inter-conversions in the catalytic modes of a plant enzyme as a cyclic additive group. In addition, we showed that DNA can be used as the platform for image coding and processing leading to DNA-coded animation by using novel PCR-based protocols. Lastly, we discussed the significance of recent attempts in the stream of natural computing and synthetic biological research, by handling DNA and related biomolecules as the media for discrete mathematical operations.

A novel improved VPN (Virtual Private Network) system based on SSL (Secure Socket Layer) protocol is discussed to overwhelm the defect of traditional SSL VPN. The concept, critical technique and working process of the geo-secured SSL based VPN are presented in details because they are very important to meet enhanced security requirement which is originally a traditional SSL VPN. Based on the GPS (Global Position System) technology a new scheme to improve the security of SSL VPN is proposed. This scheme enhances current security applications granting access to sensible information and privileges to execute orders only to entities that are in a trusted location. Moreover this system authenticate authorized user but also location of the authorized user.

The Unified Foundational Ontology (UFO-A) is a foundational ontol-ogy about endurants that has been built as a foundation for conceptual modeling, and which focuses on static aspects of endurants. In addition to UFO-A, we have the Unified Foundational Ontology-Part B (UFO-B), a formal theory dealing with the interplay between endurants and the dynamic aspects of reality (e.g., events, processes, causation, dispositions, situations). Given the objective of characterizing this interplay between endurants and perdurants, these two ontologies are meant to form an integral whole. However, currently, they diverge in the way they approach modality. While UFO-A uses a general notion of alethic modality without committing to any notion of time, UFO-B is centered around temporal aspects. As an attempt to address this issue, we here define a translation of the axioms of UFO-A to FOL, and revisit an excerpt of UFO-B in order to accommodate a partial order of time points. With the goal of producing a unified theory, we interpret the alethic modalities of necessity and contingency of UFO-A in terms of this temporal structure , paying a special attention to the interplay between the determinism of a causal nexus and the existence of counterfactual situations. The revisited UFO-B is called UFO-B and the unified theory is called UFO-AB.

Commonsense reasoning has proven exceedingly difficult both to model and to implement in artificial reasoning systems. This paper discusses some of the features of human reasoning that may account for this difficulty, surveys a number of reasoning systems and formalisms, and offers an outline of active logic, a non-classical paraconsistent logic that may be of some use in implementing commonsense reasoning.

Informal arguments that cryptographic protocols are secure can be made rigorous using inductive definitions. The approach is based on ordinary predicate calculus and copes with infinite-state systems. Proofs are generated using Isabelle/HOL. The human effort required to analyze a protocol can be as little as a week or two, yielding a proof script that takes a few minutes to run.

Protocols are inductively defined as sets of traces. A trace is a list of communication events, perhaps comprising many interleaved protocol runs. Protocol descriptions incorporate attacks and accidental losses. The model spy knows some private keys and can forge messages using components de- crypted from previous traffic.

A quotient construction defines an abstract type from a concrete type, using an equivalence relation to identify elements of the concrete type that are to be regarded as indistinguishable. The elements of a quotient type are equivalence classes: sets of equivalent concrete values. Simple techniques are presented for defining and reasoning about quotient constructions, based on a general lemma library concerning functions that operate on equivalence classes. The techniques are applied to a definition of the integers from the natural numbers, and then to the definition of a recursive datatype satisfying equational constraints.

The use of formal languages in the specification of upper ontologies helps establishing precise and unambiguous definitions for its concepts and relations. In this context, the use of formal languages with support of proof-assistant software systems has potential of aiding the specification author in various aspects. We describe a study case in which the Isabelle/HOL language and the Isabelle proof-assistant environment are used to formalize a simplified version of the UFO-A ontology of endurants, to verify and correct the original axiomatization, and to optimize the specification theory's axioms and signature.

In this paper, we analyse the structure of evolving moral dilemmas with an eye of regard for the increasing importance of the role of artificial intelligence in such context. Starting with the analysis of the famous trolley problem experiment as formulated by Philippa Foot, we consider subsequent variants of this moral dilemma conceived throughout the years, culminating with formulations of the trolley problem concerning artificial intelligence, in which self-driving vehicles will have to make life or death decisions autonomously. In doing so, we investigate the basis for the construction of dilemmatic questions both for humans and machines by considering the problem from a philosophical, social and neuroscientific perspective. After considering and analysing the trolley problem in utilitarian and deontological terms, we follow Rittel and Webber's footsteps, by highlighting the fallacies of the deontological and utilitarian traditional 'one-right-answer' approach, where a solution is undoubtedly right or wrong, and claim that moral problems are not, due to their intrinsic dilemmatic nature, resolvable. By rejecting an aut-aut approach, we find ourselves contemplating the possibility of neither approach being right in an absolute sense. Given these premises, we present a different approach on the matter, arguing for the central and creative role of the tragic as a new tool for enhancing both human and autonomous vehicles' approach to moral problems.

The hallmark of the deductive systems known as ‘conceptivist’ or ‘containment’ logics is that for all theorems of the form A→B, all atomic formulae appearing in B also appear in A. Significantly, as a consequence, the principle of Addition (the inference to A∨B from A) fails. While often billed as a formalisation of Kantian analytic judgements, once semantics were discovered for these systems, the approach was largely discounted as merely the imposition of a syntactic filter on unrelated systems. In this paper, we examine a number of prima facie unrelated deductive contexts in which Addition fails and attempt to harmonise them by developing a computational interpretation of conceptivist logics.

Computers may be thought of as engines for transforming free energy into waste heat and mathematical work. Existing electronic computers dissipate energy vastly in excess of the mean thermal energy kT, for purposes such as maintaining volatile storage devices in a bistable condition, synchronizing and standardizing signals, and maximizing switching speed. On the other hand, recent models due to Fredkin and Toffoli show that in principle a computer could compute at finite speed with zero energy dissipation and zero error. In these models, a simple assemblage of simple but idealized mechanical parts (e.g., hard spheres and flat plates) determines a ballistic trajectory isomorphic with the desired computation, a trajectory therefore not foreseen in detail by the builder of the computer. In a classical or semiclassical setting, ballistic models are unrealistic because they require the parts to be assembled with perfect precision and isolated from thermal noise, which would eventually randomize the trajectory and lead to errors. Possibly quantum effects could be exploited to prevent this undesired equipartition of the kinetic energy. Another family of models may be called Brownian computers, because they allow thermal noise to influence the trajectory so strongly that it becomes a random walk through the entire accessible (lowpotential-energy) portion of the computer's configuration space. In these computers, a simple assemblage of simple parts determines a low-energy labyrinth isomorphic to the desired computation, through which the system executes its random walk, with a slight drift velocity due to a weak driving force in the direction of forward computation. In return for their greater realism, Brownian models are more dissipative than ballistic ones: the drift velocity is proportional to the driving force, and hence the energy dissipated approaches zero only in the limit of zero speed. In this regard Brownian models resemble the traditional apparatus of thermodynamic thought experiments, where reversibility is also typically only attainable in the limit of zero speed. The enzymatic apparatus of DNA replication, transcription, and translation appear to be nature's closest approach to a Brownian computer, dissipating 20-100kT per step. Both the ballistic and Brownian computers require a change in programming style: computations must be rendered logically reversible, so that no machine state has more than one logical predecessor. In a ballistic computer, the merging of two trajectories clearly cannot be brought about by purely conservative forces; in a Brownlan computer, any extensive amount of merging of computation paths 905 0020-7748/82/1200-0905503.00/0 9 1982 Plenum Publishing Corporatiot Bennett would cause the Brownian computer to spend most of its time bogged down in extraneous predecessors of states on the intended path, unless an extra driving force of kTln2 were applied (and dissipated) at each merge point. The mathematical means of rendering a computation logically reversible (e.g., creation and annihilation of a history file) will be discussed. The old Maxwell's demon problem is discussed in the light of the relation between logical and thermodynamic reversibility: the essential irreversible step, which prevents the demon from breaking the second law, is not the making of a measurement (which in principle can be done reversibly) but rather the logically irreversible act of erasing the record of one measurement to make room for the next. Converse to the rule that logically irreversible operations on data require an entropy increase elsewhere in the computer is the fact that a tape full of zeros, or one containing some computable pseudorandom sequence such as pi, has fuel value and can be made to do useful thermodynanfic work as it randomizes itself. A tape containing an algorithmically random sequence lacks this ability.

The theorem prover Isabelle is described briefly and informally. Its historical development is traced from Edinburgh LCF to the present day. The main issues are unification, quantifiers, and the representation of inference rules. The Edinburgh Logical Framework is also described, for a comparison with Isabelle. An appendix presents several Isabelle logics, including set theory and Constructive Type Theory, with examples of theorems.

A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations, and functions and discusses interactive proofs of Cantor''s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey''s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics.

CORCORAN ON QUINE’S PHILOSOPHY OF LOGIC
Corcoran, J. 1972. Quine, Willard. Philosophy of Logic. In Philosophy of Science 39, 97–99.
WILLARD QUINE, Philosophy of logic, Harvard, 1970/1986.

This book is best regarded as a concise essay developing the personal views of a major philosopher of logic and as such it is to be welcomed by scholars in the field. It is not (and does not purport to be) a treatment of a significant portion of those philosophical problems generally thought to be germane to logic. It would be easy to list many popular topics in philosophy of logic which it does not mention. Even its "definition" of logic-"the systematic study of logical truth"-is peculiar to the author and would be regarded as inappropriately restrictive by many logicians
  There are several standard ways of defining truth using sequences. Quine’s discussions in the 1970 first printing of Philosophy of logic and in previous lectures were vitiated by mixing two. Quine’s logical Two-Method Error, which eluded Quine’s colleagues, was corrected in the 1978 sixth printing. But Quine never explicitly acknowledged, described, or even mentioned the error in print although in correspondence he did thank Corcoran for bringing it to his attention.
In regard to style one may note that the book is rich in metaphorical and sometimes even cryptic passages one of the more remarkable of which occurs in the Preface and seems to imply that deductive logic does not warrant distinctive philosophical treatment. Moreover, the author's sesquipedalian performances sometimes subvert perspicuity.

CONTENTS
INTRODUCTION
1. Tarski’s three-implication theorem.
2. Three-implication conditional-convertibility theorems.
3. Existential Import.
4. Three-implication consequent-affirmability theorems.


INTRODUCTION
The thinking in this work-in-progress began in April 1973 in the restaurant of the Albright-Knox Art Gallery in Buffalo NY. Alfred Tarski was taking me to lunch after I had given him a tour of the gallery pointing out the masterpieces he might not have heard of. I was flattered when he shared my enthusiasm for the canvases by the Buffalo artist Charles Burchfield.  https://en.wikipedia.org/wiki/Charles_E._Burchfield
It was the day after Tarski had delivered his now-famous “What are logical notions?”—the most-cited paper ever published in the journal HISTORY AND PHILOSOPHY OF LOGIC.
https://www.academia.edu/12410865/Alfred_Tarskis_What_are_Logical_Notions_Edited_and_introduced_by_John_Corcoran_

At lunch Tarski gave me a mini-lecture on what he called three-implication theorems: theorems that can be stated using cognates of the verb ‘implies’ in three different senses. He had not known that I had just published my 1973 “Meanings of implication”, which distinguish twelve senses of ‘implies’ used in the literature.
MI: https://www.academia.edu/s/643bbf0741?source=link

Tarski’s example is in Section 1 below. Since then I have collected more three-implication theorems.

The standard completeness and soundness theorems are all three-implication theorems using implications closely related to the first three in my 1973 “Meanings of implication”: (1) material implication [‘if-then’], (2) logical implication [consequence], and (3) deductive yielding [deducibility]. Completeness 213: given premises implying a conclusion implies those premises yield the conclusion. Soundness: given premises yielding a conclusion implies those premises imply the conclusion. The order in completeness is 213, in soundness 312.

One of the easiest and most classroom-appropriate is in Section 2.

My personal favorite is implicit in the theorem quoted at the end of Section 3: In order for a universalized conditional to have existential import it is necessary for the existentialization of its antecedent condition to be logically true. For any conditions S(x), P(x), if x (S(x)  P(x)) implies x (S(x) & P(x)) then x S(x) is logically true.

There are several ways of identifying three implications. Here is one: (1) the ‘if-then’ is material implication in the meta-language, (2) the universalized conditional is formal implication in the object-language, and (3) the ‘implies’ expresses logical implication relating object-language sentences. Here is another: (1) the ‘for-any-if-then’ is formal implication in the meta-language, (2) the conditional  is material implication in the object-language, and (3) the ‘implies’ expresses logical implication relating object-language sentences.

To get a three-implication sentence there are many sense of ‘implies’ and its cognates from which to choose three implications.

Section 4 gives another that will appeal to beginners.


For further discussion see John Corcoran and Hassan Masoud. 2014. Existential import today: New metatheorems; historical, philosophical, and pedagogical misconceptions. History and Philosophy of Logic. 36: 39–61. Available cost-free:
http://www.tandfonline.com/doi/full/10.1080/01445340.2014.952947

Deductive inference is usually regarded as being "tautological" or "analytical": the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of firstorder logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view and propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of "depth" or "informativeness" of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure "intelim logic", which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is "analytic" in a particularly strict sense, in that it rules out any use of "virtual information", which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed.

La logica è la scienza che si occupa di ciò che è comune e generale a tutte le branche della nostra attività conoscitiva, e quindi a tutte le discipline scientifiche, ed i suoi temi sono quindi quei temi che si ritrovano in tutte le branche dell'attività conoscitiva. La logica fa comunicare e interagire le discipline apparentemente diverse, in altre parole grazie al concetto di interdisciplinarietà, interconnette le branche conoscitive Tra i gruppi di temi che appartengono alla logica, e quindi comuni ad ogni branca dell'attività, in particolare troviamo: proposizioni, dimostrazioni e refutazioni, dibattiti. LA RICERCA DEI TEMI SULLA LOGICA NELLA STORIA La ricerca sui temi della logica è iniziata e si è sviluppata con Aristotele entro la filosofia. Nella seconda metà dell'ottocento, grazie ai matematici Boole e Cantor, scopriamo che la logica è una disciplina matematica poiché tratta oggetti di natura matematica che devono essere studiati con metodi matematici. La ricerca logica del novecento ha portato a risultati notevoli nella scienza e nella cultura attuale; la logica è infatti alla base dell'informatica. La logica è la disciplina che offre le basi teoriche concettuali e metodologiche dell'informatica.

A theory of recursive definitions has been mechanized in Isabelle''s Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs, and other computational reasoning.

Inductively defined sets are expressed as least fixedpoints, applying the Knaster-Tarski theorem over a suitable set.Recursive functions are defined by well-founded recursion and its derivatives, such as transfinite recursion.Recursive data structures are expressed by applying the Knaster-Tarski theorem to a set, such asV , that is closed under Cartesian product and disjoint sum.

Worked examples include the transitive closure of a relation, lists, variable-branching trees, and mutually recursive trees and forests. The Schröder-Bernstein theorem and the soundness of propositional logic are proved in Isabelle sessions.

Many theorems involving special functions such as ln, exp and sin can be proved automatically by MetiTarski: a resolution theorem prover modified to call a decision procedure for the theory of real closed fields. Special functions are approximated by upper and lower bounds, which are typically rational functions derived from Taylor or continued fraction expansions. The decision procedure simplifies clauses by deleting literals that are inconsistent with other algebraic facts. MetiTarski simplifies arithmetic expressions by conversion to a recursive representation, followed by flattening of nested quotients. Applications include verifying hybrid and control systems.

The impossibility of tiling the mutilated chess board has been formalized and verified using Isabelle. The formalization is concise because it is expressed using inductive definitions. The proofs are straightforward except for some lemmas concerning finite cardinalities. This exercise is an object lesson in choosing a good formalization: one at the right level of abstraction.

There is meaning in mathematics because it is knowledge. Set theory stays at the formal basis of that knowledge. Its most basic relation is the "… belongs to…" relation,. ∈ Our mathematical understanding of the world begins with the delimitation of two different entities. These delimitations follow certain rules. Strangely enough, this relation has not been formally analyzed. Besides helping us to understand this fundamental relation, its axiomatic definition eliminates paradoxes of the type analyzed in Gödel's incompleteness theorem, those due to the existence of self-reference in a logical and/or mathematical formal system.