I'm too tired (and too busy) for a proper review right now. All I'll say is that this is a great overview of some of the formal tools needed to undersI'm too tired (and too busy) for a proper review right now. All I'll say is that this is a great overview of some of the formal tools needed to understand philosophical issues: set theory, infinity, Turing machines and computability, formal semantics, probability theory. It's not a book on philosophical issues, it's a book on math and logic that is motivated by philosophical issues. The attempt is not to tackle these philosophical issues themselves, but rather to present the purely formal mathematical (not philosophical) tools needed to understand the debates going on regarding the philosophical issues. It's a sort of "mathematics for philosophers" book, and a very good one....more
I think my first real encounter of a clear abuse of Gödel's incompleteness theorem came when I was engaged (as I so often am) in the debate on religioI think my first real encounter of a clear abuse of Gödel's incompleteness theorem came when I was engaged (as I so often am) in the debate on religion, online as well as elsewhere. This was one of the former kind and in one of the lower subcategories of the bigger category of online venues for the exchange of ideas: YouTube... Some atheist or number of atheists had argued against religion, presumably (because the response regarded this aspect of the religious question, but it wouldn't surprise me much to learn that the atheist/atheists in question had in fact asked about the ethical standards of the Bible or something else completely unrelated, the intellectual integrity and rational capacity of the staunch Bible defenders most of the time leave something to be desired) specifically regarding the question of the rationality behind belief in god. The response went something along the lines of this: "Gödel proved that there are unknown/unprovable truths [he did nothing of the sort], and therefore... [something about how belief in phenomena without evidence isn't so crazy after all]". The whole thing was topped off with the brilliant argumentative tactic consisting in showing a photograph of Gödel standing next to Einstein and saying something like "Look what kind of friends he had! Kind of a smart guy that Einstein!" This feeble attempt got some responses of its own pointing out how this application of Gödel's theorem to a religious debate was... hrm, somewhat misguided (for an offense to reason of this magnitude, any adjective seems insufficient so why not use one that is so wildly insufficient as to call attention to the difficulty of finding the proper words to describe how bad it is?), though as I recall, the commenters, quite appropriately, used much harsher words. Regarding those who abused Gödel in such a horrible fashion in this particular instance, I hold little hope as to their ability to understand either the theorem itself or the actually fairly simple arguments needed to explain why it was not applicable to this situation, but in the hope that such attempts are not always entirely vain, here's a book clarifying the issues!
Franzén has written an overview of the whys and hows of Gödel's theorem in a general, fairly non-technical way, so that one can see what exactly the theorem states, what it does not state (which we will focus on a little more soon), when and where it is applicable and what general conclusions can be drawn from it on purely mathematical grounds (which does include considerations in philosophy of mathematics as these are still bound by the technical details of the mathematics, but does not include metaphorical extensions of the theorem into other fields in ways that take no considerations of the mathematics needed to prove the theorem). The focus of the book is to guide the reader through the landscape of the theorem in order to show when it can be called upon and when it can not by exhibiting a number of bad arguments that try to lean on the theorem, but fail to produce convincing (or in many cases, even sensical) arguments for their conclusions due to failures to understand the theorem, misconceptions about its reaches and consequences, and showing how these arguments go wrong. The examples are gathered into different categories: there are those that, as in my anecdote above, relate to religious debates (but the examples found in the book tend to be more about atheists misapplying the theorem to try to show how the Bible is necessarily "incomplete" if "consistent", a grave error resulting from a misunderstanding of the theorem though the Bible is quite obviously "incomplete" in the sense of not being a complete guide to the universe as some Christians like to claim it is, but not an error showing a misunderstanding of the theorem anywhere near the level exhibited by the Christians on YouTube described above), to arguments against the possibility of a Theory of Everything in physics, bold proclamations of a "post-modern" era in mathematics following Gödel's theorem (!), and various arguments about the implications of the theorem both for the supposed limits of the human mind to grasp all truths and for the limits of computers as contrasted against human minds concerning their ability of proving theorems.
None of these attempts at deriving interesting conclusions in other fields from the proof of Gödel's theorem follow at all in fact. The main points Franzén use to show this is that there is a requirement upon any system exhibiting incompleteness first that it is a formal system (in a technical sense meaning that it is a set of basic axioms with a set of rules for deriving new theorems), second that it is possible to do a certain minimal set of basic arithmetic in the system (a requirement that needs no further elaboration here other than noting that this requirement is specified exactly), and third that even when a system does exhibit these characteristics, the incompleteness is still only a property of the arithmetical component of the system, the theorem says nothing about the completeness or incompleteness of the system with regards to the other, non-arithmetical, statements included in it. Taking this as a starting point for evaluating claims of incompleteness found in other areas, or supposed philosophical implications of the theorem (of which there are many that are legitimate), Franzén goes on to show what kind of misunderstandings that seem to lie behind the abuses of this famous theorem.
We need not let the details of Franzén's investigation prolong this text needlessly, but some deliberation on the kind of things people have been found to claim in regards to the theorem deserve mention. Many seem to think that Gödel showed that the peculiar sentence "This statement is not provable in the theory PA" while shown to be indeed, not provable in PA, has somehow still been shown to be true and that it has been shown to be true in some fashion that goes beyond formalization. The view seems to be based on the observation that this formal theory has been shown to not be able to prove the statement whereas this fact is just what the statement says so we can see that it is true after all. So far so good (unless I'm misrepresenting things a bit here myself, I have to be careful not to do what the people exhibited in the book are called out for doing): Gödel did show the statement to be unprovable and if it is indeed unprovable, it is true since this is what it says. The problem with overinterpreting this is that if we do see that it is true, we do so by some further deliberation, probably by evaluating the claim and seeing that the statement's unprovability in PA is just what makes it true. This further deliberation is necessary to see this though, and Gödel's theorem by itself by no means shows that the statement is unprovable but true (the only way it could be shown to be true by a proof carried out in the system PA is if it could be proven in PA, which is exactly what can not be accomplished). So no conclusions regarding "true but unprovable statements" follow: the statement is unprovable in PA, not in any absolute sense (there is no absolute sense of being provable). Another source of confusion is that people generally fail to understand that Gödel proved only that if the theory in question is consistent, then it is incomplete, not that it is incomplete. So the question of the incompleteness of the theory depends on whether it is consistent or not, but this is something which, according to the second incompleteness theorem, the theory itself cannot prove if it is consistent. In other words, with additional reasons to suppose the theory to be consistent, we can draw the conclusion that it is indeed incomplete, but we can not prove it to be so unless we can also prove the theory's consistency, which needs another theory which will itself be incomplete if consistent and unable to prove its own consistency if consistent and so on. This all complicates matters to a degree rarely taken into account in the many attempted uses (turned abuses) of the theorem.
Franzén does an excellent job exhibiting some common (and perhaps some not so common but nevertheless sever and therefore, attention worthy) abuses and explaining carefully why they are abuses. In doing so, he also covers the landscape of related results, additional ways to prove incompleteness that do not rely upon Gödel's strange self-referential formula showing, importantly, that the theorem is not just something having to do with self-referential (always a suspect in intellectual discourse) exotic sentences never encountered outside the proof of the theorem. Though doing so is necessary to understand the theorem thoroughly enough to appreciate who and why uses of it go wrong, but Franzén tends to take these mathematical side stepping too far, going into the many (interesting but nonetheless inessential to the question of the abuses of the theorem) details of the theorem and its implications (even when it is done in a mostly informal fashion) does little to inform the reader of why many popular attempts to draw conclusions from the theorem go wrong, and the facts that this does little to inform the reader in this is evident in how Franzén uses these issues in exhibiting the failures in the abuses: not much at all. Again and again, when he demolishes yet another piece of bad writing referring to Gödel, he comes back to the main points mentioned above: the theorem only applies to formal systems powerful enough to support a certain amount of arithmetic and even so, only to the arithmetical component. The additional details of different axioms for mathematics, variants of arithmetic and so on are of course very important for understanding the implications as well as the applications of the theorem, but are only relevant to a exposition of the abuses of the theorem when such details can be used to show the errors in the abuses, otherwise they belong instead in a much more encompassing work on the reaches and limits of Gödel's theorem, a work that would not be focused on explaining how and why so many of the attempts at using Gödel's theorem outside its field fail. Such a book would be extremely interesting, but it would need to be much, much, long than the current text which does seem to want to be about the abuses. It is also clear from the text that the abuses are in focus since the rest is just there to "set the stage" and asides into the land of mathematics not directly related to any abuses of Gödel's theorem only arrive when the discussion slides into them. In these cases, Franzén should have backed off a bit more readily and kept his focus on exhibiting abuses which would, on some occasions, have been more interesting and enlightening had there been a few more pages devoted to them.
Another problem with Franzén's willingness to take up so many related issues regarding the theorem is that the reader can easily be overwhelmed by all the terminology in a book that is, after all, written for non-experts. It's even claimed to be accessible to people with no previous background in logic, a claim I'm by no so tired of commenting upon that I mostly force myself to do so due to some sense of obligation: it's technically (though not in the mathematical sense of course) true that no previous knowledge in logic is required since every bit of terminology needed to understand the argumentation is defined in the book, but considering the number of such definitions, any reader not already at least familiar with logic and Gödel's theorem is bound to be confused fairly quickly. Franzén does do a good job of commenting upon when a certain technicality is essential for understanding the rest and when it is not, but this is hardly sufficient since any reader not already familiar with the terminology will probably fairly quickly lose track of which of these terms he or she needs to remember and which only appears parenthetically. These kind of claims of the lack of a requirement of previous knowledge are so common in logic texts (and I suppose in other areas as well though I suspect it's somewhat peculiar to formal science where such claims seem necessary as to not scare away potential readers) that I've gotten used to it, but I still feel the aforementioned duty to report on them.
It is in any case a very good book and, as far as I understand, a very original one. It does an excellent job of showing how Gödel's theorem can be abused and how to respond to such abuses, but it is not the best choice for an introduction to the theorem or it's implications. It is not primarily a guide (or at least not among the best of those) to what the theorem does mean but what it does not mean....more
After spending years on my shelf and having been partially read at least once before, this book was finally finished! (I don't know why I used the pasAfter spending years on my shelf and having been partially read at least once before, this book was finally finished! (I don't know why I used the passive form there, it just felt right for some reason)
I'm glad I did finally read it, even though there were parts that were glanced through without too much attention to detail and even though I skipped the exercises that are probably needed to get a more thorough understanding of the material. I read it mostly as a way to get a good overview of the basics of typed lambda calculi, which the book supplies with a focus on the use of such formalism in the specification and analysis of programming languages. The book is divided into six major parts which all deal with increasing levels of complexity in type systems. The first part introduces untyped formal calculi, first simple arithmetic expressions, then the classic lambda calculus. After this follows a part of simply typed lambda calculi which begins with nothing but function types and later expands this with base types and different structured types such as pairs, tuples, records, lists and other common forms of types. The next major part deals with subtyping and how it interacts with the forms of types presented so far, and after that there's a part of recursive types. While the expansions to the simply typed lambda calculus introduced so far have lots of interesting uses as well as theoretical properties, neither of them introduces something really new in the expressiveness of the language, at least nothing as important as the polymorphism introduced in part five. The reason I value this so much higher is that it enables a completely new form of abstraction. All the systems introduces before this have only the basic function abstraction on the level of terms, but with polymorphism, the ability to abstract over types in terms is introduced (the first step to move away from the first corner of the Barendregt, or lambda, cube which we will talk more about soon). Polymorphism through both universal and existential quantification is presented in different forms and at the end of the part, polymorphism is combined with subtyping in a non-trivial way yielding bounded quantification. Finally, the book is ended with a short part on higher-order systems featuring type operators and kinding (a way to classify the level of types in terms of kinds in the same way that term-level expressions are classified by types) constituting another move in the Barendregt cube where we can now abstract over types in type expressions. The final move possible in the cube, yielding dependent types (abstraction over term-level expressions in types) is also briefly mentioned but not worked out formally as the rest are.
The Barendregt cube is a unifying way to look at the expressiveness possible in the different type systems presented here and elsewhere, and another interesting and unifying theme is that of the Curry-Howard isomorphism, stating that there is a correspondence between programs and proofs on the one hand and types and propositions on the other so that a program having a certain type in a formal calculus corresponds to a proof of a corresponding proposition in a logic. This is discussed at many point throughout the book, giving historical explanations of how the correspondence between specific logics and corresponding type systems have been invented/discovered (depending on your particaular view on philosophy of mathematics and/or logic).
It's a fairly theoretical text, but with many examples, the type systems are not presented as pure formalisms but are motivated through code examples in the calculi presented and at three different points in the book, attempts at formalizing object-oriented concepts in the calculi are presented with increasing sophistication as the new expressive power of the formalisms developed allows it.
As an overview of the formal specifications of programming languages, their syntax and semantics (with a special focus on type systems), along with practical motivations, some points on implementations (almost all the formalisms introduced have actual accompanying implementations to be downloaded from the book's website), proofs of metatheoretic properties and comparisons with some actual programming languages, the book could hardly be better. The only reason that I'm not giving it five stars is that I reserve that for books that are almost perfect, and this does not quite live up to such a high standard. For example, the sometimes very technical proofs are a bit too dense for my taste. Towards the end of the book, there are some attempts at giving a more intuitive sense for some of the formalisations by presenting some of the mechanisms in an anthropomorphic way, by describing a function as saying something along the lines of "give me a value of such and such type, along with a function... and I will apply the function to produce a result..." which I appreciated highly. Had the whole book attempted to present things in such a manner more, had the author stopped now and then to discuss things in a more intuitive way, then I probably would have given it five stars. As it is though, it's an excellent book that I highly recommend for anyone interested in the subject....more
This is certainly better than Budd's An Introduction to Object-Oriented Programming as an overview and explication of object-oriented features, but itThis is certainly better than Budd's An Introduction to Object-Oriented Programming as an overview and explication of object-oriented features, but it is still lacking. The presentation is a bit more formal but it falls into the same trap as so much literature on object-orientation: presenting OO as the paradigm for highly flexible programming, contrasting it with "traditional" models (meaning plain procedural programming) and treating polymorphism as something primarily and intrinsically linked to OO. It's not as bad in this respect as some of the worst offenders: other forms of polymorphism are presented and functional programming is treated to some extent, but the presentations of these are okay at best. I personally like OO quite a lot but would like to see more serious presentations, treating it the way it deserves: as one computational model our of many. Other annoyances where the many typos and typographical mishaps sometimes causing some confusion and requiring a closer look with some creative interpretation to understand what was supposed to be written on the page. A good book, but nothing great. I wonder if there are any serious, well-written presentations of object-orientated features at an introductory, non-formal level?
Anyway, I'm off to read Abadi's and Cardelli's A Theory of Objects now, a book I've started but put to the side previously and one which gives a much more formal exposition of object-oriented languages in the form of formal calculi and which also, as I recall, gives a very good, informal overview of the features in its introductory part....more
En kompakt men ändå pedagogisk introduktion till den symboliska logiken. I huvudsak behandlar boken satslogiken och predikatlogiken, men innehåller ävEn kompakt men ändå pedagogisk introduktion till den symboliska logiken. I huvudsak behandlar boken satslogiken och predikatlogiken, men innehåller även kortare avsnitt om mängdläran (här under namnet klasslogik, vilken författaren särskiljer från mängdläran) samt relationer (under namnet relationslogik).
Logiken presenteras genomgående med exempel och med pedagogiska försök (dessutom lyckade sådana) att motivera begreppen, slutsatserna, definitionerna samt att med vardagsspråkliga resonemang förklara resultat och bevis. Vid sidan av detta genomförs visserligen också ganska grundliga formaliseringar av de system som avhandlas, även några ord om metalogiska resultat gällande sådana begrepp som sundhet och fullständighet presenteras kortfattat (utan genomgång av formella bevis för dessa). Ett annat plus i kanten måste sägas utgöras av författarens tendens att nämna alternativa logiker (icke-klassiska logiker av olika slag) som för att nämna att den tolkning av t.ex. logisk giltighet och konsekvens som ges i den klassiska logiken inte utgör det enda sättet att se på begreppen. Kortfattat nämns alltså något om den filosofiska diskussionen kring logiken och de olika ståndpunkter som har intagits av "avvikande" logiska system (intuitionistisk logik exempelvis).
En till synes något underlig och avvikande detalj i boken är att utöver de förmodligen för satslogikens bevisföring obligatoriska sanningsvärdestabellerna, så är den enda bevismetoden (härledningsmetoden) som presenteras den naturliga deduktionen. I den övriga introduktionslitteraturen i ämnet verkar visserligen den naturliga deduktionen ofta nämnas, men då endast som alternativ till den till synes vanligare trädmetoden (även kallad metoden med semantiska tablåer). Detta är i alla fall min tidigare erfarenhet. Då jag läste logik som en delkurs i programmet för data- och systemvetenskap togs trädmetoden och resolutionsmetoden upp (boken, Ekenbergs och Thorbiörnsons Logikens grunder nämnde även den naturliga deduktionen men denna togs inte upp i kursen) och i den introduktionskurs i logik som jag nu har läst på filosofiska institutionen så ignoreras just kapitlen om den naturliga deduktionen i denna bok (som alltså utgör kurslitteraturen i denna delkurs) till förmån för extra stenciler om trädmetoden, vilken är den enda metod vi går igenom där. Även den bok som tidigare användes i denna kurs (Ernest Lepores Meaning and Argument) presenterade (såvitt jag minns) endast trädmetoden.
Som varande datavetare uppskattade jag att bekanta mig mer med den naturliga deduktionen då denna har intressanta kopplingar till lambda kalkylen, en koppling som (om jag inte har missförstått allt) ligger till grund för den så kallade Curry-Howard-korrespondensen. Dock är denna metod samtidigt intressant nog (nästan ironiskt) mindre mekanisk än trädmetoden och därmed svårare att tillämpa i och med att den verkar kräva mer intuition gällande tillämpningen av regler i en pågående härledning/bevis. Man tillämpar i denna metod ibland en regel för att man vill kunna härleda något ur denna senare. Detta kan tydligt kontrasteras mot trädmetoden där tillvägagångssättet i stort sätt kan beskrivas i form av en punktlista. Dock medför den naturliga deduktionens därmed antydda intuitivitet kanske även något gott, nämligen att slutledningsreglerna (och kanske i mindre utsträckning även härledningsreglerna) kommer närmare våra intuitioner av hur något kan härledas ut något annat, medan trädmetodens regler kan verkar något mer abstrakta och svårare att försvara intuitivt.
Nu är detta dock ett sidospår från den faktiska recensionen av boken, vilken måste avslutas med omdömet att detta är en mycket god introduktionsbok till logiken vilken rekommenderas för den som vill ha lite mer filosofiska motivationer för symboliken och regelsystemen och inte nöjer sig en rent formell, matematisk presentation av den samma (vilket dessvärre är det enda som ges i Ekenbergs och Thorbiörnsons bok)....more
This is a very nice treatment on basic classical logic with a strong focus on language and translation. New logical notations are always motivated byThis is a very nice treatment on basic classical logic with a strong focus on language and translation. New logical notations are always motivated by being needed to represent expressions in natural language not capable of being represented in the notation presented thus far. This creates a very nice pedagogical approach where, beginning with propositional logic, the logical language is extended in several iterations all of which start with showing how some sentences in natural language can be represented in the logical notation developed so far, and ending with some sentences not capable of being treated, leading to the next iteration in which new notational capabilities are introduces to allow further capabilities. This leads to a richer and richer logical language, from propositional logic, via monadic predicate logic (here called property predicate logic), polyadic predicate logic (relational predicate logic) up to predicate logic with identity.
The path from basic propositional logic up to predicate logic with identity seems fairly standard for introductory logic texts, but the heavy use of language not only as a tool for creating examples, a pedagogical tool, but as a way of motivating the continuing expansions of the logical language, is seemingly less common. This focus on language is pretty far reaching. Much time is spent discussing the proper logical form for various grammatical forms (the subject of definite descriptions and Russell's treatment of them in terms of logical form is one example). All of this is very interesting and might be a good way to understand the use of the logical language for those who have trouble seeing the motivations behind the formalisms when reading more strictly technical texts (I will be the first one to admit that my experience with introductory logic texts written by mathematicians can be rather puzzling), but the focus on language might be a bit too heavy for am introduction to logic. Including the appendix, the text comes to 355 pages, which seems a bit excessive considering how little of actual logic it contains. There is no discussion of proof systems other than truth trees (semantic tableaux), there is virtually nothing about semantics for logic (only a few words are dedicated to extracting counter models from open truth trees, which gives some sense of what a model is, but not a single word is dedicated to hinting at even the existence of the field of model theory), and nothing (as far as I can remember, but feel free to correct me if I am mistaken) is said about richer logical languages. The last chapter talks about verb modifiers and towards the end semms to hint a little bit towards modal logic (when talking about the “event approach” and mentioning that the example “The president possibly lied to the people” is probably best rephrased as “It is possible that the president lied to the people” before trying to figure out how it could be formalised in logic) but it is never mentioned explicitly. (Modality is mentioned in the appendix, but only as a feature of natural language, there is no mention of modal logic.)
These things single it out as an introductory text in my experience. It seems that the heavy focus on natural language and translations made sure there was no room to even mention some more advanced topics in logic, some metalogical results and some richer logical systems (I don't believe second order logic is mentioned either). This is surely a conscious decision, but I'm not sure it's the right one for an introductory course to logic, even if it is taught in a philosophy department (and surely not if it is taught in a mathematics or computer science department).
Perhaps it is better seen as a book filling a niche. Since the other introductory texts I have encountered do treat other logical systems, several different proof systems and some metalogical results (or at least mention them), there might be need for a logic text that ignores these in favor of more linguistic focus. At my university, there is a program in philosophy and linguistics and perhaps this is a perfect introductory text in logic for them.
In any case, the seemingly negative opinions expressed above are only slightly and hesitatingly negative, they should perhaps be seen more as a form of pusslement over the approach taken in this book than a dismissal of it....more
Av någon anledning bestämde jag mig sent i går natt (eller, nja, snarare tidigt på morgonen) för att det var dags att till slut gå igenom denna min föAv någon anledning bestämde jag mig sent i går natt (eller, nja, snarare tidigt på morgonen) för att det var dags att till slut gå igenom denna min första logikbok från universitetet. Boken användes som kurslitteratur i grundkursen i logik på programmet i data- och systemvetenskap och jag läste kursen för cirka 10 år sedan (jag ska avsluta min kandidatuppsats och därmed hela utbildningen vilken dag som helst nu!) Jag läste aldrig hela boken då utan endast de avsnitt som ingick i kursen (om ens dessa) och det har aldrig blivit av senare heller. I mitt trötta och förmodligen något förvirrade sinnestillstånd för ungefär ett dygn sedan fick jag syn på boken i hyllan och bestämde mig för att läsa den nu till slut. Eftersom jag har läst mycket mer logik sen dess så tänkte jag att det förmodligen skulle gå att snabbläsa denna och snabbt bedöma dess värde som introduktionslitteratur till logiken. Det intryck jag fick första gången jag gick igenom (delar av) texten var att den var väldigt opedagogisk och det är ett intryck som hänger kvar, dock blev jag något glatt överraskad av att upptäcka att boken inte alls var så dålig som jag väntade mig. I termer av vad som tas upp och hur detta beskrivs finns det nog inte mycket att klaga på, allt det viktiga från sats- och predikatlogikens grunder tas upp: definitioner av syntax och semantik med tillhörande metalogiska resultat om sundhet och fullständighet genom en samling av de förmodligen mest kända bevissystemen (sanningsvärdestabeller kommer först, axiomatiska system exemplifieras sedan genom systemet H varefter semantiska tablåer, även kallad trädmetoden, resolutionsmetoden och naturlig deduktion behandlas.
Presentationen upplevdes förmodligen som mycket klarare genom mina nu logikbekanta ögon än när jag först läste boken, jag upplevde inga otydligheter varken i språk eller definitioner. Det stora problemet som dock kvarstår är att behandlingen är precis den som jag har kommit att vänja mig vid från en (eller i detta fall, två) matematiker. Matematiker tenderar att skriva som om alla ord och begrepp kan användas fritt utan vidare utläggningar så fort de har definierats formellt. Alla olika definitioner, lemman och teorem presenteras visserligen klart, tydligt och kompakt (vilket är ett ideal värt att leva upp till i matematiska sammanhang) men för att en läsare obevandrad i logik verkligen ska hänga med och komma ihåg alla begreppen och förstå hur alla bevis fungerar så behövs mycket mer diskussion av en konceptuell karaktär. Antingen efter eller innan något samband visas formellt behövs en mer eller mindre utförlig diskussion kring varför sambandet håller, exakt hur bevisföringen fungerar och varför. Det finns vissa mindre inslag av dylikt, men det är långt ifrån den nivå av pedagogisk och konceptuell framställning som jag har vant mig vid i introduktionsböcker skrivna av filosofer. Å andra sidan medför den kompakta presentationen att fler detaljer kan behandlas, den nedan omnämnda ABC i symbolisk logik fokuserar på naturlig deduktion som bevismetod (vilket är föga oväntat då författaren av denna är en av de viktigaste figurerna i utvecklingen av metoden) och behandlar endast i senaste upplagan trädmetoden (och gör inget vidare jobb på just den punkten). I sin behandling av flera olika bevissystem lyckas Logikens grunder avverka för dessa system viktiga saker såsom normalformer (vilka är väsentliga i resolutionsmetoden) som hade varit svåra att få med (i alla fall i lika stor utsträckning) i en mer utförlig presentation av logikens grunder. I vilket fall är denna avvägning, om det nu är en medveten avvägning från författarnas sida, gjord på ett dåligt vis. I en introduktion till ett så formellt och för många onaturligt ämne som logik är det mycket viktigare att ägna så mycket tid som möjligt åt att få läsare att förstå varför man gör alla dessa definitioner, varför bevisföringen fungerar osv. än att försöka täcka så många olika aspekter som möjligt. Författarna borde ha ägnat mycket mer tid åt att förklara på ett konceptuellt och därmed pedagogiskt vis och mycket mindre åt att försöka redogöra för så pass många olika bevismetoder och utan så mycket matematik (vilket knappast är centralt för logiken i allmänhet). De matematiska exemplen dominerar boken, exemplen från vardagsspråk är få... tyvärr. En förståelse för logiken kräver att man förstår vad logiken är för. Såsom författarna här presenterar ämnet ser det ut att vara endast en aspekt av matematiken och som att det får sin motivation därifrån, vilket är en djupt missvisande bild. Logiken formaliserar resonemang inom vilket område som helst och även om matematisk bevisföring är en viktig delmängd av våra resonemang, så är de endast en delmängd och bör behandlas därefter. Försök att formalisera resonemang uttryckta i vardagsspråk är absolut centralt för en förståelse av logikens natur och den roll denna kan spela, men detta utelämnas helt och hållet.
Boken är inte alls dålig, men långt, långt ifrån det rimligaste valet för introduktionsmaterial i logik. Den intresserade bör hellre plocka upp ett exemplar av Dag Prawitz' ABC i symbolisk logik....more
An excellent book covering many different aspects of logic in an exhaustive manner. It is the first volume of two, with the somewhat misleading subtitAn excellent book covering many different aspects of logic in an exhaustive manner. It is the first volume of two, with the somewhat misleading subtitle "Introduction to Logic". It should perhaps rather have been called "Introduction to a formal treatment of logic" or something along those lines. As a first book on logic, it is not a good choice. A word might be necessary on my use of the word "formal" here. Any treatment of logic is of course in a certain sense "formal". Arguments in natural language are often translated as examples to illustrate the meaning of the logical constants. But this does not amount to a formal treatment of logic itself. This book explains the use of mathematical induction to prove things about formulas, which relies on a formal definition on the syntax of the language of logic, gives extensive treatments on logical semantics and goes into some discussions about the correspondence between the model theoretic (semantic) approach (Tarski's beautiful truth definition is there) and proof theoretic (syntactic) approach to logical inference. In this connection some metalogical results are explained.
It is written with a strong linguistic focus. The ability of the formalisms to encode natural language is always a central issue, as opposed to the situation in more mathematically inclined books on logic where the translation of natural language sentences seems to often be more of a pedagogical thing. Towards the end, after the thorough treatment of classical logic, follows a few chapters on some other topics, with a more brief treatment. A chapter on various extensions and deviations on classical logic along with an explanation of the motivations of these (again, translations of natural language sentences are in focus) comes first. Then follows one on the pragmatics on logic and language, and finally, a chapter on the formal theory of grammar with a very brief explanation of the language hierarchy initially developed by Chomsky and its connection to types of automata.
This is an great text for the reader who already has a basic understanding of classical logic and wishes to delve a bit deeper, perhaps before getting into an even more formal treatment of logic in a course on metalogic (which is exactly what I'm about to do myself in about two weeks)....more
Detta blir endast en kort notis. Jag har även läst den förra upplagan av boken samt har om denna skrivit en något längre recension vilken kan läsas häDetta blir endast en kort notis. Jag har även läst den förra upplagan av boken samt har om denna skrivit en något längre recension vilken kan läsas här.
I senaste upplagan är materialet utökat. Det enda jag verkligen la märke till (och, så vitt jag förstår, den huvudsakliga anledningen till denna nya upplaga) är att trädmetoden (semantiska tablåer) nu behandlas som härledningsmetod vid sidan av den naturliga deduktionen. Behandlingen av denna är något underlig då symboliken skiljer sig från den jag har stött på i undervisningen då jag har studerat logik (vilket jag har gjort vid flera tillfällen, vid flera olika instutitioner, med flera olika böcker som kurslitteratur). Dessutom finns ett antal allvarligare fel i presentationen av härledningsreglerna för trädmetoden.
I vilket fall märks att behandlingen är mer utförlig och trädmetoden behövdes för att göra boken mer utförlig, bättre som introduktion till logiken samt mer passande för användning som kurslitteratur. Dock är läsning av denna upplaga inte direkt nödvändig för de som redan har läst den förra, det finns för övrigt bättre källor till just trädmetoden om det är denna nyhet som lockar läsare av den tidigare upplagan att köpa även denna....more
The second volume in a series of two on logic as a tool for formalizing language, meaning and arguments. The first volume dealt with the fundamentalsThe second volume in a series of two on logic as a tool for formalizing language, meaning and arguments. The first volume dealt with the fundamentals of propositional logic and predicate logic, but it did so in a very thorough way, presenting all the intricate details of the semantics of logics and the ways in which these logics could account for the richness of natural language, making it much more than a mere first introduction to logic. After dealing with the main focus of classical logic, the first volume moved on to some more advanced logics such as second-order logic, and many-valued logic as well as the pragmatics of logical languages, but it is not until the second volume that we really encounter logics with a much higher expressive power and generality.
It starts with so called intensional logics in the form of modal logics and temporal logics and all the complications of having to deal with necessity and temporal aspects of sentences, making for a much more complex semantic framework. After dealing with the basics of different systems of intensional logic and the even more complex matters that come into play as one combines modality and quantification in predicate modal logic, we move on to Russell's theory of types and its stunning ability to generalize many different concepts in more simple logics. The theory of types is subsequently extended into an intensional theory of types and finally a two-sorted theory of types capable of dealing with even intensionality without the special treatment found in the basic intensional theory of types.
Going into the details of all these complicated systems would go much beyond the scope of this short review, suffice it to say that the authors manage to cover an impressive range of subjects of logic focused around the ability to account for the richness of natural language in a formal framework, always guided by some basic principles of representing language as closely as possible to the way in which it is actually used, never letting philosophical principles come before empirical concerns to represent the grammar of natural language as it is; and of maintaining compositionality in translation, meaning the different parts of sentences should be translated separately and then combined into a complex translation of the complete original sentence through general rules of translation, something which is not really possible (if even then) until we reach the full power of the theory of types enriched with the lambda-operator.
After the presentation of all these systems of logic, the book concludes with two chapters: one presenting Montague grammar, a system of formal grammar based on the theory of types and categorial grammar with a thorough discussion of its ability to describe the grammar of natural language; and a final chapter dealing with some recent (as of the writing of this book, in 1982) topics concerning quantifiers and semantics (specifically: the theory of generalized quantifiers, flexible categorial grammar and discourse representation theory).
To give a general overview of this volume as well as the preceding one: the reader is given a thorough presentation of some logics, focused on semantics rather than proof theory and focused on their ability to represent the grammar and semantics of natural language rather than on the meta-logical properties of the logics themselves and/or the philosophical discussions concerning these properties. Consequently, the selection of logics treated is guided toward some logics rather than other and many interesting systems of logic are not detailed. If one is looking for an overview of logics in general and the properties and characteristics of these, these books are not the best books to read. If, on the other hand one is looking for a presentation of logic as being closely tied to linguistics and wants a thorough discussion of the expressive powers of logics in terms of how good they are at representing the structure and meaning of sentenced and arguments in natural language through principled methods, this is an excellent couple of books....more
This was somewhat different from what I had expected. I thought it might be an overview of different logics with an accompanying philosophical discussThis was somewhat different from what I had expected. I thought it might be an overview of different logics with an accompanying philosophical discussion of their differences and possible justifications. In a way, this is exactly what it is, but not in the form I had anticipated.
For starters, there is almost no symbolic presentation of the different logics in this book, not even for the basic connectives for conjunction, disjunction, implication and so on. They are all presented fairly formally, but non-symbolically. I'm sure there is some pedagogical principle that in the mind of the author justified this choice and I'm not sure I disagree with the decision; too much formality can be off-putting, at least in an introductory text; but at times this makes the expression of arguments and conclusions overly long and confusing, at least some basic symbols should have been deployed.
Another thing that surprised me was that while the book deals with several different logics, it has no sign of a systematic presentation of these logics and their relation to each other. Instead, the structure of the book is mainly based on different philosophical problems arising from not only logic but also language and metaphysics with an attempt at addressing the problems through logical analysis. In this attempt, new systems of logic are presented as attempts to deal with the problems. The resulting discussion is very well presented in a sometimes heavy historical fashion with descriptions of how and when the different logics were invented and the back-and-forth between prominent philosophers on the best way to address the problems with logic (along with the author's own views), but there is something in the way this is done that confused me a bit.
On the one hand there is, as I said, no systematic approach to presenting the logics and discussing their properties in relation to each other, so it is not primarily a book on the philosophy of logics (if my use of this term is acceptable, I am an amateur after all). One could say therefore that it is more of a book on philosophical logic (meaning here, the application of logic in philosophy) and a discussion of "the right" or at least "best" way of understanding logical concepts such as implication (there author seems to take a stance against logical pluralism without ever making this explicit). On the other hand, there is no systematic approach to dealing with philosophical applications of logic either. There is no attempt to give an overview of the different areas in which philosophers try to use formal logic to solve philosophical questions.
This confusion of mine likely stems from a not very thorough understanding of either logic, philosophy of logic, nor philosophical logic (or even a good grasp of the distinction between the latter two), but a confusion it has caused me nonetheless and it is a confusion of not seeing any clear focus in the book.
That being said, it was a great and informative read with only occasional slightly heavy technical details that could be overlooked without missing the "big picture". It gave an overview of some discussion concerning some basic logical concepts and it did a good job of it. ...more