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
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
A really bizarre story with lots of strange, surreal elements. There's some fantasy in there, strange powers and magic, but there are also more subtleA really bizarre story with lots of strange, surreal elements. There's some fantasy in there, strange powers and magic, but there are also more subtle strangeness going on with lots of seemingly psychologically induced hallucinations and lots of places where the reader is left wondering what exactly was real and what just happened inside someone's head. It seems mostly focused on teenage readers, and perhaps especially female ones.
I manly borrowed it at the library for the spectacular artwork, and instantly got the feeling that the content wasn't really for me. There are bits about young girls coming of age, something which strengthened my previous judgement. In the end though, there was plenty of weirdness to make me feel it was definitely worth a read. I'd recommend it to anyone to whom this brief review seems interesting, even if you feel, like me, that you may not be in the intended demographic. The only annoying thing about it is that I found out, after finishing it, that this was a follow up to this, which I'm now going to have to go back and look for at the library. Sigh......more
This contains the first to parts of Larcenet's Ordinary Victories, translated into English from the original French. The stories are about a photograpThis contains the first to parts of Larcenet's Ordinary Victories, translated into English from the original French. The stories are about a photographer who has problems with anxiety. He's also sick of his work lately and feeling generally lost and lonely. Meeting his brother for some "big fat joints" and video games seems to bring some happiness into his life, but not much else seems able to produce that effect. After his cat gets a minor injury, he meets an interesting female vet with whom he develops a relationship which struggles a bit due to his neuroses, but seems to stabilize after a while. On the way he also visits his old parents, does a photo set with his dads' old co-workers and has the photos displayed in a show alongside one of his big idols of photography, befriends an old war veteran with a past he's unable to ignore, and has many other encounters.
The book tells a very personal moving story that really draws the reader in, while at the same time remaining completely realistic and never moving beyond picturing the everyday events of normal people. Some of the characters are unusually colorful and eccentric, but none some much that it becomes implausible. I just dove into the story and found myself unable to put it down until it was over. I'm now heading over to the next book, containing the English version of parts three and four. Expect a review of that later today!...more
I'm always a bit puzzled upon encountering a book at this level, regarding any subject within philosophy (or indeed any other field), being titled somI'm always a bit puzzled upon encountering a book at this level, regarding any subject within philosophy (or indeed any other field), being titled something along the lines of "an introduction to X" when it becomes utterly clear after reading just a short part of the first chapter that it is anything but an introduction to the subject. I've seen this before (for example with "Gamut"'s Logic, Language, and Meaning, Volume 1: Introduction to Logic) and I always come away from the reading experience feeling that while one may not absolutely need previous knowledge on the subject to understand them because they do go through the basics in the beginning, they do so at such a pace, with so little time to really grasp all the definitions, distinctions, positions and what have you, that it is hopelessly optimistic to think that any reader could understand everything without having encountered these concepts previously. This is, rather than a true introduction, a survey of the subject of epistemology that covers everything from the basics up to advanced discussions introducing the reader to the current debates going on among academic philosophers.
In any case, as a survey, it does a fantastic job. It starts with the basics: the classic definition of knowledge, the ideals of knowledge from which that definition sprung, the challenges from scepticism of different kinds (the main distinction made by Williams is between ancient scepticism's Agrippan trilemma challenging knowledge as a whole and Cartesian scepticism regarding knowledge of the external world) and moves up to modern times with the more sophisticated fallibilistic conceptions of knowledge, the problematic Gettier cases which this conception invites and the debates between internalists and externalists on the one hand, and that between foundationalists and coherentists on the other. Along the way Williams also deals with questions of the normativity of epistemology, arguments for the naturalization of epistemology, and the value of knowledge.
I've already mentioned that this book fits into the category of "sort-of-introductory-but-not-really", but something that sets it apart is that the latter part of the book contains a lot of original work regarding the challenges of scepticism where the author lays out his own suggestion for how to deal with it. This is tied to his contextualist conception of the justification of knowledge, which he poses as an alternative to both of the more traditional conceptions of foundationalism and coherentism (view which Williams reject). It's all seems very original (but keep in mind that this is coming from someone who is by no means an expert on these issues), is very advanced with a form of rigor in dealing with seemingly all relevant issues, possible objections and unwanted consequences (Williams is very keen on distancing his views from the kinds of relativism that they can seem to invite) while simultaneously being clear and pedagogical as long as the reader has the patience required to understand these issues.
In laying out his own views, Williams sometimes does a very good job of arguing for them (his contextualism seems very sensible and while I'm not sure if he really manages to dismiss charges of relativism, his conception of justification seems to fair a lot better than foundationalism and coherentism as he presents them, though I'm not sure that they get a really fair treatment) and sometimes not. An example of when he really does not do a good job of arguing for his position is in dismissing externalism without really treating it as carefully and thoroughly as one would have wished. Another example would be when he gets into the issue of truth, where I felt he did a dismal job of convincing the reader that the sort of deflationist view of truth that he seems to prefer is a reasonable one. In my view, deflationist accounts of truth are barely coherent. You often see a defence of deflationism relying on the disquotation criterion for truth (though there may be other ways which I have yet to encounter to defend deflationism), as if that was a sensible conclusion to draw from the criterion. Searle did a much better job at dealing with this issue in his The Construction of Social Reality where he showed convincingly that teh disquotation criterion actually suggests the correspondence theory of truth (or at least a specific version of that theory, which Searle lays out in a very convincing fashion). I agree wholeheartedly with Searle that (as I understand him) the disquotation criterion shows how claims of the truth of statements match the fact (a second-order fact I suppose) that the proposition expressed by the statement matches a fact in the world. Searle thinks, and I agree, that the very fact (second-order fact again) that there is such a "matching" needs a word and we might as well use "correspondence" to describe this. This is hardly central to the subject or the book though, I'd argue, very, very important to figure out correctly in laying out a particular epistemology and it seems Williams should agree considering his dismissal of epistemology as "first philosophy" (in the Cartesian tradition) i favor of a conception of the subject as something that permeates philosophy and relies upon theories of meaning and philosophy of mind (and theories of truth too, I'd argue, though Williams as a deflationist does not have a substantial theory of truth that goes beyond its dispensability so he can probably neglect to deal with it throughout his epistemological survey). I'm not sure if I treat Williams quite fairly in this area, but I think his presentation of truth and other issues in the last part of the book was shallow in a way that angered me a bit, especially since the bigger part of the book was anything but shallow.
Had the whole book kept up the level of sophistication even in dealing with the issues about which I've now complained, I would have given it five stars out of five but as it stands, it reaches four but no more (four and a half might have been more fair had that been available). Four stars is by no means bad of course, and neither is the book. It's an excellent survey of the subject of epistemology for anyone not already immersed in the issues (and probably for the experts as well) as long as they have at least a little bit of understanding of at the very least general philosophical questions (it is not by any stretch a good first book on philosophy) and probably some basic knowledge about epistemology is good too. Do read it if you have some basic understanding of the questions raised in epistemology and wish to get an excellent treatise of all the complicated issues being discussed among professional epistemologists today....more