PHIL 428/526, Pelletier

Fall 2013

 

My downloads are almost always pdfÕs.  The syllabus for this course is here.  A (tentative) order – a ÒscheduleÓ – in which we will cover the topics is here.  

 

Please continue to monitor this web page.  I will have assignments and readings posted on this page (or linked from this page).

 

The first readings are from PriestÕs book:  the Òset theory backgroundÓ and Chapter 1 ÒClassical Propositional LogicÓ.  You might also review whatever elementary logic book you used in your first symbolic logic class.   While we are on the topic of classical logic, you might also read PriestÕs Chapter 12 ÒClassical Predicate LogicÓ.  (There is somewhat more advanced material about classical logic in the Goble book, if you wanted to read it.  The course ÒscheduleÓ has details.)

 

I will be presenting material about classical logic that is not in either of these books, starting with propositional logic.  I will be interested in differing proof systems, in the notions of soundness and completeness, and (following Priest) in so-called semantic tableaux (which you may know as Truth Trees).  I will also be interested in expressive completeness, and in various difficulties there are in representing natural language in symbolic form.

 

I said in class today that IÕd post lists of Òstylistic variantsÓ of the logical connectives.  In the logic textbooks these words/phrases of natural language are normally presented as having the same Òsemantic contentÓ as the official words corresponding to the logical connectives, but differing from them in some Ònon-semanticÓ manner.  For example, but is always called a stylistic variant of and:  it is supposed to have the same ÒcontentÓ but differ by generating an expectation that the conjunction is somehow ÒunexpectedÓ.  Consider the difference between Johnny ate all the cookies and ate his entire dinner and Johnny ate all the cookies but ate his entire dinner.  A serious question to ask is whether these further so-called non-semantic things really are ÒextraneousÓ and do not contribute to the truth-conditions of the sentence they appear in. 

 

A standard description of this difference is to say that the ÒmeaningÓ is semantics whereas these other factors are due to pragmatics.  (Pragmatics being an account of Òwhy you would say such-and-so in this-or-that manner in the context so-and-soÓ.)  For instance, if you say ÒLet me show you how to hammer a nail efficientlyÓ, the semantics (the Òliteral meaningÓ) is to offer a demonstration of how to hammer a nail in an efficient manner.  But in some circumstances this might instead be a way of insulting the addressee, or of showing off, or speaking metaphorically, etc.  The literal meaning is semantics, whereas these other things that are conveyed are pragmatic effects.  A classical place to find this distinction made is H.P. GriceÕs ÒLogic and ConversationÓ, which is now probably the most well-known-and-employed-outside-of-philosophy thing about philosophy of language that there is.  Even more than Òspeech act theoryÓ.  (You can download and read it from this link).

 

 

Anyway, here are two items I have turned out to my Phil 120 class (intro to symbolic logic) giving lists of stylistic variants:  if-then and not, and and, or, biconditional.

 

We also covered stuff about semantic tableaux (truth-trees) for classical propositional logic.  Here are the branching rules in use for our signed tableaux method.  And here is an algorithm for applying the method.  Practice this with a few problems in an elementary logic book.  This is the main proof method we will use, since it is what Graham Priest uses throughout his book.  [He of course alters it to accommodate his Òfunny logicsÓ.]  Just to practice your talents in (classical logic) semantic tableaux, try these two problems.  Usually when something like this is assigned, you are asked to (i) determine whether the formula is a tautology (or whether the argument is validÉdepending on the problem), and (ii) if it is not a tautology (or is not a valid argument), to state a ÒcounterexampleÓÉthat is, state an assignment of truth-values to the atomic sentences (i.e., a row of the truth table) where the formula being tested is false (or the argument has all true premises and a false conclusion).  Try that with these two problems.

 

I also tried to show a pdf of how logics could be categorized, but my computer kept messing up the pdf display.  Here is that categorization, and here is the slide with the various types of proof methods.  <New Sept 30>: The most common form of proof method taught in elementary logic (at least in philosophy departments) is called Ònatural deductionÓ.  But that name describes a wide variety of differing systems, and is sometimes even designed to capture an entirely different notion.  Here is a paper I wrote about elementary logic textbook and what they do, trying to categorize all the different types of natural deduction.

 

In the class on Sept 16, I went over the Post Functional Completeness proof.  Only the first part of the lecture was electronic, and here it is.  If you want to see the published version of the paper that gives a proof of PostÕs Functional Completeness theorem, here that is.  I also sang the praises of a paper on Post by Alasdair Urquhart, saying youÕd be interested in the earlier parts of the paper.  Here is that paper.

 

<New Sept 30>:  The first assignment is downloadable from here.  This assignment is due on Monday October 7th at the beginning of class.  (try to write neatly!)

 

I noticed on the assignment that some people were interested in ÒreducingÓ the disjunctive normal forms that you got from applying the strategy to find an equivalent formula for any truth table.  Although this is not a topic of the course, it is interesting in itself.  And for those who want a method, here is a website that explains Karnaugh maps, which is probably the most effective way to do it Òby handÓ.  (As the author remarks, modern computerized reducing programs often have other methods.  But this is the most plausible way for non-computerized solutions of small-ish problems—ones with four variables.)  This website was for a computer science course, so it is phrased in a way somewhat different from our course. 

 

 

We started the discussion of modal logic with the description of the ÒBasic Normal LogicÓ, system K.  System K is generated from ordinary propositional logic by adding two new unary sentence connectives, the box (stupid Word doesnÕt have a square!, and so I will use ÒLÓ, which is one of the standard alternatives for the Òit is necessary thatÓ connective) and the diamond (but my Firefox wonÕt display a diamond properly, so I am going to use ÒMÓ, the usual symbol to use if you are also using LÉit means Òit is possible thatÓ), and some rules that govern them.  If one is developing this axiomatically, then the new principles are (a) the interdefinability of L and M, by using negation (Òp is necessary if and only if Âp is not possibleÓ, or in symbols: Lp iff ÂMÂp), (b) a rule of inference (ÒnecessitationÓ) which says that if a formula p is provable then its necessitation (ÒLpÓ) is also provable, and (c) an axiom (the K-axiom) that says: if a conditional is necessary, then if its antecedent is also necessary, then so is its consequent: L(p ˆ q) ˆ(Lp ˆ Lq).  (Stupid Microsoft/Firefox interface wonÕt let me use any symbols for horseshoe or arrows!  Those ˆ are all supposed to be arrows [i.e., Òif—thenÓ])

 

Priest does not do an axiomatic development of his logics, but rather uses tableaux.  These tableaux have a special symbol attached to them, indicating the ÒworldÓ in which the sentences at a node are being evaluated.  By convention he starts with 0 as the initial world.  Priest captures the interdefinability of the modal operators with his two rules that move negations to the inside of modal operators (and changing the operator).  The rule of necessitation and the K-axiom are captured by the interplay of the ÒM pÓ and the ÒL p, irjÓ rules. You should try to do some tableaux in system K, just to get the feel of it.  Priest has some problems at the end of Chapter 2 for you to try.  IÕve copied a few of them onto this page, for easy access to you.  Recall that Priest uses negation as his way of saying that a formula is false.  So for him, writing ÒÂA, 2Ó is a way of saying that A is false at world 2.  We could instead use signs for the truth values, and thus explicitly talk about truth and falsity at a world.  That is, we might state that formula as ÒA: F, 2Ó, which says that A is False at world 2.  Then we would be using signed tableaux not only for the worlds, but also for the truth values.  (From the point of view of this class, it doesnÕt matter at all which one you prefer.  We need to use signed tableaux for many-valued logic, however.)

 

Chapter 3 of Priest is about the class of Ònormal modal logicsÓ, of which K is the basic member.  All the others are generated by adding more axioms to K, until we get to the strongest of the normal systems, which is historically called S5 (or S5), but if you use the nomenclature that is now common—where you start with K and then add on names of the axioms—you might call it KT5 or one of the other equivalent combinations of axioms (Priest calls it KTB4).  Here is a writeup about these axiom systems and their names.  (Note that this writeup has the G-axiom in it, whereas Priest doesnÕt consider this axiom.  G here stands for Geach--the philosopher Peter Geach.  There is another axiom that is also called G, and is named after Gšdel.  But that does not generate a Ònormal modal logicÓ.)   Here is a diagram of the systems that are generated by these axioms, which the proper inclusion relations indicated by arrows.

 

Priest gives tableaux rules for the various normal modal logic systems that he considers.  You need to learn them.  (Of course, they donÕt include system G.  Can we think of any rules for G?)  And here are some more slides explaining the rules (also contains the tableaux rules for the non-normal logics, which weÕll discuss later).  The standard normal modal systems that everyone always talks about are:  K, KD, T (=KT), S4 (=KT4), B (=KTB), and S5 (=KT5, =KTB4, É).   Priest deals with them using the tableaux rules, building on the basic normal K.  Adding connectivity (aka ÔserialityÕ) gives us KD (Priest calls that tableaux system Ku), adding reflexivity to either K or KD (since reflexivity already implies seriality, we might as well just add it to K) yields T (the tableaux system is called Kr), adding symmetry to T yields B (the tableaux system is called Krs), and adding transitivity to T yields S4 (the tableaux system is called Krt).  One could form S5 by using Krst, but Priest instead has a simpler way of doing it using what he calls Ku.  The rules that correspond to all these names are on the slides mentioned earlier in this paragraph.  (HmmmÉthose are supposed to be Greek letters rho, tau, and sigma.  Stupid Microsoft.) 

 

Chapter 4 of Priest is about (some) Ònon-normal modal logicsÓ, including the C.I. Lewis systems S1, S2, and S3.  He also mentions systems S0.5 and S3.5—which Lewis didnÕt consider, but which were described later and put into the Lewis names in terms of strength of the systems.  Priest calls these systems N-systems, starting with the basic N and then adding conditions on the accessibility relation among possible worldsÉjust like with the normal K-systems  However, these non-normal systems are described in terms of a distinction between normal and non-normal possible worlds, and the definition of logical truth is Òtrue at all normal possible worldsÓ.  A non-normal possible world is defined as one where no necessary formula (no formula of the form Lp) is true.  These non-normal worlds do not have any worlds accessible from them.  Their tableaux systems are described as systems N, Nr, Nrt, and Nrst, corresponding to logics N, S2, S3, and S3.5 respectively.  So their rules are just like those for the normal logics with those accessibility relations, but there is a change in the application of the tableaux rules to accommodate the non-normal worlds.  Here is a writeup explaining these logics and the tableaux methods.  It also has a few problems for you to try out.  And here is a repeat of the document mentioned in the last paragraph as having the tableaux rules for both the normal and non-normal logics.

 

There are some unusual features of these non-normal logics (in addition to the weird notion of non-normal worldsÉwhich, by the way, Priest does not think is weird).  One of the weird things is that there are formulas which look like they ought to be theorems, but arenÕt.  They are of the form (p Þ q) [that Þ symbol is supposed to be the Lewis ÒfishhookÓ symbol.  Stupid Microsoft/Firefox], and they will become a theorem if some other formula – a formula that is a theorem – is added to the antecedent!!  These other theorems are traditionally called ÒT-thesesÓ and as a whole are symbolized by simply T.  So, (p Þ q) is not a theorem; but ((p&T) Þ q) is a theoremÉand all T asserts is that it is some theorem!!  Pretty weird!  S3.5 has the very unusual feature that there are theorems of the form (X v Y), where neither X nor Y are theorems, and where there is no propositional variable in common between X and Y.  (Of course, it is common for there to be theorems of the form (X v Y) in all sorts of logics, where neither X nor Y is a theorem by itself.  But pretty much they always have some variable in common.)

 

One of PriestÕs main concerns in the book is with the proper formalization of the natural language Òif—thenÓ.  Here are some slides about the Òparadoxes of the material conditionalÓ and arguments for/against using the horseshoe.  Priest is also keen to discuss whether a Òstrong implicationÓ connective of LewisÕ is adequate for representing the natural language Òif—thenÓ.  He thinks not, and here is a summary of his reasoning.  Instead, Priest is in favour of some kinds of conditional logics (particularly ceteris paribus conditionals) – although in later chapters he tries to merge this with relevant logic conditionals [a topic we wonÕt cover in this class].  Here are some slides about conditional logics.  Be sure you can apply the tableaux rules for his basic conditional logics!

 

The next topic to be covered is (propositional) many-valued logic.  This is the topic of PriestÕs Chapter 7.  Although we will cover the material Priest discusses – so you should be sure to read it – I will be introducing some other topics, such as a wider range of Òintended usesÓ for many-valued logics and tableaux methods for them.  One of the uses (perhaps the main linguistically-oriented use) of many-valued logics is to give an account of the formal properties of vagueness in language.  There are a number of places where you might read up on the background to vagueness and its logical properties.  I recommend Rosanna KeefeÕs (2000) Theories of Vagueness, which is available through our library as an electronic text: http://www.library.ualberta.ca/permalink/opac/3897993/WUAARCHIVE.

 

Assignment #2 can be downloaded from here.  It is due on Nov. 4th.  As before, you can submit it in advance (= before class) electronically, if you can do the problems electronically.  Or you can submit it at the beginning of class on the 4th.

 

The take-home midterm exam can be downloaded from here.  Please do your work independently of one another and from other sources of agency.  (You know what I mean.)  It is due back to me before I go home (which is after the Logic Reading Group meeting) on Wednesday afternoon, November 13th.  Either put it in my mailbox, or give it to me during the day, or email it to me before 5pm on that day.  This exam is worth 33% of your grade if you are in Phil 428 and 25% of your grade if you are in Phil 526.

 

Assignment #3 was sent out directly by email on Nov. 21st, and is due on Dec. 2nd.  Your final paper is due on Friday Dec. 13th before midnight.

 

<Added Nov. 25th>  Well, we agreed to modify the course requirements.  They are now: the three homework assignments, the midterm exam, and a final paper.  Each of these three items is worth one-third of your grade.

 

In discussing vagueness, I had occasion to mention a paper of Michael TyeÕs which (apparently—sometimes his writing makes details hard to discern) makes the metalanguage being used to discuss a language with vagueness have the same three values that the object language does: true, false, and ÒotherwiseÓ (or ÒgapÓ, or Òneither true nor falseÓ, or ÒborderlineÓ, or ÒindeterminateÓ, or É)  Here is that paper.  Tye has written a follow-up paper that is somewhat more accessible.  Here is that paper.

 

In that discussion, I also mentioned the position of Timothy Williamson, called Òepistemic vaguenessÓ, which makes the claim that in the world there is no vagueness: all predicates are ÒsharpÓ and have no borderline cases.  But that we perceive them to be vague because we are unable to determine what the cutoff point is.  A part of WilliamsonÕs position is that two-valued first order logic is The One True Logic, and he has argued in many places that the attempts to employ a many-valued logic are conceptually misguided.  Rob Stainton and I wrote a rebuttal to WilliamsonÕs position: here it is (for your amusement).

 

An argument against Òrealistic vaguenessÓÉthat is, vagueness-in-reality – which is the position Tye is arguing forÉwas initially given by Garreth Evans in 1978 in a one-page paper which probably has the most citations for a one-page paper (because it is so wrong).  I once wrote a paper that tried to fix up the Evans argument, and thereby become an argument against realistic vagueness.  This more general argument relies on the parametric J-operators in a many-valued logic, and demonstrated that the sorts of assumptions that believers in realistic vagueness make are inconsistent.  Here is that paper.  But since the metalanguage was classical, I suppose the style of argumentation that Tye is putting forward would find that to be a flaw in the presentation.

 

I also once gave an interpretation of what I took to be the correct logic of (epistemic) uncertainty or indeterminacy.  Here is that article.  (This journal has never put any of its articles on the webÉthis is a scan of my offprinted copy.)

 

On a totally different track, here is a paper written by me and a student when I was at Simon Fraser.  This is a logic-and-philosophy-of-language-oriented cognitive science paper on vaguenessÉwith experiments even!!

 

One topic I discussed was definite descriptions, and in particular the issue of whether RussellÕs famous arguments against treating definite descriptions as singular terms was justified.  I argued that while it may have been valid against a Meinongian-type of theory, it didnÕt really touch the various versions of singular-term theories held by Frege.  Bernie Linsky and I wrote a paper about that: here it is.  (In N. Griffin & D. Jacquette One Hundred Years After On Denoting: Russell vs. Meinong [Cambridge UP], pp. 40-64].)

 

In class I mentioned a number of topics concerning the appropriate way to formally represent constructions in English. Here are the slides with those topics. Each of the topics could form a term paper (for the grad students in the class)Éindeed, could form a pretty reasonable MA thesis.  For either project, you would take a look at the various proposals that are in the literature that are designed to avoid the problems mentioned on these slides and then evaluate which seems best (and for what reason best).  One topic in particular that takes up the last section of these slides concerns sortal predicates and sortal logics.  The last slide is a very short bibliography of work done on that topic.  Related to this is the following puzzle: The Edmonton Transit System says Ò239,800 passengers use the ETS every dayÓ.  And although ETS doesnÕt say it, it is nonetheless true that every passenger is a human.  But it is not the case that 239,800 humans use the ETS every day.  It thus seems that x can be the same human as y, but be a different passenger than y.  Such considerations form one of the basic planks in the search for a sortal logic.