Seminar LogicByTweets

Dear Twitter Followers,

I’ve been at the University Pittsburgh/PA, May 10-31, and agreed to take part in another perhaps quite time-consuming philosophical project. Hence I will not be able to continue the seminar LogicByTweets on a regular basis. Every now and then, however, I shall give a session and write a sequence of logic tweets. As before, these sequences will be thematically continuous and will be subsequently displayed here.

I’ll give notice of each session a few days in advance on Twitter.

Every once in a while, I shall also attend Twitter discussions about theoretical & practical philosophy, logic, science, politics, and law.

New account name: @PhiLoThough

Thank you for your understanding.


6. Formalism (V); Session Dec. 19, 2014

6.1 Changed plans a bit again. Since it’s our last session this year it’s perhaps useful to summarize what we have so far.

6.2 I also want to give some ideas of practical application of what we already learned from Aristotle, before we continue in January.

6.3 We said that the Syllogism of Things depends on the Syllogism of Words (5.28 f.; see 3.10.7 ff.).

6.4 We use words to classify, sort and explain the natural world. Scientific knowledge, for Aristotle then, is explanatory understanding:

6.5 That is, not merely to ‘know’ a fact incidentally & to agree that something is true, but to know ‘why’ it is the case.

6.6 And this, Aristotle thinks, can be done syllogistically: All humans are animals; all animals are mortal; so all humans are mortal.

6.7 This, of course, only works when you have sorted out the right reference class, viz. the right natural kind terms, for your deduction.

6.8 Here the distinction between genus & species comes in again (5.15, 5.21, I was a little hopplahopp there, so here’s what it means:)

[In what follows, I pick up some ideas on reduction/reductionism from a discussion with Anita Leirfall (Bergen University/Norway)].

6.9 Take the category substance/essence again (cf. 5.14), and the question of what it is for something, say F, to exists.

6.10 For Aristotle, you answer it by placing Fs in their category, C, and apply to the Fs the general account of what it is for C to exist.

6.11 Or even more generally: Check whether F falls under C. That’s what lawyers call a subsumption.

6.12 Two problems: (1) You hardly can avoid regress. Aristotle, however, rejects that there are infinitely many predicates.

[Cf. Posterior Analytics, Bk I, chs. 19-22 ].

6.13 Aristotle’s Categories, which are meant to terminate regress, are somewhat the conclusion of a somewhat ‘decisionistic’ procedure.

6.14: (2) Same problem in other gown: When we reduce an item F to another item, say, G (Aristotle calls them accidents, as opposed to categories).

6.15 All reductions face the difficulty that we hardly ever can say that Fs reduce to Gs, or to some Category, without being caught in regresses or logical circles.

6.16 This is especially true for so-called ontological reductions to ‘abstract entities’ (5.21).

6.17 Take mathematical objects (numbers, for instance). Aristotle says (Loeb edition; see 3.8.2):

“[I]f mathematical objects exist, they must be either in sensible things […], or separate from them […]; or if they are neither the one nor the other, either they do not exist at all, or they exist in some other way. Thus the point which we shall have to discuss is concerned not with their existence, but with the mode of their existence;” Metaphysics 1076a32-36.

6.18 The last sentence is certainly a stunning claim. What are we to make of it?

6.19 We skip for now the debate about mathematical Naturalism (Mill), Intuitionism (Hilbert, Bernays), Platonism (Frege), or Nominalism (Hartry Field).

6.20 And turn instead to Aristotle’s equally stunning explanation: Mathematical Objects are what mathematicians say they are (my words; cf. Metaphysics 1077b32 f.).

6.21 That’s not as silly as it seems. Recall 6.4: We use words to classify, sort and explain the natural world.

6.22 This is even truer for abstract entities, such as mathematical objects.

6.23 Hence it’s not so much a question whether certain things exist in themselves, as it were, but rather how our reason takes them to be.

6.24 This amounts neither to relativism, nor to idealism.

6.25 The problem can be exemplified by the highly contested conception of Conventionalism in the philosophy of science:

6.26 In the otherwise most instructive Cambridge Companion to Aristotle we find the following sentence (1995, 110 f.):

“The world divides (realistically, and not as a matter of mere convention) into natural kinds, and those kinds stand in relations of greater and lesser resemblance to and distinction from one another” (original parenthesis).

6.27 Did you notice the problem? The world as such certainly does not divide into natural kinds. But it is us who use natural kind terms.

6.28 It’s us who do the dividing & designating, as Aristotle himself acknowledges, when we take his categories as predicables (5.12).

6.29 The same for resemblance relations: They are the work of thought, too, viz. of sorting the world around us.

6.30 Conventionalism was en vogue in the 1930s & was advanced esp. by the Logical Positivists in the aftermath of non-Euclidean geometry.

6.31 It was Quine’s ‘Truth by Convention’ (1935) which then turned the tide. But it rested on a grave misunderstanding.

 [Reprint in: W.v.O. Quine, The Ways of Paradox, HUP 1976, ch. 11]

6.32 For nobody ever talked about ‘Truth’ by convention: Not Henri Poincaré, nor Hans Reichenbach, or his follower Adolf Grünbaum in the 1960/70s.

[An instructive recent treatment of the subject is: Yemima Ben-Menahem, Conventionalism, CUP 2006]

6.33 Of course, the world does not care how we talk about it. But only with concepts there come judgments.

6.34 And it’s only with concepts & judgments that we can be said to have the idea of an objective, independent world in the first place.

6.35 That, and not the conventionality of ‘truth,’ was the point of the Logical Positivists.

[Unfortunately, the essays in Michael Friedman’s Reconsidering Logical Positivism, CUP 1999, have had a large impact, especially on US-American Philosophy of Science, and cemented Quine’s misconception. Not to mention Friedman’s obscure ‘relativized a priori,’ which was equally influential, but unfortunate for philosophical coherence reasons. We shall say a bit more about this at a later stage].

6.36 Is that a circle again? Well, yes, of a sort. But it’s not vicious. It’s rather hermeneutic.

6.37 The question then is not whether there are conventions in science, but rather what conventions there are & whether they are useful.

6.38 Let’s leave it at that for today. It will be part of our program too.

6.39 Together with a theory of the relationship between the Syllogism of Words & the Syllogism of Things.

5. Formalism (IV); Session Dec. 16, 2014

[Today we are all with the school in Peshawar attacked by this horrible crime. Logic has only very limited access to fundamentalist religion & faith. Where there’s no reason there is no logic. But since the term λόγος also stands for ‘speech’ (4.2.2), it’s even more for us today, all over the world, to support the voice of reason. My little share in this may be this seminar here. That’s certainly not much! But let’s continue with it nevertheless]:

5.1 The ‘Syllogism of Words’ (SoW), of which we spoke at the end of the last session, is, or can be, of things too.

5.2 But while the ‘Syllogism of Things’ (SoT) deduces statements about things from things, SoW deduces meanings from meanings.

5.3 Meanings of words that designate or signify things. But not all words designate things. Words in purely formal deductions do not.

5.4 Or so it’s customarily assumed. Recall the distinction between form & content of previous sessions (2.6 ff., 3.5 ff., 3.9 ff., 3.10.4 ff.).

5.5 However, the meaning of this distinction itself is not at all clear.

5.6 Even the deduction of statements about things from things (5.2) is, of course, from statements (!) about things.

5.7 Thus it’s statements either way, that is, in both SoW and SoT.

5.8 And it’s statements that are either formal or have material content; never ‘things.’

5.9 Logic then inquires into the relationships among terms & statements, whether they be about terms & statements, or about ‘things.’

5.10 For the classification of terms Aristotle used the term Categories (κατηγορίες).

5.11 In peripatetic & scholastic scholarship the Categories were fraught with ontological connotations (4.3.1 ff.). That’s not over yet.

5.12 Most scholars today, however, take the categories as predicables, viz. as terms that classify terms (usually more or less following Kant, Critique of Pure Reason B 107 f., who had “Prädikamente,” that is, ‘Predicaments,’ according to the Cambridge Edition of Kant’s Works; see also Prolegomena, § 39: Plus “Prädikabilien”).

5.13 Aristotle’s Categories list types of predicates, i.e., types/kinds of expressions/terms that can be predicated of something.

5.14 These categories are: Substance/Essence (what), Quantity (how much, how large), Quality (what sort of thing), Relation (half, double, greater than), Location (where), Time (when), Position (also: posture, stance, attitude), State/Condition/Possession (how circumstanced), Action/Activity, and Passivity/Affection.

5.14.1 Aristotle names somewhat different categories at different times and different places:

5.14.2 Compare Categories 1b25-2a4 with e.g. Topica 103b20-25 & Metaphysics 1003b5 ff.

5.15 Generally, though, Categories in Aristotle are somewhat highest genera in the game of predication.

5.16 That is, each category designates one possible relationship between predicate & subject.

5.17 That is, some predications say of their subject ‘what it is,’ others what it is like, again others how much or how heavy it is, etc.

5.18 Accordingly, Aristotle is quickly to note: None of the category terms has any truth value, viz. is true or false (Categories 2a5 ff.).

5.19 Truth values can only be generated by combination of these terms to statements. That’s especially true for Substance.

5.19.1 Take the word ‘tree.’ There’s nothing true or false in the term itself. Only when we combine it with other predicates we get a statement that can be true/false (and like before in 5.8: Only statements can be true or false; never ‘things;’ cf. Metaphysics Bk VII, ch 4).

5.19.2 For example: This tree is a plant (substance) which is green (quality) and stands on a hill (location). [For those metaphysicians or ontologists among us who are searching for a ‘deeper’ sense of Substance than a plant; see Metaphysics 1032a19].

5.20 With this in hand, there is no need to speculate metaphysically about substance, or so-called substantial forms (4.3.2 ff).

5.21 Nor do we need an ontology of ‘abstract entities.’ We can safely leave it to the natural sciences to develop the resp. genera.

5.22 And as Aristotle himself noted: Existence (or Being) is not a genus; Metaphysics 998b21-24.

5.23 Nor is ‘existence’ a predicate; we come back to this when we study Kant on logical formalism (for those already curious: see Critique of Pure Reason B 626 f.).

5.24 This does not mean that we don’t need any philosophy at all! Philosophy is first & foremost an endeavor about words & interpretation.

5.25 And philosophy in the analytic tradition has dealt with just this: With the semantics & syntax of (scientific) language.

5.26 Here then are we back to the Syllogism of Words. It’s about the logical operations of conjunction, negation & conditionals.

5.27 And last, but not least, about quantification; the ontological philosopher’s “everything” when universally quantifying over ‘things.’

5.28 As you might have guessed then, the conditions of the Syllogism of Words are the preconditions of the Syllogism of Things.

5.29 This will be our program. In the next few sessions, however, we’ll be still concerned with some history (e.g. Descartes, Kant, Frege).

4. Formalism (III); Session Dec. 12, 2014

4.1 Changed plans a bit. We’ll be concerned with the history of syllogism for a few more sessions before we get to logical formalism itself.

4.2 Now to the passage of Aristotle, Prior Analytics 24b18-20 (see 3.9 ff.), and the exercise from the end of last session.

4.2.1 It’s central term is Logos: λόγος (thanks to @cathyby & @SteveCooke for helping me with the inserting of Greek letters on Twitter).

4.2.2 The Greek term λόγος stands for reason/ratio, as well as for ‘word’ and ‘speech.’

4.2.3 So we are back to the relationship between thought and language (cf. 2.1 ff.).

4.2.4 The term ‘thing’ then in the Oxford Translation (3.9.2; 3.10 ff.) must be taken metaphorically for “the subject matter spoken of.”

[Hope my English is correct here, but I think you can gather my point. Generally, I welcome suggestions that make my English more precise].

4.2.5 Accordingly, they have revised the translation in 1995. Now the passage is about statements, as I indicated in last session’s addendum.

4.3. The general problem is that translations influence philosophical scholarship. And they have done so throughout the ages.

4.3.1. In peripatetic scholarship our passage was largely taken as a substantial claim concerning ‘things.’

4.3.2 Or ‘substance.’ The medieval theory of Substantial Forms (later revitalized by Leibniz mainly on theistic grounds) was of this kind.

4.3.3 Substantial forms, however, are spurious. They are neither form, nor substance, but something rather occult.

4.3.4 That’s not something we’re dealing with in logic. Descartes, e.g., saw substantial forms as impositions of mind onto matter.

4.3.5 He vigorously polemicized against “all sorts of strange objects which have no resemblance to what is perceived by the senses such as ‘prime matter,’ ‘substantial forms’ and the whole range of qualities that people […] introduce, all of which are harder to understand than the things they are supposed to explain;”

[Principles of Philosophy, IV § 201; translation John Cottingham et al. (eds.), The Philosophical Writings of Descartes, vol. I, CUP 1985].

4.3.6 There are problems with the invocation of sense perception as the basis of knowledge as well. But this leads us too far afield.

4.3.7 In Leibniz, at any rate, substantial forms reappeared as somewhat the world soul in all matter, grounded—guess what?—in God.

4.3.8 In this respect, Leibniz is a clear regression behind Descartes. It impeded the development of German philosophy for quite a while.

4.3.9 Despite his invention of the infinitesimal calculus which he purportedly developed independently of Sir Isaac Newton.

4.4 Anyhow, we considered these intricacies only to exemplify how translations may influence philosophical scholarship (4.3).

4.4.1 Having ‘things’ for ‘words’ in our Aristotle passage may well have led to what we called the ‘Syllogism of Things’ in 3.10.7.

4.4.2 But we must postpone our inquiry into the Syllogism of Things (3.10.9).

4.4.3 In our next session we continue with the ‘Syllogism of Words’ and finish our introduction of Aristotle.

Addendum. For the Leibnizians among us who insist on Leibniz as the inventor of the Calculus (4.3.9), a good and thorough account gives:

A. Rupert Hall, Philosophers at War: The Quarrel Between Newton and Leibniz, CUP 1980. Hall presents the evidence and lets the reader judge for herself.

3. Formalism (II); Session Dec. 9, 2014

3.1 After our last session a colleague asked: How in the world can you recommend Frege to beginners? (see 2.9 ff.)

3.2 Well, as you will find out, reading the originals is in no way more difficult than reading secondary literature. The contrary is the case!

3.3 As Bertrand Russell once remarked: Don’t read current handbooks & treatises; turn to the classics. See his Problems of Philosophy, Bibliographical Note.

3.4 It’s even somewhat a cool experience to have a real classic book in hand. Thus there’s nothing to fear.

3.5 Let’s now return to logical Formalism & to the distinction between Form & Content. It goes back to Aristotle.

3.6 And let’s have Aristotle speak for himself (see 3.3).

3.7 But that’s not at all an easy task. The problem is in the translations of what Aristotle supposedly said.

3.7.1 There are other problems, e.g., whether Aristotle is the author of what we take to be his works. But we don’t get into this here.

3.7.2 Readers who are interested in provenance may consult the Cambridge Companion to Aristotle, for instance.

3.8 Back to translation. A general remark: Whenever you are able to read the original language, do not read translations.

3.8.1 Translations always make choices as to meanings of expressions, hence different translations sometimes present very different content.

3.8.2 I often consult the Aristotle translations of the Loeb Classical Library of Harvard UP, 1930s ff., because Loeb also includes the Greek.

3.8.3 More popular is the ‘Oxford Translation,’ a revised version of which was edited by Jonathan Barnes, 2 vols., Princeton UP 1984.

3.8.4 Pagination is the same; references are usually of this sort: page number, column number, line number.

3.8.5 So a typical reference like 24a10-15 is: Page 24, column ‘a’ (left-hand column), lines 10 to 15 (you may see ‘a’ also in superscript)

3.9. Let’s turn to Form & Content then: Aristotle invented what he called a ‘syllogism,’ or what we today call a logical or formal deduction

3.9.1 Aristotle says according to the Loeb translation:

“A syllogism is a form of words in which, when certain assumptions are made, something other than what has been assumed necessarily follows from the fact that the assumptions are such;” Prior Analytics 24b18-20.

3.9.2 According to the Oxford Translation (OxTrans) of the same passage, Aristotle says:

“A deduction is an argument in which, certain things being supposed, something else different from the things supposed follows of necessity because of their being so.”

3.10 Did you notice the difference? (Not that of syllogism/deduction; that’s the same for our purposes)

3.10.1 For our purposes decisive is: OxTrans has “things” for “words” & “assumptions” in Loeb. Philosophically, this is a huge difference!

3.10.2 OxTrans has “things being supposed” from which other “things” follow because of the “being so” of the supposed things.

3.10.3 According to Loeb, by contrast, Aristotle is rather concerned with “words” & “assumptions” & other assumptions that follow from them.

3.10.4 Put otherwise: The OxTrans passage is concerned with the deduction of material matters while the Loeb passage is rather formal.

3.10.5 You can see the distinction between Form & Content at work right here. The irony is that we didn’t even start with it yet.

3.10.6 But we are already confronted with the intricacies of ordinary language in formulating the subject of formal language (see 2.4 ff.).

3.10.7 And hence we may term our subject now as the difference between the ‘Syllogism of Words’ and the ‘Syllogism of Things

3.10.8 The ‘Syllogism of Words’ will concern us in the next session, Friday, Dec 12.

3.10.9 The ‘Syllogism of Things,’ that is, deductions in the demonstrative sciences (physics etc.) will follow at a later stage.

Addendum: In later printings of OxTrans our passage has been revised. Since 1995, it’s rather statements too: See here.

All works of Aristotle in the Loeb Classical Library can be accessed online here. The Prior Analytics in particular is here.

Exercise: Try to figure out what Aristotle’s original meaning was. I shall disclose it in the next session as well.

2. Formalism (I); Session Dec. 5, 2014

2.1 Ordinary language is vague. And since we use language to communicate thought, communicated thought is often vague and muddled as well.

2.2 Donald Davidson is perhaps right that speaking a language is not a trait we can lose while retaining the power of thought & judgment.

[see, e.g., Davidson, Inquiries into Truth & Interpretation, OUP 1984, 185]

2.3 That’s controversial, though. Steven Pinker, e.g., believes in ‘Mentalese,’ the language of thought, independent of spoken words.

2.3.1 Pinker is a prolific popular writer, his books are easy reads. He’s not so much an academic philosopher, though.

[See, e.g., Simon Blackburn’s review of Pinker’s The Blank Slate in the New Scientist of September 7, 2002. Simon Blackburn, by the way, has a nice & entertaining website at Cambridge]

2.3.2 On Mentalese, check Pinker’s The Language Instinct, 1994, ch. 3 & The Stuff of Thought, 2007, and judge for yourself. But maybe also consult, as an instructive contrast, Wilfrid Sellar’s ‘Myth of Jones,’ in his Empiricism & the Philosophy of Mind of 1956, esp. §§ 56 ff.

2.4 Either way, it can’t perhaps hurt if we do something about language’s vagueness. That’s one purpose of logic.

2.5 Its tool, as it were: Formalization. It was Aristotle, who discovered the formal nature of logical derivation:

2.6 Certain statements can be derived from others on the basis of their formal structure alone, independently of their specific content.

2.7 Thus the distinction between Form & Content. The idea is that inquiring into formal structure may also enlighten our everyday communication.

2.8 Yet we must use vague ordinary language to create the precise formal language of logic. That’s not as silly as it sounds.

2.9. Even Gottlob Frege, founder of modern mathematical logic, to whom we shall return, didn’t build up his symbolic language from scratch.

2.9.1 Rather it’s been a reconstruction of natural language. And Frege hoped that it may in turn also help to cure its vagaries of vagueness.

[see Frege, ‘Der Gedanke’ (1918); Engl in: MIND 65 (1956), 289-311]

2.9.2 Please read Frege’s ‘Der Gedanke.’ It’s easy; no math in it at all. We shall discuss it in January or February.

2.9.3 There are other translations (one by P. Geach); we shall discuss that too. But the one in 2.9.1 is available online.

2.9.4 See also Preface of Frege’s ‘Begriffsschrift’ (1879), where he is confident that his ‘ideography’ can clarify philosopher idioms:

2.9.5 The description of this definitely worthwhile task is also not too difficult & gives a good first impression of Frege’s Platonism.

2.9.6 It’s only 4 pages. Translation in Jean van Heijenoort (ed.), From Frege to Gödel, HUP 1971, 5-8. They have it in your local library.

[I also found an online scan from the van Heijenoort book here. You may find others].

2.10. We shall return to it after we have discussed the beginnings of logic (from Aristotle to Kant) in the next few sessions.

1. Introduction; Session Dec. 1, 2014

1.1 Logic is a tool, as it were, for deriving conclusions from accepted assumptions.

1.2 Here the first problem shows up: What’s an accepted assumption? Such assumptions can be ‘logical’ themselves, or they can be ‘non-logical.’

1.3 ‘Non-logical’ is not pejorative, as common speech sometimes suggests, but just means extra-logical.

1.4 ‘Extra-logical’ is everything beyond the formal machinery of logic. Physics, e.g., is extra-logical (non-logical) in this sense.

1.5 Physics uses logic. This, however, does not make its truths logical truths.

[A nice description of the relation Logic/Science gives W.v.O. Quine, Mathematical Logic, HUP 1981, 7]

1.6 Logical truth is only that which is derived by formal logical means, that is, by certain accepted ‘rules of inference.’

1.7 It is assumed (cf. 1.1) that these formal rules, to which we come later in detail, lead to logically true conclusions.

1.8 No logic, however, can derive from a sequence of completely false factual, e.g. physical, premises a formally true conclusion.

1.9 So for a true scientific conclusion we need both factually true premises and a consistent sequence of formal logical inferences.

1.10 Application: A critical test of any scientific theory is its accuracy in predicting phenomena before they are observed.

1.11 Any such prediction must involve the application of the formal rules of logical inference.