Formal Logic

17 Sep 2016

[I'll update this post as I find things that are interesting in logic]

I'm actually familiar with some of the things here: https://www.youtube.com/watch?v=tpeeTHNVnxQ and am watching the second part now: https://www.youtube.com/watch?v=VOhGLnxJt4E. (Joel David Thomkins 'The Continuum Hypothesis and Other Set-Theoretic Ideas for Non-Set-Theorists), which makes me glad that I have taken courses in analysis.

I've glanced at some literature on the algebra of cardinals at the University of Iowa library which I would like to understand better.

I tried reading Hermann Weyl's The Continuum Hypothesis last summer and a lot of it went over my head. Sitting in on a formal logic course on Irving Copi's Symbolic Logic (I got the 5th Edition for like $9 online) has been helpful and I'm hoping to get back to Weyl's book eventually.

I've taken two analysis courses and two abstract algebra courses and have done homework assignments on the Riemann Zeta function, Gamma function, and have heard about elliptic curves in a lecture from a course that I was taking for a grade. This is farther than I ever thought I would get in math, so I am happy about that. What I like about formal logic is that it can give me a way back into math, with stronger foundations: Irving Copi's formal logic goes into calculus and algebra by the end, so I would have a way to "stay alive" in mathematics. If I am not accepted to any graduate courses in logic then I would be able to say that I have given it a good run in math and would feel accomplished in my work.

I did touch on some Set Theoretical basics with my '14-'15 Algebra I students, so I'm glad that they'll have some access to Set Theory.

I learned that DeMorgan's Laws for Logic give an analog for the way that I need to remember to explain semantic incompleteness. I had previously noted:

Week of 9/12/16: It is the case that in any language, there is always something that cannot be expressed. It is not the case that there is some thing that cannot be expressed in any language; for, to name it would be to invalidate the reason we were searching for it. - (On the notion of semantic completeness)

This can be expressed in formal logic as: 

¬∃x∀yφ(x,y)  = ∀x∃y¬φ(x,y)

I am thinking of the left side as:                                       and the right side as:

"It is not the case that there is a subset of the                   "For every subset of the real line there

real line which is such that for every language,          =   exists a language such that it is not the

the specific subset of the real line is inexpressible"         case that the subset of the real line is 

                                                                                       inexpressible."

Now, the left side is what we were given from the notion of semantic completeness. It is interesting that this is logically equivalent to saying that for every subset of the real line, there is some language in which the subset can be expressed.

The error I was tending to make in natural language was to say the negative case of the Week of 9/12/16 Quote: "It is not the case that there is some thing that cannot be expressed in any language..." Using formal logic keeps this straight in my head.

Quote of the Week

Week of 9/5/16 “It is striking that we are less likely to criticize violinists, say, than political philosophers, for failing to provide justice-promoting guidance, as if being interested in identifying truths about justice meant that one was more rather than less culpable for failing to tell us how to bring it about… But I find it hard to feel more impatient with political philosophers than with those who show no interest in justice at all” Adam Swift's The Value of Philosophy in Nonideal Circumstances, 367.

Week of 9/12/16: It is the case that in any language, there is always something that cannot be expressed. It is not the case that there is some thing that cannot be expressed in any language; for, to name it would be to invalidate the reason we were searching for it. - (On the notion of semantic completeness)

Derrida's 'Of Grammatology'

Gayatri Spivak’s translation Of Grammatology (1974) of Derrida’s De la Grammatologie (1967)

Derrida is considering Saussure’s linguistics and says that Peirce had some things right in linguistics, then talks about Rousseau, Claude Lèvi-Strauss, presence, metaphysics, and morality. The people he’s writing about that stuck out to me are Claude Lévi-Strauss, Rousseau, Saussure, Husserl, Hegel, Diderot, and Bergson.

Main Idea: Derrida says that metaphysics of presence cannot account for the spacing within writing. That is, if I print a written document, the "actual meanings" of the words on the page do not account for how the shape of the page, formatting, type of ink and paper, and shape of paper affect the meaning of the document as a whole. This has implications for the relationship between speech and writing: while each has characteristics that are unique to it, there is no sharp divide between the two. How would our lives (or, more exactly, our writings) be different if, instead of writing on rectangles, we wrote on circles or S-shaped paper? How would our lives be different ("how would the meanings of our writings change...") if, instead of writing upright we always had to stand on our hands to write as a social norm? The "literal meaning" (even these will have to be discarded) of the word "blue" will still mean what "blue" means regardless of the shape of the paper it is written on, or whether it is written by someone who is upright or standing on their hands. However, since metaphysics of presence can't account for spacing (I want to use the word style), repetition does not produce the same meaning time and again, but rather, a simulacrum is produced. In this way, there is always a trace of ineffability within articulation; of absence within presence. So "blue" is not a standalone referent from the signifier in the process of evocation and how we write or say the word (upright? upside-down? Is it spray painted?)... how we talk about it is tied up with what it is.  

Why it's useful: This forces us to consider whether, in language, it is the ideas themselves, reduced to simple (logical) statements, that are most important in describing what is being told. In a way, Derrida is saying that if you only focus on logic, you drain speech and logic of the character, musicality, and the things that make language and communication worth actually hearing or listening to. This is called Derrida's critique of logocentrism. I like this for my own life because it forces me to consider that to tell a good story, it's not about focusing solely on translating the plot into formal logic, but rather, the ideas and logical connectives become emergent from within a discourse through the act of explaining and gesturing. I think that this holds true for what masters of a discipline or sport would tell you as well: if you focus only on the finish line, you probably won't run a very good race! If you focus only on the degree, you probably won't get a very good education. If you only think about getting the dive over with, ya might not look so graceful in the air...

Key Terms

Differance: Tied to the notion that to mean something, it must differ from other things and defer to what system we use to eff something effable (i.e., “able to be grokked”; it’s really hard to explain what the “eff” in ineffable means[1]). Both difference and the supplement are emergent from Derrida’s consideration of the relationship between speech and writing.

The supplement: tied to the notion of evil, imitation, presence nature, “the light of day,”, inside versus outside, our relationship to signs and referents, our believing that we have a lack, the possibility of a guardrail in our thinking, and getting ahead of ourselves in our thinking. Heidegger also said in Being and Time that we exist as fallen, and as such tend to get ahead of ourselves. I did like the quote “As soon as man comes to life, he is ready to die,” some of the things he says about fallenness, and the anticipation of death serving as an individualizing principle, but that’s about as much as I want to say about that man’s work. A tangential point is the relationship one has to the footnotes of a book gets better and better the more one reads the book because the index and footnotes are getting closer – the first full sentence I understood in German was from Being and Time! So that was cool. There’s also a German dictionary of relevant terms in the HarperCollins edition. In getting ahead of ourselves, we can come to care more about the signs for things than the things themselves, and I think that’s part of what Derrida is saying about perversion. Anways, supplementarity is the play between substance and absence that no metaphysical or ontological concept can comprehend (266). 

Key Quotes: I see p. 180 & 181 as the “heart” of the text

Auto-effection is a universal structure of experience. All living things have the power of an auto-affection. And only a being capable of symbolizing, that is to say of auto-effecting, may let itself be affected by the other in general. Auto-effection is the condition of an experience in general. This possibility – another name for “life” – is a general structure articulated by the history of life, and provides space for complex and hierarchical operations. Auto-affection, the as-for-itself or for-itself – subjectivity – gains in power and in its mastery of the other to the extent that its power of repetition idealizes itself. Here idealization is the movement by which sensory exteriority, that which affects me or serves me as signifier, submits itself to my power of repetition, to what thenceforward appears to me as my spontaneity and escapes me less and less… Conversation is, then, a communication between two absolute origins that, if one may risk this formula, auto-effect reciprocally, repeating as immediate echo the auto-affection produced by the other. Immediacy is here the myth of consciousness… As soon as nonpresence comes to be felt within the voice [vowel] itself – and is at least pre-felt from the very threshold of articulation and diacriticity – writing is in some way fissured in its value (translated in 1974 from Derrida 1967, 180 – 181, – italics added by me!).

Notable WTF Moments from Rousseau

But in the north, where the inhabitants consume a great deal off a barren soil, men, subject to so many needs, are easily irritated. Everything that happens around them disturbs them. As they subsist only through effort, the poorer they are the more firmly they hold to the little they have. To approach them is to make an attempt on their lives. This accounts for their irascible temper, so quick to turn in fury against everything that offends them (Derrida 1967 in Spivak 1974, 244).  

The country is not a matter of indifference in the education [culture] of man; it is only in temperate climes that he comes to be everything he can be. The disadvantages of extreme climes are easily seen. A man is not planted in one place like a tree, to stay there the rest of his life, and to pass from one extreme to another you must travel twice as far as he who starts half-way…. A Frenchman can live in New Guinea or in Lapland, but a negro cannot live in Tornea nor a Samoyed in Benin. It seems also as if the brain were less perfectly organized in the two extremes. Neither the Negroes nor the Laps have the sense of the Europeans. So if I want my pupil to be a citizen of the world I will choose him in the temperate zone, in France for example, rather than elsewhere. In the north with its barren soil men devour much food, in the fertile south they eat little. This produces another difference: the one is industrious, the other contemplative (p. 27; italics added by Derrida) (on p. 242 in Spivak’s translation of Derrida).

Hopefully we have learned by now that if you call another culture primitive or basic you are only expressing your own ignorance of something beyond yourself!

A good quote from Rousseau: “He who imagines nothing senses no-one but himself; he is alone in the midst of humankind” (this is Rousseau on page 203 in Spivak’s translation Of Grammatology).