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Yet More on Logic: Quantification

March 26, 2011 by Seth Paskin 11 Comments

Against both my better judgment and the hue and cry of many, I will continue my semi-informed-by-past-years-of-studying "exposition" of predicate logic which I started here.  If I accomplish nothing else, I will give Burl something to complain about for the next week or so.

In the previous installment, we talked about how syllogistic statements about "all x's" assert the truth of a conditional statement.  "All dogs bark" asserts that for all x's, if x is a dog, then x barks.  Formally expressed, that's:

∀x(Dog(x) –> Barks(x))

or something similar.  It doesn't say anything about whether there actually are any dogs.  Additionally, the 'For all...' symbol - ∀ - doesn't allow you to say anything about only some dogs.  Let us address that issue.

The ∀ symbol is called a quantifier.  In fact, it's called the universal quantifier.  Can you guess why?  Read the Wikipedia entry on quantification if you'd like more detail.  There is another quantifier for when you want to say 'some', which is called the existential quantifier.  Its symbol looks like this -  ∃.  Let's see it in action.

If we take the statement above "All dogs bark" and instead assert that "Some dogs bark," we have to say:

∃x(Dog(x) AND Barks(x))

You don't use the "if-then" arrow here, because you're not saying "There is something and, if it's a dog, then it barks." That sounds like it means the same as "all dogs bark." Instead of "if-then," we use "AND".   Now, this statement is compatible with there being only one dog that barks, or all of the dogs barking, or just some of them. It doesn't tell you, for any given dog you meet, whether it barks or not.

Just as you can assert that a proposition is true and its opposite is false, you can also restate the false proposition negatively to make it true.  For example, if you say that 'All dogs bark' is false, you can also restate that as 'Not all dogs bark' and say that it is true.  This type of negation can be captured using symbolism as well (I'll use the tilde "~" to represent "not"):

∀x(Dog(x) –> Barks(x))      is false

~∀x(Dog(x) –> Barks(x))   is true

So when you say 'not all dogs bark', you are kind of saying that while some might, some definitely don't.  It turns out negating a ∀ (universal) statement is the equivalent of asserting a related ∃ (existential) statement. "Not all dogs bark" is the same as "Some dogs don't bark," or:

~∀x(Dog(x) –> Barks(x)) ...is equivalent to   ∃x(Dog(x) AND ~Barks(x))

To reiterate: the universal statement doesn't actually state the existence of anything, because it's a conditional:  if something is a dog, it will bark.  The negated universal statement which is an existential statement does (implicitly) assert the existence of something.

This is major difference between the two statements that we'll explore further in a future post.

--Initiated by Seth & rescued by Mark from Seth's confusion and drunkenness

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Filed Under: PEL's Notes Tagged With: logic, philosophy blog

Comments

  1. Burl says

    March 29, 2011 at 5:11 am

    Hey Guys,

    I promise to try to refrain from further anti-analytic rants. For a bit more insight into my thought:

    If God is Love and Love is blind and
    Burl is (legally) blind, then Burl is God.

    I must remain a skeptic of logic, since it can abstract fallacy in this argument, whereas in real experience, I cannot!

    I’m very accustomed to no one bothering with anything I say – it sums my life up quite well! It’s a main reason why I left a perfectly cush, tenured associate professorship. My students hated structural engineering and fellow faculty never really appreciated my admonitions that the university is more than a personal money-trough for self-gorging.

    Whereas I knew my dogs at home were likewise predisposed to ignore me, I found their daily companionship far superior in Quality.

    Reply
    • Snorre says

      October 19, 2015 at 9:56 am

      I think that’s wrong. ..

      If God is Love, and Love is Blind… then we can say that God is Blind.

      premis : G is L
      premis : L is B
      conclusion: G is B…

      so far,so good. .. we can loose the ” Love” part from here on.

      To now say that since Burl is Blind, that makes him God is wrong.

      G is B
      Burl is B

      therefore: G is Burl is equal to saying ” since both Beth and Ruth is historians, they are the same person.

      Either I misunderstood you, or you misunderstood syllogistic reasoning. Because thats not how its suppose to be used. ..

      what say u ?????

      Reply
  2. Tom McDonald says

    March 29, 2011 at 10:42 am

    It sounds like Seth is on here to why Kant came to the conclusion that being is not a predicate, amongst other interesting conclusions in the history of philosophy.

    It seems to be in the nature of logic that universals, e.g., “∀x(a(x) –> b(x)),” are necessarily empty of existential content, i.e., that they can only point us to or disclose existential content in being negated, for example “∃x(a(x) AND ~b(x)),” which implies that we must encounter the particular individual “a” to understand whether “b” or “~b” is true of it.

    If this is right, it could be the way by which Anglo analytical formalist philosophy can be led out of “the empty night of the supersensible other-worldly beyond” and brought “into the spiritual daylight of the present.” That is, be brought from fixation on abstracted mechanical rules (scientism) to the actual, existential conditions of reason in the present time.

    Cheers,
    Tom

    Reply
  3. Geoff says

    March 29, 2011 at 5:19 pm

    Burl :
    For a bit more insight into my thought:
    If God is Love and Love is blind and
    Burl is (legally) blind, then Burl is God.
    I must remain a skeptic of logic, since it can abstract fallacy in this argument, whereas in real experience, I cannot!

    Yeah, but natural language is the source of the fallacy in the first place.

    Abstracting a fallacy doesn’t give credence to the fallacy. The logic is simply the structure of the argument – the equivocation is in the language which hardly seems the fault of logic. If we reject the premise the logical form hardly seems to matter.

    Reply
    • Mark Linsenmayer says

      March 29, 2011 at 8:15 pm

      Citing logical argument forms is useful when arguing adversarially, such as on debate team in school or when writing a paper of that same ilk (e.g. contemporary moral problems class in undergrad). When you really want to get clear why you believe what you believe, you spell out the premises explicitly and show how they lead to the conclusion. Of course, then you have to argue for each premise, or argue for it’s being so basic or already agreed upon that it doesn’t require argument in this context. Then you can map the counters to your position by pointing out which premise they run against. It all makes it very organized, for an academic exercise.

      For more casual philosophizing, I think a background in the explicit rules is good; it’s one of the things that keeps your thoughts from going over ground that doesn’t need going over and wasting time, but while linguistic clarifications are often necessary, actually spelling out formulas to check their validity is hardly ever called for, in my limited experience.

      Reply
  4. Burl says

    March 30, 2011 at 6:15 am

    I do appreciate the need to have a good grasp of logic for philosophy work. However, why do so many former mathematicians turned philosopher (which is most of the biggies) nevertheless write stuff that is deemed great, but which others will easily find fallacious in logic?

    I just came across a great little Alfie-byte from _ Modes of Thought_ that seems to capture in English the Hindu ‘tat-twam-asi’ (thou art that) even better than Alan Watts or Ram Das:

    “I am in the room, and the room is an item in my present experience. But my present experience is what I now am.”

    ANW uses this as an exemplification of mutual immanence as in “my soul is in the world just as the world is in my soul.”

    How would logic be applied to his ‘room’ quote?

    (BTW, we can also see this quote is analogous to panenthism.)

    Reply
  5. Seth Paskin says

    March 30, 2011 at 8:51 am

    Tom is right, ultimately I want to talk about the challenge of treating existence as a predicate. Using logical formalism is one way to approach that problem that draws it in stark contrast.
    I’d also note that formal logic is a tool that can be used to supplement philosophical analysis. Logic <> Philosophy. To think that philosophy can only be done through formalism or completely without it are two wrong positions on either end of the same spectrum. Let the tool fit the job: if the problem you are trying to solve requires the use of a tool: use it. If not, don’t.
    –seth

    Reply
  6. Burl says

    March 30, 2011 at 9:27 am

    “I want to talk about the challenge of treating existence as a predicate.”

    Would you elaborate on this?

    Reply
  7. Burl says

    March 30, 2011 at 9:50 am

    I mean, is existence itself the subject to be predicated upon. or do you mean to speak of existence as a plurality of subjects? Is the nature (the stuff of) existence being, becoming, experience, process, material, mental, monistic, dualistic, pluralistic…, and so, are the predicates to be nouns, verbs, adverbs…?

    Reply
    • Seth Paskin says

      March 30, 2011 at 10:51 pm

      Burl–
      The question has been around for a long time and it’s vexing. If I say ‘Burl is retired’, it makes sense that I am predicating of you that you are retired. Same with ‘is smart’, ‘lives in the US’, ‘is writing’, etc. But if I want to say that you exist, I wouldn’t say ‘Burl is existing’. Nor would I say simply ‘Burl is’.

      In fact, I would say instead that ‘Burl exists’. Which sounds like an action, but I don’t really mean it in the same way that I mean ‘Burl writes’ or ‘Burl reads’. It seems like when I say ‘Burl exists’, I am predicating of you that you have the property of existence (so to speak), in the same way that when I say ‘Burl is tall’ I am predicating of you that you have the property of being tall.

      But ‘Burl is…’ implies that you exist. It’s like it is embedded in the “is” and we shouldn’t need to use ‘exists’ as a predicate. Another way to think of it is that one can say that ‘to be’ is always ‘to be something’.

      So is existence something you predicate of things the way you do other properties like size, color, shape, attitude, etc? Or is it something else? Frege & Russell famously tackled this issue and the logical formalism I’m trying to build is to help draw out the distinction and make it more clear.
      –seth

      Reply
  8. Anh-vu Doan says

    March 30, 2011 at 3:15 pm

    Noo! Keep talking about logic! I love this stuff. I think it’s good to learn logic, because it clarifies the laws of thought, it delimits the boundaries of good thinking. That is, as we can’t help but to reason about things, it’s incumbent upon us to study HOW we reason in a systematic manner. This is the purpose of logic as I understand it.

    Reply

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