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Some more on logic

March 20, 2011 by Seth Paskin 12 Comments

So Matt Teichman was kind enough to post a primer on basic logic, showing with syllogisms how informal logical inference was turned into formal notation by Frege and thus predicate calculus was born.  There is a wealth of stuff to learn about the predicate calculus and many serious logicians (as well as frustrated mathematicians) have developed and extended systems in a number of different ways.

One of the things that was interesting about developments before and around the time we were in grad school was how people got wrapped around the axle on the implications of formalism for 'the real world'.  Mark pointed this out in his post about 'The True' and we discussed it when talking about Frege:  what kind of object was Truth in his ontology and why didn't he seem to care that much?

What happened in the 20th century was that you had people that continued with the formal endeavor without regard to ontology, metaphysics and epistemology.  You also had people that would call out the 'philosophical' consequences of formal systems, which some of the formalists cared about and some didn't.  Then you had people like David Lewis who thought that the fact that we could create formal systems implied that the corresponding ontologies must exist.  Witness the birth of possible world semantics!

So how did it get to that?  Basically, predicate calculus requires you to abstract from the concrete thing being spoken of and make a 'predicate' or property of whatever identifies it.  So focusing on an object:

A person -> a thing that is a person ->an x that is a person -> an x that has the property of 'personness' ->Person(x)

So "Person(x)" means something like 'an x that has the property of 'personness' (or 'being a person').  It's called Predicate Calculus because we are predicating of x that it is a person.  This notation of Predicate(x) is pretty common.  When the predicate is abbreviated to a letter like P, often the parenthesis are left off, e.g. Px.

So let's return to Matt, who left his exposition at the formal representation of a syllogism.  Let's look at his formalism of the first assertion:

All people drink water = ∀x(Person(x) –> DrinksWater(x))

We know that 'Person(x)' means 'x is a person' and 'DrinksWater(x)' means 'x drinks water'.  What we need to understand is the the upside down "A" - ∀ and the arrow "->".   Here's the deal:

  • The ∀ represents "All...".  Often when we speak of propositions, we will say 'For all...'
  • The -> means 'if...then...'.  That is, whatever comes before the -> is the 'if...' part and what comes after is the 'then...' part.  E.g.  'a -> b' = 'if a then b'.  An 'if...then...' statement is called a conditional statement.

So one way to read the right half of Matt's equation is 'For all x's, if x is a person then x drinks water.'  What we are really saying with the proposition is that it is true that if x is a person, then x drinks water.  The truth of the assertion is expressed indirectly in the "For all x's" part.  We are basically saying that it's true for every x, therefore it must be true for any particular x.

Note that what we are asserting with the proposition is the truth of the 'if...then...' statement, not that any such x that is a person exists.  Note too that the structure of the proposition doesn't tell you whether the assertion is valid.  Matt pointed this out - the proposition asserts something as true; whether it actually is true needs to be verified.  Formally,∀x(Person(x) –> DrinksWater(x)) has the same structure as :

∀x(Unicorn(x) –> Onlyallowsvirginstoride(x))

Both propositions are asserting the truth of conditional statements.  If you are a person, you drink water.  If you are a unicorn, you only allow virgins to ride you.  In both cases, the soundness of the statement would need to be established through some kind of empirical process, probably starting with establishing the existence of either persons or unicorns.  So what if you want to assert something about existence or soundness?  Well, things get tricky, which we'll explore in another post shortly.

--seth  [ed. note:  changed "validity" to "soundness" 3/26/11]

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Comments

  1. Geoff says

    March 20, 2011 at 10:29 pm

    “In both cases, the validity of the statement would need to be established through some kind of empirical process”

    Being relatively new to any knind formal logic, a question.

    I thought that ‘validity’ was something to do with the logical form, and ‘soundness’ was what you got after empirical verification.

    So
    ∀x(Person(x) –> DrinksWater(x)) is valid and sound

    while
    ∀x(Unicorn(x) –> Onlyallowsvirginstoride(x)) is valid but not sound

    Or doess predicate logic use different descriptions for these properties?

    Reply
  2. Tom McDonald says

    March 20, 2011 at 10:44 pm

    So I would infer then that

    ( ∀x(Unicorn(x) ⇒ Onlyallowsvirginstoride(x)) )
    ⇒
    ( ∀x(Virgin(x) ⇒ OnlyAllowedToRideUnicornOneTime(x)) )

    Though perhaps this depends on the actual referents of some of the terms ; )

    I’m a little confused about “validity” though… I thought validity was a formal property of inference depending on the consistent use of terms in a proposition, and truth or falsity was a matter of empirical verification. At least the latter is what I recall pickup up from Wittgenstein’s Tractatus.

    Reply
  3. Wes Alwan says

    March 21, 2011 at 2:04 am

    Yeah validity is independent of the truth (or empirical verification) of premises.

    Reply
  4. Burl says

    March 21, 2011 at 4:00 am

    Logic is the formalization of a system that in its entirety represents the fallacy of misplaced concreteness – mistaking that which is abstracted of reality with reality.

    Let it die with the losers who bought into it to prop up tenured careers promulgating dessicated verbiage in lieu of wisdom – you know, the ugly shit you guys had the good sense to run away from in order to think like human creatures.

    It is far more important for a proposition to be interesting than that it be provably true, as Whitehead would say. Nature is organic, and the objectivity that logic abstracts from it, and then substitutes for it, is vacuous actuality – quality-less. At its best, it’s the stuff of a tightly-written computer program.

    Just my opinion, of course.

    Reply
    • Anh-vu Doan says

      March 22, 2011 at 12:39 am

      I would disagree! I feel that logic reflects how the world must be – what is valid logically reflects how certain processes in nature must actually BE. That the reason why valid statements in logic are valid is because something in the way the universe is structured compels the to be so. We see that even animals obey logical reasoning and inference – this is why a dog chasing a rabbit down a forked path rules out the path that does not have the scent of a rabbit.

      I suppose the difference is whether one wants to understand true things about the way the world really works, or if one is interested in amusing oneself with speculative discussions.

      Reply
  5. Seth Paskin says

    March 21, 2011 at 6:23 am

    Yes, that should have read ‘soundness’ instead of ‘validity’, I’ll edit the original post when I get a chance.

    Reply
  6. Tom McDonald says

    March 21, 2011 at 9:29 am

    @ Burl:

    Aye, ye speaketh words after my own heart! Although I would make the dialectical point that if we did not have a negative moment of “abstraction” (such as formalists always give us) then we could not appreciate the return of thought to reality in a more self-grasping fashion. On this point, Hilary Putnam has noted that the advent and growth of computer programming may actually be helping to diminish the formalist desire to view human thought on the model of the computer, i.e., the failure of symbolic logic in artificial intelligence helps to turn our own thought toward the organic and the whole. I hope he is right, but in any case this is one of the most interesting areas for contemporary philosophical inquiry.

    Cheers,
    Tom

    Reply
  7. Burl says

    March 21, 2011 at 12:08 pm

    Tom

    I have always avoided logical symbols – sane for subtitled movies.

    Lo and behold, no sooner than I posted my opinion above did I read Chapter 7 of Rescher’s _Process Metaphysics_ wherein he spends the whole time saying the same thing. I am brainwashed on process!

    Rescher is a refreshing turn a bit away from ANW. David Buchanan would do well to read Resher’s book to see that Pirsig’s levels of Quality pattern, if seen as different types of process, find great support.

    Reply
  8. Burl says

    March 21, 2011 at 2:27 pm

    For fans of analistic philosophy and things Quine, the appendix to Rescher’s above noted book is devoted to developing a logical analysis termed ‘semantics of process’ which can overcome some problems with the standard semantics of objects-and-attributes. It replaces adjectival attributes with adverbs. He shows how to prove non-existents (namely, Pegasus) exist.

    Being full of symbols, I sjipped it all.

    Reply
  9. Burl says

    March 22, 2011 at 4:13 am

    *I suppose the difference is whether one wants to understand true things about the way the world really works, or if one is interested in amusing oneself with speculative discussions.*

    The logical positivists and all of analytic philodophy, and thus all of the tenured philosophocologists of the last century said NO to metaphysics but yes to Aristotelian logic. This logic is based on a substance metaphysics that way overthrown just as these brilliant men set about building their vapid tenured lives of word-chess.

    Reply
    • Alex says

      March 26, 2011 at 1:21 pm

      Burl, what you say about analytic philosophy incoherent (perhaps studying formal logic would help you clarify what you are saying) and ill-informed. First, the research programme of analytic philosophy came out of a rejection of Aristotelian logic for the more powerful quantificational logic of Frege. But that is really only a minor factual point. What is more frustrating is your idea that analytic philosophy as it is practiced today is basically a continuation of the anti-metaphysical logical positivists. However, analytic philosophers see as one of the greatest triumphs of the tradition the wholesale rejection of logical positivism by even the people who were hardcore logical positivists.

      In any case, sorry for the somewhat hostile tone. I too dislike the type of philosophy you bitch about. It’s just that it isn’t what most “analytic” philosophers do. If you care to find out how vague the term “analytic philosophy” really is, and how varying their opinions on what philosophy ought to be are, this article by the very non-boring Jerry Fodor is a great resource:

      http://www.lrb.co.uk/v26/n20/jerry-fodor/waters-water-everywhere

      Reply
  10. Alex says

    March 26, 2011 at 1:34 pm

    By the way, I would say that the opinions Fodor expresses aren’t in the minority as he says that they are. The majority of anglo-american philosophers who specialize in just about anything besides metaphysics, epistemology or perhaps philosophy of language would be sympathetic with Fodor’s views, and this group of people would very likely exceed 50% of the people who fall under the umbrella of analytic philosophy.

    Reply

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