{"id":754,"date":"2009-03-17T19:27:03","date_gmt":"2009-03-17T19:27:03","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2009\/03\/17\/mr-spock-is-not-logical-book-draft-excerpt\/"},"modified":"2009-03-17T19:27:03","modified_gmt":"2009-03-17T19:27:03","slug":"mr-spock-is-not-logical-book-draft-excerpt","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2009\/03\/17\/mr-spock-is-not-logical-book-draft-excerpt\/","title":{"rendered":"Mr. Spock is Not Logical (book draft excerpt)"},"content":{"rendered":"

As I mentioned, I’ll be posting drafts of various sections of my book here on the blog. This is a rough draft of the introduction to a chapter on logic. I would be extremely greatful for comments, critiques, and corrections.<\/em><\/p>\n

I’m a big science fiction fan. In fact, my whole family is pretty
\nmuch a gaggle of sci-fi geeks. When I was growing up, every
\nSaturday at 6pm was Star Trek time, when a local channel show
\nre-runs of the original series. When Saturday came around, we
\nalways made sure we were home by 6, and we’d all gather in front of
\nthe TV to watch Trek. But there’s one one thing about Star Trek for
\nwhich I’ll never forgive Gene Roddenberry or Star Trek:
\n“Logic”. As in, Mr. Spock saying “But that would
\nnot be logical.”.\n<\/p>\n

The reason that this bugs me so much is because it’s taught a
\nhuge number of people that “logical” means the same
\nthing as “reasonable”. Almost every time I hear anyone
\nsay that something is logical, they don’t mean that it’s logical –
\nin fact, they mean something almost exactly opposite – that it
\nseems correct based on intuition and common sense.\n<\/p>\n

If you’re being strict about the definition, then saying that
\nsomething is logical by itself is an almost meaningless
\nstatement. Because what it means for some statement to be
\nlogical is really that that statement is inferable
\nfrom a set of axioms in some formal reasoning system. If you don’t
\nknow what formal system, and you don’t know what axioms, then the
\nstatement that something is logical is absolutely meaningless. And
\neven if you do know what system and what axioms you’re talking
\nabout, the things that people often call “logical” are
\nnot things that are actually inferable from the axioms.\n<\/p>\n

<\/p>\n

Logic, in the sense that we generally talk about it, isn’t
\nreally one thing. Logic is a name for the general family of formal
\nproof systems with inference rules. There are many logics, and a
\nstatement that is a valid inference (meaning that it is logical)
\nin one system may not be valid in another. To give you a very
\nsimple example, think about a statement like “The house on the corner
\nis red”. Most people would say that it’s logical<\/em>
\nthat that statement is either true or false: after all, either the house
\nis<\/em> red, or the house isn’t<\/em> red. In fact, most
\npeople would agree that the statement “Either the house is red, or it
\nisn’t red” must be<\/em> true.\n<\/p>\n

In the most common logic, called predicate
\nlogic<\/em>, that’s absolutely correct. The original
\nstatement is either true or false; the statement with an “or” in
\nit must be true. But in another common logic, called
\nintuitionistic<\/em> logic, that’s not<\/em>
\ntrue. In intuitionistic logic, there are three possible truth
\nvalues: something can be true (which means that there is a proof
\nthat it is true); something can be false (which means that there
\nis a proof that it is false); and something can be unknown so far
\n(which means that there’s no proof either way).\n<\/p>\n

In addition to having different ways of defining what’s true
\nor provable, different logics can describe different things. Our
\ngood old familiar predicate logic is awful at describing things
\ninvolving time – there’s really no good particularly good way in
\npredicate logic to say “I’m always hungry at 6pm”. But
\nthere are other logics, called temporal
\nlogics<\/em> which are designed specifically for making
\nstatements about time. We’ll look at temporal logics later. For
\nnow, we’ll stick with simple familiar logics.<\/p>\n<\/p>\n

So What is Logic?<\/h3>\n

A logic is a formal symbolic system, which consists of:<\/p>\n

    \n
  1. A set of atoms<\/em>, which are the objects
    \nthat the logic can reason about.<\/li>\n
  2. A set of rules describing how you can form statements
    \nin the logic (the syntax<\/em> of the logic).<\/li>\n
  3. A system of inference rules<\/em>
    \nfor mechanically discovering
    \nnew true statements using known true statements. <\/p>\n<\/li>\n
  4. A model<\/em> which describes how the atoms and predicates
    \nin the logic map onto a real, consistent set of objects and properties.<\/li>\n<\/ol>\n

    The key part of that definition is the
    \nmechanical<\/em> nature of inference. What logic does is
    \nprovide a completely mechanical system for determining the truth
    \nor falsehood of a statement given a set of known truths. In
    \nlogic, you don’t need to know what something means in order to
    \ndetermine if it’s true!<\/em>. As long as the logic has<\/em> a
    \nvalid model, you don’t need to know what the model is<\/em> to
    \nbe able to do valid reasoning in that logic.<\/p>\n

    The easiest way to get a sense of how that can possibly work
    \nis to use an example. We’ll start with one simple logic, and show
    \nhow it can be used in a mechanical fashion to deduce true statements
    \n– without knowing what those statements mean. For now, we won’t even
    \nreally define the logic formally, but instead just rely on intuition.
    \nMost arguments that we hear day to day are based informally on a logic
    \ncalled predicate logic<\/em>; to be more specific, they’re
    \nmostly first order predicate logic.<\/p>\n

    In predicate logic, we’ve got a collection of objects which we
    \ncan reason about, which we usually call
    \natoms<\/em>. To say anything about objects, we use
    \npredicates. Predicates are statements that assert some property
    \nabout on object, or some relationship between objects. For
    \nexample, if I had a pet dog named Joe, we could make statements
    \nabout him like Dog(\"Joe\")<\/code>,
    \nwhich would say “Joe is a dog.”. Or we could form statements about
    \nspecific relationships: Likes(\"Joe\",
    \n\"Rex\")<\/code> is a logical way of saying “Joe
    \nlikes Rex”. <\/p>\n

    We can also form general statements. For example, if Joe likes
    \nall other dogs, we can say that in logic: (∀x)
    \nDog(x) ⇒ Likes(\"Joe\", x)<\/code>. The
    \nupside down “A” stands for “for all”; the statement
    \nsays “For all x, if x is a dog, then Joe likes x.”<\/p>\n

    Inference Rules<\/h3>\n

    Where things get interesting is the inference rules. Inference
    \nrules describe how to perform reasoning in the logic – which is another
    \nway of saying that they describe how the logic can allow you to figure
    \nout what’s true or false, based on reasoning starting from an initial
    \nset of given facts.<\/p>\n

    Inference rules are usually written as sequents<\/em>,
    \nwhich we’ll get to in another section; for now, we’ll
    \nstick with informal descriptions.<\/p>\n

    The simplest inference rules allow you to just manipulate
    \nsimple statements. For example, if you know A ∧
    \nB<\/code> is true, then you know A<\/code> is
    \ntrue.<\/p>\n

    Another group of inference rules combine similar
    \nstatements to derive new facts. For example, the most famous rule
    \nof logic is called modus ponens<\/em>: if you know
    \nthat the statement P ⇒ Q<\/code> is true,
    \nand you also know that P<\/code> is true, then you
    \ncan infer that Q<\/code> must be true.<\/p>\n

    More interesting rules allow you to do things like work from
    \nthe general to the specific: if you know that ∀
    \nx: P(x)<\/code>, and “A” is an atom, then you can
    \ninfer P(\"A\")<\/code>.<\/p>\n

    Yet other rules allow you to transform statements. For example,
    \nif you know that ∃x : P(x)<\/code>, then
    \nyou can infer that ¬∀x: ¬P(x)<\/code>.\n<\/p>\n

    With the rules we’ve looked at so far, we can build an example of
    \nwhat I meant by totally mechanical inference.<\/p>\n

    Let’s suppose we have a bunch of atoms, “a”,
    \n“b”, “c”, …, and two predicates, P
    \nand Q.<\/p>\n

    We know a few simple facts:<\/p>\n

      \n
    • P(\"a\", \"b\")<\/code><\/li>\n
    • P(\"b\", \"c\")<\/code><\/li>\n
    • ∀x, y, z: P(x,y) ∧ P(y,z)
      \n⇒ Q(x,z)<\/code><\/li>\n<\/ul>\n

      What can we infer using this? Using a general-to-specific
      \ninference, we can say P(\"a\",
      \n\"b\") ∧ P(\"a\", \"b\")
      \n⇒ Q(\"a\", \"c\")<\/code><\/p>\n

      Then, we can combine P(\"a\",
      \n\"b\")<\/code> and P(\"b\",
      \n\"c\")<\/code> to infer
      \nP(\"a\", \"b\") ∧
      \nP(\"b\", \"c\")<\/code>. (Remember, we’re
      \nbeing totally mechanical, so if we want to use the implication, we
      \nneed to exactly match its left-hand side, so we need to do an
      \ninference to get the “and” statement.)<\/p>\n

      Finally, we can now use modus ponens to infer
      \nQ(\"a\",\"c\")<\/code>. We
      \nhave no idea<\/em> what the atoms a, b, and c are; we have
      \nno idea what the predicates P and Q mean. But we’ve been able to
      \ninfer true statements.<\/p>\n

      So what do the statements mean? That depends on the model. For a given set of symbolic statements, you can use more than one model – so long as each model is valid, the meanings of the inferences will be valid in all models assigned to the statements. (We’ll talk more about models in section …) In this case, we could use several different models; I’ll show two examples:<\/p>\n

        \n
      1. “a” could be 1, “b” could be 2,
        \nand “c” could be 3, with P(x,y) meaning “x is
        \n1 plus y”, and Q(x,y) meaning “x is 2 plus y”.
        \nThen we would have used the fact that 2 is 1+1 and 3 is 1+2 to infer
        \nthat 3 is 2+1.,\/p><\/li>\n
      2. “a” could be my father, Irving;
        \n“b” could be me, and “c” could be my
        \nson Aaron, with P(x,y) meaning “x is the father of
        \ny” and Q(x,y) meaning “x is the grandfather of
        \ny”. Then we would have used the fact that Irving is my
        \nfather, and I am Aaron’s father to infer that Irving is
        \nAaron’s grandfather.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

        As I mentioned, I’ll be posting drafts of various sections of my book here on the blog. This is a rough draft of the introduction to a chapter on logic. I would be extremely greatful for comments, critiques, and corrections. I’m a big science fiction fan. In fact, my whole family is pretty much a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[33],"tags":[193,221],"class_list":["post-754","post","type-post","status-publish","format-standard","hentry","category-logic","tag-logic-2","tag-rant"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-ca","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/754","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=754"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/754\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=754"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=754"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=754"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}