This is another great basics topic, and it's also one of my pet peeves. In general, I'm a big
science fiction fan, and I grew up in a house where every saturday at 6pm, we all gathered in front
of the TV to watch Star Trek. But one thing which Star Trek contributed to our vocabulary, for which
I will never forgive Gene Rodenberry, is "Logic". As in, Mr. Spock saying "But that would not be
logical.".
The reason that this bugs me so much is because it's taught a huge number of people that
"logical" means the same thing as "reasonable". Almost every time I hear anyone say
that something is logical, they don't mean that it's logical - in fact, they mean something
almost exactly opposite - that it seems correct based on intuition and common sense.
If you're being strict about the definition, then saying that something is logical by itself is an almost meaningless statement. Because what it means for some statement to be "logical" is really
that that statement is inferable from a set of axioms in some formal reasoning system. If you don't know what formal system, and you don't know what axioms, then the statement that something is logical is absolutely meaningless. And even if you do know what system and
what axioms you're talking about, the things that people often call "logical" are not things that are actually inferable from the axioms.
Logic, in the sense that we generally talk about it, isn't really one thing. Logic is a
name for the general family of formal proof systems with inference rules. There are many
logics, and a statement that is a valid inference (is logical) in one system may not be valid in
another. To give you a very simple example, most people are familiar with the fact that in logic, if
you have a statement "A", then either the statement "A or not A" must be true. In the most common
simple logic, called propositional logic, that's a tautology - that is, a statement which is
always true by definition. But in another common and useful logic - intuitionistic logic - "A or not A" is not necessarily true. You cannot infer anything about whether it's true or false without proving whether A is true or false.
To give another example: the most common logic that we use in arguments is called first order predicate logic (FOPL). FOPL is a very useful logic for things like geometric proofs. But it's
absolutely awful at talking about time. In FOPL, there's no good way to say something
like "I won't be hungry until 6pm tonight." that really captures the temporal meaning of that statement. But there are several kinds of logic that are very good at that kind of statement - but they're not particularly useful for the kinds of things that FOPL is good at.
So what is a logic? A typical formulation would be that a logic is a formal symbolic
system which consists of:
- A way of writing a set of statements (the syntax of the logic); and
- A system of rules for performing mechanical inferences over those statements.
The easiest way to get a sense of this is to look at one familiar logic: the first order predicate logic (FOPL). The first order predicate logic is the most common logic that we really use; it's the one that we're generally using implicitly when we write things like proofs in geometry.
Logicians tend to use a very strange method of describing the syntax of logical statements. I'm going to ignore that, and just walk through the syntax informally. FOPL has five kinds of basic things that are put together to form statements. As I go through the syntax, I'll give some examples based on reasoning about my family.
- A constant is a particular object, number, or value which can be reasoned
about using the logic. In reasoning about my family, the constants will be the names
of members of my family, the places we live, and so on. I'll write constants as either
numbers, or quoted words. - A variable is a symbol which represents a value. Variables can be used in
the logic to reason about things like universal properties - if every object has
a property (like, for example, every person has a father), there's a way of using
a variable to say that in the logic. - A predicate is something which allows you to make statements about
objects and variables. A predicate is written as an uppercase identifier,
with the objects it's talking about following inside parens. For example, I
can say that my father is Irving using a predicate named "Father":Father("Irving", "Mark"). - Quantifiers are things that introduce new variables. For a statement to be valid
every variable in that statement must have been introduced by a quantifier. There
are two quantifiers in FOPL: ∀ (for all, the universal quantifier, which is
used to make statements about all possible constants); and ∃ (there exists, the existential quantifier, which is used to make statements that there is some constant
for which a statement is true). - An operator is something that modifies or connects sentence(s). There are five
operators in FOPL. Four of them connect pairs of statements: (A ∧ B (and), A ∨ B (or), A ⇒ B (implies), A ⇔ B (if and only of). The fifth one negates a statement: ¬ A.
The meanings of the statements are:
- Predicate Statement
- A predicate with its parameters filled in with either constants or variables.
- And statement
- Two sentences joined by ∧.
A ∧ Bis true if/f both A and B are true. - Or statement
- Two sentences joined by ∨.
A ∨ Bis true if/f either A or B is true. - Implication statement
- Two sentences joined by ⇒.
A ⇒ Bis true if/f when A is true, B is also true, and when B is false, A is also false. - If/f statement
- Two sentences joined by ⇔.
A ⇔ Bis true if/f(A ⇒ B) ∧ B ⇒ A)is true. - Universal Statement
- A sentence preceeded by the universal quantifier and a variable:
∀x:A(x). A
universal statement is true if any constant substituted for the variable will result
in a true statement. - Existential Statement
- A sentence preceeded by the existential quantifier and a variable:
∃x:A(x). An
existential statement is true if there is at least one constant that can be substituted for the variable that will result in a true statement - Parenthesized Statement
- Any statement surrounded by parens. The only meaning of parens is grouping.
The meanings of the different statements can be briefly described as follows:
- Each constant represents some specific entity or object which
the logic is going to be reasoned about. So, for example, if I wanted
to do reasoning about my family, the atoms would be me, my wife, my
children, etc. - A predicate statement expresses a property of the atoms
that are its parameters. Continuing with the example of my family, I
could write statements likeFather("Mark","Aaron"),Father("Mark","Rebecca"),Spouse("Mark","Jennifer"). - ∧ statements combine two statements; they're true when both of the
member statements are true.Spouse("Mark","Jennifer") ∧ Father("Mark","Aaron") ∧ Mother("Jennifer","Aaron"). - The ∨ connector works in basically the same way as ∧, except that it's
true when either of its component statements are true.Father("Mark","Aaron") ∨ Father("Jennifer","Aaron") - ¬ is logical negation: ¬X is true when X is false.
¬Father("Jennifer","Aaron"). - ⇒ is an implication statement:
A ⇒ Bmeans that if A is true, then B must be true; if B is false, then A must also be false. (Note the reversal there - if A is false, it says nothing about whether or not B is true, and if B is true, it says nothing about whether or not A is true.) For example,Spouse("Mark","Jennifer") ⇒ Spouse("Jennifer","Mark")(If Mark is Jennifer's spouse, then Jennifer is Mark's spouse.) - ∀ and ∃ statements are where it gets interesting. ∀ is read "For all", and ∃ is read "there exists". For example,
∀c : (∃p : Father(p,c))(For every person, there is a person who
is their father.);∃f: Father("Mark",f)(There is someone whose father is Mark.)
What I've gone through so far is not yet a logic. It's just a language for writing
statements. What makes it into a logic is the addition of inference rules, which
give you a way of using statements that are known to be true, and using them to
infer other true statements. I'm not going to go through the entire set of inference
rules allowed in FOPL in detail, but I'll give you a couple of examples, followed by the full list of rules.
- If we know that
P(x) ∧ Q(x)is true, then we can infer thatP(x)must be
true. - If we know that
P(x) ⇒ Q(x)is true, and we also knowP(x)is true, then we can infer thatQ(x)is also true. - If we know that
∀x: P(x)is true, and"a"is a constant, then we can infer thatP("a")is true. - If
xis a constant, and we know thatP("a")is true, then
we can infer that&exists;x:P(x)is true.
The rules are divided into two groups. One is a set of equivalences - if you know something on one side of the ≡ sign, then you can infer whatever is on the other side. The second set of rules is implications: if know you know the left side, then you can infer the right.
The
equivalence rules are:
- ¬∀x:P(x) ≡ ∃x:¬P(x)
- ¬∃x:P(x) ≡ ∀x:¬P(x)
- ∀x:(∀y: P(x,y)) ≡ ∀y:(∀x:P(x,y))
- ∃x:(∃y: P(x,y)) ≡ ∃y:(∃x:P(x,y))
- ∀x:P(x) ∧ ∀x:Q(x) ≡ ∀x:P(x)∧Q(x)
- ∃x:P(x) ∨ ∃x:Q(x) ≡ ∃x:P(x)∨Q(x)
And the implication rules are:
- ∃x : ∀y: P(x,y) → ∀y : ∃x: P(x,y)
- ∀x: P(x) ∨ ∀x: Q(x) → ∀x: P(x) ∨ Q(x)
- ∃x:(P(x) ∧ Q(x)) → ∃x:P(x) ∧ ∃x:Q(x)
- ∃x:P(x) ∧ ∀x:Q(x) → ∃x:(P(x) ∧ Q(x))
- ∀x:P(x) → P(c) (where c is a constant)
- P(c) → ∃x:P(x) (where c is a constant, and x is not an unquantified
variable in P(c))
To reason with a logic, you start with a set of axioms - that is, a set of statements that you know are true even though you don't have a proof. Given those axioms, we say that a statement can be proven if there is
some way of applying the inference rules to generate the statement.
So, once again with an example from my family. Here's a set of axioms about
my family.
1: Father("Mark","Rebecca")
2: Mother("Jennifer","Rebecca")
3: Father("Irving","Mark")
4: Mother("Gail","Mark")
5: Father("Robert", "Irving")
6: Mother("Anna", "Irving")
7: ∀a, ∀b:(Father(a,b) ∨ Mother(a,b)) ⇒ Parent(a,b)
8: ∀g,∀c : (∃p : Parent(g,p) ∧ Parent(p,c)) ⇒ Grandparent(g, c)
Now, suppose we want to prove that Irving is Rebecca's grandparent.
- Since we know by statement 1 that
Father("Mark","Rebecca"), we can inferParent("Mark","Rebecca"). We'll call this inference I1. - Since we know by statement 3 that
Father("Irving","Mark"), we can infer
Parent("Irving","Mark"). We'll call this inference I2. - Since we know by I1 and I2 that
Parent(Irving,Mark)andParent(Mark,Rebecca), we can inferParent(Irving,Mark)∧Parent(Mark,Rebecca). We'll call this inference I3. - Since by I3, we know
Parent(Irving,Mark)∧Parent(Mark,Rebecca), using
statement 8, we can inferGrandparent(Irving,Rebecca)
That chain of inferences is a proof in the first order predicate logic. A very important thing to notice is that the proof is entirely symbolic: we don't need to know what the atoms represent, or what the predicates mean! The inference process
in logic is purely symbolic, and can be done with absolutely no clue at all about what the statements that you're proving mean. It's all a mechanical process
of working from the premises using the inference rules. Given the right set of premises,
you can prove almost any statement; given a choice of both logics and premises, you can prove absolutely any statement.
So when someone says, a la Mr. Spock, that something is logical, the correct thing to do is to whack them in the head with a logic textbook for saying something nonsensical.
- Log in to post comments
It is a great post! People use the words logic, reason and rational all interchangeably.
Your explaination about logic and various sytems of logic is very helpful. Maybe I suggest you to write about the differences between logic, reason and rationality?
Also, exploring the connection between various logic systems and computer languages (or theories) will be just great!
From Dr. Who, Destiny of the Daleks (1979, Tom Baker's era)
The Doctor: All elephants are pink. Nellie is an elephant. Therefore Nellie is pink. Logical?
Davros: Perfectly.
The Doctor: Do you know what a human would say to that?
Side character: ...but...elephants *aren't* pink.
Davros: Bah. Humans have no concept of logic.
"A robot is logical but not reasonable," said Asimov in The Caves of Steel.
Your contrasting propositional logic with intuitionistic logic is idiosyncratic, somewhat akin to contrasting compact cars with red cars. Propositional logics are those that ignore the internal structure of propositions except for how they are built out of other propositions using logical connectives. Intuitionistic logics are those that stem from the philosophy of L.E. Brouwer. There are intuitionistic propositional logics and classical propositional logics, intuitionistic predicate logics and classical predicate logics, etc.
Star Trek at 6:00 in my household, too. Except, given that I was born in 1984, it was TNG. I'll always remember that, because my little brother and I wouldn't turn on the TV until our parents got home. We couldn't have been any older than 4 and 6 or 5 and 7.
So we ended up missing the beginning of quite a few episodes, only to see the middle and the end. But that's OK! The credits in TNG went on and on for about 10 minutes into the show, so we knew that we hadn't really missed all that much if we turned on the show and the credits were still at the bottom of the screen.
Also, the best characters star trek ever created were Data and Spock. Iconic.
My main beef with teaching logic is that it's done in geometry class -- there should be minimal proofs in high school math for the mean student (brainiacs are a different matter). They should teach logic in English class, since most chains of reasoning the mean person is most likely to encounter, as well as those they're mostly to construct themselves, are verbal and have zero to do with math.
English composition classes don't care about logic, reason, etc. -- as long as you express yourself well, you're fine. Who cares if it makes any sense?
Agnostic:
I agree with you absolutely 100%. I suffered and struggled my way through high school geometry, because they didn't teach logic - they just expected that we would be able to recognize a proof. I just absolutely could not tell when I'd written a good proof. I hated it, and it was nearly enough to put me off math!
Then, after I got to college, just for the fun of it, I signed up for a logic course, and it was amazing - totally eye-opening. Suddenly, the things that had seemed to be totally beyond me made perfect sense.
It's infuriating to me that we continually expect students in math, in english, in compositional writing, to understand logic, but we never bother to actually teach it to them.
Basic logic - just enough basic propositional and predicate logic to be able to understand proofs, and distinguish between valid arguments and invalid arguments - should be part of very basic math - no later than 7th or 8th grade. It's not that hard, and that's the age where we expect them to start using it - we just don't bother to teach it.
In follow-up to the first (Anonbymnous) comment: what are good answers to these questions:
(1) Why should I be logical?
(2) Why should I be reasonable?
(3) Why should I be rational?
I feel you've gone over this before, but I'm feel I'm still struggling to understand.
You say 'Given the right set of premises, you can prove almost any statement; given a choice of both logics and premises, you can prove absolutely any statement.'
On what level does Godel's Theorem enter into this?
Great post.
Even though Spock wasn't really logical he was a TV hero for those of us who enjoy(ed) maths & science. The only one, unless you include the robot from Lost in Space :o)
Thanks again for the best blog on scienceblogs.
--------------------------
To: Jonathan Vos Post
To answer your questions don't we have to agree a common meaning for the words?
Also, I suspect, that even if we agree a common meaning what is reasonable and rational for me will often conflict with other peoples reasonable and rational.
Godel's Theorem says that given any sufficiently complex set of axioms (and the logic system they're in) there will be some statements which are true, but which cannot be proven. I believe that the cutoff for being "sufficiently complex" is being able to express arithmetic, but it's been a while since I worked with this. The last time I dealt with a formal expression of arithmetic we couldn't, for example, prove that a + b = b + a. That could be added as another axiom, but then there would just be some other statement which couldn't be proven.
The reason that Mark's statement about being able to prove anything given a choice of axioms and logics is still true is that he didn't say that he was able to prove everything all at once. By first choosing a logic which is able to express the statement you wish to prove, it shouldn't be hard to pick a set of premises which let you prove it. In fact, it's trivially easy if you let yourself add the statement itself as an axiom, but thats no fun.
Godel's Theorem becomes a problem when you try to prove everything from one set of axioms, like Russell and Whitehead were trying to do.
Godel's Theorem says that given any sufficiently complex set of axioms (and the logic system they're in) there will be some statements which are true, but which cannot be proven.
I don't mean to go after you, but I hate when people say this about Godel's Incompleteness Theorem. What is this "true"? What is produced is a statement which is consistent but not provable. To evaluate its truth requires a model, and Incompleteness says that there will be multiple models of the axioms. The statement will be true in some and false in others.
To fully place Antendren's comment in the contexts of Mathematical Logic AND common usage, requires an explanation of 5 different domains of usage of "truth" that each have their own paradigms, standards of proof, protocols, and history:
(1) Axiomatic Proof (from Euclid through Godel and beyond);
(2) Empirical Proof (Scientific Method);
(3) PoliticoLegal (O.J. Simpson was "Not Guilty" by Criminal Law; elected politician claims mandate from constituency);
(4) Aesthetic Truth (Symphony, painting, poem is beautiful or ugly to you regardless of critics and other audiences);
(5) Revealed Truth (religious/spiritual/paranormal experience subjectively true to you, incommuncable to others).
I believe that many of the problems of the world come from people inappropriately using the assumptions of one of these in the context of another. This occors with, for instance, attempts to leglislate the value of Pi; to reject axiomatic proof as not verifiable in the "real world"; to prove that a work of art is ugly or should be censored; to legislate that the United States is "A Christian Nation"; to ban the teaching of Evolution in a public school system.
Scientists and Mathematicians can be elected heads of state (we have a Ph.D. Physicist in Congress now, by the way); poets and novelists playwrights can be heads of state, as with Mao Tse-Tung, Benjamin Disraeli, Václav Havel, but do not rewrite the nation as if it were a poem, play, novel, science paper, or axiomatic proof.
Leaders in each of the 5 domains can be dangerous if they enter another domain with hammer in hand, seking the familar nails. It is generally a mistake, albeit often made by media, to assume that a Nobel laureate, film celebrity, sports hero, spiritual leader, or regent is magically qualified to solve probelsm in another domain. "What is truth"? is not just a line from a Pilate/Christ trial, but a legitimate metaphysical question whenever the term is used without restriction to its domain.
"I believe that the cutoff for being "sufficiently complex" is being able to express arithmetic, but it's been a while since I worked with this."
The word that Gödel uses in the German original of his famous paper is "reichhaltig" and he does not explain what it means. A correct English translation would be "rich in content" and not complex. There is a vast literature on what exactly this means and as far as I know nobody has managed to produce a formal definition of what exactly a sufficiently "reichhaltig" system is. If one investigates different formal systems as to their susceptibility to Gödel's theorem then FOPL is not Gödel susceptible. Arithmetic with only one operator (either addition or mutiplication) is not Gödel susceptible but arithmetic with both operators is.
A small post script, if you write Gödel's name on an English keyboard without umlauts then the correct spelling is Goedel and not Godel!
Mark I hate to ruin a great post on the basics of "formal logic" with a quibble about your outrage concerning Mr Spock's apparent missuse of the word logical. In fact logic comes from the Greek logos, which means reason and to use logical to mean thinking or arguing reasonably is perfectly correct. That is the reason why logical used in your sense is called "formal logic" to distinguish it from the more general and wishy washy meaning.
Hey, Mark!
Did you ever check out this nice piece of math logic?
http://uscountvotes.org/ucvAnalysis/US/exit-polls/ESI/ESI-hypothesis-il…
I think some kind of prize is offered to anyone who can invalidate it.
Foxy:
Gödel's incompleteness results were about working within a specific formal system. A formal system is a complete logic - that is, a syntax for making statements, along with its inference rules and a set of axioms.
You can construct numerous different logics, each of which can be used for different kinds of reasoning, and which can be used for getting different kinds of results. Given a particular desired result, you can construct a logic in which that result is inferable in a small number of steps.
What Gödel said was that given a single formal system, either that system will be incomplete (there will be statements that are obviously true but which cannot be proven) or inconsistent (there will be statements that are neither true nor false. The idea behind that was that people thought that they could design a single formal system where every statement was either true or false, and the truth or falsehood of every statement was provable. So it's about starting from a single very general logic, and probing the outer limits of that logic; whereas what I'm talking about is creating a specific custom logic whose purpose is to make a specific statement provable.
OmenPigeon:
You can formulate the cutoff point for a logical system where incompleteness hits in terms of arithmetic - if your logic can support the Peano axioms, then you've got enough power. (And if you have the peano axioms, it's actually not that hard to prove things like a+b=b+a. The hard part is creating a logic where you can prove the Peano axioms without just accepting them as axioms :-).
Well as long as we speak of logic, let me plug in that Jaynes probability theory is rooted in a generalizations of logic which rests on Cox's theorem. This theory bridges together axiomatic logic (as the limit when probability p=1 or p=0) with scientific induction and Spock's common sense logic
"Presburger arithmetic is the first-order theory of the natural numbers containing addition but no multiplication. It is therefore not as powerful as Peano arithmetic. However, it is interesting because unlike the case of Peano arithmetic, there exists an algorithm that can decide if any given statement in Presburger arithmetic is true (Presburger 1929). No such algorithm exists for general arithmetic as a consequence of Robinson and Tarski's negative answer to the decision problem. Presburger (1929) also proved that his arithmetic is consistent (does not contain contradictions) and complete (every statement can either be proven or disproven), which is false for Peano arithmetic as a consequence of Gödel's incompleteness theorem...."
Weisstein, Eric W. "Presburger Arithmetic." From MathWorld--A Wolfram Web Resource.
Nice exposition, but it completely misses the point of the character Dr Spock, which was as much about being passionless and intellectual, as about reaching rigorous conlusions.
Given that there were some discussions recently on ScienceBlogs about the use of modal logic in a "proof", I would find it interesting if you did a follow-up post on modal logic some time.
Mark:
Wonderful post, great coments -- might add a few (not-so-great) ones in a while if things settle down here, but couldn't resist:
David:
Dr. Spock = pediatrician, great at raising and educating babies to adulthood
MR. Spock = science officer, not quite as good at raising and educating James T. Kirk
Oups, seems my other message didn't make it. I just wanted to say that, Spock's fuzzy kind of logic isn't that unacceptable, considering Jayes' version of probability theory which
bridges logic and probabilities through Cox's theorem.
Eliezer Yudkowsky defends quite convincingly the validity of reasoning through Bayesian probability as an extension to logic with vague theories. Just like Spock would have done.
Great thread; and Mark, Jonathan and Antendren addresses some of my pet peeves too. "What is truth"? is [ ...] a legitimate metaphysical question whenever the term is used without restriction to its domain." Exactly.
Though I will probably forever be confused about the exact meaning of "syntax" and "semantics". Here the syntax seems to be all of "a complete logic - that is, a syntax for making statements, along with its inference rules and a set of axioms." Perhaps Mark has material for another Basic post here?
Nat:
I'm quite sure you are saying something interesting. Unfortunate you don't explain what it is, to me at least. How does intuitionist logic go about to not "ignore the internal structure of propositions", and what is its connection to Brouwer's philosophy?
Of many characters that exhibited growth, the one that started out with less of humanity was the most interesting. But indeed Spock and Data stood out against such as Worf, The Doctor, and Seven-of-Nine, the first for integrating his humanity, the second for defining his.
My own eye-opener was when axiomatization was introduced - I hadn't taken time to find out how (some parts of) math was founded. But after that it was a struggle to get rid of the pleasant illusion that everything needs an axiomatic. ;-)
What would have helped me a lot is the part of how to "distinguish between valid arguments and invalid arguments", since there are fallacies in "reasonable" argumentation not covered by FOPL.
Thony:
The correct spelling is more flexible IMHO. Diacritics may change recognition, pronunciation and meaning a lot, but not always. For example, I'm perfectly happy to see "Torbjorn", "Godel", and "Mobius". At least the swedish name, it doesn't make much of a pronunciation difference. (And if we nitpick, here "oe" isn't the correct sound either.) Maybe austrians (Gödel) and germans (Möbius) can weigh in.
Of course, in some cases it can make an additional improvement, for example when "Erik Häger" and "Erik Hager" attend the same conference.
Scientists and Mathematicians can be elected heads of state (we have a Ph.D. Physicist in Congress now, by the way)
Thatcher and Merkel don't have an especially good record at running their respective countries...
The correct spelling is more flexible IMHO. Diacritics may change recognition, pronunciation and meaning a lot, but not always. For example, I'm perfectly happy to see "Torbjorn", "Godel", and "Mobius". At least the swedish name, it doesn't make much of a pronunciation difference. (And if we nitpick, here "oe" isn't the correct sound either.) Maybe austrians (Gödel) and germans (Möbius) can weigh in.
In German, the spellings with e are seen as variants of the spellings with umlauts. It's different in Nordic and Finno-Ugric languages, where the umlauted letters are seen as letters in their own right. For example, I've seen a German math professor write about his "Erdoes number," even though in Hungarian ö is never oe.
"The correct spelling is more flexible IMHO. Diacritics may change recognition, pronunciation and meaning a lot, but not always. For example, I'm perfectly happy to see "Torbjorn", "Godel", and "Mobius". At least the swedish name, it doesn't make much of a pronunciation difference. (And if we nitpick, here "oe" isn't the correct sound either.) Maybe austrians (Gödel) and germans (Möbius) can weigh in."
Although I have lived in Sweden I can't comment on the Swedish use of diacritics or pronounciation, however in German there is an official grammatical rule which says that ö=oe, ä=ae and ü=ue.
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Having a training in science gave/gives them an advantage when trying to discern facts, as compared to lawyers who judge by rhetorical skill.
Personally, I think Margaret T did an OK job for the UK (yes I am a Briton); it may just be that her aims weren't yours, so what you see as rational/reasonable Mrs Thatcher saw as wrongheaded.
It is what makes logic based on axioms interesting, define your axioms appropriately and you can "prove" most anything.
"official grammatical rule"
That should of course read orthographical and not grammatical rule. This rule is valid in Germany, Austria and Switzerland as there are tripartite political agreements between these countries, which guarantee the teaching of a single uniform High German in the schools of all three countries.
"austrians (Gödel)"
Technically Gödel was Czech and not Austrian. Born in Brno in the time of the "K und K" (the Austro-Hungarian Empire) he was compulsorily assigned Czech citizenship when the "K und K" was broken up, something he resented deeply and complained bitterly about on several occasions. In 1948 he became a naturalised American citizen.
Torbjörn: 'semantics' and 'syntax' are both terms out of linguistics, ultimately, and refer to entirely different fields of knowledge. The Wikipedia article for 'Semantics' has a concise summary in the lead paragraph of the difference between the fields.
Do math/CS people ever even have to DEAL with semantics? Syntax is a concern, obviously, WHENEVER you're constructing representational systems. But it doesn't seem like semantics is something that hard-sciences people deal with much. (The limit of my education here is that i'm a CS minor; it certainly never came up there in any of my math OR CS classes.)
--Adrienne
Oops. As an update to my last comment, if i'd bothered scrolling down, i'd have seen things about applications of semantics in math and CS. I note in particular: "In computer science, considered in part as an application of mathematical logic, semantics reflects the meaning of programs." This is a new one on me. As i say, i have only a MINOR in CS, but we never used the word 'semantics' at all.
--Adrienne
Adrienne, a CS major would need to take a programming languages theory course to deal with semantics, which is usually an elective course.
Torbjörn Larsson asks: "How does intuitionist logic go about to not 'ignore the internal structure of propositions'"
In essentially the same way that non-Intuitionistic logics do: by taking that internal structure into account, such as by studying the nature of predicates and their quantification.
"what is its connection to Brouwer's philosophy"
Brouwer rejected the Law of the Excluded Middle, which is why P or not-P isn't a tautology in Intuitionistic Logic, and a complete formalization of Intuitionistic Logic was first published by Brouwer's student Arend Heyting. Read Paolo Mancosu's From Brouwer to Hilbert for more details.
Ah yes, that is entirely correct - umlauts vs letters with diaeresis - åäö is placed last in the swedish alphabet far from ao, in contrast to vw which is consecutive letters. That explains any difference in spelling conventions.
Which makes the spelling of Gödel what? "Kurt Gödel (IPA: [kurt gøËdl]) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic - January 14, 1978 Princeton, New Jersey) was an Austrian logician, mathematician, and philosopher of mathematics". [The IPA convention doesn't seem to help, btw. And the local rule, the tripartite agreement, should take presedence anyway.)
Should it be Goedel (german, modern austrian) or Godel (perhaps old (austria-)hungarian)? Forced czech - my guess is that Gödel would strongly prefer old austrian spelling, whatever it was. But that should in any case now be Goedel by convention, and who cares about what the historian thinks? ;-)
Btw, if "oe" is a strict convention on much of the continent (and perhaps here too), I will try to make a point of using it myself. I have to remember the Hungarian contra-convention though.
Yes, and Mark has described some uses earlier, which is why I mentioned it. But I'm pretty much in your position - my education in math didn't stretch into these descriptions either.
And as you can see, Wikipedia doesn't help. The definition is pretty much recursive and empty, since if you follow through the brief description to formal languages, it immediately points back to the semantics article. :-)
Semantics (and/or pragmatics) is interesting since it is about applications of theories (or software), if I understand it correctly. For example, a Hamiltonian description can be used both for classical and quantum systems. Another example, the possible configurations of world lines for a point particle in QM are analogous to the possible configurations of strings in string theory. And so on.
Nat:
Thank you for your answer! I'm not sure the description was as exciting as it looked earlier.
"This rule is valid in Germany, Austria and Switzerland as there are tripartite political agreements between these countries, which guarantee the teaching of a single uniform High German in the schools of all three countries."
"Should it be Goedel (german, modern austrian) or Godel (perhaps old (austria-)hungarian)? Forced czech - my guess is that Gödel would strongly prefer old austrian spelling, whatever it was."
The first such agreement unifying the teaching of German came into force in 1900 therefore preceeding Gödel's birth by six years.
I liked Marc post on "logical", as I had a similar discussion with a collegue last week on why logicians (we both used to work in automated theorem proving) never use the word "illogical" ("unlogisch") while everyone else does.
Our conclusion was that we like inconsistent ("widersprüchlich") more, as it is more precise and usually captures what is meant when people say illogical.
Which brings me to some problem with Marcs definition of logic: I am under the impression that people here define logic as something consisting only of a language's syntax and some inference rules. But I have never seen any definition of a logic that did not also include a full definition of the *semantics* of the language's constants (such as quantors and operators). Logicians like to prove the properties of their logics, such as completeness and soundness. There is, in general, no way of knowing whwther a statement of a language can be true or false, or a theorem or inconsistent, without a definition of the language's semantics and they way it can be interpreted.
And, as I thought before reading Marcs article, a statement in any logic or language should not be called *logical* if it is provably false, i.e., has no interpretation that makes it true. Is Marc's use of the word, which disregards semantics, more appropriate?
Regards,
Karsten
Speaking of Spock, his ancestry (fictionally) includes Sherlock Holmes. Sir Arthur Conan Doyle's most famous creation was based on an actual professor, was an expert in several fields, published, and specialized in acute scientific observation and longs chains of causal reasoning.
The misanthropic physician "House" as a fictional (television) character has similarities to Holmes and Spock. Indeed, his creator says that the name "House" is a pun on "Homes ~ Holmes."
In all of these cases, the primary dramatic conflict is between an exaggerated use of logic in the protagonist versus disbelief and outrage by more normal humans who insist that logic is insufficient to understand the human world in depth and context.
Now that we have a post on logic, is anyone planning to do a set of Basics posts on common fallacies? You know, like the warning signs in Carl Sagan's baloney detection kit?
I've given up telling people that 'data' is plural. Even to computer heads. I've given up telling people that 'input' is a noun. For the record, you enter a datum. English could be such a nice language if it weren't for people.
Blake Stacey:
I believe Bronze Dog already has that covered, in the form of his Doggerel entries. Look him up. ^_^
Stephen Uitti:
Actually, you're wrong. Data is both plural and singular. Input is a verb and a noun. You see, we didn't discover English inscribed on a tablet on a moon of Saturn, given to us by an unknown divine intelligence. We created it, bit by bit, year by year. You know how Middle English looks? How you can't, you know, read it? How even the English of the time around the founding of America is hard to read, full of words we don't use and grammar that's fallen into disuse? There's a reason for that, summarized by three simple words:
LANGUAGES CHANGE, RETARD.
This public service announcment has been brought to you by the Language Log Society Against Small-Minded Prescriptivism.
Xanthir:
Please, no need to call people retards. I'd really like to keep things civil around here; there's no need to go calling names. Even in response to grammar flames.
Gotcha. Prescriptivists get under my skin; there is always an undercurrent of "I'm better than you are, peasant." But I can hold my tongue. You can edit my comment if you wish to remove that word. If you do, please change the preceding sentence to say, "two simple words". ^_^
According to the Vulcan dictionary (http://www.starbase-10.de/vld/) the Vulcan word 'olozhikaik' (translated - 'logical'), includes 'reasonable' as part of its definition (of relating to, in accordance with, or of the nature of logic; based on earlier or otherwise known statements, events, or conditions; reasonable).
Expecting English to fully capture the nuances of Vulcan is just not olozhikaik.