For this class you will merely need to be able to read outloud sentences of first order logic, and have some intuitive grasp of what they mean. Luckily, developing the ability to read the sentences out loud really helps with the ability to understand the language.
How to read logical notation. The following contains all the information you need to be able to read out sentences of first order logic.
(" x) = "for all x,"
Now we'll give a bunch of sentences in the language with their English language translation.
(" x)(Sx --> Px) =
($ x)(Sx & Px) =
($ x)(Sx & ~Px) =
~($ x)(Sx & Px) =
(" x)~(Sx & Px) =
The above correspond, respectively, to the syllogistic figures A, I,O, and E.
In using logic to translate natural language, philosophers and linguists diverge in one small way. Philosophers will ususally let the letters such as "P" and "S" stand for natural language words via a translation manual, while linguists will often use natural language words themselves. For example, if a philosopher wanted to represent the argument, "Every person is a mortal. Socrates is a person. Therefore, Socrates is a mortal," she would first translate "man," "mortal" and "Socrates" into symbols of the predicate logic, perhaps in this manner.
Let "person" translate into "P,"
Then the argument can be translated into this.
(" x)(Px --> Mx)
In Linguistics literature you are much more likely to see something like this
(" x)(person'(x) --> mortal'(x))
Here "Socrates" has been translated to avoid confusion, but "person" and "mortal" are only translated by putting the dash after them. We will follow the linguists usage when possible, as I think it keeps us a little bit closer to actual human languages.
Again, all students of this class will need to do is to be able to read the sentences of logic outloud and have some grasp of what they are communicating. The harder tasks of translating natural language into logic, utilizing the logical formulas in proofs, and obtaining mathematical results about logics, are tasks for classes in logic, mathematics, computer science, and linguistic semantics. Reading out the sentences following the linguistics conventions is just as easy, using the method above.
For example we can read (" x)(person'(x) --> mortal'(x)) as "for all x, (if x has the property person' then x has the property mortal')." When you get more comfortable with this it will be easy to shorten "has the property" in the above to something more natural language, such as "for all x, (if x is a person, then x is a mortal)." This doesn't always work though, since logic predicates can be translates of many nouns, adjectives, and verbs. (quick rules of thumb, if the predicate is a mass noun, change "has the property" to "is," if the predicate is a singular count noun, change "has the property" to "is a," if the predicate is an intransitive verb then simply delete "has the property". More than one place predicates get more complicated.)
(" x) = "for all x,"
make clear what semantic value is and why Frege's project
required thinking about it.
DEFINITION: The semantic value of any expression is that feature of it that determines whether sentences in which it occurs are true or false.
make clear the relation between truth value and reference
show how saturated, unsaturated stuff doesn't really matter
and how notion of concept is unclear and unhelpful in Miller's
discussion of the functional role of quantifiers.
explain in terms of set of ordered pairs
thesis 6: Functions are extensional: if function f and function
g have the same extension, then f = g.
note-quantifier stuff bothers me and is very confusing-- if a concept is itself a function from objects to truth values, then yes the characterization works, but this is not how most of us think of concepts. . . Miller's talk of the quantifier's extension is also very confusing. Here's how a quantifier works,
(forall x) G[x] = T iff for any alpha in the domain
of discourse "G[alpha]" = true.
Like the function (forall x), a quantifier is a function that
takes in functions from objects to truth values, and spits out
a truth value.
Mark Twain example is hazy because Miller doesn't give
the best version of the argument.
(II) One can know the meaning of a sentence without knowing that it is true. One can know the meaning of a predicate without knowing everything in the extension of the predicate (or knowing the function Frege describes). This is good reason to think that meaning isn't just reference.
(III) On the other hand, one can know the reference of an expression and know that the referenent in question is the refererent of the expression in question and still not grasp the meaning. red car example.
note: argument in 2.1.3 committs de-dicto/de-re fallacy in
almost ironic manner.
make clear what it means to know sense and not know semantic
thesis 16: Substitution of one expression in a sentence with another that has the same sense will leave the sense of the sentence unchanged.
talk about semi-ambiguity of English word "thought,"
between act of thinking and content of thought.