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,"
($ x) = "there exists an x such that,"
~(A) = "It is not the case that, (A)"
(A --> B) = "(If A, then B)"
(A v B) = "(A, or B)"
(A & B) = "(A, and B)"
(Px) = "x has the property P"
(Rxy) = "x is in the relation R to y"
(a = b) = "a is equal to b"

Now we'll give a bunch of sentences in the language with their English language translation.

(" x)(Sx --> Px) =
 "for all x, (if x has the property S, then x has the property P)."
 (another way to say "Every S is a P")

($ x)(Sx & Px) =
 "there exists an x such that, (x has the property S, and x has the property P)"
 (another way to say "Some S is a P")

($ x)(Sx & ~Px) =
 "there exists an x such that, (x has the property S, and it is not the case that (x has the property P))"
 (another way to say "Some S is not a P")

~($ x)(Sx & Px) =
 "it is not the case that, (x has the property S, and x has the property P)"

(" x)~(Sx & Px) =
 "for all x, it is not the case that, (x has the property S, and x has the property P)"
 (two ways to say "No S is a P")

The above correspond, respectively, to the syllogistic figures A, I,O, and E.

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Brief note-

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,"
Let "mortal" translate into "M," and
Let "Socrates" translate into "s."

Then the argument can be translated into this.

(" x)(Px --> Mx)
Ps
therefore, Ms.

In Linguistics literature you are much more likely to see something like this

(" x)(person'(x) --> mortal'(x))
person'(s)
therefore, mortal'(s).

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.)

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Now we will focus on understanding what sentences of logic mean. The primary notion relevant to meaning in first order logic is the conditions under which one of the formulas are true.

(" x) = "for all x,"
($ x) = "there exists an x such that,"
~(A) is false when A is true, and ttrue when A is false.
(A --> B) is false only when A is true and B is false, and true in all other circumstances.
(A v B) = "(A, or B)"
(A & B) = "(A, and B)"
(Px) = "x has the property P"
(Rxy) = "x is in the relation R to y"
(a = b) = "a is equal to b"

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1.4

make clear what semantic value is and why Frege's project required thinking about it.
reason for this has to do with notion of validity being important.

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
thesis 1: The semantic value of a sentence is its truth-value (true or false).
thesis 2:  The semantic value of a complex expression is determined by the semantic values of its parts.
thesis 3: Substitution of a constituent of a sentence with another that has te same semantic value will leave the semantic value (i.e. truth-value)of the sentence unchanged.
thesis 4: The semantic value of a proper name is the object that it refers to or stands for.

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 what's going on with the distinction between the list of names verses sentence (what does is or is a do).
1.6
thesis 5: The semantic value of a predicate is a function

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.
thesis 7:  The semantic value of a predicate is a first-level function from objects to truth-values; the semantic value of a sentential connective is a first-level function from truth-values to truth values; the semantic value of a quantifier is a second-level function from concepts to truth values.

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.
                       F iff for some alpha in the domain of discourse "G[alpha]" = false.

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.
 

Why sense?

Mark Twain example is hazy because Miller doesn't give the best version of the argument.
(I)  My version of the substitution argument--

  1. It's possible for "Frank believes that Cicero is happy" to be true and "Frank believes that Tully is happy" to be false.
  2. If it's possible for one sentence to be true and another false, then the first sentence does not logically entail the second one.  (by any definition of logical entailment).
  3. So "Frank believes that Cicero is happy" does not logically entail "Frank believes that Tully is happy."
  4. Whether or not a sentence A, logically entails a sentence B depends upon the meanings of the words in the sentence and the way those words are put together.
  5. Since the words in "Frank believes that Cicero is happy" are put together in exaclty the same manner as those of "Frank believes that Tully is happy" it must be the case that some words in the two sentences have different meanings.
  6. Clearly, then "Cicero" must mean something different than "Tully."
  7. Since the two words refer to the same thing, meaning can't be the same as reference.

(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.
2.2
sense determines semantic value given the way the world is!!! (re Christian's question 1)
thesis 8: The sense of an expression is that ingredient of its meaning that determines its semantic value.

make clear what it means to know sense and not know semantic value.
thesis 9: It is possible to know the sense of an expression without knowing its semantic value.
thesis 10:  The sense of an expression is what someone who understands the expression grasps.
thesis 11: The sense of a complex expression is determined by the senses of its constituents (Compositionality of sense).
thesis 12:  If someone grasps the senses of two expressions, and the two expressions actually have the same sense, then she must know that the two expressions have the same sense (Transparency of sense).
thesis 13:  An expression can have a sense even if it lacks a semantic value.
thesis 14:  A sentence that contains an expression that lacks a semantic value is neither true nor false.
thesis 15:  In a belief context, the (indirect) reference of a proper name is its customary sense.
 

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.
thesis 17:  The sense of a sentence is a thought.
thesis 18:  The semantic value of an expression is no part of what someone who understands the expression grasps.

2.3
thesis 19: Sense is objective: grasping a sense is not a matter of having ideas, mental images, or private psychological items.
thesis 20:  The sense of an expression is normative: it constitutes a normative constraint that determines which uses of that expression are correct and which are incorrect.