An Exegesis of Russell and Frege
When people tell people that I’m philosophy major they usually ask two questions: 1. Why would you ever want to study philosophy? and 2. What will you ever to do with a philosophy degree? The way these questions are frame usually come across as a straw-man argument connoting to such a pursuit is a fruitless endeavor. Despite societal preconceived notions of philosopher they fail to see major contributions that they have given society from governments, ethics, and technology. Most people are not aware that without the advancement that were made in the philosophy of language, we wouldn't have the knowledge to build computers. Perhaps computers could be attributed as the major contributors to the advancements that we have seen in our society. Computers have infinitely impacted the way that we approach the world, its impacted the food we eat, security (for better and worse), scholarly pursuits, social networking, etc. its hard to imagine a world that doesn't have computers; without computers we wouldn't have all the inventions and services that have come about because of them. It was the work of four philosophers / logicians: Gottfried Leibniz, Gottlob Frege, Bertrand Russell, and Alan Turning.
Leibniz invented the binary number system which allows all number to be expressed in terms of 0's and 1's. By expressing the numbers as 0's and 1's all you need is a register which has only has to process two options: on and off. But programming in bits is very slow and tedious: failing to get input the right bit in a code that consist of thousands of switches will cause the entire program to fail. Once more programming in machine language is very repetitious. When writing a program in macing language you would see that you reuse a lot of the same code over and over which creates more data for the programmer to work with. Instead of typing the same code over and over wouldn't be easier to use a sign to substitute for entirety of the code? Programmers needed a language that was logically consistent, but also flexible to allow for defining new data types
Frege work at the turn of the 19th century invented predicate calculus which allows computer programers to represent lines of machine code as sign which could then be treated as an abstract object, this type of programming came known as object-orient programing. In object orient programming there are predefined objects, and also there are object/data types that can be defined. By allowing programers to define the different data types allowed for flexibility to create an infinite amount of programs, all the while minting a degree of rigidity. Contained within the program there are arguments that use different methods and expressions, which are predefined i.e. If, then, else, while, equal, and etc. By using objects the programmer capability is greatly enhanced as the programmer doesn't need to remember lines of code which takes more time, because it requires much more input, but also is more prone to error; object-orient language compiles the machine code into objects which are easier to manipulate. It was the work of Frege and Russell which made it possible for programmers to construct long and complicated algorithms which has catapulted us into the information age.
With Leibniz’s advancement in mathematics and Frege's and Russell advancement in logic and language we were able to create machines, and use those machines to create even more complicated machines. In this paper we will see how predicate calculus was invented and how it impacted the world we live in today. But in order to do this we will first need to examine the naive view of philosophy of language were its thought that things are given names through ostension, however there are a few problems with this method which I will raise. Next we will see how Frege invented predicate calculus in his attempt to place mathematics within the realm of analytical philosophy. As the father of analytical philosophy Frege wrote three extensive books concerning predicate calculus, but not before his second book came out did the young philosopher Bertrand Russell discovered a flaw in his system: On a small postcard Frege saw his entire life work shattered. In this paper we will see what was written on that postcard, and see how Russell improved on Frege's philosophy of language
I. Naive view of philosophy of language
The naivepread notion or idea about the meaning is that words and expression get their arbitrarilypicking out things in the world. Words formed together to form expression and that is how we understand language. Take the simple proposition, “the horse is brown”. The proposition is meaningful because the strings of marks make a meaningful sentence and the horse is predicated by the category of brown, though the sentence doesn't refer to a particular horse, its still meaningful because we know the classes of horse which refers to a four-legged herbivore. Suppose we substitute horse with horsespam, “the horsespam is brown” , in this sentence a horsespam is predicated by the category of brown, but the object horsespam doesn't refer to anything and thus making the sentence meaningless. Suppose I have a sentence “hormena iasdf ijsdfaf” The marks in this sentence don't have a sense or a reference. So in order for a proposition to makes any sense you need syntax, but the terms of the sentence also needs a reference. In the first sentence “the horse is brown” the parts that makes proposition meaningful referring to a particular mammal belonging to the class of brown things. In order for the proposition to make sense its seems that it must have both a sense and a reference. At first glance that we could choose to use anything in place of a horse like the made up “harrpieam” to refer to the object of a horse, as long as we are in all in agreement on the choice of the word. In this way the referential thoery sees words are more akin to labels; these symbols designate, label or denote to things that are actually cointain in our world.
The naïve view at first glance looks very appeasing, because it seems that we can just arbitrarily pick out a name to refer to a particular object. This gives significance to a statement by its ability to refer to things or state of affairs that we see in the world. Our ability to understand language in terms of our ability to understand how words refer to things, seems very natural and indeed commonsensical. Indeed it seems ridiculous to deny that there is a relationship between are words and how they extend to object in our world. However this commonsensical view falls apart when it comes to negative existential like the horsespam.
There are some words in our language that don't refer or denote actual objects yet the words are meaningful for example unicorns, Santa Clause. Etc. In reality there is no such thing as a class of mammals that resemble a horse that has a horn between the eyes. Another example as an objection to the naïve view can be seen with the statement: I didn't see nobody. “Nobody” doesn't denote or refer to anything in the physical world, because its an idea rather than a particular object. Along those same lines consider the simple subject predicate sentence, “Byron is chubby”. The name Byron of course refers to the author of this paper, but what does “chubby” refer too? To the best of my knowledge (which isn't saying much) I have never seen or heard of a benchmark, or gold standard when it comes to the word “chubby” , but the predicate does accurately describe me.
Quine points out that there are some words that are grammatically are nouns but don't denote anything, for example the words: “behalf” or “sake”[i]. “I received the award on Jenny's behalf” or “I did that for my child's sake”. I'm sure if you closely inspected Jenny you will not find that she owns a “behalf”, and nor is there any ontological existing “sake”. Yet both sake and behalf our nouns that don't refer to existential objects. In fact these nouns don't really make any sense unless they are being used to construct an expression, and when they are added to a sentence they add more meaning to the expression. Also, there are words that are not nouns but don't refer to anything e.g.: “yes, no, many, the, is, etc” Yet if any of these words and other like them, were removed from our language, it would make it nearly impossible to communicate.
Another objection to the referential theory is there are some terms that are co-refering, though they are not synonyms. For example, Superman and Clark Kent are two different names both referring the same person. In this sense X = Y as long as X and Y share all of same properties. The classic example of co-refering terms is the Greek astronomical observation of the planet Venus. At first astronmers thought the the morning star Phosphorus was different of the evening star Hesperus, though can't be a ontological distinction, but both Phosphorus and Hesperus have different meaning.
What we can take away from the referential theory is that marks and noises when structured right are meaningful. The fact that we can understand these marks and noises and interpreted them as being meaningful, is amazing ability unique to humans. Some words don't refer to anything but when comprised in a sentence they become meaningful. II. Gottlob Frege It could be said that Aristotle was the uncontested king of logic for almost two thousand years. It wasn't until Gottlob Frege came around and discovered predicate calculus in his search to ground mathematics in logic did Aristotle loose his thrown. As a mathematician and a philosopher Frege work was to ground mathematics, in analytical logic, Frege discovery of predicate calculus was a by-product of his pursuit into grounding his mathematics. In this sens it could be compared to the discovery of the “new world” by Christoper Columbus. Columbus set out across the pacific ocean to find a quicker trade route to India; instead he discovered a route to the American's inhabited with millions of people. Though Columbus was never successful in finding a route to India he did however expand our geographical knowledge of the world. In this sense Frege sought out to ground mathematics, however the closest he came was by positing a third realm or operators and numbers that exists. In this realm the number two traces to the class of all things that come in pairs. The number three would contain in the class of the number of thrice. Frege in an effort to give some ontological meaning to theses classes posited that there was a third realm. This realm would exist outside of space and time where these “thought's” are waiting to be found. Whether or not this third realm exist won't be explored in this paper, however to his credit Frege view were not that far removed from the dualist thinking of the time, who posited a distinction between body and mind. Later on we will see that this his method of isomorphism will end up being the weakest link, which Russell exposed on a little postcard addressed to Frege. II.I. Sense and Reference In order to solve the problems with language of the naïve view, Frege set out in his Begriffsschrift, to create a system of notation and logic that universally analyze any language. Frege's hopes was to place mathematics within the realm of formal logic. In order to complete this feat Frege had to show that an object can have a different sense, as long as language is referring to the same thing, we can talk about the relationship the sign has for an object. Frege uses the example of the morning star and the evening star. The morning star is different then evening star in the sense that its dependent on what time of day and the star is being experiencing . To say the morning star equals the morning star m = m seems trivially true, according to Kant m = m is a priori and can be known analytically, but to say that m = e isn't known to be a prori truth, because both morning star and the evening star have a different senses. Even though the morning star and evening star have a different sense they are both referring to the same thing the planet Venus, so it is the case that m = e. What Frege was concerned with isn't the arbitrary designation of signs to objects, but rather he was interested in the relationship of the signs to the objects they were designating, and how those objects can be predicated. Frege argues that a propositions have both a sign, a sense and a reference take the proposition, Socrates is bald. The sign or the subject of the proposition is Socrates who is being predicated on being bald The sense in this proposition enables us to refer to a particular philosopher of Greek antiquity; how that proposition is interpreted from person to person can vary from person to person depending on what they know about the philosopher Socrates. Some people will hear Socrates and think of the old man that used to question the social elite who felt threaten by his questioning and sentenced him to death, while to other Socrates refers to the philosopher who taught Plato, and championed the dialectical thinking. Though both senses are correct some people would only commit to the former definition of Socrates and deny the latter definition because they are unaware that Socrates is the teacher of Plato, so how is it possible that we are able to refer to the same person? In each sense of the word Socrates are not using the same marks or signs, though each sense is referring to the same Hellenistic philosopher. Yet despite these differences we are still able to speak the same language and refer to the same person. Even though the sense is different, people are still able to talk to each other because they are refering to the same person. So with the proposition Socrates is bald, the sign are the words that are contained in the expression, the sense is the various meaning that are attached to the word Socrates, and the reference refers to the philosopher Socrates. It could be said that a proposition is meaningful, as long as its in an expression and contain a proper name, however there are times that proposition suffer from referential failure. It would be nice if every proposition that has a definite sense belong to a definite sign, but Its not always the case that a proposition has a reference though it may have a sense. For example suppose I say “the greatest philosopher since Socrates”, is a proposition that has a sign, sense, but the proposition fails to refer to a particular philosopher. In instances like this Frege would argue that the sentence doesn't have any meaning because its not referring to a particular philosopher. So its not always the case that a sign and sense correspondingly have a reference.
Frege points out that we have signs for signs: when we introduce a new word or to indicate when someone is speaking we use double quotations “a”. But suppose I'm quoting another writer within double quotations, I would use single quotations to show that speakers are different in each case. Unless we are Bill Cosby (a comedy sketch he did where he explicitly made sounds, and use sign language to denote periods, commas question marks etc.) we don't use sign for signs explicitly when we are speaking about other people quotes, though its quit clear in reported that speaking words “do king words do not have their customary reference but designate what is usually their sense”[ii] In order to have an expression in reported speech, Frege makes a distinction between “customary from the indirect reference of a word; and its customary sense from its indirect sense. The indirect reference of a word is accordingly its customary sense.” Frege further this distinction by making a distinction between a thought and an idea.
An idea is different than the sign and the sense of a proposition, because an idea is subjective and dependent upon our impressions of a particular thought. If there is a proposition whose object is perceivable to the senses, than that object brings about an internal image which we have interacted with both internal and externally. Each time the object is brought present to the mind, the idea oscillates according the person who is experience the object. According to Frege, each idea is saturated with feelings that change over time for every individual so its not always the case that you have the same idea about the same object. Each sense isn't always connected to an idea, but the sense of an object varies from man to man and even the same man may not have the same idea about the same object. To illustrate the point that there are different ideas associated with the same sense Frege thinks that a painter, a horseman and a zoologist would have a different idea with the name of 'Bucephalus'. A painter idea of Bucephalus might have the sense of the many paintings and statues that show Alexander riding the now famous horse; a horseman on the other hand might have the sense that Bucephalus must have had certain qualities as to have Alexander pick him out of all the potential candidates. Though to a zoologist who isn't to familiar with the history of Bucephalus they might only know that Bucephalus is an instance of horse which is a single toed mammal contained with the class of Equus ferus caballus. In each case the painter, horseman, and zoologist were referring to the same object, but each had a different idea about the subject. What this shows is that there is a difference between an idea and the sign's sense or thought; the sign's sense is said to is objective which is common property that is passed down from generation to generation.[iii] In consideration of these facts there shouldn't be an argument when we are speaking about sense of an object, but an idea on the other hand isn't objective, and must belong to a person at a particular time. “Si duo idem faciunt, non est idem. If two persons picture the same thing, each still has his own idea.” [iv] To help illustrate this point Frege uses an example of a telescope that is pointed at the moon. The telescope which is pointed in the right direction which collects the visible light and magnifies it could be said to be that of the sense, or the thought; because the telescope is stationary numerous people can peer refer the telescope and confer about certain distinguishing marks which are seen on the surface of the moon. The light reflected from the moon could then be said to be objective to the lens of the telescope, but how that light is interpreted in our occipital lobe and the sensations and memories that arise with the sensation of the moon would be subjective from observer to observer. The person who is experiencing the sensation of seeing the light from the telescope is subjective and could then be compared to that of an idea. The problem with ideas is that they are not objective, if each observer relayed the qualia or the experience of moonless we could never establish any objectivity. What Frege is interested in is in thoughts which for him are universal truths that exist in an independent realm like that of Plato's heaven which exist independent of us. To be more exact in our language Frege proposes the following “To make short and exact expressions possible, let the following phraseology be established:
A proper name (word, sign, sign combination, expression) expresses its sense, stands for or designates its reference. By means of a sign we express its sense and designate its reference”.[v] Frege then asks the question how is it the case that the sign moon refer to the planetary object that circles the earth? Frege argues that the sign moon is a presuppositions of the object and that if we were to talk about the moon as an idea, we would be explicit, “my idea of the moon” implies that its subjective to the way that I perceive the moon. Frege goes on to say that be doesn't need to argue the presupposition of the moon, because all he needs to show is justification of using the sign the moon to refer the planetary object.
So far we have seen that there can be a different sense for a term though they have the same reference, but what about declarative sentences that have different sense but refer to the same thing? When you have a declarative sentence that has the same reference but different sense, we see because of the law of identity , if we were to swap out the terms in the sentence with different words that have the same reference or relationship, but have a different sense, would have no bearing on the reference of the declarative sentence. e.g “The morning star is a planetary object that is illuminated by the sun” is the same thought as “The evening star is a planetary object that is illuminated by the sun”. Both sentences are referring to Venus, for some people they would be okay with the reference of the former sentence being true, while they would maintain that the latter declarative sentence is false because they don't know the morning star and the evening star are both referring to Venus. But because the mornings and the evening star are the same thing what is ever true is true of the other, the logical structure of each proposition are the same. The syntactical structure of the sentence changes which express a different idea, the logical structure of the sentence remains unchanged and the terms of the sentence are changed though they have the same meaning.
How does Frege handle sentences that have no reference? For example the sentence, “Nikki went to the river to have a drink” has a sense, but doesn't have a reference. The object Nikki could be a human or a dog, but being able to determine the truth value of the sentence one would need to know whether or not Nikki existed, and if so, is she is predicated on the fact that she is a being who“went to the the river to drink”. Being able to determine the truth value of this sentence depends on knowing whether Nikki refers to an individual who is predicated by the class of being thirsty. Even though this sentence has a sense to advance the reference of the sentance isn't required to understand the thought of the sentence. Just because the sentence doesn't have a references, the thought still remains the same, however because the proposition doesn't refer to anything the thought loses value. In cases like this Frege didn't think the truth value could be determine giving a null value. But, the fact that we expect the sentence to have a reference shows that we are not satisfied with the thought, because of its referential failure. But isn’t the thought sufficient enough in understanding the proposition? Or do we need not only the sense but the reference in order to determine the truth value of a proposition? Without a reference there is noway to determine whether its true of false that Nikki went to the river to get a drink. Without a reference, the thought expressed could only be appreciated in aesthetically which doesn't tell us anything about the real world, so in order to be objective or as scientific in our language we need to be as exact as possible always striving for the truth values of a proposition which requires it to have a reference. To find a truth value of a proposition we see that the sentence must have a reference as well as the components of the sentence must also serve to refer[vi]. If a proposition has reference then circumstances of whether a proposition is true or false can be mapped to either the true or the false. But the truth value doesn't add anything to the thought, “By combining subject and predicate, one reaches only a thought, never passes from sense to reference, never from a thought to its truth value. One moves at the same level but never advances from one level to the next. A truth value cannot be a part of a thought, any more than, say, the Sun can, for it is not a sense but an object.”[vii]

Proposition and expressions In Frege logic the proposition is an expression that can be seen in terms of what its denotes. These expression include (a) First order objects, this includes simple names of objects, number, names etc (b) complex terms or second order thoughts, which require you to understand first world objects. And (c) sentences which include both (b) and (c)[viii] . Expression than use both a and c and can bee seen in terms of how they function. For example 1. (1) I have two dollars and you gave me two more. This expression function as a binary function which can be expressed in simple terms of 2 + 2 .. In the function 2 + 2 can be seen structured as ( ) + ( ). Where any variable, or number can be used in between the parentheses. In this expression the addition sign is being used as the logical “and”operator. Along those same lines if you takes the mathematical operator 22, this can be seen as the function equivalent of ( )2. When proposition are expressed which have the properties of both (b) and (c) and they make a claim than the are in a form of an arguments by deriving a conclusion. Argument can trace the truth values of proposition as either being the True or the False. For instance take sentence (d) from expression and add the following argument (2)I have two dollars and you gave me two more. Therefore I have four dollars. By adding the last part to the end of this expression it turns the expression into an argument that can be test as being either the True of the False. If we instead used the expression (1) and added Therefore I have eight dollars to give. (3) I have two dollars and you gave me two more. Therefore I have eight dollars. If you are using the expression (3) you can trace the truth value of that to the category of quadruples which can than be traced to the things that things that are in form of quadruples, but the category of octuplets in proposition (3) is different then the category of quadruples, giving the expression in (3) would traced to the truth value False. The sentence is using first order objects that can be empirically verified, that is we could take the four dollars and view them in terms of (a) but its also can be seen in terms of (b) and (c) which contains of secondary thoughts what require you to understand the idea of addition. In Frege's world these secondary objects have always existed they are just around to be discovered, but they have always existed. Other concepts that we have discovered include things like multiplication, lesser than, greater than etc. Frege thinks that any object that a concepts traces to “the True” falls under a concept. Try as you might is will never be the case that (3) will ever trace to the concept of the “true”. In Frege's notation when we express things in terms of function we use the lowercase f( ) and the uppercase F( ) to talk about concepts. In example (d) we can see this expressed in the function of f(e)(x+y=4) but when used to understand a concept of predicates of that of (f) it becomes a concept of prime number would be expressed as in example of (f) would be the second prime number is P(2)[ix]. Frege system works well for objects that have definite descriptions. If I asked the question what is the capital of Utah? This identifies that there is a place called Utah and contained within that place is a place has the property of being the capital of Utah. But it doesn't work well for proper nouns. Frege's systems works well for things that are singular and that have definite descriptions. For Frege everything has a definite description and if it doesn't than it doesn't refer to anything. So in the proposition that Socrates is bald, Socrates refers to the philosopher Socrates and the extension is that Socrates is bald. The sense of the proposition would be x is bald. The reference of a grammatical predicate is a concept, but the concept is incomplete if it doesn't trace to a truth value. What happens is the sentence is the subject and predicate as terms are incomplete but when they are joined together they complete the expression.
III. Bertrand Russell On Denoting
In an his attempt to find a home for mathematics Frege's isomorphic system numbers are categorized into certain sets, the number two mapped to the category of couples, and the the number three mapped to the category of triples. At first glance Frege's theory of language seem quit eloquent in building a foundation for numbers by categorizing. On a postcard addressed to Frege, the philosopher Bertrand Russell recognized flaw in Frege logic, but not right before his second volume of the Grundgesetze was going to go to press in 1903, Frege had his entire life work demolished an instance. In an attempt to defend his Begriffsschrift published an ad hoc defense as an appendix, however it didn't receive a warm reception from the community.[x] The paradox that arises for the referential theory of Frege is what to do about the category of predicates who are not members of themselves? For example if I use the proposition to denote “the horse is in the field” the word “horse” refer to the category of mammals who are four-legged and have a single toe. But its not like the word “horse” has any of the properties of the categories of horses, because its just a word used to categorize a set of objects. Since the word “horse” doesn't belong to the category of horses, it can be categorize into the set of members who are not members of themselves, but this creates a contradiction. For is you say there is a set that “is a set that is not a member of itself”, but what about the predicate, “the set that is not a member of itself”, does it belong to its own set or to that of another? If the predicate is a member of itself, then its contradicting the definition of the category. On the other hand if you say that a set exist that is not a member of themselves would be qualifying the category, by the definition the predicate into existence. A more vivid way of demonstrating Russel's paradox is by using Russel's barber paradox: In a town there exist one male barber who gives shaves to men who don't shave themselves giving us two categories of men 1. Men who shave themselves and, 2. Those who go to a barber to get a shave. But what about the barber, who is suppose to give him a shave? On the surface there doesn't seem to be a problem answering the question it seems natural the barber gives a shave to himself but when you if the barber gives himself a shave, he would be contradicting the definition of what it means to be the town barber. If the barber does shave himself, then the barber (himself) must not shave himself. If the barber does not shave himself, then the barber (himself) must shave himself.
Russell's paradox was just one of the problems with Frege's referential theory Russell exposed, another problem for Frege's problems was finding out the the truth value of negative existential. For Frege questions that contain negative existential were meaningless, and don't have a truth value. For logicians its somewhat embarrassing to come to a proposition and not be able to discern a truth value, so Russell's referential theory address the issues of negative existentials which are things that don't exist I.e. unicorn, nobody, god, Zeus etc. . The idea of a unicorn refers to four-legged, single toed mammal that resembles a horse, but is bestowed with a magical horn between the eyes. The thought of a unicorn means something, but how can something that doesn't exist mean something when it doesn't even exist? Frege attempt to solve this problem was to deny the third law of logic, the “law of the excluded middle”, (P ∨ ~P) which states that either a proposition is true or the negation of it its true, but you cannot have a null value it either has to be true or false but it can't be both which would be a violation the first law of logic the “Law of contradiction”, which states that a proposition cannot be both true and false at the same time. ¬(P ∨ ¬P). Frege thought that it didn't make sense to give a truth value to something that doesn't have negative existential. Frege argument for a null truth value violates the third law of logic “the excluded middle”: A proposition can either be true or false. Usually its the case that denying one of the basic laws of logic shows the structural integrity of a theory is lacking and when there is a violation you know something philosophically fishy is going on. Russell thought that it was a good idea for a person to keep a stack of paradox to think about in his paper on Denoting he gave three (or four) language puzzles and how his philosophy of language solves them.

III.I Reference Failure: The first problem deals with proposition that don't refer to anything. Russell uses the example

1. “The King of France is bald”.

At first glance it seems apparent (1) is false, but why? Is it false that the King of France has hair or because there isn't a King of France? For Frege this proposition would be meaningless and couldn't be logically analyze thus giving a null value. To Frege credit it makes more sense to refrain from giving a truth to a proposition, as opposed to posting that the proposition is false because intuitively feels so. Russell provides an analysis that demonstrates why the proposition is false. Russell thinks that the proposition is actually making trio of generalizations about (1). a. At least one person is presently king of France.
b. At most one person is presently king of France.
c. And if such a king exists he is predicated by the category of bald.

The statement is making an existential and universal claim, and we that in analysis the term king disappears, since the king of France doesn't exist it thus making the entire conjunct false.
(∃x)(Kx & (∀y)(Ky → y=x)& Bx)
In English the notion reads: “There is an x such that x is king of France and for all y, if y is king of France, then y is identical with x and x is bald.”When we see the statement in logical notation its easier to see that the proposition is false because of the existential claim. We see the expression “the king of France” just simply disappears. There is no sense-reference distinction as in Frege.

Though Frege invented his own notation, most logic classes around the world are using the same notation for term logic that Peter of Spain invented in 12th century.[xi] Russell introduced two new quantifier (∃x), read “there is an x such that…” and the universal quantifier (∀x), read “for every x, x is such that ”. What is great about this notation is that its is more accurate and precise about our ordinary language, because it eliminates both the ambiguities that arise from the scope and the reference. Before Russell when you read the sentence “Every boy loves some girl” was ambiguous. This could either mean roughly, “There is one girl who is very popular because everybody loves her” and “Each boy loves a some girl or other.” What Russell's notation does is clears up the ambiguity, the sentence can be seen more clearly as follows:
(∃x)(Gx &(∀y)(By  Lyx))
In English this notation reads, “there exists an x such that x is a girl and for every y that is a boy, that y loves x.”
(∀y)(By  (∃x)(Gx & Lyx))
In English this notation reads: for all y such that y is a boy, there exists an x such that x is a girl and y loves x.[xii]

III.II Negative Existential

Russel uses the sentence
1. “The Golden Mountain does not exist” to demonstrate the paradox that arises for the referential theory. Again we have a meaningful proposition that intuitively seem false, however on closer examination it seems there is more then meets the eyes. Though its true that (1) doesn't epistemically exist, however it still seem that the Golden Mountain must have some reality because we can use the term in a proposition. But it doesn't make sense to commit to the ontology of something that doesn't exist. In an effort to solve this problem the philosopher Alexius Meinong argued the Golden Mountain doesn't exist but is said to subsist. For Meinong there are two modes of being: existence and subsistence. Things that are existence are things that exist in space-time, whereas things that don't exist in space-time are said to subsist. The only problem with this you could say that that Golden Mountain doesn't subsist, does this mean that it subsist to its subsist? On and on the subsistence can be said to go onto an infinite regress. Russell objects to Meinong language claims in two ways: 1. It illogical to talk about the properties of things that are impossible to exist, e.g. round square. 2. If a round-square really have the properties have being round and square, its shouldn't only subsist but it should also exist, but this creates and absurdity. What we see is that existence isn't a property of an object and thus can't make a relational predicate but its a quantifier. In notation
~(∃x)(GMx & (∀y) (GMy → y=x))
The universal claim that the mountain that is golden is categorizing the mountain into the set of things that are golden and that what is ever true of y is also true of x. We see the universal claim is within the scope of the existential claim that states there exist at least one mountain and at most one mountain that is Golden. By using notation we see the proposition more clearly by seeing the negation which would read, “its not the case” or “its false that there exists a x such that its as the monadic predicate GM and for every x its is y and y has the relationship of GM. In notation we see the grammatical form of the proposition is misleading and by using notation we see the parts of the sentence more clearly. In the logical notation we see that if any proposition of the conjunct is false it makes the entire expression false, however we see the scope of the negation covers both the existential and universal proposition. Since the conjunct is false, the negation makes the expression true under Russelian analysis. III.III Identity statements. & substitution.

The last paradox that Russell gives us is really two problems combined into one proposition: the problem of identity and the substitution. Leibniz law of identity states that whatever property that x has and if y is identical to x then whatever is true of x is also true of y (x = y) Thus if there are terms that have different meaning but have the same relationship then they can be substituted in proposition. By being able to substitute terms for variable you can give a better analysis of the truth and falsity of a proposition, without altering the truth of falsity of the proposition. If George wanted know whether “Scott was the author of Waverley; and in fact Scott was the author of
Waverley. Hence we may substitute Scott for the author of `Waverley', and thereby prove that George IV wished to know whether Scott was Scott. Yet an interest in the law of identity can hardly be attributed to the first gentleman of Europe”.[xiii] Apparently this was a jab at the not so popular prince George , who apparently didn't know that the famous Sir Walter Scott had wrote the Waverley chronicles. “the author of Waverley is Scott”.
This is a simple statement where is the subject denotes the Author, which is predicated as be Scottish. Notice that before the subject the author is the word “the” which implicitly means : a. At least one person authored the book Waverley, and b. At most one person authored the book Waverley, and c. Whomever authored Waverley is Scotch

What we see is the proposition is again making its making a trio of generalizations. If there existed more then one author the word “the” would need to be changed in front of the subject. But since the scope of the quantifier is limited by both a. & b. The conjuncts of a. & b. make the first part of the expression existential claim that something exist, the set of a. & b. the conjunct of c. We see that individually all of a - c are necessary conditions in order to form the proposition of (1). We see jointly that there are enough necessary and sufficient conditions in order to form the proposition of (1). In notation:
a. (∃x)(WWx)
b. (∃x)(WWx & (∀y)(WWy → y=x)
c. (∀y)(WWy → y=x) & x=Scott)

Thus giving the completed proposition from 1.

1. (∃x)(WWx & (∀y)(WWy → y=x) & x=Scott)

Upon analysis we that the the grammatical form of proposition 5. is much different then its logical form. The grammatical form doesn't explicitly quantify over the object of the proposition . Another example of this is the expression “I saw nobody”. The grammatical form leads us to believe that I saw one person, however nobody is not a subject of the proposition but is a quantifier. In logical form the equivalent would be “I didn't see not one person” ~(∃x)(Aix) or the equivalent of “there is no one that I saw” (∀y) ~(Aix)[xiv]. In grammatical form the expression is confusing but when we see it in logical notation we see its has the same truth value though it can be expressed differently.

The second part of the puzzle deals with the problem of substitution
2. “George IV wanted to know whether Scott is Scott.”
If substitution is possible then changing the terms of an expression can be substituted without changing the meaning of the terms. When we switch the subject “the author of Waverley” with “Scott” we see that the substitution changes the meaning of the expression. So that the proposition would then read “the author of Waverley is Scott” and George the IV wanted to know whether Scott was Scott. There are two different logical reading of the statement about George IV. :
Primary Occurrence d. (∃x)(WWx & (∀y)(WWy → y=x) & GIV wanted to know whether x=Scott)

The scope of the universal quantifier binds the conjunct of the expression that GIV wanted to know whether wanted to know whether x=Scott, not whether Scott was Scott. Secondary occurrence: e. GIV wanted to know whether (∃x)(WWx & (∀y) (WWy → y=x) & x=Scott)

So how does this solve the puzzles? By changing the quantifier we see there are no no singular term left to substitute.

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[ i ]. Quine W.V. Words and objects Cambridge MA. MIT Press
[ ii ]. Frege Gottlob, On sense and reference [As reprinted in A.W. Moore (ed.) Meaning and Reference. Oxford: Oxford University Press.] pg. 24
[ iii ]. Ibid P. 25
[ iv ]. Ibid
[ v ]. Ibid p.26
[ vi ]. Ibid p. 28
[ vii ]. Ibid p. 30
[ viii ]. Gottlob Frege The Stanford Philsophy Encloypedia http://plato.stanford.edu/entries/frege/
[ ix ]. Ibid p. 1
[ x ]. Godehard Link (2004), One hundred years of Russell's paradox, p. 350, ISBN 978-3-11-017438-0
[ xi ]. http://en.wikipedia.org/wiki/Peter_of_Spain
[ xii ]. Searle, John Lecture he gave at UC. Berkely in Podcast Lecture 12 http://itunes.apple.com/us/itunes-u/philosophy-133-001-fall-2010/id391538483 on 10/5/10
[ xiii ]. Russell,, Bertrand: On Denoting pg. 7
[ xiv ]. Lycan WIlliamPhilosophy of Language: A Contemporary Introduction (Routledge Publishers, 1999), xvi + 243 pp. 17