> orng bjbjVV 8r<r<H%><<<PPP8\LP*d00"RRR---)))))))$@,.*<-----*.$RR*/!/!/!-<R<R)/!-)/!/!V'@(RC[()*0*(x//((/<(4--/!-----**/!---*----/--------- : handout/homework 2 - A Fregean functional semantics, part 1: functions, names, and predicates
In the first handout we gave rough translations of the parts of speech of first order predicate logic in the following manner.
((x) = for all x, ((x) = there exists an x such that, (meaning, there exists at least one x such that)((A) = It is not the case that, (A) (A ( B) = (If A, then B) (A ( B) = (A, or B) (A ( B) = (A, and B) (Rx) = x has the property R, (Rxy) = x is in the relation R to y (Rxyz) = w, x, y, and z are in the relation R
(Rxyzz1) = x, y, z, and z1 are in the relation R
Etc. (one can do this for any finite number of variables or constants)
We did not rigorously define a grammar for such a language though.
Why do such a thing? Remember that Frege developed his language in the hope that one could formalize mathematical reasoning so that each step in a mathematical proof is shown to be clearly unproblematic. Then the hope was that difficult mathematical theorems could be shown to be provable from clear premises that are obviously true. To the extent that one can do these two things (provide a logic where each step in a proof is clearly unproblematic and also specify unproblematic premises), then one will have shown that the mathematical theory in question is not philosophically problematic.
But this will only work if the grammar for the logic can itself be characterized such that the question of whether a sentence is a sentence of the logical language is itself clearly unproblematic. The limiting case of unproblematic here (and in the case of a system of proof) is if we can specify dumb mechanical procedures that determine whether a string of symbols is a sentence of the language or not (or a proper step in a proof, for that matter). This is always possible for a logical language if one can provide a recursive specification of the set of sentences in the language.
Moreover, in order to come up with a precise notion of proof, it is absolutely essential that certain forms of ambiguity are prohibited. For example, consider the following English language sentence.
Im going to the store, and shes buying smokes, or Im watching T.V.
This sentence is ambiguous. If we use parentheses to disambiguate, on the one hand it could mean (Im going to the store) and (shes buying smokes or Im watching T.V). From this reading of the sentence we would know that Im going to the store. If we parsed the sentence as (Im going to the store and shes buying smokes) or (Im watching T.V), we wouldnt know that Im going to the store.
By formalizing the logical languages, we prohibit all such ambiguities.
Vocabulary:
All capital English letters (A, B, C. . . .Z), as well as all numerically subscripted capital Zs (Z1,Z2, Z3. . .), are predicates, each one of one or more places.
The lower case English letters (a, b, c, d), as well as all numerically subscripted lower case ds (d1,d2, d3. . .), are proper names.
The lower case English letters (x, y, z), as well as numerically subscripted lower case zs (z1,z2, z3. . .), are variables.
(, (, (, and ( are propositional connectives (respectively: negation, conditional, disjunction, and conjunction).
( and ( are quantifiers (respectively: universal and existential).
( and ) are parentheses.
Formation rules:
(1) If is an n-place predicate and every one of 1. . . .n is either a variables or proper name, then (1. . . .n) is a sentence of L.
(2) If G is a sentence of L, then (G is a sentence of L.
(3) If G and F are sentences of L, then (G ( F) is a sentence of L.
(4) If G and F are sentences of L, then (G ( F) is a sentence of L.
(5) If G and F are sentences of L, then (G (F) is a sentence of L.
(6) If G is a wff and is a propositional variable, then ((G) is a sentence of L.
(7) If G is a wff and is a propositional variable, then ((G) is a sentence of L.
(8) All and only the sentences of L of L are generated by the above 7 rules.
If we were to spend the time checking, we would see that all of the logical sentences given in the previous handout can be constructed from the above vocabulary and rules.
The logical tradition that stems from Frege and others is philosophically interesting for a number of reasons. First and foremost, philosophers have always been interested in codifying valid argumentation. This is because philosophy itself consists in using reason to produce arguments concerning the things that centrally concern us as humans. However, the Fregean revolution is of special interest to philosophers of language since Freges language was the first to be provably compositional, in that (when understood via Freges semantic theory) it satisfies Millers
Thesis 2: The semantic value of a complex expression is determined by the semantic value of its parts.
Remember that the semantic value of an expression is defined as that feature of it which determines whether sentences in which it occurs are true or false. When put together with Thesis 2, this means that the truth or falsity of a sentence is a function of properties of parts of that sentence.
Part of Freges genius is that his semantic theory for his logic showed in detail how this is the case (and well learn this in these handouts). Again though, why does this matter? From the perspective of the philosophy of mathematics and logic, it matters because logically valid proofs must show how it is not logically possible for the premises to be true and the conclusions false, and this ends up requiring a theory of how the logical parts of speech (the logical operators above) contribute to the truth or falsity of sentences in which they occur.
From a broader perspective, compositionality has even more interest. In Descartes Discourse on Method the actual argument given for the conclusion that the mind/soul is different from the physical brain involves language. Descartes argues that the brain behaves purely mechanistically, like a complicated machine. But since human language is not something a machine could do, this proves that the human mind is not purely mechanistic. Therefore the human mind is more than just the brain. And the reason Descartes thought that human language was non-mechanical was because of the massive creativity involved in language use. Most sentences we understand and speak have never been uttered before, and will never be uttered again. Yet, for the most part, without effort we manage to use and understand language. Descartes thought it impossible that a piece of biological clockwork (the brain) could do anything like this.
One way to respond to Descartes is to argue that the linguistic creativity he noticed is not quite as creative as he thought. In particular, if there are a finite number of words, a finite number of syntactic principles by which those words and the resulting phrases can be combined, and a finite number of semantic principles governing how meaning of phrases and sentences is a function of those words and how they are combined, then it seems more plausible that a machine-like entity (assuming with Descartes that the brain is machine-like) could master language. But Freges syntactic requirements for his logical language combined with his form of compositionality (the way in which Thesis 2 is worked out in his semantic theory) actually ends up showing in detail a language with a finite number of words, a finite number of syntactic principles by which those words and the resulting phrases can be combined, and a finite number of semantic principles governing how meaning of phrases and sentences is a function of those words and how they are combined.
So if one could plausibly argue that human languages are similar enough to languages like Freges, then one would have a response to Descartes. This is in part why so much contemporary philosophy of language has been a footnote to Frege.
One more piece of the puzzle- to answer Descartes challenge, it must be the case that the meaning (that which we understand) of sentences is provably compositional. But even if Freges theory of semantic content works, we just have that truth or falsity is compositional. Remember that the semantic content of a sentence for Frege is just truth or falsity (this is Millers Thesis 1). Well, arguably, part of the understanding of a sentence is the understanding of what it is for that sentence to be true or false. But then if truth or falsity itself is a function of the semantic content of the expressions of sentences, a big part of our understanding will be grasp of the rules that determine how this works.
Here we want to see how Freges language is compositional. Then we will look at Freges own argument that the semantic account for his language as a language of mathematical proof is insufficient.
In mathematics, when we talk about some things being determined by other things, we usually are talking about functions. So Thesis 2 could have said something like, The semantic value of a complex expression is a function of the semantic value of its parts. Unfortunately, at the time Frege was writing, mathematics did not have a well worked out theory of functions (that we do is probably in part due to Freges advances), so we have to reconstruct his views a little bit here.
In particular, today mathematical functions can be understood either extensionally or intentionally (we will see in a few days how this very distinction follows from Freges insights). Intensionally, a function is a rule or procedure that takes you from one ordered group of values to another value. So consider the following two functions:
f(x) = (x + 4) 2,
g(x) = (x +6) 4.
Intuitively, they are different functions because they give different procedures for solving them. So:
f(5) = (5 + 4) 2
9 2
7
g(5) = (5 + 6) 4
11 4
7
So we can say that by the intensional notion of functions, f and g are different functions.
However, there is clearly a sense in which they are the same functions. For any numerical input, f and g return the same values. By the extensional notion of functions, when this happens we say that the two functions are actually one and the same, that we have one function characterized differently. When we think this way, we characterize functions just in terms of the set of inputs and outputs, for one-place functions like f and g, this will be a set of ordered pairs. If f and g are defined on the natural numbers ({0,1,2,3,4. . . .) then we have:
f = g = {<0,2>,<1,3>,<2,4>,<3,5>. . . .}
So extensionally, functions can be understood as just a set of inputs and outputs. Freges theory of semantic value is extensional in this sense, which is why we have:
Thesis 6: Functions are extensional: if function f and g give the extension, then f = g.
When modern mathematicians are being clear they almost always treat functions extensionally, referring explicitly to rules or procedures when intensional functions are being compared. For Frege, what is amazing is that his basic insights into the semantics of his language can be presented entirely in terms of extensional functions. Again, the basic compositionality claim (Thesis 2) can be understood as the claim that the truth or falsity of a sentence is entirely a function of the semantic value of the parts of that sentence, and (for the language presented above) one can present Freges theory entirely in terms of our modern extensional notion of function. Thus, we get:
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.
Function, function, function. Pretty cool. Of course this only works if one can actually specify the functions in question. But for the above language, we can.
To do this we will consider part of the fragment of the above language we gave in the previous handout. That is, our language will be just like the one defined above, except that the only proper names are c, m, and d, and the only predicates are M, F, H, C, T, and O. For simplicity, we will also amend the above to be able to stipulate that M, F, H, and C are one-placed predicates, in that they can only bind to one proper name or variable at a time (e.g. Mc), and T and O are two place predicates (e.g. Tdc). Here we will be able to show how if one knows the semantic values of the names and predicates then one can determine whether sentences in which they occur are true or false. First we have:
Thesis 4: The semantic value of a proper name is the object which it refers to or stands for.
So lets assume that the semantic value of c is Charlie, m is Mary, and d is Dan. We can write this functionally.
sv(c) = Charlie
sv(m) = Mary
sv(d) = Dan
Again, we are pretending here that Charlie, Mary, and Dan are actual objects in the world. The semantic values of the names c, m, and d are the things to which those names refer.
Then, the semantic values of predicates are functions from objects to truth values. Again, since we understand these extensionally, we can treat one place predicates as ordered pairs. For simplicitys sake lets assume the concrete objects in the universe just consist in Charlie, Mary, Dan, and a short, canine hermaphrodite we can call Frank, but who has no name in the language (surely most things in the universe dont have names in any language).
sv(M) = {,,,}
sv(F) = {,,,}
sv(H) = {,,,}
sv(C) = {,,,}
From this one can determine the semantic values of an atomic formula (sentence with no quantifiers or propositional variables) by applying the semantic value of the predicate (which is a function) to the semantic value of the names. That is sv(Md) is equal to sv(M)(sv(d)). Since the semantic value of d is Dan, and the semantic value of M takes Dan to true, we know that the semantic value of Md is True. We can write this as a derivation in the following manner.
1. sv(Md) is = sv(M)(sv(d)) by Freges theory
2. sv(M)(sv(d)) = sv(M)Dan by the model
3. sv(M)Dan = True by the model
Likewise, we can derive the semantic value of Hc as follows.
1. sv(Hc) is = sv(H)(sv(c)) by Freges theory
2. sv(H)(sv(c)) = sv(H)Charlie by the model
3. sv(H)Charlie = False by the model
-------------
Excercise1
Write out the above kind of derivations Mc, Fd, Hf, and Cm. Note that each one will have three steps, of the form,
1. sv() is = sv()(sv()) by Frege s theory
2. sv()(sv()) = sv() by the model
3. sv() = by the model,
Where is a predicate of the formal language, is a name of the formal language, names one of the objects denoting the names, and is True or False.
-------------
All of the mathematical functions discussed above have been one-place functions, that is functions that take in one argument to return a value. However, most interesting mathematical functions take more than one argument. For example the addition and multiplication functions take two arguments and return values. And just like extensional one-place functions can be understood in terms of ordered pairs, two-place functions can be understood in terms of ordered triplets (e.g. + = {<0,0,0>, <0,1,1>, <1,1,2>, <1,0,1>, <0,2,2>, <1,2,3>, <2,2,4>, <2,1,3>, <2,0,3>, <0,3,3>. . .}). In fact any n place extensional function (function that takes n number of inputs) can be represented as a set of n+1-tuples in this manner.
From this we can see how n-place predicates are handled in the Fregean way. Lets say that T intuitively means is taller than and that Dan is taller than Mary, who taller than Charlie, who is taller than Frank. Then, the semantic value for T is the following.
sv(T) = {,,,,,,,,,,,,,,,}
Lets say that S intuitively means is older than and that Mary is older than Dan, who is older than Frank, who is older than Charlie. Then, the semantic value for O is the following.
sv(O) = {,,,,,,,,,,,,,,,}
But then we can do derivations just like we did above. Consider.
1. sv(Ocd) is = sv(O)(sv(c),sv(d)) by Freges theory
2. sv(O)(sv(c),sv(d)) = sv(O)(Charlie,Dan) by the model
3. sv(O)(Charlie,Dan) = False by the model
Thus, since sv() is determined by applying the semantic value of to the ordered pair containing and s semantic values, every such derivation will look like the following,
1. sv() is = sv()(sv(),sv()) by Frege s theory
2. sv()(sv(),sv()) = sv()(,) by the model
3. sv()(,) = by the model,
Where is a two-place predicate of the language, and are names of the language, is equal to the semantic value of , is equal to the semantic value of , and is either truth or falsity.
I realize that sv() is = sv()(sv(),sv()) is a mouthful. Think about addition again. When we represent addition as a set of ordered three-tuples (+ = {<0,0,0>, <0,1,1>, <1,1,2>, <1,0,1>, <0,2,2>, <1,2,3>, <2,2,4>, <2,1,3>, <2,0,3>, <0,3,3>. . .}) what we are saying is that + is applied to any of the first two members uniquely yields the third member. So the fact that <0,1,1> is a member of the set consisting of addition is the fact that +(<0,1>) = 1. Likewise, the fact that is a member of sv(0) is the fact that sv(0)(Charlie, Dan) = False. Thats all were doing in the derivations above.
-------------
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-------------
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PAGE 2
Miller, 11.
[citation needed] Modern philosophy of language arguably begins with Descartes argument.
Incidentally, this was the reason Cartesians thought that animals did not have minds. Since language was the reason minds are different from mere brains, and animals did not possess language, it followed that animals did not possess minds.
[should note with reference to Freges Begrifschrift (sp?) earlier] It is important to note that the actual logical symbols Frege used are completely different from the symbols that have become standard in modern logic. But in the other important respects, the language is the same.
Miller, 16. Miller notes that this isnt quite right as a sketch of Freges views, since understanding functions this way is post-Fregean. As I understand Frege, he explicitly only thought of functions intensionally. This being said, once we understand them extensionally we can understand the simplicity and systematicity of Freges semantic theory much more clearly.
Footnote to self- interesting reason course of values for reference of predicates undermines compositionality if you dont have multiple domains- with extensional function reading you get the anti-extension too and hence the whole domain. Frege had a fixed domain so the difference didnt matter. Some theorems here?)
Miller, 12.
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