Download handout/homework 2 - A Fregean functional semantics, part 1

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.
I’m going to the store, and she’s buying smokes, or I’m watching T.V.
This sentence is ambiguous. If we use parentheses to disambiguate, on the one hand it could
mean “(I’m going to the store) and (she’s buying smokes or I’m watching T.V).” From this
reading of the sentence we would know that I’m going to the store. If we parsed the sentence as
“(I’m going to the store and she’s buying smokes) or (I’m watching T.V),” we wouldn’t know
that I’m going to the store.
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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 Φ is an n-place predicate and every one of α1. . . .αn is either a variables or proper
name, then Φ(α1. . . .αn) is a sentence of L.
(2) If  is a sentence of L, then  is a sentence of L.
(3) If and  are sentences of L, then (  is a sentence of L.
(4) If and  are sentences of L, then (  is a sentence of L.
(5) If and  are sentences of L, then ( is a sentence of L.
(6) If  is a wff and α is a propositional variable, then α( is a sentence of L.
(7) If  is a wff and α is a propositional variable, then α( 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 Frege’s language was the
first to be provably compositional, in that (when understood via Frege’s semantic theory) it
satisfies Miller’s
Thesis 2: The semantic value of a complex expression is determined by the semantic value
of its parts.1
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 Frege’s genius is that his semantic theory for his logic showed in detail how this is the
case (and we’ll learn this in these handouts). Again though, why does this matter? From the
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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
Method2 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.3
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 Frege’s
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
Frege’s, 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 Frege’s
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 Miller’s 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 Frege’s language4 is compositional. Then we will look at Frege’s own
argument that the semantic account for his language as a language of mathematical proof is
insufficient.
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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 Frege’s 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 Frege’s 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
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g(5) =
(5 + 6) – 4
11 – 4
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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. Frege’s 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.5
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
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(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 Frege’s 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 truthvalues; the semantic value of a sentential connective is a first-level function from truthvalues 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 predicates6 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.7
So let’s 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
simplicity’s sake let’s 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 don’t have names in any language).
sv(M) = {<Charlie, True>,<Dan, True>,<Frank, True>,<Mary, False>}
sv(F) = {<Charlie, False>,<Dan, False>,<Frank, True>,<Mary, True>}
sv(H) = {<Charlie, False>,<Dan, True>,<Frank, False>,<Mary, True>}
sv(C) = {<Charlie, True>,<Dan, False>,<Frank, True>,<Mary, False>}
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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))
2. sv(M)(sv(d)) = sv(M)Dan
3. sv(M)Dan = True
by Frege’s theory
by the model
by the model
Likewise, we can derive the semantic value of Hc as follows.
1. sv(Hc) is = sv(H)(sv(c))
by Frege’s 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(Φα) is = sv(Φ)(sv(α))
2. sv(Φ)(sv(α)) = sv(Φ)Γ
3. sv(Φ)Γ = Σ
by Frege’s theory
by the model
by the model,
Where Φ is a predicate of the formal language, α is a name of the formal language, Γ names
one of the objects denoting the names, and Σ 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. Let’s 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) = {<Charlie, Charlie, False>,<Charlie, Dan, False>,<Charlie, Frank, True>,<Charlie,
Mary, False>,<Dan, Charlie, True>,<Dan, Dan, False>,<Dan, Frank, True>,<Dan, Mary,
True>,<Frank, Charlie, False>,<Frank, Dan, False>,<Frank, Frank, False>,<Frank, Mary,
False>,<Mary, Charlie, True>,<Mary, Dan, False>,<Mary, Frank, True>,<Marie, Mary,
False>}
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Let’s 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) = {<Charlie, Charlie, False>,<Charlie, Dan, False>,<Charlie, Frank,
False>,<Charlie, Mary, False>,<Dan, Charlie, True>,<Dan, Dan, False>,<Dan, Frank,
True>,<Dan, Mary, False>,<Frank, Charlie, True>,<Frank, Dan, False>,<Frank, Frank,
False>,<Frank, Mary, False>,<Mary, Charlie, True>,<Mary, Dan, True>,<Mary, Frank,
True>,<Marie, Mary, False>}
But then we can do derivations just like we did above. Consider.
by Frege’s theory
by the model
by the model
1. sv(Ocd) is = sv(O)(sv(c),sv(d))
2. sv(O)(sv(c),sv(d)) = sv(O)(Charlie,Dan)
3. sv(O)(Charlie,Dan) = False
Thus, since sv(Φαβ) is determined by applying the semantic value of Φ to the ordered pair
containing α and β’s semantic values, every such derivation will look like the following,
1. sv(Φαβ) is = sv(Φ)(sv(α),sv(β))
2. sv(Φ)(sv(Φα),sv(Φβ)) = sv(Φ)(Σ,Ψ)
3. sv(Φ)(Σ,Ψ) = Γ
by Frege’s theory
by the model
by the model,
Where Φ is a two-place predicate of the language, α and β are names of the language, Σ is
equal to the semantic value of α, Ψ is equal to the semantic value of β, and Γ is either truth or
falsity.
I realize that sv(Φαβ) is = sv(Φ)(sv(α),sv(β)) 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 <Charlie, Dan, False> is a member of sv(0) is the fact that sv(0)(Charlie, Dan) =
False. That’s all we’re doing in the derivations above.
------------Exercise 2
Do the above type derivations and the model above to determine the truth or falsity of the
English language sentences “Charlie is taller than Mary,” “Dan is taller than Frank,” “Frank is
older than Charlie,” “Mary is older than Dan.” Your derivations must be of the same form as
the above.
------------1
Miller, 11.
[citation needed] Modern philosophy of language arguably begins with Descartes’ argument.
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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.
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[should note with reference to Frege’s 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.
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Miller, 16. Miller notes that this isn’t quite right as a sketch of Frege’s 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 Frege’s semantic
theory much more clearly.
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Footnote to self- interesting reason “course of values” for reference of predicates undermines compositionality if
you don’t 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 didn’t matter. Some theorems here?)
7
Miller, 12.
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