• Frege’s Criticism of Statements as Sayings
• Entertaining Thought Without Assertion
  • Propositional Questions
  • Thought Without Assertion
  • Conditional Judgment
• Denial is Not a Distinct form of Judgment
  • The Varieties of Negative Thought
  • Negating the Predicate or Copula?
  • Entertaining Negative Thoughts Without Denial
• Many Judgments Lack a Definable Subject/Predicate Form
  • Conditional Judgments
  • Judgments of Fact
  • Active/Passive Transformations
• Naming’s Limited Role in Judgment
  • Coreferring Names Can Differ in Cognitive Significance
  • The Distinctive Semantic Role of Predicates
  • The Distinctive Semantic Role of Predicates is Essential for Judgment
• The Distinction Between Thoughts and Ideas
• Statements Involving Multiple Generality
• Summary

It is useful to introduce Frege’s thought in opposition to the traditional subject/predicate analysis of statement that originates with Aristotle’s Prior Analytics. I will present a much-simplified version of the subject/predicate analysis, indeed a straw man, to serve as Frege’s stalking horse. The number of valid inferences that a logic can represent is constrained by the conception of statement it involves. A logic’s treatment of valid inference depends on what it takes to be the logically relevant structure of the premises and conclusion. The difficulties encountered by the old logic in handling certain forms of valid inference were due to the conception of statement it embodied. Progress in logic required that we radically rethink the traditional conception of statement. Every major theme of Frege’s philosophy of language emerges out of his criticism of the traditional conception of a statement.

Frege’s Criticism of Statements As Sayings

On the old logic, statements are understood as sayings. One part of the statement is what we want to talk about and the other part is what we want to say about it. Thus the old logic distinguished between the subject expression and predicate expression, on the one hand, and their denotation, subject and predicate, on the other. Subject and Predicate expressions are conceived to be just names for different things. The subject is the thing the statement is about and the predicate is the thing that gets said of the subject. The copula is the expression of saying. As opposed to Aristotle, Kant would replace talk of saying by talk of judgment.

“Socrates sits”, on the subject/predicate analysis, says that sitting is predicated of Socrates. Though the copula is not explicitly represented by surface grammar “Socrates sits” can always be transformed into “Socrates is sitting” or “Socrates is a sitter”, etc. Thus on the subject/predicate analysis, the copula is conceived to be prevalent throughout all judgments.

According to the old logic, there are two forms of saying or judgment: affirmation and denial. “Socrates sits” says of Socrates that he sits. Or to put it another way, the statement affirms sitting of Socrates. In making statements not only do we affirm predicates of things but some time we also deny predicates of things. The statement “Socrates is not standing”, for instance, denies standing of Socrates. Denial is conceived to be an act of judgment distinct from affirmation. Judgment is understood as the joining or holding apart of ideas.

Frege’s criticisms:

1. We can entertain a thought without assertion
2. Negation can be involved in an unasserted thought. We have a notion of negation priori to denial
3. Many thoughts do not have a definable subject/predicate form
4. Naming has only a limited role in judgment
5. The distinction between thoughts and ideas
6. Inferences involving multiple generality

Entertaining Thought Without Assertion

On the subject/ predicate analysis, the subject is what the judgment is about and the predicate is what gets said of the subject. The copula is the expression of a judgment. The expression of judgment is thus conceived to be a constituent of the content of that judgment. Frege doubts, however, that this can be coherently maintained and thus doubts that the copula could have the expressive role assigned to it by the old logic.

Propositional Questions

Frege considers propositional questions such as:

• Did the Greeks defeat the Persians at Platea?

Propositional questions are to be answered by a yes or a no depending on whether the thought is true or false in contrast to “who,” “what,” “where,” and “when” questions such as “Who’s on first?”. Compare the propositional question to the assertion:

• The Greeks defeated the Persians at Platea.

The two sentences express the same thought but only the latter asserts it. Frege’s observation is that we must be able to grasp the sense of a question, or no answer is possible at all. This requires that we be able to entertain the thought before forming a judgment. We thus must be able to entertain the content of a judgment before judging that content, and so the content of the judgment must be specified independently of any expression of judgment.

Thought Without Assertion

The point about propositional questions is an instance of a more general point. Frege observes that we can entertain a thought without judging it to be the case. Thus, for instance, we can coherently wonder whether it will rain tomorrow without thereby judging that it will rain tomorrow.

Conditional Judgment

Conditional judgments also underscore Frege’s observation that we can entertain thoughts without judging or asserting them-that we must distinguish the act of judgment from its subject matter. Consider the sentence:

• Schmidt is the author of the incompleteness theorem.

When embedded in the antecedent of a conditional judgment or assertion, one does not thereby judge or assert that Schmidt is the author of the incompleteness theorem. One can coherently judge or assert that if Schmidt is the author of the incompleteness theorem, then Gödel is a fraud while at the same time judging or asserting that Gödel and not Schmidt is the author of the incompleteness theorem.

The subject/predicate analysis obscures or conflates the distinction between the act of judgment and its content by representing the act of judgment as part of the content (by virtue of the copula, “saying that”). Frege thus distinguishes the force of a statement (the act of judgment) and its content (its subject matter). The distinction between the act of judgment and its content or subject matter is explicitly representing in the symbolism of the Begriffsschrift by the judgment stroke and the content stroke:

Denial Is Not a Distinct Form of Judgment

The subject/predicate analysis distinguishes two types of saying or judgment: affirmation and denial. The traditional logician seems to hold that denial is distinct from affirmation or only arise in the act of denying.

The Varieties of Negative Thought

Frege invites us to consider the following types of examples:

• Zeus is immortal
• Zeus is not mortal
• Zeus lives forever

Which is the affirmation and which is the denial? What is the real distinction between affirmation and denial? We cannot say, and Frege suggests that this because the categorical distinction between affirmation and denial is illusory.

Negating the Predicate or Copula?

In denial is the “not’ a constituent of the predicate or the copula? Is there one form of denial or two:

• S is-not P;
• S is not-P?

Thus, for instance, Kant distinguishes:

• Zeus is-not mortal (negative)
• Zeus is not-mortal (infinitive)

and tries to explain the difference in vain (cf. the Jasche Logic, the Blomberg Logic).

Negative judgments for Kant are the expression of denial. Kant quite naturally held that the distinct act of denial is expressed by attaching negation to the copula, the expression of judgment. He correctly reasons that were negation attached to the predicate it would modify the content of the judgment and not the act of judgment. Judgments where negation attaches to the predicate he deems infinite judgments where infinite judgments are distinct in subject matter from affirmative judgments and distinct from negative judgments in the act of expression (denial). What is the difference between negative and infinite judgments? Kant says of infinite judgments that they affirm of the subject that it does not fall under the predicate concept. So to judge that Zeus is non-moral is to affirm of Zeus that mortality is not predicated of him. But how is this different from denying mortality of Zeus? Kant claims that negative judgments merely exclude the subject from the range of things that fall under the predicate whereas infinite judgments positively assert that the subject lies somewhere in the infinite space beyond the range of things that the predicate properly applies to. Infinite judgments are, in effect, positive judgments that the range of applicability of the predicate is restricted in not applying to the subject. But surely that is at the very least an immediate consequence of the corresponding denial if indeed we can mark a genuine distinction in content. For isn’t to say that mortality is restricted in excluding Zeus just to say that Zeus is not mortal?

Entertaining Negative Thoughts Without Denial

Frege also observes that there can be negative thoughts as well as denials. We can entertain the thought that Zeus is not mortal without judging or asserting that Zeus is not mortal. As such the negation must be part of content of the thought as opposed a distinct act of judgment.

Frege’s point is that denial is not an act distinct from judgment or assertion. Rather, denial just is judging or asserting the negation of a thought. The is reflected in the symbolism of the Begiffsschrift where the negation sign is not a distinct judgment stroke but is part of the content stroke:

Many Judgments Lack a Definable Subject/Predicate Form

The traditional logician holds that all judgment can be understood in terms of predicating something of the subject. Frege observes, however, that many judgments lack a definable subject/predicate form.

Conditional Judgments

The idea that a saying is simply predicating something of the subject doesn’t work because of the occurrence of multiple subjects. Conditional constructions thus pose a problem for the traditional logician. Consider:

• If Schmidt is the author of the incompleteness theorem, then Gödel is a fraud.

This sentence is of the form:

• If x is y, then z is w.

What is the subject of the conditional assertion, Schmidt or Gödel? We cannot say, because conditional judgments lack a well-defined subject/predicate form.

Judgments of Fact

Consider the following two judgments:

• Archimedes perished at the capture of Syracuse.
• The death of Archimedes at the capture of Syracuse is a fact.

Intuitively each expresses the same thought, but on the subject/predicate analysis they express distinct judgments. The first says of Archimedes that he perished at the capture of Syracuse. The second says of the death of Archimedes at the capture of Syracuse that it is a fact. Notice the subject of the second statement represents the whole content of the judgment, and the predicate merely converts that content into a form capable of judgment. (This example thus also supports Frege’s distinction between force and content.)

Active/Passive Transformations

Consider the following pair of judgments:

• Wellington defeated Napoleon.
• Napoleon was defeated by Wellington.

Are they equatable? Notice one is true, if both are, and each participates in the same pattern of valid inference. But the old logic would have them mean very different things. One predicates something of Wellington, while the other predicates something of Napoleon. Frege contends that active/passive transformations merely provide different grammatical access to the same thought. Logic, insofar as it is interested in the truth-conditioning structure of thought-insofar as it is interested in the content of judgment that is relevant to valid inference, must transcend the subject/predicate analysis. Frege contends that the difference between active and passive constructions is not a difference in content, but is merely rhetorical. Frege thus distinguishes between the content of a judgment and its color. Consider:

• Edgar will come out and Bernice will not.
• Edgar will come out but Bernice will not.

According to Frege, the but-statement just puts the thought differently without any difference in the content of the judgment or assertion

If one is true, both are:

• p and q
• p but q

The but-statement doesn’t modify the thought in itself but functions as commentary on the thought, for example, as the expression of surprise.

Naming’s limited role in judgment

According to the old logic, all the subject expression does is pick out something to be said about.

Coreferring Names Can Differ in Cognitive Significance

Take two subject expressions:

• Socrates is sitting.
• The teacher of Plato is sitting.

By the old logic they should say the same thing. But there is a difference. One may rationally judge or assert one without the other since the identification of Socrates with the teacher of Plato depends on the thought, unstated, that Socrates is the teacher of Plato. Frege contends that the contribution of the subject to the thought expressed isn’t exhausted by the referent. A name also has a sense. A sense is the way the referent is presented-its mode of presentation. (“Sense and Meaning”)

The Distinctive Semantic Role of Predicates

Consider the sentence:

• Socrates is mortal.

On the subject/predicate analysis, “Socrates” and “mortal” are both names of things or objects. “Mortal” names the thing that is said of Socrates. But consider:

• “Socrates” is a name of Socrates.
• *”Mortal” is a name of mortal.

The second of these cannot be said because it lacks sense-it is ungrammatical.

• “Mortal” is a name of mortality.

can be said, but the predicate, here, has been severely modified. “Mortality” is a nominalization. The predicate has been transformed into a noun phrase. It is in virtue of this transformation that we have produced a grammatical statement. Names are noun phrases while predicate expressions just aren’t.

• ”+” is a name for addition.

expresses the pretheoretical position that “+” stands for the addition function. But “+” doesn’t function grammatically in the same way as “addition”.

• *”+” is a name for +.

is nonsense.

• 2 + 2 = 4

is grammatical while

• *2 addition 2 = 4

is nonsense. Frege notes the analogy between function expressions and predicate expressions and assimilates predicates to the class of function expressions. (“Function and Concept”)

The Distinctive Semantic Role of Predicates is Essential for Judgment

There is a more fundamental difficulty. Merely naming two things (Socrates and mortality) is not yet to form a judgment or even to express a content capable of judgment. The question of truth simply cannot arise. Predicates must have a grammatical and semantic role distinct from naming if judgment is to be possible.

The Distinction Between Thoughts and Ideas

On the doctrine of sayings, expressions are conventionally associated with ideas that stand in natural presenting relations to their referents. These natural (as opposed to conventional) relations are typically conceived to be resemblance relations. Thus, for example, it is in virtue of the fact that the idea of Socrates is more similar to Socrates than to Plato that it refers to Socrates and not Plato. Judgment is a matter of the joining or holding apart of ideas. Affirming sitting of Socrates is understood to be the joining of the ideas of Socrates and sitting, whereas denying standing of Socrates is understood as the holding apart of the ideas of Socrates and standing.

Berkeley complains that only ideas can resemble ideas and so no idea can represent any non-mental thing. He thus derives support for his empirical idealism from this aspect of the doctrine of sayings.

Frege contends that ideas aren’t available for any intersubjective comparison. (Objectivity as the criterion of intersubjectivity.) Two of us literally can’t experience the same complex of ideas, but if thoughts are the bearers of truth or falsity, then all of us must be capable of literally entertaining the same thought. Depends on truths and falsehoods being among the things we want to call thoughts. Frege’s fundamental insight here is that a thought is something for which the question of truth or falsity may arise.

Statements Involving Multiple Generality

A natural account of the structure of sentences is to conceive of them as built out of their component expression in linear order, the order in which they are uttered in time. Of course there are restrictions as to what expressions are to count as meaningful. The traditional logician embraced this idea and ran into difficulty with multiple generality. Consider the ambiguity of:

• Everyone loves someone.

Two readings:

• Everyone loves someone or another.
• There is some particular person that everyone loves.

The most natural reading of “Everyone loves someone” is the first. The second reading would typically be expressed by the passive construction:

• Someone is loved by everyone.

But how, on the subject–predicate analysis, could there be such an ambiguity? There is no lexical ambiguity—the component expressions have unique meanings. And if statements are built out of their comonent expressions in linear order, then this leaves no room for any structural ambiguity.

The traditional logician conceived of “Everyone loves someone” as simultaneously constructed out of the expressions “everyone”, “loves”, and “someone”, and tried to account for the difference in terms of various doctrines concerning suppositio. Suppositio were the different ways in whcih an expression named its object. This led to ever increasing distinctions and complexity of rules.

Some remarks:

1. The number of statements to be handled is determined by the conception of statement.
2. There is a lot of language involved in the syllogistic system-complex jargon and rules.
3. There are large structures of generality that cannot be handled by the syllogistic system-generality statements involving relations.
4. No mathematical inferences can be handled through syllogisms.

Frege on generality:

• Everyone is such that [he is mortal].

Frege’s analysis breaks generality statements into two parts:

1. clause with a pronoun (a condition specifying clause); and
2. a generality prefix.

The generality prefix takes the clause and makes it into a statement. On Frege’s view, “Everyone is mortal” expresses the thought that everything that falls under the concept of humanity also falls under the concept of mortality. Similarly “Someone is mortal” expresses the thought that there is at least one thing that falls under the concept of humanity that also falls under the concept of mortality.

Consider Frege’s analysis of “everyone loves someone”. On the first reading:

• Everyone is such that [someone is such that [the first person loves the second]

where “the first person” and “the second” are pronoun analogues. (The artificial pronouns are the variables of quantification.) The inner generality prefix takes the two-place condition specifying clause and makes it into another one-place condition specifying clause:

• The first person loves the second

becomes

• The first person loves someone.

The point is clearer if we follow Frege in moving beyond natural language. The two-place condition specifying clause:

• x loves y

is transformed into a one-place condition specifying clause by the addition of the inner generality prefix:

• Some person y is such that [x loves y].

which is itself transformed into a statement by the addition of the outer generality prefix:

• Every person x is such that [some person y is such that [x loves y]]

Notice that the generality statement could have been constructed in a different way beginning with the two-place condition-specifying clause:

• x loves y

is again transformed into a one-place condition specifying clause but this time by prefixing “everyone”:

• Every person x is such that [x loves y]

which is in turn transformed into a statement with a judgeable content by prefixing “someone”:

• Some person y is such that [every person x is such that [x loves y]]

For Frege the content of a generality statement is determined by two things:

1. the compositional history of the statement; and
2. the meaning of the generality prefix.

In the symbolism of the Begriffsschrift, unlike statements in natural language, the compositional history of a statement is explicitly represented by its grammar. It is the differencce in compositional history that accounts for the ambiguity of “Everyone loves someone”. The logical structure of a statement is not linear; the logical structure of a statement, as constituted by its compoisitional history, is rather a tree.

As presented so far, the generality prefix determines what we are talking about—the range of things that are under consideration (for every whole number…, for every person x…, etc.). Notice:

• Every person x is such that [x is mortal].

Can be further regimented as:

• Every x is such that [if x is a person then x is mortal].

We have switched the function of delimiting the scope of the variable from the generality prefix to the antecedent of the conditional clause. Similarly:

• Some person x is such that [x is mortal].

is regimented as:

• Some x is such that [x is a person and x is mortal].

Unlike universal generalizations, the scope of the variable in existential generalization is delimited by a conjunct of the condition-specifying clause. We thus arrive at the modern conception of the quantifier.

Summary

1. Statements cannot be understood as sayings. According to Frege, they are the expression of thought where a thought is something for which the question of truth can arise.
2. Progress in logic depends on being able to reduce language to a small number of idioms that can be regulated and systematized.
3. Mathematical practice can reveal how to systematize an artificial idiom for language.