• The Project
• The Primacy of Judgmeent
• Compositionality and the Identity of Thought
• Subject and Predicate, Argument and Function
• The Fomalism
  • The Primitives
  • The Definitions of Conjunction and Disjunction
  • Quantification
• Summary

The Project

Frege’s logic is post-psychologistic. Thoughts are not ideas. Whereas ideas are subjective, thoughts are objective. They are thus like physical bodies except that they are nonspatial like ideas. Judgment and assertion insofar as they express thoughts are governed by the norm of truth. A judgment or assertion is correct, in the semantically relevant sense, just in case it is true. Logic is concerned with the Laws of Truth and not the Laws of Mind. Frege doesn’t employ the concept of truth in his Laws of Truth. To maintain that a thought is true is simply to judge or assert that thought. Thus the notion of truth is redundant as well as primitive philosophically. It is no use to speculate what the truth of a true thought consists in. If neither traditional grammar nor psychology provide a foundation for logic what does?

Frege contends that there is no need to appeal to any special science to provide a foundation for logic. Logic is valid through the sheer conceptual structure of its syntax. The grammar of natural language does, however, provide clues.

Frege was led to his logical investigations by the demands of his mathematical research. Frege, following Kant, describes his logic as formal logic. By this he does not mean that it is the study of an unintepreted calculus, rather that by means of it valid inferences may be performed without supplementation of intuition. Frege distinguished between two kinds of truths:

• truths that can be proved by pure logic
• truths whose proof requires the supplementation of empirical intuition.

(In the Foundations, Frege will propose a more complex taxonomy.) Frege had sought to establish that arithmetic belonged in the former class of truths. However, he found himself hampered by the language of arithmetic. Though the notation for the material content of arithmetic was well developed, its formal component, that which gives expression to logical relations, was undeveloped. Moreover, natural language proved too imprecise for this purpose. Frege was thus led to his conceptual notation (the Begriffsschrift, literally “the concept script”), in order to determine whether arithmetic could be derived from the laws of logic.

Two observations:

• Given this project, he could not follow Boole in extending the conceptual content of arithmetical symbols to express logical relations. A logical notation must receive independent development, if one is to show that arithmetic can be derived from logic.
• Nonetheless mathematical practice could reveal how to regulate and systematize a notation adequate for the purposes of logic. The most important effect of this in Frege’s logic is his replacement of talk of “subject” and “predicate” with that of “argument” and “function”. Another important effect of this in Frege’s logic is his use of variables in the analysis of generality.

A conceptual notation adequate to Frege’s purposes had to satisfy a number of stringent requirements:

1. Every expression must be unambiguous, otherwise there is the danger of apparently valid arguments being invalid because of the fallacy of equivocation
2. All assumptions must be explicitly stated since any implicit assumption might be the tacit product of intuition
3. Only content relevant to logical inference must be epressed. (Thus the difference in color between “and” and “but” would not be expressed since these expressions are logically equivalent.)
4. The modes of inference must be as simple and few, and must be syntactically definted so that adeherence to grammar will ensure the correctness of reasoning.
5. The logical relations among statment must be visually evident. To this end, Frege exploited the two-dimensonality of a writing surface devising a two-dimensional display with the intention of making the notation as easy to read as possible.

Unlike the first four requirements, the last is not motivated by philosophical or logical concerns but is purely pragmatic. The pragmatic end was both a success and a failure. It is a success, since once the two-dimensional conceptual notation is learned, the validity of an inference can be visually determined. It is a failure, since the novelty and unfamiliarity of the two-dimensional notation proved to be an obstacle to the reading and understanding of Frege’s work.

The Primacy of Judgment

The traditional logician took it for granted that subject and predicate terms could be understood independently of their role in judgment and that judgment could be explained in terms of the joining or holding apart of the ideas they denote. A theory of judgment consists of an account of how the mind combines these ideas and how the resulting judgment is correct or incorrect depending which ideas are combined and how. A theory of valid inference, in turn, explains how the logical connections between judgments depend on how their constituent ideas are combined.

Frege, following Kant, rejects this order of explanation. Kant claimed that the fundamental unit of cognition or understanding is judgment: “all acts of understanding can be reduced to judgments, the understanding may be defined as the faculty of judgment.” Subject and predicate terms are understood solely in terms of their role in judgment. A concept for Kant is a predicate of a possible judgment which is why “the only use which the understanding can make of concepts is to form judgments by them.” One always begins with the content of a possible judgment, and anything else has a content only insofar as it contributes to the content of a possible judgment.

Reviewing his life’s work in 1919, Frege writes:

What is distinctive about my conception of logic is that I begin by giving pride of place to the content of the word “true”, and then immediately go on to introduce a thought as that to which the question “Is it true?” is in principle applicable. So I do not begin with concepts and put them together to form a thought or judgment: I come by the parts of thought by the analysis of the thought. (“Notes for L. Darmstädter”)

Just as Kant claims that judgment is the minimum unit of cognition or understanding, Frege contends that a thought, that for which the question of truth arises, is the minimum with which one can do something—perform an act of assertion or judgment that is normatively evaluable in terms of the notions of truth or falsity. Thus in the Begriffsschrift and the Grundlagen a thought is described as “the content of a possible judgment” or “a judgeable content”. The content of subsentential expressions are arrived at by analyzing the thought expressed, and such expressions have a content only insofar as they contribute the truth conditioning structure of the thought. In a response to a review of the Begriffsschrift (“Boole’s Logical Calculus and the Begriffsschrift”) Frege summarizes his procedure thus:

I start out from judgments and their contents, and not from concepts … instead of putting a judgment together out of an individual as subject and an already previously formed concept as predicate, we do the opposite and arrive at a concept by splitting up the content of a possible judgment.

(It is for this reason that Frege will later express dissatisfaction with his description of his logic as a Begriffsschrift or concept-script which perhaps suggests that concepts and not judgeable contents are the primary unit of logical analysis.)

The thought expressed by a judgment or assertion is that part of its content that is relevant to the Laws of Truth (its conceptual content). A thought is that for which the question of truth arises and a thought’s identity is what it takes to make it true. What makes a sentence true or false for Frege is not so much the set of truth-determining circumstances (truth-conditions, in contemporary philosophical parlance), but the way such circumstances are structured.

On the old logic, the copula was the expression of jdugment and was understood to be prevalent throughout all judgment. This obscured or conflated what, for Frge, was an important distinction between the act of judgment (or force of assertion) and the content of judgment (or asssertion). This is explicitly represnted in the symbolism of the Begriffsschrift. The content of a judgment is represented by the content stroke:

Compositionality and the Identity of Thought

In the Begriffsschrift the identity of thought just is the identity of logical power. Two sentences express the same thought or share the same conceptual content just in case they participate in the same pattern of valid inference. If, for every possible valid inference in which a sentence participates, it is possible to substitute another sentence for it without affecting the validity of the inference, then the two sentences share the same conceptual content or express the same thought. (In the Begriffsschrift content is thus conceived as substitutional invariance of inferential role.) Logically equivalent sentences share the same conceptual content. Thus all logical truths are the expression of a single thought.

Frege’s doctrine concerning the identity of thoughts shifts in “Sense and Reference”. Sentences express the same thought just in case they are equivalent in cognitive value.

Implicit in Frege’s statement of the identity of thought, is his principle of compositionality. Let S and S’ be sentences, then Frege claims that:

S and S’ express the same thought just in case for all arguments in which S participates, the result of substituting S’ in for at least one occurrence of S in the argument does not affect the validity of the argument.

The principle can be generalized to subsentential expressions:

e and e’ share the same conceptual content just in case for all arguments in which some sentence with e as constituent occurs, S[…e…], the result of substituting e’ in for at least one occurrence of e, S[…e’…], does not affect the validity of the argument.

The principle can be generalized further to divorce it from the doctrine, peculiar to the **Begriffsschrift*, that conceptual contents are identified with inferential role:

e and e’ share the same conceptual content just in case for all sentences, S[…e…], in which e occurs, the result of substituting the e in for e’, S[…e’…] is such that S[…e…] and S[…e’…] express the same thought.

In general then the meaning of a complex expression depends upon:

• the logico-syntactic structure of the complex expression;
• the general meaning of the mode of combination;
• the conceptual content of each component.

The compositionality principle in its most general form is the guiding principle of Frege’s logic and philosophy of language. Two motivations for the compositionality principle in Frege’s Logic:

1. In making the compositional structure of a sentence explicit (which reflects the truth-conditioning structure of the thought expressed) one reveals the logical connections with other sentences.
2. There are indefinitely many logical truths and no way to list them all. If complex sentences are systematically constructed out of simpler expressions, then the entire structure of logical space can be described simply in terms of a finite list of axioms and a single rule of inference.

Subject and Predicate, Argument and Function

In the preface to the Begriffschrift, Frege suggests that traditional logic has been hampered by adhering to ordinary language and grammar too closely reamrking that “[i]n particular, I believe that the replacement of the concepts subject and predicate by argument and function will prove itself in the long run”. This is a prime example of how mathematical practice can regulate and systematize a notation adequate to logic.

Frege maintains that active/passive transformations such as:

• At Platea the Greeks defeated the Persians
• At Platea the Persians were defeated by the Greeks

are logically equivalent—each can be substituted for the other in an argument without affecting the validity of the argument. They thus share the same content, but according to the traditional logician, they say quite different things, the first says of the Greeks that they defeated the Persians at Platea and the second says of the Persians that they were defeated by the Greeks at Platea. If, however, logic is solely concerned with that aspect of meaning relevant to truth and logical inference, they should be treated as equivalent in content. In this way, the traditional distinction between subject and predicate is inadequate for the formulation of a conceptual notation for logic. In its stead, Frege proposes to replace talk of subjects and predicates with talk of arguments and functions. We will return to Frege’s functional analysis of a statement when we discuss generality statements since Frege is only led to discern semantically significant subsentential structure in his analysis of generality statements.

The Formalism

Frege systematized the Laws of Truth in his artificial language, the Begriffsschrift (concept-script). The Laws of Truth were represented by nine axioms, a single rule of inference (detachment), and a substitution rule which though not explicitly stated may be reconstructed from the structure of substitutions he licenses. Frege’s logic comprises what we would now describe as the propositional calculus, and first- and second-order logic with identity. Frege’s axioms are not the axiom schemata of modern logic. Axiom schemata are formulated in terms of schematic letters and represent an infinite number of axioms that result from substituting well-formed sentences in the artificial language for the schematic letters. There are no axiom schemata in Frege’s logic. Frege’s laws say something about things in general in a general way.

Frege’s logic thus has a normative and a descriptive dimension. Normative in that the laws are prescriptions governing the expression of thought in judgment and assertion, descriptive in the sense that the laws are taken to describe the most general features of the world—its logical structure. Thus Frege describes “fundamental principles of thought…in order to transform them into rules of use of our signs.” Thoughts thus have a prelinguistic structure to which their verbal expression is responsible. (Frege’s platonism receives its mature expression in the “Thought”.) Sometimes Frege uses the term the Laws of Truth to denote the descriptive aspect of logic and the Laws of Thought to denote its normative aspect.

The Primitives

Frege’s logic has two sentential connectives, negation and the conditional:

Notice, the sign for negation is appended to the content stroke and not the judgment stroke. The horizontal stroke preceding the negation sign is content stroke of the negated sentence and the horizontal stroke after the negation sign is the content stroke of the embedded sentence. Thus implicit in Frege’s syntax is his doctrine that negation is a feature of a judgment’s subject matter and not an act distinct from judgment or assertion. There are two cases to consider:

1. A is affirmed
2. A is denied

Negation is understood as excluding the first case. One may affirm the negation of A just in case one denies A.

Consider the conditional:

What follows the bottom content stroke after the vertical stroke is the antecedent of the conditional and what follows the top content stroke after the vertical stroke is the consequent of the conditional.

Four cases to consider:

1. A is affirmed, B is affirmed
2. A is affirmed, B is denied
3. A is denied, B is affirmed
4. A is denied, B is denied

Frege’s conditional is understood as excluding the second case, the case where A is affirmed and B is denied. One may not affirm “If A, then B” if A is affirmed and B is denied but may affirm it otherwise. Difference between Frege’s conditional and conditionals in natural language. Frege notes that his conditional differs in meaning from conditionals in natural language which he takes to express a causal or subjunctive connection between antecedent and consequent. This needn’t be the case in the symbolism of the Begriffsschrift. “If Gödel is the author of the incompleteness theorem, then Socrates is wise” may be true, though there is no connection whatsoever between the wisdom of Socrates and the author of the incompleteness theorem. While Frege perhaps overestimates the causal import of natural language conditionals, he was perhaps right in denying that they are truth-functional.

Frege here anticipates the truth-tabular representation of the meaning of logical connectives with one important difference—Frege intends his remarks as an informal explication of the meaning of an interpreted symbol and not, as in its modern presentation, as a stipulation as to the meaning of an uninterpreted symbol. For the most part, the metalogical perspective common to modern presentations of logic is foreign to Frege. (There are, however, metalogical proofs in The Basic Laws of Arithmetic, for example, the proof of referentiality of extension terms.) One may object that Frege’s explanation proceeds in terms of the notions of affirmation and denial and not in terms of truth and falsity. But according to Frege, to recognize a thought as true is simply to judge or assert it. Frege’s conception of truth as redundant is implicit in his procedure in the Begriffsschrift.

Frege’s single rule of inference is modus ponens or detachment. There are four cases to consider:

1. A is affirmed, B is affirmed
2. A is affirmed, B is denied
3. A is denied, B is affirmed
4. A is denied, B is denied

Frege explains the validity of this inference as follows: The assertion of the conditional excludes the second case and the assertion of A excludes the third and fourth cases. That leaves the first case according to which B is affirmed. Thus B follows of necessity from the conceptual structure of the contents of the premises. The analysis of complex thoughts into their constituent thoughts and the manner in which they are combined allows Frege to explicitly display the logical connections between them.

The Definitions of Conjunction and Disjunction

Frege goes on to explain how the other logical connectives can be represented in terms of negation and the conditional. Consider disjunction, “or”. Frege notes that in natural language disjunction receives an inclusive and exclusive sense. Exclusive disjunction is typically signaled by the construction “either…or…”. Thus to assert that “A or B” in the exclusive sense is to assert that either A obtains or B but not both. Consider again the four relevant cases:

1. A is affirmed, B is affirmed
2. A is affirmed, B is denied
3. A is denied, B is affirmed
4. A is denied, B is denied

Exclusive disjunction excludes the first and fourth cases. Inclusive disjunction on the other hand excludes only the fourth case. Thus to assert “A or B” in the inclusive sense is to claim that at least A or B obtains and possibly both.

Inclusive disjunction may be represented in Frege’s symbolism as:

This asserts that the case where A is denied and the negation of B is affirmed does not obtain. It thus excludes the fourth case.

Consider:

This asserts that the case where B is affirmed and the negation of A is denied does not obtain. Which is just to say that the possibility of affirming both A and B does not exist or that A and B exclude each other. That is, the fourth case is excluded.

Exclusive disjunction can thus be represented as meaning:

and

But how is conjunction to be represented in the notation of the Begriffsschrift? “A and B” may be affirmed just in case A may be affirmed and B may be affirmed. A conjunction thus excludes the second through fourth cases. Notice these are the cases under which:

may be affirmed, so the conjunction may be represented as:

which explicitly excludes these cases. Putting this together “Either A or B” is represented either as:

or equivalently:

Quantification

The portion of Frege’s logic that we have been discussing so far constitutes the propositional calculus. At this point, Frege has only been concerned with sentences and their combinations into complex sentences. Frege is only led to discern semantically significant sub-sentential structure through his analysis of generality. It is only by explicitly representing the logical form of generality statements can the logical connections between them be made explicit. It is because Frege is led to discern the parts of a thought in his account of the validity of inferences involving statements of generality, that he refers to generality as a method of concept formation (thus reflecting his doctrine of the primacy of judgment).

Essential to his account of generality is his functional analysis of the conceptual content of sentences (which he understands as mapping the truth-conditioning structure of thought). Consider the sentence:

Socrates is wise

It is possible to substitute the expression “Plato” for the constituent expression “Socrates” to arrive at another meaningful sentence:

Plato is wise.

By a functional expression, Frege means that part of the sentence that is invariant under the substitution, e.g.:

( )is wise.

Those expressions that are substituted one for another are the arguments of the function. Formalizing this, we get the following grammatical criterion of functional expressions:

Let S[…e…] be a complex expression with at least one occurrence of the constituent expression e. Let e and e’ be expressions that belong to the same syntactic category. S[…( )…] is a functional expression just in case it remains invariant under the substitution of e’ for at least one occurrence of e while preserving well-formedness.

There is a problem with Frege’s criterion: it overgenerates. Specifically, expressions which are not functional expressions count as functional expressions by this criterion. Consider the sentences:

Socrates is wise.

Socrates is bald.

Since “Socrates” remains invariant under a substitution that preserves well-formedness, it is a functional expression. In later work, when Frege draws the distinction between complete and incomplete expressions, this will provide him with the resources to ammend his criterion so that it does not overgenerate in this way:

Let S[…e…] be a complex complete expression with at least one occurrence of the constituent expression e. Let e and e’ be complete expressions that belong to the same syntactic category. S[…( )…] is a functional expression just in case it remains invariant under the substitution of e’ for at least one occurrence of e while preserving well-formedness.

The thought expressed by a sentence doesn’t necessarily have a unique functional analysis. Consider:

Cato killed Cato.

If we substitute the expression “Cicero” for the first occurrence of the expression “Cato” to arrive at

Cicero killed Cato

the resulting functional expression, that which is invariant under the substitution, is the expression:

( ) killed Cato

which can be understood as expressing the property of having killed Cato. However, if we substitute “Cicero” for the second occurrence of “Cato” to arrive at:

Cato killed Cicero

and the resulting functional expression:

Cato killed ( )

expresses the property of having been killed by Cato. Moreover, if we substitute the expression “Cicero” in for both occurrences of “Cato” we arrive at the functional expression:

( ) killed ( )

which expresses the property of having killed oneself. Exactly how a thought is functionally decomposed is only important in the analysis of generality.

A number of points are worth making:

1. Frege is self-consciously extending the domain of application of the notion of a function as it is found in analysis. Predicate expressions are understood as a subclass of the class of functional expressions and the contents of predicates (concepts in Frege’s special sense of the term) are understood as a subclass of the class of functions. Not only is Frege led to his work in logic by the demands of his mathematical research, but reflection on mathematical practice can reveal the methods required for the logical analysis of statements.
2. This is a nice example of Frege’s procedure in the grammatical analysis of sentences. For Frege a syntactic category is an equivalence class of expressions under the relation of well-formedness. Two expressions belong to the same syntactic category just in case one is substitutable for the other while preserving well-formedness. For Frege, names and sentences are the primitive syntactic categories and all others can be derived in terms of these. Thus predicate expressions (understood as a subclass of the class of functional expressions) can be understood as mapping names onto sentences, N/S. Though Frege does not explicitly do so, this can be formalized and the resulting theory is known as category theory. Category theory is equivalent to the phrase structure component of modern syntactic theories.
3. If “Socrates” is the argument of the functional expression “( ) is wise” what is its value? Frege is inexplicit on this point in the Begriffsschrift and the notion of a concept (i.e., the content of a predicate conceived as a functional expression) is successively refined throughout his philosophical career. In “Function and Concept”, Frege explicitly claims that a concept is a function from an object to a truth value. This doctrine is implicit in his procedure in the Begriffsschrift.

Explicit statements of generality are made in terms of a generality prefix or quantifier. Frege’s logic makes use of a single quantifier, the universal quantifier, whose order is determined by the type of variable it binds, but he explains how existential quantification may be explained in terms of it. In his logical notation a quantifier is represented by a concavity in the content stroke and the variable that it binds occurs as a Gothic letter within the concavity:

Frege explains the quantified statement:

as meaning that the functional expression “Φ” is true of whatever argument is substituted for the occurrence of the variable in “Φ(a)”. It can thus be read as “for every a ‘Φ(a)’ is true”, or simply “for every a Φ(a)”“.

Frege remarks that it would be a mistake to conceive of the quantifier as an argument of the functional expression. Consider:

• The number 20 can be represented as the sum of four squares.
• Every positive integer can be represented as the sum of four squares.

It might appear that the quantified statement was arrived at by substituting “every positive integer” in for “the number twenty”. Frege observes that these expressions denote concepts of different ranks. Let functions that take objects as arguments be level 1 functions. A level 2 function would be a function that takes level 1 functions as arguments. A level 3 function would be a function that takes a level 2 function as its argument, and so on. There thus emerges a hierarchy of functions whose level depends on the level of its argument. According to Frege, quantifiers are second level functions; they map level 1 functions onto truth-values. The quantified assertion maps the function Φ onto the True just in case for every argument that may be substituted in for the variable a is true. Thus though the surface grammar of the two statements:

• The number 20 can be represented as the sum of four squares.
• Every positive integer can be represented as the sum of four squares.

are apparently similar, this obscures the syntactic and logical difference between the expressions “The number twenty” and “every positive integer”. Whereas the former belongs to the syntactic category N, the latter belongs to the syntactic category N/S/S, that is a quantifier is a syntactic category that maps predicate expressions, N/S onto sentences S.

The negation of the quantified statement is written be writing the negation sign immediately to the left of the concavity:

This asserts that it is not the case for every a Φ(a). It is thus not the case that for every argument that may be substituted for the variable “Φ(a)” is true. For some argument that may be substituted in for the variable “Φ(a)” is false. The assertion thus has the same conceptual content as the assertion that some a is not Φ. If the negation sign is written to the right of the concavity conceptual content of the judgment shifts:

This asserts that for every a it is not the case that Φ(a). The assertion is true just in case for every argument that may be substituted in for the variable it is not the case that “Φ(a)”. This is just another way of asserting that no a is Φ. The denial of this statement has the same conceptual content of an existential generalization:

This asserts that it is not the case that for every a it is not the case that Φ(a). Or that it is not the case that no a is Φ which is another way of saying that there exists an a such that Φ(a).

The variable is unrestricted in scope. Any object whatsoever may be assigned as a value of the variable. Sometimes, however, we wish to restrict the domain of quantification as in the expression “every positive integer”. To do so, the domain restricting condition is placed in the antecedent of the conditional. Let P be a functional expression that maps positive integers onto the true, and let X be the functional expression “is the sum of four squares”, then conceptual content of the assertion that every positive integer is the sum of four squares can be represented as:

This asserts that whatever a may be the case where “X(a)” is denied and “P(a)” is affirmed does not occur. Or in other words, for every object whatsoever, if it is a positive integer, then it is the sum of four squares.

The assertion:

asserts that whatever a may be, the case where “not X(a)” is denied and “P(a)”” is affirmed does not occur, or that X(a) and P(a) cannot be affirmed together. It thus translates “No P is an X”.

is the denial of:

and means that some P is not X.

denies that no P is X and therefore means that some P are X.

Frege thus claims to be able to reconstruct the traditional square of opposition:

Summary

1. Logic is concerned with the Laws of Truth and not the Laws of Mind.
2. Mathematical practice can reveal how to regulate and systematize a notation embodying the Laws of Truth.
3. A thought is that for which the question of truth arises.
4. The content of subsentential expressions is understood solely in terms of their contribution to the truth-conditioning structure of the thought expressed.
5. Logical connections between sentences (the pattern of valid inferences in which they participate) depends on and is derived from the logical connections between the thoughts expressed.
6. In Logic it is necessary to discern the semantically significant structure of a sentence (that which corresponds to the truth-conditioning structure of the thought expressed) in order to make explicit the logical connections between sentences and the thoughts they express.