Rudolf Carnap’s Notes for Symbolic Logic is a compact technical set of notes rather than an introductory textbook. The text itself states that it is not intended as an introduction to symbolic logic; it offers a survey of a formal symbolism and its rules, with little or no explanatory prose. Detailed explanations are deferred to cited works: Carnap’s Abriss der Logistik and Logical Syntax of Language, Hilbert and Ackermann’s Grundzüge der theoretischen Logik, Lewis and Langford’s Symbolic Logic, and Whitehead and Russell’s Principia Mathematica. The notes are therefore best understood as a synoptic reference aid for readers already familiar with formal logic.
Use of Letters
The notes begin by classifying symbolic letters according to their role and logical level. Sentential symbols are divided into constants such as \(A, B, C\ldots\) and variables such as \(p, q, r\ldots\). Individual symbols occupy level 0, with constants \(a, b, c\ldots\) and variables \(x, y, z, u\ldots\). One-place predicates occupy level 1, with constants \(P, Q\ldots\) and variables \(F, G\ldots\); two-place predicates likewise occupy level 1, with constants \(R, S\ldots\) and variables \(H, K\ldots\). Higher-level predicates, such as predicates of predicates, are indicated by superscripts and further letter conventions.
The notes give elementary examples of sentences using this notation. \(P(b)\) is read as “the object \(b\) has the property \(P\)” or “\(b\) is \(P\).” \(R(a,b)\) says that the relation \(R\) holds between \(a\) and \(b\). Expressions such as \({}^2P(Q)\) illustrate a property of a property: the property \(Q\) has the higher-level property \({}^2P\). These examples establish the basic typed notation that the rest of the notes employ.
Lower Functional Calculus
The section on the lower functional calculus introduces universal and existential sentences, distinguishing open from closed sentences. A formula with a free variable, such as \(P(x)\), is open; when the variable is bound by a quantifier, the sentence is closed. The notes display the anatomy of quantified sentences: an operator such as \((x)\) or \((\exists x)\), a bound variable, and an operand. Thus \((x)P(x)\) is the universal sentence “every object is \(P\),” and \((\exists x)P(x)\) is the existential sentence “some object is \(P\)” or “there is an object which is \(P\).” The notes treat the lower functional calculus as including the sentential calculus.
Formative Rules
The notes set out formation rules for sentences of the lower functional calculus, including the sentential calculus. A sentence may be a sentential symbol; a full sentence of a predicate, i.e. an atomic sentence; or an identity sentence. Molecular sentences are formed by negation, disjunction, conjunction, implication, and equivalence. General sentences are formed by prefixing a universal or existential quantifier to an expression containing an individual variable. The notes give schematic forms and examples, such as \((x)((P(x)) \lor (R(x,b)))\) and \((\exists x)((y)(R(x,y)))\), to show how quantifiers combine with sentential connectives and predicates.
Conventions for Avoiding Brackets
The notes then list conventions for omitting brackets without ambiguity. Brackets may be dropped when the enclosed expression is an atomic sentence occurring as a member of a sentential connection; when a negation occurs as a member of any sentential connection; when a disjunction or conjunction occurs as a member of an implication or equivalence; and in further analogous scope conventions. These rules are purely notational: they specify how to read formulas economically while preserving the intended grouping of connectives and quantifiers.
Overall, Notes for Symbolic Logic is a concise formal manual. It presents notation, levels of symbols, quantification, sentence formation, and bracketing conventions, while relying on other works for explanation and development. Its scope is the symbolism and rules of the lower functional calculus, including the sentential calculus, rather than philosophical exposition or a narrative account of logic.
Truth-Conditions and Validity
The notes proceed to define truth-conditions for sentences within this formal system. For atomic sentences, truth is taken as primitive, corresponding to a state of affairs. For molecular sentences, truth-conditions are defined recursively based on the truth-values of their components and the meaning of the logical connectives. For instance, a negation \(\neg A\) is true if and only if \(A\) is false; a disjunction \(A \lor B\) is true if and only if \(A\) is true or \(B\) is true (or both).
The treatment of quantified sentences also receives a truth-conditional definition. A universal sentence \((x)P(x)\) is true if and only if \(P(a)\) is true for every object \(a\) in the domain of discourse. An existential sentence \((\exists x)P(x)\) is true if and only if \(P(a)\) is true for at least one object \(a\) in the domain. These definitions establish the semantic basis for evaluating the truth of any well-formed sentence in the system.
Following the definitions of truth, the notes introduce the concept of validity. A sentence is valid if it is true under all possible interpretations (i.e., for all assignments of truth-values to its atomic components and all possible domains and interpretations of its predicates). This notion of validity is central to logical inference, as valid sentences represent logical truths that hold irrespective of empirical facts. The notes provide examples of valid sentences, such as instances of the law of excluded middle or tautologies involving quantifiers.
Rules of Inference
The notes then outline rules of inference, which specify how to derive new sentences from existing ones while preserving truth. These rules are presented as a formal calculus for deduction. They include standard rules like Modus Ponens, which allows the inference of \(B\) from \(A\) and \(A \rightarrow B\), and rules for introducing and eliminating quantifiers. For example, universal instantiation permits inferring \(P(a)\) from \((x)P(x)\), and existential generalization permits inferring \((\exists x)P(x)\) from \(P(a)\).
The emphasis is on the syntactic manipulation of symbols according to these rules, ensuring that each step in a derivation is formally justified. The goal is to construct proofs, which are sequences of sentences where each sentence is either an assumption or derived from preceding sentences by an inference rule. The notes demonstrate how to construct simple proofs, illustrating the application of the inference rules to establish the validity of arguments or the truth of certain logical statements. The focus remains on the mechanics of formal deduction rather than a philosophical justification of the rules themselves.
Identity
A dedicated section addresses the concept of identity within the formal system. The identity predicate, typically denoted by \(=\), is introduced to express that two individual terms refer to the same object. The notes present the axioms governing identity, which include reflexivity (\((x)(x=x)\)) and the principle of indiscernibility of identicals. The latter states that if two objects are identical, then they share all properties: \((x)(y)((x=y) \rightarrow (P(x) \rightarrow P(y)))\). These axioms provide the formal foundation for reasoning about equality between objects within the logical framework. The inclusion of identity allows for a more expressive language, enabling the formulation of statements that distinguish between objects or assert their sameness.
Description
The notes also touch upon definite descriptions, which are expressions of the form “the so-and-so.” While not fully developed, the treatment of descriptions indicates how such expressions can be analyzed within the framework of symbolic logic, typically by translating them into quantified statements that assert the unique existence of an object satisfying a certain property. This brief inclusion suggests an awareness of the need to handle complex linguistic structures within the formal system, even if the detailed mechanisms are left to more extensive treatments. The overall aim is to provide a foundational understanding of the symbolic tools necessary for precise logical analysis.
