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From a single proposition p we form its negation not-p. From two propositions p and q we form various further propositions, e.g., p or q, p and q, etc.; similarly (though less often) with three or more propositions. For each such method of forming propositions it is common to speak of a certain truth-function as corresponding to it. On the one hand, there is a function which takes senses of propositions as inputs and returns the sense of a proposition as output. On the other hand, there is a truth-function, i.e., a function taking some finite number of truth-values as arguments and giving a single truth-value as value, which is representable by a truth-table or truth-diagram. This chapter discusses the relationship between the two.

This chapter explores Wittgenstein’s remarks on ‘the general form of the proposition’ and argues that these too are informed by his desire to show that the impression that there are substantial ‘logical truths’ is an illusion. It explores Wittgenstein’s understanding of logical inference as ‘unpacking’, his analysis of truth-functional connectives using truth-tables, his account of logical propositions as tautologies and contradictions, and the philosophical significance for Wittgenstein of providing a ‘topic-neutral’ account of logical inference and the logical connectives. Drawing on retrospective remarks of Wittgenstein’s, an explanation is offered on how he could have failed to recognize that his notion of the general form of the proposition is laden with problematic metaphysical commitments.

This chapter takes up an important and disputed issue in the interpretation of Wittgenstein's Tractatus Logico-Philosophicus, and uses ideas articulated in the previous chapter to shed light on it. Truth-functions are central to Wittgenstein's thought; they play the crucial role not only in accounting for logic but also in explaining the possibility of any language or thought beyond the most elementary. Yet his account of the truth functions is by no means easy to understand. In particular, Wittgenstein calls them operations, and insists that they must be sharply distinguished from functions; yet most of what he says about them seems to apply equally to functions. When Wittgenstein uses the word ‘function’ he is following Russell, and speaking of Russellian propositional functions. Reading him in this way can make sense of the contrasts he draws between functions and operations. This enables us to achieve a better understanding of his notion of an operation. On the basis of this understanding, it is argued that operations in his sense cannot be identified with ordinary mathematical functions or with Fregean functions. The account that Wittgenstein gives of them cannot be separated from his general project of reconceiving the way in which language and thought represent the world; we cannot understand it in other terms.

The discussion of Wittgenstein's account of quantification in Chapter 20 left unaddressed what sort of theory of types it commits us to. To answer this question we need to look in more detail at the motivation for believing in logical types at all. That motivation derives from Russell's paradox, the problem which had originally attracted Wittgenstein's notice back in 1909. This chapter discusses Russell's theory of types, Wittgenstein's vicious circle principle, types as classes of propositions, types and molecular propositions, types and generality, uniting generality and truth-functions, the general form of proposition, and unsayability.

The notion of operation plays a pivotal role in the symbolism of the Tractatus Logico-Philosophicus: on the one hand, truth-functions are based on truth-operations; on the other, numbers are exponents of an operation. Considering that operations seem to be so central, it is amazing to notice how little is understood of Ludwig Wittgenstein's remarks: not enough attention has been paid in the past to the curious piece of symbolism of 6.01. What Wittgenstein says about the notion of operation very much resembles informal explanations of the notion of operator. Two differences with the set-theoretic notion of function were mentioned in Chapter 1: firstly, an operator is defined by describing how it transforms an input without defining the set of inputs, that is without defining its domain. Secondly, there is no restriction on the domain of some operators.

Introduces the criticisms put forward by philosophers such as Donald Davidson and Richard Rorty to the idea that one thing might represent another: that thoughts, utterances, and inscriptions are said to have content by virtue of their power to represent reality; and that those that do the job accurately are true, they correspond to the facts, or mirror reality—they are representations of reality. The author then outlines the deductive proof that he will present in the book to show that Davidson's and Rorty's criticisms are unfounded. The content of the proof is based on the work of Kurt Gödel and W. V. Quine, and demonstrates conclusively that (i) any supposedly non‐truth‐functional operation must satisfy an exacting logical condition in order to avoid collapsing into a truth‐function, and (ii) any theory of facts, states of affairs, situations, or propositions must satisfy a corresponding condition if such entities are not to collapse into a unity. The three main sections of the chapter: examine the case against representations made by Davidson and Rorty, including Davidson's notorious argument against facts; discuss Rorty's critique of representational philosophy (his anti‐representationalism); and consider collapsing (or slingshot) arguments that discredit the existence of facts (and are used in the critique of the fact–representation distinction), as put forward by Gottlob Frege, and developed by Alonzo Church, Quine, Davidson, and Gödel. The last section of the chapter outlines the structure of the book.

The view that the ordinary conditional ‘if p then q’ has the same meaning as the material conditional of standard truth‐functional logic confronts the more intuitive view that it imports, as part of its meaning, some ground‐consequent relation between antecedent and consequent clauses. Grice's powerful defence of the former position demonstrates that the operation of certain effective general principles of assertive discourse would normally ensure that the truth‐functional conditional carried a non‐conventional consequentialist implication. A defence of the contrary view rests on the fact that ‘p, so q’, which incontestably carries a consequentialist implication as part of its meaning, differs from ‘if p then q’ only in that the former conjoins asserted, the latter unasserted, propositions. There is plenty of occasion for the use of both ‘so’ and ‘if’. The balance of argument favours the consequentialist position.

This chapter argues that Wittgenstein’s account of how we know that a proposition is a logical consequence of other propositions provides the main motivation for his flagship doctrine that every proposition is a truth function of elementary propositions, as well as for his extraordinary demands on the propositions that serve as elementary—that they are independent of one another and that their constituents refer to simple items. It then considers the metaphysical and epistemological questions raised by the kind of analysis that, according to Wittgenstein, must be possible, concentrating on the relationship between logical structure and use. The chapter ends by considering the task of extending to non-elementary propositions the pictorial account of how propositions represent.

This chapter discusses truth functions and the concept of the material conditional. It introduces the criteria of assertability or acceptability; notes alogical aspects of conditionals as they are used in everyday discourse; and distinguishes among truth, validity, and soundness in the context of argument evaluation.

Chapter 3 is a discussion of Lotze’s early metaphysics and logic. One section is devoted to Lotze’s 1841 Metaphysik, an innovative text in the criticism of the metaphysical tradition and Hegelianism. Another section deals with Lotze’s early philosophy of physiology or the life sciences, especially his influential critique of the concept of Lebenskraft and his attempt to reconcile mechanical and teleological explanation. Still another section deals with Lotze’s 1843 Logik, attempting to specify how this text anticipates later developments in logic.