Online research in philosophy

The founder of modern logic and grandfather of analytic philosophy was 70 years old when he published his paper 'Der Gedanke' (The Thought ) in 1918. This essay contains some of Gottlob Frege's deepest and most provocative reflections on the concept of truth, and it will play a prominent role in my lectures. The plan for my lectures is as follows. What is it that is (primarily) true or false? 'Thoughts', is Frege's answer. In §1, I shall explain and defend this answer. In §2, I shall briefly consider his enthymematic argument for the conclusion that the word 'true' resists any attempt at defining it. In §3, I shall discuss his thesis that the thought that things are thus and so is identical with the thought that it is true that things are thus and so. The reasons we are offered for this thesis will be found wanting. In §4, I shall comment extensively on Frege's claim that, in a non-formal language like the one I am currently trying to speak, we can say whatever we want to say without ever using the word 'true' or any of its synonyms. I will reject the propositional-redundancy claim, endorse the assertive-redundancy claim and deny the connection Frege ascribes to them. In his classic 1892 paper 'Über Sinn und Bedeutung' (On Sense and Signification) Frege argues that truth-values are objects. In §5, I shall scrutinize his argument. In §6, I will show that in Frege's ideography (Begriffsschrift) truth, far from being redundant, is omnipresent. The final §7 is again on truth-bearers, this time as a topic in the theory of intentionality and in metaphysics. In the course of discussing Frege's views on the objecthood, the objectivity of thoughts and the timelessness of truth(s), I will plead for a somewhat mitigated Platonism.

Frege’s logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the “neologicist” approach of Hale & Wright. Less attention has been given to Frege’s extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the extension of a concept C, together with extra-logical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the so-called Hume’s Principle and its connections to the root of the contradiction in Frege’s system.

There are many different approaches to the logic of truth. We could agree with Tarski, that the appropriate way to formalise a truth predicate is in a hierarchy, in which the truth predicate in one language can apply only to sentences from another language. Or, we could attempt to do without type restrictions on the truth predicate. Bradwardine’s theory of truth takes the second of these options: it is type-free, and admits sentences which say of themselves that they are not true to be well-formed. We could take the behaviour of the paradoxes such as the liar to motivate a revision of the basic logic of propositional inference, to allow for truth-value gaps or gluts [9, 11, 15]. On the other hand, we could take it that the paradoxes are no reason to revise our account of the basic laws of logic: a novel account of the behaviour of the truth predicate is what is required. Bradwardine’s account, as elaborated by Read, takes this second option.1 Finally.

In this paper I present an abstract theory of senses, thoughts, and truth, inspired by ideas of Frege. "Inspired" because for the most part I shall not pretend to interpret Frege in a literal sense, but, rather, develop some of his ideas in ways that seem to me to preserve important aspects of them. Senses are characterized as identifying properties; i.e., roughly, as properties that apply, in virtue of their logical structure, to exactly one thing, if they apply to anything at all. When Frege's analysis of sentences in terms of function and arguments is combined with his analysis of quantification as higher-order predication, all sentences (formal and informal) can be analyzed in various ways as a function (predicate) applied to one or more arguments. This allows for an abstract characterization of thoughts as senses that combine other senses in a uniform way, and whose truth derives from their instantiation by corresponding items of reality.

Transparency is the following (alleged) property of truth: if one possesses the concept of truth, then to assert, believe, inquire whether it is true that S just is to assert, believe, inquire whether S (and conversely). It might appear (as it did to Frege in 'Thoughts') that if truth ascriptions were transparent, then the truth predicate must be redundant; but the fact that some truth ascriptions are not transparent-for instance, those that quantify over, name, or describe the proposition(s) to which truth is ascribed-shows that the truth predicate could not be redundant. It is argued that the apparent paradox is resolved by treating content as more basic than truth (and arguing, accordingly, that content cannot be explained, even in part, in terms of truth conditions). This strategy is illustrated by three candidate analyses, each of which treats the truth predicate as non-redundant but can, nevertheless, account for transparency.

For Frege’s general views about truth the standard reference is the first couple of pages of ‘The Thought’. Less attention has been paid to a short passage in ‘On Sense and Reference’ – in, fact, only one paragraph long – where Frege argues indirectly for the view that the relation between the thought and the True is an instance of the relation between sense and reference. He argues for this by discrediting the alternative view that it is an instance of the relation between “subject and predicate”. Here is the paragraph: One might be tempted to regard the relation of the thought to the True not as that of sense to reference, but rather as that of subject to predicate. One can, indeed, say: ‘The thought, that 5 is a prime number, is true’. But closer examination shows that nothing more has been said than in the simple sentence ‘5 is a prime number’. The truth claim arises in each case from the form of the declarative sentence, and when the latter lacks its usual force, e.g. in the mouth of an actor upon stage, even the sentence ‘The thought that 5 is a prime number is true’ contains only a thought, and indeed the same thought as the simple ‘5 is a prime number’. It follows that the relation of the thought to the True may not be compared with that of subject and predicate. Subject and predicate (understood in the logical sense) are indeed elements of thoughts; they stand on the same level for knowledge. By combining subject and predicate, one reaches only a thought, never passes from sense to reference, never from thought to its truth value. One moves at the same level but never advances from one level to the next. A truth value cannot be part of a thought, any more than, say, the Sun can, for it is not a sense but an object. (Frege 1892, pp 34-35). Two subordinate, but still major, positive ideas are expressed in this passage.

The syntax of Frege's scientific language iscommonly taken to be characterized by two oddities:the representation of the intended illocutionary roleof sentences by a special sign, the judgement-stroke,and the treatment of sentences as a species ofsingular terms. In this paper, an alternative view isdefended. The main theses are: (i) the syntax ofFrege's scientific language aims at an explication ofthe logical form of judgements; (ii) thejudgement-stroke is, therefore, a truth-operator, nota pragmatic operator; (iii) in Frege's first system,` ' expresses that the circumstance is a fact, and in his second system that thetruth-value - is the True; (iv) in bothsystems, the judgement-stroke is construed as a signsui generis, not as a genuine predicate; (v) itscounterpart in natural language is the syntactic ``formof assertoric sentences'', not the (redundant)truth-predicate; (vi) neither in Frege's first nor inhis second system sentences are treated as singular terms.

Does Frege have a metatheory for his logic? There is an obvious and uncontroversial sense in which he does. Frege introduces and discusses his new logic in natural language; he argues, in response to criticisms of Begriffsschrift, that his logic is superior to Boole's by discussing formal features of both systems. In so far as the enterprise of using natural language to introduce, discuss, and argue about features of a formal system is metatheoretic, there can be no doubt: Frege has a metatheory. There is also an obvious and uncontroversial sense in which Frege does not have a metatheory for his logic. The model theoretic semantics with which we are familiar today are a post-Fregean development. The question I address in this paper is, does Frege have a metatheory in the following sense: do his justifications of his basic laws and rules of inference employ, or even require, ineliminable use of a truth predicate and metalinguistic variables? My answer is ‘no’ on both counts. I argue that Frege neither uses, nor has any need to use, a truth predicate or metalinguistic variables in his justifications of his basic laws and rules of inference. Quine's famous explanation of the need for semantic ascent simply does not apply to Frege's logic. The purpose of the discussions that are typically understood as constituting Frege's metatheory is, rather, elucidatory. And once we see what the aim of these particular elucidations is, we can explain Frege's otherwise puzzling eschewal of the truth predicate in his discussions of the justification of the laws and rules of inference.

It is commonly assumed that the conception of truth defended by Frege in his mature period is characterized by the view that truth is not the property denoted by the predicate 'is true', but the object named by true sentences. In the present paper, I wish to make plausible an alternative interpretation according to which Frege's conception is characterized by the view that truth is what is expressed in natural language by the "form of the assertoric sentence". So construed, truth is neither an object (like the True) nor a property (like the Bedeutung of the predicate 'is true') but something of a very special kind that belongs to the same logical category as the logical relations (like subsumption). The main argument justifying this interpretation is that Frege's explication of truth does not hold of the True, but only of truth, considered as what is expressed by the form of the assertoric sentence.

I argued that Frege does not have a metatheory in the following sense: the justifications he offers for his basic laws and rules of inference neither employ nor require a truth-predicate or metalinguistic variables. In ‘Does Frege Use a Truth-predicate in his "Justification" of the Laws of Logic?’, Dirk Greimann disputes this. As Greimann interprets Frege, (i) Frege's remarks commit him to giving a metatheoretic justification of the basic laws and rules of his logic, and (ii) Frege actually gives such a justification in the early sections of Grundgesetze—although the truth-predicate that Frege employs is a non-standard one: it is neither a predicate that holds of all and only true sentences nor a predicate that holds of all and only true thoughts. I argue that Greimann's interpretation is not, in the end, true to the text, and that his non-standard view of what is required of a Tarskian truth-predicate is ultimately not viable. CiteULike    Connotea    Del.icio.us    What's this?