Response‐dispositional (RD) properties are standardly defined as those that involve an object's appearing thus or thus to some perceptually well‐equipped observer under specified epistemic conditions. The paradigm instance is that of colour or other such Lockean “secondary qualities”, as distinct from those—like shape and size—that pertain to the object itself, quite apart from anyone's perception. This idea has lately been thought to offer a promising alternative to the deadlocked dispute between hard‐line ‘metaphysical’ realists and subjectivists, projectivists, social constructivists, or hard‐line (...) anti‐realists. A chief source text is Plato's Euthyphro, where the issue is posed in ethical terms: do the gods infallibly approve virtuous acts on account of their divine moral omniscience or are virtuous acts just those the gods approve? Among the areas proposed as amenable to an RD approach are epistemology, ethics, political theory, and philosophy of mathematics. It is claimed that by making due allowance for the involvement of normalised or optimised human responses one can steer a course between the twin poles of an objectivist realism that places truth beyond our cognitive grasp and an epistemic conception that confines truth within the limits of humanly attainable proof, knowledge, or verification. Here I argue—on the contrary—that RD approaches can be shown to offer nothing more than a variant of the same old realist versus anti‐realist dilemma. That is, they work out either as a trivial (tautologous) claim that ‘truth’ simply equates with ‘best judgement’ in the ideal (quasi‐objective) limit or as the claim—advanced by anti‐realists like Michael Dummett—that we cannot form any adequate conception of objective (recognition‐transcendent) truths. After looking at this issue in various contexts of debate, I conclude that one useful (if pyrrhic) outcome is to demonstrate the non‐availability of any middle‐ground stance. We are left with the strictly unavoidable choice between a realist or objectivist approach and one that assimilates truth to the consensus of accredited best opinion. This latter amounts to a roundabout, elaborately qualified version of the anti‐realist case. (shrink)

Arithmetical self-reference through diagonalization is compared with self-recognition in a mirror, in a series of diagrams that show the structure and main stages of construction of self-referential sentences. A Gödel code is compared with a mirror, Gödel numbers with mirror images, numerical reference to arithmetical formulas with using a mirror to see things indirectly, self-reference with looking at one’s own image, and arithmetical provability of self-reference with recognition of the mirror image. The comparison turns arithmetical self-reference into an idealized model (...) of self-recognition and the conception(s) of self based on that capacity. (shrink)

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...) with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)

Gödel's two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot (...) prove that the system itself is consistent (assuming it is indeed consistent). These results have had a great impact on the philosophy of mathematics and logic. There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more controversial. The present entry surveys the two incompleteness theorems and various issues surrounding them. (shrink)