From Logic to Mathematics

2020 ◽  
pp. 197-208
Author(s):  
Jared Warren

Part II (chapters 3-7) of the book developed and defended an inferentialist/conventionalist theory of logic. In this, the opening chapter of part III, it is explained why the extension of part II’s approach from logic to mathematics faces significant philosophical challenges. The first major challenge concerns the ontological commitments of mathematics. It is received wisdom in philosophy that existence claims cannot be analytic or trivially true, making it difficult to see how a conventionalist account of mathematics could possibly be viable. The second major challenge concerns mathematical truth. Unlike (first-order) logical truth, mathematical truth, even in basic arithmetic, is computationally rich. There are serious challenges for conventionalists in trying to capture our intuition that mathematical truth is fully determinate, in light of this feature.

PARADIGMI ◽  
2012 ◽  
pp. 69-77
Author(s):  
Carla Bagnoli

This paper argues that the most innovative aspect of Kant's ethical theory is not afirst-order normative ethics, even though the importance and long-lasting mark ofKant's ethics of autonomy cannot be questioned. Rather, it consists in a constructivistaccount of moral cognition. This claim may be perplexing in more than one way, sinceconstructivism is often characterized both as a first-order account of moral judgmentsand as a retreat from epistemological and ontological commitments. This characterizationis misleading in general, and mistaken for Kant's constructivism in particular.Kant's constructivism is a methodological claim about the authority and productivefunction of reason and an epistemological claim about the nature of moral cognitions.


1958 ◽  
Vol 23 (4) ◽  
pp. 417-419 ◽  
Author(s):  
R. L. Goodstein

Mr. L. J. Cohen's interesting example of a logical truth of indirect discourse appears to be capable of a simple formalisation and proof in a variant of first order predicate calculus. His example has the form:If A says that anything which B says is false, and B says that something which A says is true, then something which A says is false and something which B says is true.Let ‘A says x’ be formalised by ‘A(x)’ and let assertions of truth and falsehood be formalised as in the following table.We treat both variables x and predicates A (x) as sentences and add to the familiar axioms and inference rules of predicate logic a rule permitting the inference of A(p) from (x)A(x), where p is a closed sentence.We have to prove that from


Author(s):  
Jared Warren

What is the source of logical and mathematical truth? This volume revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. In Shadows of Syntax, Jared Warren offers the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. He argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims. In Part I, Warren explains exactly what conventionalism amounts to and what linguistic conventions are. Part II develops an unrestricted inferentialist theory of the meanings of logical constants that leads to logical conventionalism. This conventionalist theory is elaborated in discussions of logical pluralism, the epistemology of logic, and of the influential objections that led to the historical demise of conventionalism. Part III aims to extend conventionalism from logic to mathematics. Unlike logic, mathematics involves both ontological commitments and a rich notion of truth that cannot be generated by any algorithmic process. To address these issues Warren develops conventionalist-friendly but independently plausible theories of both metaontology and mathematical truth. Finally, Part IV steps back to address big picture worries and meta-worries about conventionalism. This book develops and defends a unified theory of logic and mathematics according to which logical and mathematical truths are reflections of our linguistic rules, mere shadows of syntax.


Author(s):  
George S. Boolos

This chapter outlines an argument to the effect that there is no reduction of arithmetical truth to logical truth, where “logic” is understood to be elementary (first-order) logic, or any system of logic whose theses form an effectively generable set. It suggests, however, that it leaves open the possibility of a significant reduction of arithmetic to something that might be called a system of logic. By investigating metatheoretic differences between first- and second-order logic, it explores the extent to which second-order logic might play a role in facilitating such a reduction.


1990 ◽  
Vol 55 (3) ◽  
pp. 1315-1319 ◽  
Author(s):  
Michael Hallett

Author(s):  
Roger Wertheimer

If logical truth is truth due solely to syntactic form, then mathematics is distinct from logic, even if all mathematical truths are derivable from definitions and logical principles. This is often obscured by the plausibility of the Synonymy Substitution Principle that is implicit in the Fregean conception of analyticity: viz., that synonyms are intersubstitutable without altering sentence sense. Now, unlike logical truth, mathematical truth is not due to syntax, so synonym interchange in mathematical truths preserves sentence syntax, sense, and mathematical necessity. Mathematical necessity, therefore, differs from both logical and lexical necessity.


Author(s):  
Alex Orenstein

Quine is the foremost representative of naturalism in the second half of the twentieth century. His naturalism consists of an insistence upon a close connection or alliance between philosophical views and those of the natural sciences. Philosophy so construed is an activity within nature wherein nature examines itself. This contrasts with views which distinguish philosophy from science and place philosophy in a special transcendent position for gaining special knowledge. The methods of science are empirical; so Quine, who operates within a scientific perspective, is an empiricist, but with a difference. Traditional empiricism, as in Locke, Berkeley, Hume and some twentieth-century forms, takes impressions, ideas or sense-data as the basic units of thought. Quine’s empiricism, by contrast, takes account of the theoretical as well as the observational facets of science. The unit of empirical significance is not simple impressions (ideas) or even isolated individual observation sentences, but systems of beliefs. The broad theoretical constraints for choice between theories, such as explanatory power, parsimony, precision and so on, are foremost in this empiricism. He is a fallibilist, since he holds that each individual belief in a system is in principle revisable. Quine proposes a new conception of observation sentences, a naturalized account of our knowledge of the external world, including a rejection of a priori knowledge, and he extends the same empiricist and fallibilist account to our knowledge of logic and mathematics. Quine confines logic to first-order logic and clearly demarcates it from set theory and mathematics. These are all empirical subjects when empiricism is understood in its Quinian form. They are internal to our system of beliefs that make up the natural sciences. The language of first-order logic serves as a canonical notation in which to express our ontological commitments. The slogan ‘To be is to be the value of a variable’ ([1953] 1961: 15) encapsulates this project. Deciding which ontology to accept is also carried out within the naturalistic constraints of empirical science – our ontological commitments should be to those objects to which the best scientific theories commit us. On this basis Quine’s own commitments are to physical objects and sets. Quine is a physicalist and a Platonist, since the best sciences require physical objects and the mathematics involved in the sciences requires abstract objects, namely, sets. The theory of reference (which includes notions such as reference, truth and logical truth) is sharply demarcated from the theory of meaning (which includes notions such as meaning, synonymy, the analytic–synthetic distinction and necessity). Quine is the leading critic of notions from the theory of meaning, arguing that attempts to make the distinction between merely linguistic (analytic) truths and more substantive (synthetic) truths has failed. They do not meet the standards of precision which scientific and philosophical theories adhere to and which are adhered to in the theory of reference. He explores the limits of an empirical theory of language and offers a thesis of the indeterminacy of translation as further criticism of the theory of meaning.


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