The Missing Links – February 10, 2013

  • A self-described lesbian leftist professor describes her conversion at Christianity Today.  “I continued reading the Bible, all the while fighting the idea that it was inspired. But the Bible got to be bigger inside me than I. It overflowed into my world. I fought against it with all my might. Then, one Sunday morning, I rose from the bed of my lesbian lover, and an hour later sat in a pew at the Syracuse Reformed Presbyterian Church.”
  • a Liberal-Democrat Member of Parliament and former minister, explaining why she voted against the redefinition of marriage in the British Parliament on February 5.   “My concern, however, is that by moving to a definition of marriage that no longer requires sexual difference, we will, over time, ultimately decouple the definition of marriage from family life altogether. I doubt that this change will be immediate. It will be gradual, as perceptions of what marriage is and is for shift. But we can already see the foundations for this shift in the debate about same-sex marriage. Those who argue for a change in the law do so by saying that surely marriage is just about love between two people and so is of nobody else’s business. Once the concept of marriage has become established in social consciousness as an entirely private matter about love and commitment alone, without any link to family, I fear that it will accelerate changes already occurring that makes family life more unstable.”
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Philosophy Word of the Day – Gödel’s Incompleteness Theorem

By the early part of the twentieth century, the work of mathematical logicians such as Gottlob Frege, Bertrand Russell, and Alfred North Whitehead had honed the axiomatic method into an almost machine-like technique of producing mathematical theorems from carefully stated first principles (axioms) by means of clear logical rules of inference. In 1931, however, Kurt Gödel (1906–1978), an Austrian logician, uncovered a surprising limitation inherent in any axiomatic system intended to produce theorems expressing the familiar mathematical properties of integer arithmetic.

Gödel developed a method, whose reach was slightly extended by J. Barkley Rosser in 1936, that shows how, given any such (consistent) system of axioms, one can produce a true proposition about integers that the axiomatic system itself cannot produce as a theorem. Gödel’s incompleteness result follows: Unless the axioms of arithmetic are inconsistent (self-contradictory), not all arithmetical truths can be deduced in such machine-like fashion from any fixed set of axioms. This result, that here consistency implies “incompleteness,” has striking implications not only for mathematical logic, but also for machine-learning (artificial intelligence) and epistemology, although its precise significance is still debated. (continue article)

— W. M. Priestly, Encyclopedia of Science and Religion



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Philosophy Word of the Day – Enthymeme

An argument with an unstated premiss or an unstated conclusion.  This accords with the seventeenth-century definition of an enthymeme as ‘a syllogism complete in the mind and incomplete in expression,’ e.g., ‘If it is raining, I will take my umbrella; therefore, I will take my umbrella.’

Here, the premiss ‘It is raining’ is not stated, perhaps because on the particular occasion it will be too obvious for words.

In some enthymemes, it is the conclusion that is not explicitly stated, and again the reason may be that it is obvious.  For instance:  ‘It is raining, and if it is raining I will take my umbrella.’  At that point enough has been said for the purposes of normal conversation.

It may happen that both the conclusion and some premiss is implicit.  The statement: ‘Either he is a rogue, or I will eat my hat’ can be understood as an enthymeme with a suppressed premise: ‘I will not eat my heat’ and a suppressed conclusion: ‘He is a rogue.’

The Penguin Dictionary of Philosophy (Penguin Books, 2005), 189-190.

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Philosophy Word of the Day — Modus Ponens/Modus Tollens

Modus Ponens: (Latin for: mood that affirms.)  In its basic form, an argument that runs ‘If p, then q. p.  Therefore q.’

For example,

If today is Tuesday, then I will go to work.

Today is Tuesday.

Therefore, I will go to work.

Modus Tollens: (Latin for: mood that denies.)  In its basic form, an argument that runs ‘If p, then q.  But not-q.  Therefore not-p.’

For example,

If I miss the train, I take a taxi.

I didn’t take a taxi.

Therefore, I didn’t miss the train.

Definitions, but not examples, taken from Antony Flew, ed., A Dictionary of Philosophy, rev. 2nd ed., 236.

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Philosophy Word of the Day — Dialectic

The portrait of G.W.F. Hegel (1770-1831); Stee...
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“In ancient Greece, dialectic was a form of reasoning that proceeded by question and answer, used by Plato.  In later antiquity and the Middle Ages, the term was often used to mean simply logic, but Kant applied it to arguments showing that principles of science have contradictory aspects.  Hegel thought that all logic and world history itself followed a dialectical path, in which internal contradictions were transcended, but gave rise to new contradictions that themselves required resolution.  Marx and Engels gave Hegel’s idea of dialectic a material basis; hence dialectical materialism.”

—  Peter Singer, The Oxford Companion to Philosophy, 198.

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Philosophy Word of the Day – Logically Perfect Language

Natural languages may be thought in various ways to be ‘logically imperfect.’  Certain grammatical forms may mislead us about logical form; thus, ‘It is raining’ looks as if it refers to something (‘it’).  More radically, certain concepts may even involve us in contradiction or incoherence.  For example, Tarski argued that the ordinary concept ‘true’ did this, since it generated such paradoxes as the liar.

A logically perfect language would be one lacking these faults, as well, perhaps, as some other ‘defects’, such as ambiguity and redundancy.  Frege attempted to create such a language (the Begriffsschrift [concept writing or concept notation]), in which to couch the truths of logic and mathematics.  Rather later, the Logical Positivists were interested in the idea of a logically with which to express the whole of natural science.

Dr. Roger Teichmann in The Oxford Companion to Philosophy

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Book Review – Logicomix: An Epic Search for Truth


  • Paperback: 352 pages
  • Publisher: Bloomsbury USA (September 29, 2009)
  • Official Logicomix Website
  • Amazon

The wedding of philosophical mathematics and a graphic novel seems an ideal marriage. What in other forms of media could be an overlong, tedious tale, can spring to life and breathe in an illustrated story. That’s my impression after reading Logicomix: An Epic Search for Truth. Logicomix tells the story of mathematician and philosopher Bertrand Russell’s decades-long search to establish an unshakeable foundation for mathematics in logic. Russell narrates his own life story and describes his quest, which began in childhood and continued well into his adult life. Along the way, he interacts with some of the greatest logicians, mathematicians, and philosophers of the early twentieth century—Ludwig Wittgenstein, Georg Cantor, G. E. Moore, David Hilbert, Kurt Gödel, Alfred Whitehead, and Gottlob Frege.

While the quest comprises the main storyline, the authors also shed light on Russell’s private life, which had its fair share of drama and conflict. Having a family history of mental illness, one of his greatest ongoing fears was losing his mind. (On a side note, nearly all of the thinkers mentioned above contended with a serious mental illness.) Russell was married four times, though the book only features his first two wives, and his well known affairs are hinted at. Though his contributions to mathematics and logic were considerable, he came to see his original quest as a failure, especially in light of Wittgenstein’s criticisms and Gödel’s Incompleteness Theorem (which the authors explain in the text and in a helpful glossary of terms in the back).

At the same time, the authors suggest that Russell was admired for his pacifism, his endeavoring to apply reason and logic to every area of human activity, and his ideas, which influenced a younger generation of mathematicians such as Alan Turing, who helped break the German “Enigma” code in World War II and whose work importantly influenced the development of the digital computer.

An interesting and intentionally ironic literary device used by the writers is inserting themselves into the story, so that the reader follows the process (interspersed occasionally) of the writing of Logicomix along with Russell’s story. Logicomix is thus self-referential, which characteristic also lies at the heart of “Russell’s Paradox” (also nicely described in the glossary).

At the end of his life, the Russell of Logicomix arrives at the conclusion that much of human nature and behavior can’t be explained or captured by logic, and that no single system can encompass the multi-faceted nature of reality. He declares, “If even in logic and mathematics, the paragons of certainty, we cannot have perfect assurances of reason, then even less can this be achieved in the messy business of human affairs—either private, or public! . . . Wittgenstein has a point, you see: ‘All the facts of science are not enough to understand the world’s meaning.’” (p. 296).

While the authors admit that they have taken some liberties with the reconstruction of Russell’s life, the major characters “are based as closely as possible on their real-life counterparts” and no liberties were taken with “the content of the great adventure of ideas which forms our main plot.” (p. 315, 316). I was pleased to know this, since a mainly fictional account wouldn’t have interested me nearly as much.

Logicomix succeeds, in my view, in shaping a fascinating story out of complex and abstract ideas. The story is epic, recounting some of the most important events in philosophy and mathematics in the last century, while also capturing the very human face of that unfolding drama. Who said comic books couldn’t be educational?

Thanks to Bloomsbury USA for this review copy.

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Philosophy Word of the Day – The Analytic/Synthetic Distinction

“Analytic” sentences, such as “Ophthalmologists are doctors,” are those whose truth seems to be knowable by knowing the meanings of the constituent words alone, unlike the more usual “synthetic” ones, such as “Ophthalmologists are ill-humored,” whose truth is knowable by both knowing the meaning of the words and something about the world.

Beginning with Frege, many philosophers hoped to show that knowledge of logic and mathematics and other apparently a priori domains, such as much of philosophy and the foundations of science, could be shown to be analytic by careful “conceptual analysis.” This project encountered a number of problems that have seemed so intractable as to lead some philosophers, particularly Quine, to doubt the reality of the distinction. There have been a number of interesting reactions to this scepticism, both in philosophy and in linguistics, but it has yet to be shown that the distinction will ever be able to ground the a priori in the way that philosophers had hoped.

(Via Stanford Encyclopedia of Philosophy)

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Philosophy Word of the Day – Logical Positivism

AJ Ayer
Image by Pickersgill Reef via Flickr

Twentieth-century philosophical movement that used a strict principle of verifiability to reject as meaningless the non-empirical statements of metaphysics, theology, and ethics. Under the influence of Hume, Russell, and the early Wittgenstein, the logical positivists regarded as meaningful only statements reporting empirical observations, taken together with the tautologies of logic and mathematics. Prominent logical positivists included members of the Vienna Circle and Ayer.

(Via Philosophy Pages)

Early critics of logical positivism said that its fundamental tenets could not themselves be formulated in a way that was clearly consistent. The verifiability criterion of meaning did not seem verifiable; but neither was it simply a logical tautology, since it had implications for the practice of science and the empirical truth of other statements. This presented severe problems for the logical consistency of the theory. Another problem was that, while positive existential claims (“there is at least one human being”) and negative universal claims (“not all ravens are black”) allow for clear methods of verification (find a human or a non-black raven), negative existential claims and positive universal claims do not allow for verification.

(Via Wikipedia)

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Philosophy Word of the Day – Validity and Soundness

An argument is valid when it is impossible for its premises to be true and its conclusion to be false, or, to put it another way, if the premises were true the conclusion would have to be true, or again, the conclusion follows necessarily from the premises.

An argument can be valid even though the premises are false. Note, for example, that the conclusion of the following argument would have to be true if the premises were true, (even though they are, in fact, false):

→All fire-breathing rabbits live on Earth
→All humans are fire-breathing rabbits
→(Therefore,) all humans live on Earth

The argument, however, is not sound. In order for a deductive argument to be sound, it must not only be valid, the premises must be true as well.

(Via Wikipedia)

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