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Papastephanou, eds. The World and the Teacher: prospects and challenges for teacher education in an age of cosmopolitanization, Ethics and Global Politics 5 4 , Educational Studies in Japan 13 2 in press. Critical Horizons 20 1 , Educational Philosophy and Theory 50 14 , Knowledge Cultures 6 3 , Studies in Philosophy and Education 37 1 , Educational Philosophy and Theory 50 4 , Papastephanou and Z. Educational Philosophy and Theory 49 14 , Education Sciences, 7 1 , Metaphilosophy, 47 , Semiotica , 3: Ethics and Education 10, 3: Psychosociological Issues in Human Resource Management 3, 2: Ethics and Education 9, 2: Hyperbole aside, what both Bergson and Santayana are pointing to is the frequently underrated and misunderstood role of imagination in the production of almost all major systems of philosophy.

Great philosophers from Plato and Aristotle to Kant and Nietzsche and Bergson himself have never been simply logic mills or thinking machines, but bold thinkers with an imaginative "feel" for abstract reality. In this respect, even if we dismiss the assessments of Bergson and Santayana as extravagant, we can still accept Lucretius as a bona fide philosopher and not just as a poetical embellisher and interpreter. Every philosopher has strengths and weaknesses; those of Lucretius are conspicuous. In addition to his powerful imagination, his main strength not surprisingly is his verbal skill and force of expression.

He is one of the most quotable of philosophers, with a flair for striking images and tightly packed statements. A few samples:. It would be wearisome to run through the whole list. Of all Lucretius' intellectual strengths, perhaps none is more characteristic or stands out more impressively than his hard, clear commitment to naturalism. Throughout the poem he consistently attacks supernatural explanations of phenomena and resists the temptation to give in to some form of natural religion or "scientific" supernaturalism.

The world, he argues, was not created by divine intelligence, nor is it imbued with any form of mind or purpose. Instead, it must be understood as an entirely natural phenomenon, the outcome of a random though statistically inevitable and lawful process. In short, whatever happens in the universe is not the product of design, but part of an ongoing sequence of purely physical events. Lucretius' principal philosophical shortcoming is that not only will he occasionally follow Epicurean doctrine to the point of absurdity e. As Bergson points out, these howlers can usually be attributed to the defective method of ancient science, which, because it did not require that hypotheses be confirmed by experimentation, allowed even the wildest conjectures to pass as plausible truths.

One further problem is that, for all his reliance on naturalistic explanations and his attempted reduction of metaphysics to physics, Lucretius at times seems to back away, if only ever so slightly, from a purely materialist world view. Indeed in his effusive descriptions of the creative power of nature, effectively symbolized by the figure of Venus, he seems almost like Bergson to postulate an immaterial life-force surging through the universe and operating above or beyond raw nature.

To read this romantic streak into him is clearly a mistake. Lucretius remains a thorough-going naturalist.

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Yet when his verse is in high gear, one almost gets the impression that somewhere inside this staunchly scientific, fiercely anti-religious poet there is a romantic nature-worshipper screaming to get out. Lucretius' literary influence has been long-lasting and widespread, especially among poets with epic ambitions or cosmological interests, from Virgil and Milton to Whitman and Wordsworth.

Not surprisingly, as one of the main proponents and principal sources of Epicurean thought, his philosophical influence has also been considerable. The extent of his communication with and influence on his contemporaries, including other Epicurean writers, is not known. What is known is that by the end of the first century A. De Rerum Natura was hardly read and its author had already begun a long, slow descent into philosophical oblivion.

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It was not until the Renaissance, with the recovery of lost Lucretian manuscripts, that a true revival of the poet became possible. It is probably an exaggeration to say that the restoration and study of Lucretius' poem was crucial to the rise of Renaissance "new philosophy" and the birth of modern science. On the other hand, one must not ignore its importance as a spur to innovative sixteenth- and seventeenth-century scientific thought and cosmological speculation.

Lucretius' influence on early modern thought is most directly visible in the work of the French scientist and neo-Epicurean philosopher Pierre Gassendi In Gassendi published his Syntagma Philosophiae Epicuri , a theoretical refinement and elaboration of Epicurean science. A Catholic priest with a remarkably independent mind, Gassendi seemingly had no problem reconciling his personal philosophical commitment to atomism and materialism with his Christian beliefs in the immortality of the soul and the doctrine of divine providence.

Every modern reader of De Rerum Natura has been struck by the extent to which Lucretius seems to have anticipated modern evolutionary theories in the fields of geology, biology, and sociology. However, to acknowledge this connection is not to say that the poet deserves accredited status as some kind of scientific "evolutionist" or pre-Darwinian precursor. It is merely to point out that, however we choose to define and evaluate its influence, De Rerum Natura was from the 17 th century onward a massive cultural presence and hence a ready source of evolutionary ideas.

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The poem formed part of the cultural heritage and intellectual background of virtually every evolutionary theorist in Europe from Lamarck to Herbert Spencer whose hedonistic ethics also owed a debt to the poet - including though he claimed never to have read Lucretius' epic Darwin himself. Bergson's early study of Lucretius obviously played an important role in the foundation and development of his own philosophy. In Bergson published Creative Evolution , outlining his bold, new vitalistic theory of evolution, in opposition to both the earlier vitalism of Lamarck and the naturalism of Darwin, and Spencer.

Bergson's evolutionary philosophy influenced the later "process" philosophy of Alfred North Whitehead and the teleological scientific theories of Pierre Teilhard de Chardin , with the interesting result that it is possible to trace out a fairly direct, if unlikely, line of descent from Greek atomism through the pagan anti-spiritualist Lucretius to the Catholic naturalist Gassendi and then on, via the Jewish-Catholic Bergson, to the highly abstract theism of Whitehead and the "spiritualized" evolutionism of Father Teilhard.

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That Lucretius' ideas wound up two thousand years after his death influencing those of a godly British mathematical theorist and a highly original and even eccentric French scientist-priest is remarkable testimony to their durability, adaptability, and persuasive power. In conclusion, it seems fair to say that, far from being a mere conduit for earlier Greek thought, the poet Titus Lucretius Carus was a bold innovator and original thinker who fully deserves the appellation of philosopher.

While his literary fame clearly and properly comes first, and although his philosophical reputation is based largely and again properly on his role as one of the principle sources and prime exponents of Epicureanism, his own ideas, especially his evolutionary theories and his entirely naturalistic explanation of all universal phenomena, have exerted a long and important influence on western science and philosophy and should not be underestimated. Both date from the 9th century. Recently, however, scholars have deciphered a much older and previously illegible manuscript, consisting of papyri discovered in Herculaneum and possibly dating from as early as the first century AD.

All other Lucretian manuscripts date from the 15th and 16th century and are based on the one no longer extant discovered in a monastery by the Italian humanist Poggio Bracciolini in David Simpson Email: dsimpson condor. Logical consequence is arguably the central concept of logic.

The primary aim of logic is to tell us what follows logically from what. In order to simplify matters we take the logical consequence relation to hold for sentences rather than for abstract propositions, facts, state of affairs, etc. Correspondingly, logical consequence is a relation between a given class of sentences and the sentences that logically follow. One sentence is said to be a logical consequence of a set of sentences, if and only if, in virtue of logic alone, it is impossible for the sentences in the set to be all true without the other sentence being true as well.

If sentence X is a logical consequence of a set of sentences K, then we may say that K implies or entails X, or that one may correctly infer the truth of X from the truth of the sentences in K. For example, Kelly is not at work is a logical consequence of Kelly is not both at home and at work and Kelly is at home. The central question to be investigated here is: What conditions must be met in order for a sentence to be a logical consequence of others? One popular answer derives from the work of Alfred Tarski, one of the preeminent logicians of the twentieth century, in his famous paper, "The Concept of Logical Consequence.

Accordingly, we begin by examining the common concept focusing on Tarski's observations of the criteria by which we intuitively judge what follows from what and which Tarski thinks must be reflected in any theory of logical consequence. Then two theoretical definitions of logical consequence are introduced: the model theoretic and the deductive theoretic definitions.

They represent two major approaches to making the common concept of logical consequence more precise. The article concludes by highlighting considerations relevant to evaluating model theoretic and deductive theoretic characterizations of logical consequence. For a given language, a sentence is said to be a logical consequence of a set of sentences, if and only if, in virtue of logic alone, the sentence must be true if every sentence in the set were to be true. This corresponds to the ordinary notion of a sentence "logically following" from others.

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Logicians have attempted to make the ordinary concept more precise relative to a given language L by sketching a deductive system for L, or by formalizing the intended semantics for L. Any adequate precise characterization of logical consequence must reflect its salient features such as those highlighted by Alfred Tarski: 1 that the logical consequence relation is formal, that is, depends on the forms of the sentences involved, 2 that the relation is a priori, that is, it is possible to determine whether or not it holds without appeal to sense-experience, and 3 that the relation has a modal element.

For more comprehensive presentations of the two definitions of logical consequence, as well as further critical discussion, see the entries Logical Consequence, Model-Theoretic Conceptions and Logical Consequence, Deductive-Theoretic Conceptions. Tarski begins his article, "On the Concept of Logical Consequence," by noting a challenge confronting the project of making precise the common concept of logical consequence. The concept of logical consequence is one of those whose introduction into a field of strict formal investigation was not a matter of arbitrary decision on the part of this or that investigator; in defining this concept efforts were made to adhere to the common usage of the language of everyday life.

But these efforts have been confronted with the difficulties which usually present themselves in such cases. With respect to the clarity of its content the common concept of consequence is in no way superior to other concepts of everyday language. Its extension is not sharply bounded and its usage fluctuates.

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Any attempt to bring into harmony all possible vague, sometimes contradictory, tendencies which are connected with the use of this concept, is certainly doomed to failure. We must reconcile ourselves from the start to the fact that every precise definition of this concept will show arbitrary features to a greater or less degree. Tarski , p. Not every feature of the technical account will be reflected in the ordinary concept, and we should not expect any clarification of the concept to reflect each and every deployment of it in everyday language and life.

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Nevertheless, despite its vagueness, Tarski believes that there are identifiable, essential features of the common concept of logical consequence. From an intuitive standpoint, it can never happen that both the class K consists of only true sentences and the sentence X is false. Moreover, since we are concerned here with the concept of logical, that is, formal consequence, and thus with a relation which is to be uniquely determined by the form of the sentences between which it holds, this relation cannot be influenced in any way by empirical knowledge, and in particular by knowledge of the objects to which the sentence X or the sentences of class K refer.

The consequence relation cannot be affected by replacing designations of the objects referred to in these sentences by the designations of any other objects. Tarski , pp. According to Tarski, the logical consequence relation as it is employed by typical reasoners is 1 necessary, 2 formal, and 3 not influenced by empirical knowledge.

I now elaborate on 1 - 3 in order to shape two preliminary characterizations of logical consequence. Tarski countenances an implicit modal notion in the common concept of logical consequence. If X is a logical consequence of K, then not only is it the case that not all of the elements of K are true and X is false, but also this is necessarily the case. That is, X follows from K only if it is not possible for all of the sentences in K to be true with X false. For example, the supposition that All West High School students are football fans and that Kelly is not a West High School student does not rule out the possibility that Kelly is a football fan.

Hence, the sentences All West High School students are football fans and Kelly is not a West High School student do not entail Kelly is not a football fan , even if she, in fact, isn't a football fan. Also, Most of Kelly's male classmates are football fans does not entail Most of Kelly's classmates are football fans. What if the majority of Kelly's class is composed of females who are not fond of football?

We said above that Kelly is not both at home and at work and Kelly is at home jointly imply Kelly is not at work. Note that it doesn't seem possible for the first two sentences to be true and Kelly is not at work false.