Main Page - Revision history

Hannibal at 14:13, 20 March 2015

Hannibal — Fri, 20 Mar 2015 14:13:04 GMT

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-	This site is part of Candle In The Dark, a series of interconnected websites devoted to philosophy and science. See also: http://editthis.info/scientificmetho and http://www.edithis.info/criticalthought	+	This site is part of Candle In The Dark, a series of interconnected websites devoted to philosophy and science. See also: http://editthis.info/scientificmetho, http://editthis.info/atheology and http://www.edithis.info/criticalthought

	This site is intended as a resource for those interested in learning Logic. While the author of this page will take a scholarly approach to the subject, this page will approach the subject in a more user-friendly manner than available at Wikipedia, at times even using colloquial language. The site will offer a full [[Course in Logic 101]], based on two of the most reliable texts on logic: Copi and Cohen's ''Introduction to Logic (11th Edition)'' and Hurely's ''A Concise Introduction to Logic (7th Edition)''. The [[Course in Logic 101\|course]] will cover the basics of Logic, [[Classical Logic]], [[Propositional Logic]], [[Predicate Logic]], [[Modal Logic]] and [[Inductive Logic]] including a discussion of [[Inductive Logic\|Bayesian Theory]]. Those interested in taking the [[Course in Logic 101\|course]] can consult that page for the proper order of reading the pages of this site.		This site is intended as a resource for those interested in learning Logic. While the author of this page will take a scholarly approach to the subject, this page will approach the subject in a more user-friendly manner than available at Wikipedia, at times even using colloquial language. The site will offer a full [[Course in Logic 101]], based on two of the most reliable texts on logic: Copi and Cohen's ''Introduction to Logic (11th Edition)'' and Hurely's ''A Concise Introduction to Logic (7th Edition)''. The [[Course in Logic 101\|course]] will cover the basics of Logic, [[Classical Logic]], [[Propositional Logic]], [[Predicate Logic]], [[Modal Logic]] and [[Inductive Logic]] including a discussion of [[Inductive Logic\|Bayesian Theory]]. Those interested in taking the [[Course in Logic 101\|course]] can consult that page for the proper order of reading the pages of this site.

Hannibal at 13:00, 6 October 2012

Hannibal — Sat, 06 Oct 2012 13:00:31 GMT

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	:* '''[[Completeness]]''', which means that there are no true sentences in the system that cannot, at least in principle, be proved using the derivation rules (and axioms, if any) of the system.		:* '''[[Completeness]]''', which means that there are no true sentences in the system that cannot, at least in principle, be proved using the derivation rules (and axioms, if any) of the system.

-	~~Not all~~ systems achieve all three virtues. It has been proven by [[Kurt Godel]] that a system with enough axioms and/or rules of derivation to derive the principles of arithmetic cannot be both consistent and complete. This is called [[Godel's Incompleteness Theorem]].	+	Good luck finding systems that achieve all three virtues. It has been proven by [[Kurt Godel]] that a system with enough axioms and/or rules of derivation to derive the principles of arithmetic cannot be both consistent and complete. This is called [[Godel's Incompleteness Theorem]].

	===Deductive and inductive reasoning===		===Deductive and inductive reasoning===

Hannibal: /* Review */

Hannibal — Wed, 14 Apr 2010 19:03:32 GMT

Review

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	==Review==		==Review==

-	If this all seems a blur at this point, never fear	+	If this all seems a blur at this point, never fear<BR>
		+
	http://i602.photobucket.com/albums/tt102/hanniballecturer/candleinthedark/drsmith_peaceful.jpg		http://i602.photobucket.com/albums/tt102/hanniballecturer/candleinthedark/drsmith_peaceful.jpg

Hannibal: /* Review */

Hannibal — Wed, 14 Apr 2010 19:03:07 GMT

Review

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	If this all seems a blur at this point, never fear		If this all seems a blur at this point, never fear
-		+	http://i602.photobucket.com/albums/tt102/hanniballecturer/candleinthedark/drsmith_peaceful.jpg
-	http://~~www~~.~~candleinthedark~~.com/drsmith_peaceful.jpg	+

	Smith is here.		Smith is here.
-		+
-	~~[[Media:smith.ogg]]~~	+

	Every key point on this page will be explained, in detail, in the following sections. Students following the [[Course in Logic 101]] should proceed to the section entitled: [[The Laws of Classical Logic]]		Every key point on this page will be explained, in detail, in the following sections. Students following the [[Course in Logic 101]] should proceed to the section entitled: [[The Laws of Classical Logic]]

Hannibal: /* Relation to other sciences */

Hannibal — Wed, 14 Apr 2010 18:59:37 GMT

Relation to other sciences

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	===Relation to other sciences===		===Relation to other sciences===

-	Closely related to logic is semantics, or the philosophy of language, which concerns the meaning of the words and sentences; [[~~epistemlogy~~\|epistemology]], or the theory of knowledge, which concerns the conditions under which assertions are true; and the psychology of reasoning, which concerns the mental processes involved in reasoning. Logic, however, is generally understood to describe reasoning in a prescriptive manner (i.e. it describes how reasoning ought to take place), whereas psychology is descriptive and therefore more inclusive vis-a-vis the various methods of judgment humans actually use.	+	Closely related to logic is semantics, or the philosophy of language, which concerns the meaning of the words and sentences; [[epistemology\|epistemology]], or the theory of knowledge, which concerns the conditions under which assertions are true; and the psychology of reasoning, which concerns the mental processes involved in reasoning. Logic, however, is generally understood to describe reasoning in a prescriptive manner (i.e. it describes how reasoning ought to take place), whereas psychology is descriptive and therefore more inclusive vis-a-vis the various methods of judgment humans actually use.

	===Definition of an Argument===		===Definition of an Argument===

Hannibal: /* Relation to other sciences */

Hannibal — Wed, 14 Apr 2010 18:59:09 GMT

Relation to other sciences

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	===Relation to other sciences===		===Relation to other sciences===

-	Closely related to logic is semantics, or the philosophy of language, which concerns the meaning of the words and sentences; epistemology, or the theory of knowledge, which concerns the conditions under which assertions are true; and the psychology of reasoning, which concerns the mental processes involved in reasoning. Logic, however, is generally understood to describe reasoning in a prescriptive manner (i.e. it describes how reasoning ought to take place), whereas psychology is descriptive and therefore more inclusive vis-a-vis the various methods of judgment humans actually use.	+	Closely related to logic is semantics, or the philosophy of language, which concerns the meaning of the words and sentences; [[epistemlogy\|epistemology]], or the theory of knowledge, which concerns the conditions under which assertions are true; and the psychology of reasoning, which concerns the mental processes involved in reasoning. Logic, however, is generally understood to describe reasoning in a prescriptive manner (i.e. it describes how reasoning ought to take place), whereas psychology is descriptive and therefore more inclusive vis-a-vis the various methods of judgment humans actually use.

	===Definition of an Argument===		===Definition of an Argument===

Hannibal: /* Nature of logic */

Hannibal — Fri, 29 Jan 2010 20:41:18 GMT

Nature of logic

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-	'''Logic''', (from the Greek word λόγο&sigmaf; (''logos''), originally meaning ''the word'', but also referring to ''speech'' or ''reason'') is the science of evaluating the reasoning within [[Argument\|arguments]]. It ~~is a method consisting of~~ a set of rules that tell us when an [[argument\|argument's]] premises support their conclusion (Hofweber 2004). Logic arose from a concern with correctness of [[argument]]ation (See [[Aristotle]]).	+	'''Logic''', (from the Greek word λόγο&sigmaf; (''logos''), originally meaning ''the word'', but also referring to ''speech'' or ''reason'') is the science of evaluating the reasoning within [[Argument\|arguments]]. It refers to any a set of rules that tell us when an [[argument\|argument's]] premises support their conclusion (Hofweber 2004). Logic arose from a concern with correctness of [[argument]]ation (See [[Aristotle]]).

	Surprised? People usually come to hold, through their lack of any formal training in logic, that logic is far more than it really is. Therefore, a further understanding of just what logic is, can be enhanced by delineating it from what it is not:		Surprised? People usually come to hold, through their lack of any formal training in logic, that logic is far more than it really is. Therefore, a further understanding of just what logic is, can be enhanced by delineating it from what it is not:

Hannibal: /* Modal logic */

Hannibal — Fri, 29 Jan 2010 19:50:01 GMT

Modal logic

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	===Modal logic===		===Modal logic===

-	~~[[File:~~http://www.thadguy.com/wp-content/uploads/2007/07/modal-logic-can-solve-all-problems.png]]	+	http://www.thadguy.com/wp-content/uploads/2007/07/modal-logic-can-solve-all-problems.png

	The concepts behind Modal Logic date back to [[Aristotle]], who was concerned with the [[alethic modalities]] of necessity and possibility, which he observed to be dual in the sense of [[De Morgan duality]]. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatisations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include [[deontic logic]] and [[epistemic logic]]. The seminal work of [[Arthur Prior]] applied the same formal language to treat [[temporal logic]] and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of [[frame semantics]] which revolutionized the formal technology available to modal logicians and gave a new [[graph theory\|graph-theoretic]] way of looking at modality that has driven many applications in [[computational linguistics]] and [[computer science]], such as [[dynamic logic]].		The concepts behind Modal Logic date back to [[Aristotle]], who was concerned with the [[alethic modalities]] of necessity and possibility, which he observed to be dual in the sense of [[De Morgan duality]]. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatisations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include [[deontic logic]] and [[epistemic logic]]. The seminal work of [[Arthur Prior]] applied the same formal language to treat [[temporal logic]] and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of [[frame semantics]] which revolutionized the formal technology available to modal logicians and gave a new [[graph theory\|graph-theoretic]] way of looking at modality that has driven many applications in [[computational linguistics]] and [[computer science]], such as [[dynamic logic]].

Hannibal at 19:49, 29 January 2010

Hannibal — Fri, 29 Jan 2010 19:49:36 GMT

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	===Modal logic===		===Modal logic===

-	~~(Note, this section is merely ripped from WIKI, I will remove it when I get a chance to update this section on my own)~~	+	[[File:http://www.thadguy.com/wp-content/uploads/2007/07/modal-logic-can-solve-all-problems.png]]
-		+
-	~~In language,~~ [[~~modality]] deals with the phenomenon that subparts of a sentence may have their semantics modified by special verbs or~~ modal ~~particles. For example, "''We go to the games''"~~ can ~~be modified to give "''We should go to the games''", and "''We can go to the games''"" and perhaps "''We will go to the games''"~~. ~~More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied.~~	+	The concepts behind Modal Logic date back to [[Aristotle]], who was concerned with the [[alethic modalities]] of necessity and possibility, which he observed to be dual in the sense of [[De Morgan duality]]. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatisations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include [[deontic logic]] and [[epistemic logic]]. The seminal work of [[Arthur Prior]] applied the same formal language to treat [[temporal logic]] and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of [[frame semantics]] which revolutionized the formal technology available to modal logicians and gave a new [[graph theory\|graph-theoretic]] way of looking at modality that has driven many applications in [[computational linguistics]] and [[computer science]], such as [[dynamic logic]].
-		+
-	The ~~logical study of modality dates~~ back to [[Aristotle]], who was concerned with the [[alethic modalities]] of necessity and possibility, which he observed to be dual in the sense of [[De Morgan duality]]. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatisations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include [[deontic logic]] and [[epistemic logic]]. The seminal work of [[Arthur Prior]] applied the same formal language to treat [[temporal logic]] and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of [[frame semantics]] which revolutionized the formal technology available to modal logicians and gave a new [[graph theory\|graph-theoretic]] way of looking at modality that has driven many applications in [[computational linguistics]] and [[computer science]], such as [[dynamic logic]].	+

	==Review==		==Review==

Hannibal at 15:25, 16 May 2009

Hannibal — Sat, 16 May 2009 15:25:26 GMT

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	which is invalid, because there are no connectors between the premises and between the premises and the conclusion.		which is invalid, because there are no connectors between the premises and between the premises and the conclusion.

-	Therefore, the need for a ''First Order'', or [[Predicate Logic]] became apparent. The new ingredient of first-order logic not found in propositional logic is [[Predicate Logic\|quantification]]: where Ï� is any (well-formed) formula, the new constructions ~~â��x Ï�~~ and ~~â��x Ï� â��~~ read "for all x, Ï�" and "for some x, Ï�" ~~â��~~ are introduced, where x is an individual variable whose range is the set of individuals of some given universe of discourse (or domain). For example, if the universe consists solely of people, then x ranges over people. ~~For~~ convenience, we write Ï� as Ï�(x) to show that it contains only the variable x free and, for b a member of the universe, we let Ï�[b] express that b satisfies (i.e. has the property expressed by) Ï�. Then ~~â��x Ï�~~(x) states that Ï�[b] is true for every b in the universe, and ~~â��x Ï�~~(x) means that there is a b (in the universe) such that Ï�[b] holds.	+	Therefore, the need for a ''First Order'', or [[Predicate Logic]] became apparent. The new ingredient of first-order logic not found in propositional logic is [[Predicate Logic\|quantification]]: where φ is any (well-formed) formula, the new constructions ∀x φ and ∃x φ — read "for all x, φ" and "for some x, φ" — are introduced, where x is an individual variable whose range is the set of individuals of some given universe of discourse (or domain). For example, if the universe consists solely of people, then x ranges over people. or convenience, we write φ as φ(x) to show that it contains only the variable x free and, for b a member of the universe, we let φ[b] express that b satisfies (i.e. has the property expressed by) φ. Then ∀x φ(x) states that φ[b] is true for every b in the universe, and ∃x φ(x) means that there is a b (in the universe) such that φ[b] holds.

	The argument about Socrates can be formalized in first-order logic as follows. Let the universe of discourse be the set of all people, living and deceased, and let Man(x) be a predicate (which, informally, means that the person represented by variable x is a man) and Mortal(x) be a second predicate. Then the argument above becomes		The argument about Socrates can be formalized in first-order logic as follows. Let the universe of discourse be the set of all people, living and deceased, and let Man(x) be a predicate (which, informally, means that the person represented by variable x is a man) and Mortal(x) be a second predicate. Then the argument above becomes

-	* ~~â��~~ x (Man(x) ~~â��~~ Mortal(x))	+	* ∀ x (Man(x) → Mortal(x))
	* Man(Socrates)		* Man(Socrates)
-	* ~~â�´~~ Mortal(Socrates)	+	* ∴ Mortal(Socrates)
-		+
-	A literal translation of the first line would be "For all x, if x is described by 'Man', x must also be described by 'Mortal'." The second line states that the predicate "Man" applies to Socrates, and the third line translates to "Therefore, the description 'Mortal' applies to Socrates."	+

		+	A literal translation of the first line would be "For all x, if x is described by 'Man', x must also be described by 'Mortal'." The second line states that the predicate "Man" applies to Socrates, and the third line translates to "Therefore, the description 'Mortal' applies to Socrates."

	===Modal logic===		===Modal logic===