The Art Of Controversy by Arthur Schopenhauer.

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Three types of arguments:

Retroduction:

1. All men are mortal. [Generalization]
2. Socrates is mortal. [Particular Observation]

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3. Hence, Socrates is a man. [Particular Occurrence]

Induction:

1. Socrates is mortal. [Particular Observation]
2. Socrates is a man. [Particular Occurrence]

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3. Hence, all men are mortal. [Generalization]

Deduction:

1. All men are mortal. [Generalization]
2. Socrates is a man. [Particular Occurrence]

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3. Hence, Socrates is mortal. [Particular Observation]

From Bruce E. R. Thompson's "Introduction to Fallacies Page" with modifications in word choice to make unambiguous the characterization. He uses the word "rule" where I use "generalization." I did that in order to keep an eye to translation into formal logic. In keeping with my formal logic bias, I named two particularitie: I replaced "case" with "particular occurrence" and "result" with "particular observation."

What do you think about this schema?

NOTE: I updated the links on my del.icio.us page with regard to this Community.

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Premiss or premise: a note on spelling...

From what I can tell, "premiss" is a more unambiguous spelling, but since both words mean what amounts to the same thing etymologically, denotationally, and conotationally speaking, there seems to be no other reason to use one word (or spelling) over another. The following is the result of a small inquiry made into the matter:

Bruce E. R. Thompson states on his "Introduction to Fallacies Page:"

Alert readers will notice that on this website I use the spelling "premiss" and "premisses" rather than "premise" and "premises." If correctness is to be measured by etymology rather than common practice, these spellings are preferable to the more common spellings. The word "premiss," as it was originally used in English, is derived from a medieval Latin word that refers (just as one would expect) to the portion of an argument in which justification and reasons are offered in support of a conclusion. The word "premise" is derived from a French legal term meaning "aforesaid," and in English has come to be used (usually in the plural) to refer, rather legalistically, to property (and its appurtenances), in such phrases as "to occupy the premises" and "to vacate the premises." The two words are etymologically unrelated to each other, sharing only coincidentally the same pronunciation. However confusion over the spelling of the two words is also long-standing. Even a hundred years ago Charles Peirce felt he had to explain his preference for the older (and in his view more correct) spelling, just as I am doing now.

Dictionary.com describes the etymology "premise" as:

Word History: Why do we call a single building the premises? To answer this question, we must go back to the Middle Ages. But first, let it be noted that premises comes from the past participle praemissa, which is both a feminine singular and a neuter plural form of the Latin verb praemittere, “to send in advance, utter by way of preface, place in front, prefix.” In Medieval Latin the feminine form praemissa was used as a term in logic, for which we still use the term premise descended from the Medieval Latin word (first recorded in a work composed before 1380). Medieval Latin praemissa in the plural meant “things mentioned before” and was used in legal documents, almost always in the plural, a use that was followed in Old French and Middle English, both of which borrowed the word from Latin. A more specific legal sense in Middle English, “that property, collectively, which is specified in the beginning of a legal document and which is conveyed, as by grant,” was also always in the plural in Middle English and later Modern English. And so it remained when this sense was extended to mean “a house or building with its grounds or appurtenances,” a usage first recorded before 1730.

What do you think?

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A Philosophy 124 exercise
00.|
00.|-
01.|| P [Assumption]
00.||-
02.||| Q [Assumption]
00.|||-
03.||| P [Reit: 1]
04.|| Q->P [-> Intro: 2-3]
05.| P->(Q->P)) [-> Intro: 1-4]

But naraht1 pointed out that the same proof, if logial equivilances were allowed, could be proved thus:

~(P -> (Q -> P)) <=>
P ^ ~( Q -> P) <=>
P ^ (Q ^ ~P) <=>
P ^ ( ~P ^ Q) <=>
(P ^ ~P) ^Q <=>
False ^ Q =>
False.

QED (P-> (Q-> P)

Originally posted on February 16, 2005 to my personal journal.

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Another Philosophy 124 exercise
( The smallest solutionCollapse )

( How I solved itCollapse )

Note: Ø = Contradiction [Damn Unicode issues]

Sorry it looks like a crappy version of Fitch. I hope this helps. I really miss natural deduction systems like this. When you start getting serious with logic you switch back to those annoying axiomatic deduction systems and are reduced to using nothing but material conditionals as your sentential logic connective symbols and modus ponens is your only real rule.

This was originally posted on February 7, 2005 to my personal journal.

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This is an exercise from my Philosophy 126 class on Goedel's Incompleteness Theorem and Logic
A proof of two Arithmetic truths:</strong> The point of this exercise is to do arithmetic through deduction using a modified formal language of first-order logic (adding the Dedikind-Peano Axioms and Axioms of Equality to the formal language).

├_N ~(0⁽³⁾ = 0⁽¹⁾)

├_N (0 = 0)

1.	(\-/x1) ~(x1’= 0)	(\-/x1) ~A12(f11 (x1), a1)	[N1*]
2.	(\-/x2) ~(x2’ = 0)	(\-/x1) ~A12 (f11 (x2), a1)	[N1*]
3.	x3 = x3	A12 (x3, x3)	[E7]
4.	(\-/x3) (x3 = x3)	(\-/x3) A12 (x3, x3)	3: Generalization
5.	(\-/x3) (x3 = x3) → (0 = 0)	(\-/x3) A12 (x3, x3) → A12 (a1, a1)	(4:) [K4]
6.	(0 = 0)	A12 (a1, a1)	4, 5: Modus Ponens
7.	(\-/x1) (\-/x2) (~(x1 = x2) → ~(x1’ = x2’))	(\-/x1) (\-/x2) (~ A12 (x1, x2) → ~(f11 (x1), f11 (x2)))	[N2*]
8.	(\-/x1) ("x2) (~(x1 = x2) → ~(x1’ = x2’)) → (~(0(2) = 0) → ~(0(3) = 0(1)))	(\-/x1) ("x2) (~ A12 (x1, x2) → ~(f11 (x1), f11 (x2))) → (~ A12 (f11 (f11 (a1)), a1) → ~ A12 (f11 (f11 (f11 (a1))), f11 (a1)))	(7:) [K5]
9.	(\-/x1) ~(x1’ = 0) → ~(0(2) = 0)	(\-/x1) ~A12 (f11 (x2), a1) → ~ A12 (f11 (f11 (a1)), a1)	(1, 8:) [K5]
10.	~(0(2) = 0)	~ A12 (f11 (f11 (a1)), a1)	1, 9: Modus Ponens
11.	~(0(2) = 0) → ~(0(3) = 0(1))	~ A12 (f11 (f11 (a1)), a1) → ~ A12 (f11 (f11 (f11 (a1))), f11 (a1))	7, 8: Modus Ponens
12.	~(0(3) = 0(1))	~ A12 (f11 (f11 (f11 (a1))), f11 (a1))	10, 11: Modus Ponens
Thus:	(0 = 0)	A12 (a1, 1)	(6:) Derivative conclusion
Thus:	~(0(3) = 0(1))	~ A12 (f11 (f11 (f11 (a1))), f11 (a1))	(12:) Derivative conclusion

These proofs were originally posted on November 2, 2003 to my personal journal.

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