User:Philogo

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[edit] Quote de jour

It is all the same to me...whether it is your own opinion or not. It is the argument itself that I wish to probe, though it may turn out that both I who question and you who answer are equally under scrutiny.

Plato, Protagoras, 333c

The Philosophy Tripos Philosophy is a study of problems which are ultimate and very general, concerned with the nature of reality, knowledge, mind, language, and value. In university courses it is studied in a manner which lays considerable emphasis on precise and careful argument. In the earlier stages of the Cambridge course, the central elements are logic, metaphysics, ethics, and philosophy of mind; later on attention is also paid to political philosophy, philosophy of science, and aesthetics. As the course proceeds the number of optional elements increases, so that in Part II there are no compulsory subjects.
...
3. Logic
Candidates are asked to answer three questions out of at least eight set, including at least one from each section.
Section A: Formal Logic
Basic concepts: formalized languages; object-language and metalanguage; use and mention; validity, implication and consistency. Propositional logic: truth-functions, tautologies, proof; soundness and completeness. Introduction to predicate logic: the language of quantifiers and variables; validity and counterexamples; elements of the logic of identity. Classes and relations; elements of probability calculus.
Section B: Philosophical logic
Deductive and inductive arguments. Problems of translation between natural and formal languages. Sentences, statements and propositions. The theory of descriptions. Necessity, analyticity and the a priori. Meaning and verification. [University of Cambridge]

[edit] int link

dirty sandbox

[edit] Recommended Reading

[edit] try harvard

(Smith 2006, p. 25)

[edit] Citation template styles

A summary of the syntax of all Harvard citation templates is at Wikipedia:Citation templates.

The Harvard citation templates available for use can be divided into two groups, depending on the format used for displaying page numbers. One style displays page numbers using p., creating a citation that looks like (Blust 1999, p. 12). The second style uses a colon, as for example (Blust 1999:12).

The actual text that you would write in order to display this on the page, including the wikitext of the templates, would look like this:

…creating a citation that looks like {{Harv|Blust|1999|p=12}}. The second style uses a colon, as for example {{Harvcol|Blust|1999|p=12}}.

[edit] With p.

Three templates are currently available using this style: {{Harvard citation}}, {{Harv}} and {{Harvnb}}. Except for their different names, the {{Harv}} and {{Harvard citation}} templates are identical. The name of the former has been shortened, making it quicker and easier to type. The third template, {{Harvnb}}, does not display the surrounding parentheses (brackets), so the "nb" in its name is an abbreviation for "no brackets." This style is very useful for placing more than one citation inside a single set of parentheses, which is often done when listing a number of related articles. An example of using a number of citations together inside parentheses might look like this:

"Although most scholars accept the blahblah theory, a few have criticized its main points (e.g. Blust 1999, Tsuchida & Yamada 1991, and Peiros 2004)."

The actual text that you would write in order to display this on the page, including the wikitext of the templates, would look like this:

"Although most scholars accept the blahblah theory, a few have criticized its main points (e.g. {{Harvnb|Blust|1999}}, {{Harvnb|Tsuchida|Yamada|1991}}, and {{Harvnb|Peiros|2004}})."

[edit] my try harvard

(Blust 1999, p. 12).

[edit] try refs

vanilla reference:

[1]

Same ref used twice or more The first time a reference appears in the article, you can give it a simple name in the <ref> code:[2] The second time you use the same reference in the article, you need only to create a short cut instead of typing it all out again: [2]

My try:

The first time [3] The second time [3]

end my try1 my try 2

The first time [4] The second time [4]

end mytry2

my try 3

The first time Mumford 1999[4] The second time Mumford 1999[4]

end mytry3 my try 4

The first time Mumford 1999[5] The second time Mumford 1999[5]

end my try 4


Cite a book like this:-

{{cite book | last = Mumford | first = David | authorlink = David Mumford | title = The Red Book of Varieties and Schemes | publisher = [[Springer-Verlag]] | series = Lecture notes in mathematics 1358 | year = 1999 | doi = 10.1007/b62130 | isbn = 354063293X }}

which will appear like this: Mumford, David (1999). The Red Book of Varieties and Schemes, Lecture notes in mathematics 1358. Springer-Verlag. DOI:10.1007/b62130. ISBN 354063293X. 

the follwing skips some parms:- Mumford, xxx (1999). my title, Lecture notes in mathematics 1358. Springer-Verlag. </nowiki>Template:Unreferenced

read about it here:- Template:Citation/doc

eg an edited book looks like this:

Bidamon, Emma Smith (March 27, 1876), “Letter to Emma S. Pilgrim”, in Vogel, Dan, Early Mormon Documents, vol. 1, Signature Books, 1996, ISBN 1-56085-072-8 

eg make a referenced citation like this:-

<ref>{{Citation | last=Bidamon | first=Emma Smith | author-link=Emma Hale Smith | chapter=Letter to Emma S. Pilgrim | date=[[March 27]] [[1876]] | year=1876 | editor-surname=Vogel | editor-first=Dan | title=Early Mormon Documents | volume=1 | publisher=Signature Books | publication-date=1996 | isbn=1-56085-072-8 }}</ref>

and it will look like this:-[6]

eg make a named referenced citation

<ref name=Bidamon1876>{{Citation | last=Bidamon | first=Emma Smith | author-link=Emma Hale Smith | chapter=Letter to Emma S. Pilgrim | date=[[March 27]] [[1876]] | year=1876 | editor-surname=Vogel | editor-first=Dan | title=Early Mormon Documents | volume=1 | publisher=Signature Books | publication-date=1996 | isbn=1-56085-072-8 }}</ref>

and it will look like this: ref name=Bidamon1876>Bidamon, Emma Smith (March 27, 1876), “Letter to Emma S. Pilgrim”, in Vogel, Dan, Early Mormon Documents, vol. 1, Signature Books, 1996, ISBN 1-56085-072-8 </ref>







London

Contents




[edit] user subpages:








  1. My Articles
  2. Articles of interest
  3. Articles I have edited
  4. Articles of concern
  5. links
  6. logic articles
  7. Mathematical Induction
  8. my page called new page
WikiProject Philosophy This user is a member of the WikiProject Philosophy.




This user is a member of the Logic task force




~(p&~p) This user studies the concept of Logic



[edit] Templates:

Template:User WP Logic‎
Template:User WP Philosophy


+++

Socrates
Philosophy Portal
This article is within the scope of the WikiProject Philosophy, which collaborates on articles related to philosophy. To participate, you can edit this article or visit the project page for more details.
??? This article has not yet received a rating on the quality scale.
??? This article has not yet received an importance rating on the importance scale.

+++
Template:Logic navigation


[edit] links

Wikipedia:How to rename (move) a page
Wikipedia:Cheatsheet Portal:Philosophy and so are: click here for list-> [1]



Philosophy Quick Topic Guide
Wikipedia:WikiProject Logic
Wikipedia:WikiProject Philosophy
Wikipedia:Manual of Style

Table of logic symbols


Atomic sentence#Further reading
Wikipedia:How to edit a page
[[Wikipedia:Manual of Style#Italics]] >Wikipedia:Manual of Style#Italics is a link to a section within another page.

[[#Links]] >#Links is a link to another section on the current page.

http://en.wikibooks.org/wiki/Logic/First-Order_Logic http://www.answers.com/topic/first-order-predicate-calculus?cat=technology

[edit] subpages

[edit] wiki help

Hello, Philogo, and welcome to Wikipedia! Thank you for your contributions so far. Here are a few good links for newcomers:

If you need help, post a question at the Help Desk or ask me on my talk page.

[edit] dirty sandbox

User talk:Philogo#propositions

Section linking


my wiki help

my wiki help

my links

http://en.wikipedia.org/wiki/Philosophy_of_logic

Philosophy_of_logic

Philosophy of logic

Tarski's definition of Truth

Philosophy of logic#Plato's Beard & The problem of non-being

Philosophy of logic: Plato's Beard & The problem of non-being

Plato's Beard & The problem of non-being (Philosophy of logic)

Plato's Beard & The problem of non-being in Philosophy of logic

Philosophy of logic#The analytic/synthetic distinction

/new page

same new page

Wikipedia:Administrators' noticeboard/Incidents


  1. Sandoxes

Article text.[7]

Article text.[1]

To reuse the same note, reuse the name with a trailing slash:

Article text.[7]

hullo [8]

##[[/sandbox|sand chars]].

    1. sand chars.


    1. Sandbox.
    2. LogUserBox.


|}

    1. test word paste.
    2. Argument (Logic)
    3. test table

test only:

{{3O|article}}

 There is a dispute about this article and an editor has requested a third opinion.
If you would like to contribute to the discussion, please see the talk page.

{{Refimprove|date=October 2007}}

 This article needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (October 2007)

{{Original research|date=May 2008}}</nowiki>

 This article or section may contain original research or unverified claims.
Please improve the article by adding references. See the talk page for details. (May 2008)
  1. A list of proposals

[edit] A list of proposals

This section is just a list of the proposed sentences for the lede. Please add your own proposal below. Tomorrow (Friday) we can begin to vote. Please vote here, but discuss above or below this section to avoid clutter.

1) Its investigations are, unlike those of religion or superstition, wedded to reason, striving to make no unexamined assumptions or make assertions purely on the basis of faith, revelation or analogy.

2) Philosophy is thinking clearly and carefully about the most fundamental questions. Most philosophers agree that rational discourse is the proper method to address these questions.

3) The distinctive mark of philosophy is its explicitly rational and critical way of proceeding and its systematic nature.

4)<namely that the use of reason is not merely part of it, but distinguishes philosophy, or is central to it.> see above - (talk) 14:15, 5 June 2008 (UTC)

5) Its investigations use reason, striving to make no unexamined assumptions.

6) No additional phrase

7) Philosophy is a study of problems which are ultimate and very general, concerned with the nature of reality, knowledge, mind, language, and value. In university courses it is studied in a manner which lays considerable emphasis on precise and careful argument. The central elements are logic,epistemology, metaphysics, philosophy of language, ethics, philosophy of mind, political philosophy, philosophy of science, philosophy of logic, and aesthetics.

(based on University of Cambridge Phil Dpt prospectus)--Philogo 00:50, 6 June 2008 (UTC)

8) We will have no sentences trying to define what philosophy is, or what its methods are, or the equivalent, in this article. --Philogo 00:53, 6 June 2008 (UTC)

9) There is no agreed definition of the word philosophy, despite many attempts to provide one, see Talk:Philosophy/Quotations

[edit] drafts

[edit] pol

philosophy of logic

[edit] Truth, Propositions and Meaning

Since Since logic refers to truth and falsity it presupposes the meaningfulness of these terms and so philosophy of logic has concerned itself with the correct analysis of the meaning of these terms. It is as well to begin with making some distinctions based on Wolfram 1989, (Chapter 2 Section1), and introduce and some terminology as used in this article • A: This toucan can catch a can. • B: If you have a bucket, then you have a pail. • C: I promise to be good. • D: He is grnd. • E: Are you happy? • F: Cats blows the wind • G: This stone is thinking about Vienna • H: This circle is square Words. Word-tokens and word-types and word-meanings. Word-tokens A word-token is a sequence of characters (or units of speech). The sequence of characters A (above) contains five word-tokens Word-types A word-type is an identical sequence of characters (or units of speech). The sequence of characters A (above) contains four word-types (the word-token can occurring twice) Word-meanings The sequence of characters A (above) contains four word-meanings. Although it contains only four word-types, the two occurrences of the word-token can have different meanings. Consider the sequence of characters labeled B above. On the assumption that bucket and spade mean the same, B contains nine word-tokens, six-word-types,

Sentences In grammar a sentence can be a declaration, an explanation, a question, a command. [Truth, Propositions and Meaning]

Since logic refers to truth and falsity it presupposes the meaningfulness of these terms and so philosophy of logic has concerned itself with the correct analysis of the meaning of these terms. It is as well to begin with making some distinctions based on Wolfram 1989, (Chapter 2 Section1), and introduce and some terminology as used in this article

  • A: This toucan can catch a can.
  • B: If you have a bucket, then you have a pail.
  • C: I promise to be good.
  • D: He is grnd.
  • E: Are you happy?
  • F: Cats blows the wind
  • G: This stone is thinking about Vienna
  • H: This circle is square
  • I: The author of Waverly is dead
  • J: The author of Ivanhoe is dead
  • K: I am under six foot tall
  • L: I am over six foot tall
  • M: The conducter is a bachelor
  • N: The conducotr is married

Words.
Word-tokens and word-types and word-meanings.

Word-tokens
A word-token is a sequence of characters (or units of speech).
The sequence of characters A (above) contains five word-tokens

Word-types
A word-type is an identical sequence of characters (or units of speech).
The sequence of characters A (above) contains four word-types (the word-token can occurring twice)

Word-meanings
The sequence of characters A (above) contains four word-meanings.
Although it contains only four word-types, the two occurrences of the word-token can have different meanings.
Consider the sequence of characters labeled B above.
On the assumption that bucket and spade mean the same, B contains nine word-tokens, six-word-types,


Sentences
In grammar a sentence can be a declaration, an explanation, a question, a command. In logic a declarative sentence is considered to be a sentence that can be used to communicate truth. Some sentences which are grammatically declarative are not logically so.
Meaningful Declarative-sentences A declarative-sentence is a ...

Sentence-tokens
A sentence-token is a sequence of meaningful word-tokens.
The sequence of characters D (above) is not a sentence-token because grnd is not a meaningful word-token.

Declarative-sentence-tokens
A declarative-sentence-token is a sentence-token which that can be used to communicate truth or convey information.
The sequence of characters E (above) is not a declarative-sentence-token because it interrogative not declarative.
Meaningful-declarative-sentence-tokens
A meaningful-declarative-sentence-token is a declarative-sentence-token which has meaning.
The sequence of characters F (above) (Cats blows the wind) is not a meaningful-declarative-sentence-token because it is grammatically ill-formed
The sequence of characters G above ( This stone is thinking about Vienna) is not a meaningful-declarative-sentence-token because thinking cannot be predicated of a stone
The sequence of characters H (above) (This circle is square) is not a meaningful-declarative-sentence-token because it is internally inconsistent

Nonsense- declarative-sentence-token
A nonsense-declarative-sentence-token is a declarative-sentence-token which is not a meaningful-declarative-sentence-token.
The sequences of characters F, G & H above are nonsense-declarative-sentence-token because they are declarative-sentence-token but not meaningful-declarative-sentence-tokens


Propositions
A meaningful-declarative-sentence-token expresses a proposition.
Two meaningful-declarative-sentence-tokens which have the same meaning express the same proposition.
The two sequences of characters I: (The author of Waverly is dead) and J (The author of Ivanhoe is dead) have different meanings and therefore express different propositions.

Statements
The concept of a statement was introduced by John Stebbing in the 1950s.
Two meaningful-declarative-sentence-tokens which say the same thing of the same object(s) make the same statement.
On the assumption tha the same person wrote Waverly and Ivanhoe, the two sequences of characters I: (The author of Waverly is dead) and J (The author of Ivanhoe is dead) make the same statement but express differnet propositions.
The pairs of sentence-tokens K,L & M,N have different meanings, but they are not necessarily contradictory, since K& L may have been assertd by different people, and M&N nay have been asserted about different conductors.

What there examples show is that we cannot identify that which is true or false (the statement) with the sentence used in making it; for the same sentence may be used to make different statements, some of them true and some of them false.


(Stebbing 1952)

[edit] Truth

[edit] Tarski's definition of Truth

...

[edit] Propositions

see also Willard Van Orman Quine, Proposition

[edit] pol glossary

  • Word.
    • 1. (word-token) an individual instance of a word.
    • 2. (word-type1) word-tokens are of the same word-type if they are typographically identical
    • 3. (wordtype2) word-tokens are of the same word-type if they are typographically identical and have the same meaning
  • Sentence. (varied usage) Series of words bounded by full stops, etc. and distinguished into sentence-token and sentence-type.
  • Token. (word, sentence, proposition, statement). Individual instance of a word &c., a particular.
  • Particular. Individual such as material object, event person.: distinguished from non-particulars by feature that there can be two particulars which are indistinguishable except for their location in time and space.
  • word-token: A word-token is a sequence of characters (or units of speech).
  • word-type. A word-type is an identical sequence of characters (or units of speech).

A word-token is a sequence of characters (or units of speech).
A word-type is an identical sequence of characters (or units of speech).
The sequence of characters A (above) contains four word-types (the word-token can occurring twice)

  • Word-meanings
  • Meaningful Declarative-sentences: A declarative-sentence is a ...
  • Sentence-tokens. A sentence-token is a sequence of meaningful word-tokens.
  • Declarative-sentence-tokens: A declarative-sentence-token is a sentence-token which that can be used to communicate truth or convey information.
  • Meaningful-declarative-sentence-tokens. A meaningful-declarative-sentence-token is a declarative-sentence-token which has meaning.
  • Nonsense- declarative-sentence-token. A nonsense-declarative-sentence-token is a declarative-sentence-token which is not a meaningful-declarative-sentence-token.
  • Propositions. A meaningful-declarative-sentence-token expresses a proposition. Two meaningful-declarative-sentence-tokens which have the same meaning express the same proposition.
  • Statements. Two meaningful-declarative-sentence-tokens which say the same thing of the same object(s) make the same statement.

[edit] pol citations

Wolfram (1989). Philosophical Logic: an introduction. Routledge. DOI:1. ISBN 0 415 02317 3 (0 415 02318 (pbk)). 

[edit] quotes

http://www.hf.uio.no/ifikk/filosofi/njpl/vol1no1/meaning/node2.html:

Let us first look at the term proposition. It has its origin in the Gr. , used by Aristotle in the Prior Analytics, the third part of the Organon. It was translated, apparently by Cicero, into Lat. propositio, which has its modern counterparts in It.\ proposizione, Eng. proposition and Ger. Satz. In the old, traditional use of the word proposition, propositions are the things that we prove. We talk about proposition and proof, of course, in mathematics: we put up a proposition and let it be followed by its proof. In particular, the premises and conclusion of an inference were propositions in this old terminology. It was the standard use of the word up to the last century. And it is this use which is retained in mathematics, where a theorem is sometimes called a proposition, sometimes a theorem. Thus we have two words for the things that we prove, proposition and theorem. The word proposition, Gr. , comes from Aristotle and has dominated the logical tradition, whereas the word theorem, Gr. , is in Euclid, I believe, and has dominated the mathematical tradition. With Kant, something important happened, namely, that the term judgement, Ger. Urteil, came to be used instead of proposition. Perhaps one reason is that proposition, or a word with that stem, at least, simply does not exist in German: the corresponding German word would be Lehrsatz, or simply Satz. Be that as it may, what happened with Kant and the ensuing German philosophical tradition was that the word judgement came to replace the word proposition. Thus, in that tradition, a proof, Ger. Beweis, is always a proof of a judgement. In particular, the premises and conclusion of a logical inference are always called judgements. And it was the judgements, or the categorical judgements, rather, which were divided into affirmations and denials, whereas earlier it was the propositions which were so divided. The term judgement also has a long history. It is the Gr. , translated into Lat. judicium, It. giudizio, Eng. judgement, and Ger. Urteil. Now, since it has as long a history as the word proposition, these two were also previously used in parallel. The traditional way of relating the notions of judgement and proposition was by saying that a proposition is the verbal expression of a judgement. This is, as far as I know, how the notions of proposition and judgement were related during the scholastic period, and it is something which is repeated in the Port Royal Logic, for instance. You still find it repeated by Brentano in this century. Now, this means that, when, in German philosophy beginning with Kant, what was previously called a proposition came to be called a judgement, the term judgement acquired a double meaning. It came to be used, on the one hand, for the act of judging, just as before, and, on the other hand, it came to be used instead of the old proposition. Of course, when you say that a proposition is the verbal expression of a judgement, you mean by judgement the act of judging, the mental act of judging in scholastic terms, and the proposition is the verbal expression by means of which you make the mental judgement public, so to say. That is, I think, how one thought about it. Thus, with Kant, the term judgement became ambiguous between the act of judging and that which is judged, or the judgement made, if you prefer. German has here the excellent expression gefälltes Urteil, which has no good counterpart in English.

[edit] formal language

A formal language is a language in which an expression’s grammaticality and interpretation (if any) are determined by precisely defined rules that appeal only to the form or shape of the symbols that constitute it (rather than, for example, to the intention of the speaker). (Audi, Robert (1999). The Cambridge Dictionary of Philosophy, Second Edition. Cambridge UP. ISBN 10 0-521-63722-8. )

Formal languages are sometimes referred to as artificial languages or formalised languages.

In Logic the idea of constructing such languages is at least as early as Leibniz but the first successful attempt was the Begriffsschrift ("concept writing") of Gottlob Frege. (Mates, Benson (1972). Elementary Logic publisher = OUP. )

[9]

A formal language is characterised by its vocabulary and syntax. (Gamut, L.T.F (1991). Logic Language and Meaning, Vol. 1 Introduction to Logic publisher= The University of Chicago Press. ISBN 0 -226-28085-3. ) ).

[10]

[11]


 The vocabulary sets out the basic expressions; the syntax set out how the composite expressions can be built up from the basic expressions.  The vocabulary (basic expressions) of languages of Logic typically consists of constants, variables and the auxiliary signs.  The constants are either logical constants or non-logical constants.  The non-logical constants have no fixed meaning ("semantic content") in a language except under

[edit] mess

OK Hans. I found a lede which appeared non-sensical with no citations. I replaced it with the best definition I could lay my hands on at the time and carefully cited it. I gave another citation which supported the first. I assumed the lede would be edited and built upon.

My appetite is somewhat whetted. I read elsewhere:

The theory that we consider in this chapter is based on the following simple abstract notions.
An alphabet is a set of symbols.
A string over an alphabet is a sequence of symbols taken from that alphabet.
A formal language over an alphabet is a set of strings over that alphabet.
http://www.cs.princeton.edu/introcs/71language/

and

Symbol
A character, glyph, mark.
An abstract entity that has no meaning by itself, often called uninterpreted.
Letters from various alphabets, digits and special characters are the most commonly used symbols.

Alphabet
A finite set of symbols.
An alphabet is often denoted by sigma, yet can be given any name.
B = {0, 1} Says B is an alphabet of two symbols, 0 and 1.
C = {a, b, c} Says C is an alphabet of three symbols, a, b and c.
Sometimes space and comma are in an alphabet while other times they
are meta symbols used for descriptions.

String also called a Word
A finite sequence of symbols from an alphabet.
01110 and 111 are strings from the alphabet B above.
aaabccc and b are strings from the alphabet C above.
A null string is a string with no symbols, usually denoted by epsilon.
The null string has length zero.
The null string is usually denoted epsilon.
Vertical bars around a string indicate the length of a string expressed as a natural number. For example |00100| = 5, |aab| = 3, | epsilon | = 0

Formal Language, also called a Language [emphasis addeded]
A set of strings from an alphabet.
The set may be empty, finite or infinite.
There are many ways to define a language. See definitions below.
There are many classifications for languages. See definitions below.

http://www.cs.umbc.edu/help/theory/lang_def.shtml

Putting these latter together we have:-

A formal langauge is a set of finite sequences of symbols from a finite set of symbols, where a symbol is either a character, glyphs or mark OR abstract entitity that has no meaning by itself OR a letter from one of various alphabets.

The definitions of alphabet and word are familiar enough. What is surprising is this concept of formal language does not include any rules of syntax. It does not exclude them so I must suppose that SOME formal langauges are defined by rules of syntax as well as an alphabets (otherwise known as a vocabulary I believe) and our old friends used in Logic are examples of the latter - formal languages with a syntax/rules of grammar. --Philogo 23:51, 27 May 2008 (UTC)

[edit] Interpretation Functions

Interpretation Functions

Interpretation by Substitution (p87)

The interpretation of quantifiers by substitution

Definition:
A valuation for a language L of predicate logic is a function with the sentences in L as its domain and {1, 0} ({True, False} ) as its range Interpretations by means of assignments (p 94)

[edit] references

  1. ^ hullo
  2. ^ a b Details of ref here
  3. ^ a b Details of blah here
  4. ^ a b c d Mumford, David (1999). The Red Book of Varieties and Schemes, Lecture notes in mathematics 1358. Springer-Verlag. DOI:10.1007/b62130. ISBN 354063293X. 
  5. ^ a b Mumford4, David (1999). The Red Book of Varieties and Schemes, Lecture notes in mathematics 1358. Springer-Verlag. DOI:10.1007/b62130. ISBN 354063293X. 
  6. ^ Bidamon, Emma Smith (March 27, 1876), “Letter to Emma S. Pilgrim”, in Vogel, Dan, Early Mormon Documents, vol. 1, Signature Books, 1996, ISBN 1-56085-072-8 
  7. ^ a b Link text, additional text.
  8. ^ Hullo to you
  9. ^ (Mates, Benson (1972). Elementary Logic publisher = OUP. )
  10. ^ Gamut, L.T.F (1991). Logic Language and Meaning, Vol 1 Introduction to Logic publisher= The University of Chicago Press. ISBN 0 -226-28085-3. 
  11. ^ name= Gamutv11991 Gamut, L.T.F (1991). Logic Language and Meaning, Vol. 1 Introduction to Logic publisher= The University of Chicago Press. ISBN 0 -226-28085-3.