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Logic and Philosophy of Logic

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Since 1998 I carry out independent research on the logic operators and in particular of "if then" and "if and only if then". Given my difficulty when a student, and the difficulties of my students to understand mainly the operator "if then" and the concepts of sufficient condition and necessary condition in it implicit, and given my passion for logic, neuroscience and psychology, I spent over 20 years researching how these concepts arise in reality, and how our minds create their abstract models.
The study of the 4 cards test of Peter Wason (1966), is an excellent didactic example to explain the logical infrastructure of sufficient condition, and always use with my students. But in spite of literary research, not having met a specimen, equally excellent in explaining the necessary condition, I started a thorough study on the issue that has led me not only to devise a new specific test for the necessary condition, but also to find out what I think is a millenary gap in on the o ... (read more)
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Some words in my paper:

T(hj|ei)--fuzzy truth function of a predicate hj.

T(hj)--logical probability or  average thue-value of a predicate hj.

Popper defined Testing severity and Verisimilitude (1963/2005, 526, 534). Since Logical Probability and Statistical Probability are not well distinguished by him, his definitions are not satisfactory. The author suggests defining log [1/T(hj)] as testing severity, and T(hj|ei)/T(hj) as verisimilitude. In terms of Likelihood method, P(ei| hi is true)/P(ei) =T(hj|ei)/T(hj) is also called standard likelihood. So, we may say Semantic information = log (Standard likelihood) = log (Verisimilitude)=Testing severity - Relative deviation
 If negative verisimilitude for lies or wrong predictions is expected, one may also define verisimilitude by log [T(hj|ei)/T(hj)]. 

The figure 8 in the paper shows how positive and negative degrees of believe affect thruthlikeness. 

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1 Turing and the Myth of Universality

There is a strong, not to say absolute belief in the consistency of Turing's thesis, which can be, informally, expressed as such: what a computer can do, any other computer can.

Let us start with the simplest expression of all:

1) 0+1=1

It will be obvious to anyone that any computer worth its silicone, or any other material substrate, will be able to compute (1).

What does that say about universal computing?

Well, that's just it, really. It does not say anything at all. All it shows is that, once the problem has been solved, or at least, put ... (read more)

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Could anyone explain the difference between being part and being member (if any)? References to existing literature are welcome. 
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The challenge is very simple: Give a full and explicit proof of the existence of Infinity.There is only one restriction: it is not allowed to refer to "proofs" already known. If you believe Cantor has proved Infinity, you cannot just refer to his work, but have to state explicitly what and how you think he has proved it.
I wish you luck.
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[The idea is to start with a concrete situation, like a mother preparing a meal for an extended family, and discovering numbers in their different form: natural, whole, negative, rational (including radicals), real, imaginary, complex, etc...

Mother is of course an archetype and can include many generations of mothers. There are 12 family members and, to make things simpler, they all eat the same amount of food.  Some kind of grain.]

1) Mother knows how much grain she needs to cook for all of them. She just keeps taking handfuls of grain and putting them in the cooking pan until she is satisfied that it will be enough. She has no way of knowing or naming exact quantities. Her experience as a cook is sufficient for the task. She can also enumerate each family member by name, including herself, while grabbing grain, since she also knows how much each member approximately eats. To be sure she does not forget anyone, or count somebody twice, she starts with Father, then herself, and then with ... (read more)
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Preliminary Observation
Set Theory is believed to be the foundation of Mathematics, the theory from which everything mathematical would be deduced. It sounds like a metaphysical prejudice to my ears: how could a mathematical theory ever found mathematics? What would then found Set Theory itself? Its axioms? They are all of a mathematical nature, so that would not work. We would need non-mathematical axioms to found Set Theory, before it ever could found Mathematics. Is that even possible? This is what I intend to research in this thread. But please, bear with me, there is no royal road to the foundation of the foundation of Mathematics. If there even exists such a thing.

Axiom of Choice and Well-Ordering Principle
The universal consensus is that WOP relies on AC. I have the strong impression that it is in fact the other way around.
When I look at the incredibly complicated "proof that every set can be well-ordered" by Zermelo (1904), I cannot escape the feeling that he would not be ab ... (read more)
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Reference and Self-Reference
The philosophical theme of reference is no doubt a wide and deep ocean. My attempts at presenting a new perspective on the subject can certainly not be considered as the final word on the subject. [see my thread Truth and Necessity]. Reading Quine "The Ways of Paradox" (1966), I realized that my (Strawsonian) conception that language does not refer poses special problems when the objects of reference are themselves linguistic elements. As Juergen Habermas would say, (natural) language is its own metalanguage, and just like our mind, seems to be able to look upon itself.
Self-reference is not only the source of many antinomies, it could mean the negation of my analysis as a whole: if language does not refer, how could it ever self-refer? My aim is quite simple. I will try to show that self-reference is not possible. The line of argumentation is easy to follow: no self-reference without reference. And since reference is not a linguistic property but an action un ... (read more)
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What is your opinion about the IF-logic? Jaakko Hintikka has claimed, that it should replace FOL, and I can see reasons for that. The main reason for that is, in my opinion, that IFL allows us to use such combinations of quantifiers, that FOL doesn't allow. The greater expressive power of IFL is brought by the use of signaling prefixes and Henkin's prefixes. The main idea is, that IFL allows independence of quantifiers, which is completly natural idea. Actually it seems to be less natural to not allow different sorts of combinations of dependences between quantifiers. 
What is your opinion about this?
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Logic covers vast areas of philosophy. It would be unfair to say mind is only a fraction of logic, empirically, the mind is causal. Immanuel Kant says in his 'Critique of Practical Reason' it is a priori and causal. Unfortunately he could not back this argument. Dummet's equation can be cracked by logic, if it is given, in rational terms. Logic given a priori, in empirical application is therefore causal.


To those interested in the philosophy of logic:


If everything goes right 2017 should see the release of a first of its kind book, Philosophical Perceptions on Logic and Order by IGI Global publishers (, a set of readings to accompany material presented in mainly introduction to logic courses, but also to those interested in the subject generally.


This work is being prepared based on my observation that students in logic courses, as well as many instructors teaching them, are clueless about the central philosophies driving the discipline. The generally prevailing view is “this is the way it is”, referring to the systems and methods of thinking in the course. These are mechanistically presented as the way of describing the world. After being told about what logically supposedly is, they are told that observations about their environment are put into relationships called “arguments”, where an emerging statement can be evaluated as to it having a ... (read more)

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A proposition is said to be necessarily true if it is true in all possible worlds.
I would not know how to refute such an affirmation but I do wonder whether that proves the existence of any necessarily true proposition. After all, is it not possible that there are no such propositions for the simple reason that we could always imagine a world where a proposition, true in all others, would then be false?
Let us take a very likely candidate to modal necessity, the proposition "a=a". Can we imagine a world where that would not be true?
Even Alice's world, with all its indifference to logic rules would seem to sustain this inexorable truth: a thing is equal to itself, for however long it exists in one and the same state. Changing the subject from 'thing' to 'state' does not alter this necessity.

Still, imagine a world where, just like within the core of a living star, or even better, the condensed matter right before the Big Bang, or in a Black Hole, all things are in perpetual change from on ... (read more)
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Tarski’s convention T: condition beta. South American Journal of Logic. 1, 3–32.

John Corcoran and Leonardo Weber


HISTORICAL NOTE: This paper is the culmination of a years-long joint effort by the two authors. A preliminary report appeared in 2013: Corcoran-Weber, Bulletin of Symbolic Logic, 19 (2013) 510–11. Their co-operative work was conducted by email dialogue in which each author’s work was developed and corrected by the other. Each section went through several iterations. The final version was the result of dozens of reciprocal exchanges; it is impossible to allocate credit. Each author learned from and taught the other. During this time they consulted several other scholars including the Tarski experts David Hitchcock, James Smith, and Albert Visser.

The senior author expresses his deep gratitude to the junior author. Moreover the senior author acknowledges publicly what he has already said privately, viz. that without the junior author’s help and mastery of ... (read more)

REQUEST: Please send errors, omissions, and suggestions. I am especially interested in citations made in non-English publications.

Some of the entries have already been found to be flawed. For example, Tarski’s expression ‘materially adequate’ was misinterpreted in at least one article and it was misused in another where ‘materially correct’ should have been used. This “session” provides an opportunity to bring more flaws to light.


Acknowledgements: Each of these entries was presented at meetings of The Buffalo Logic Dictionary Project sponsored by The Buffalo Logic Colloquium. The members of the colloquium read drafts before the meetings and were generous with corrections, objections, and suggestions. Usually one 90-minute meeting was devoted to one entry although in some cases, for example, “axiomatic method”, took more than one meeting. Moreover, about half of the entries are rewrites of similarly named entries in the 1995 first edition.

I am trying to start a discussion for teaching INSEPARABILITY OF LOGIC AND ETHICS. A COLLEAGUE WROTE: I'm going to be teaching your "Inseparability of Logic and Ethics" in a couple weeks. I was wondering if you had any tips on doing so or thoughts about points to emphasize. I've always loved the paper and found your pedagogical techniques quite helpful.
MY ADVICE TO MY COLLEAGUE: First, before assigning the paper to be read, ask the students to look up “ethics” and “logic” in a dictionary or other reference work and then to write a paragraph on what the two have to do with each other. Second, after the students were supposed to have read the paper, ask them what they got out of it. Just let them talk and prompt them where necessary. No contentiousness. Third, read the first page aloud to them and see what happens. As you go read chunks aloud and ask questions—just like I did teaching you Tarski’s truth-definition paper. Fourth, go around the clas ... (read more)



Corcoran’s 2009 ARISTOTLE’S DEMONSTRATIVE LOGIC deals decisively with several issues that had previously been handled by vague speculation and dogmatic pontification if at all. One possible example: Corcoran [2009, p. 13] proves conclusively that the imperfect syllogisms Baroco and Bocardo—which Aristotle completed indirectly [by reductio-ad-impossible]—cannot be completed directly. More generally, Corcoran shows that no valid premise-conclusion argument, regardless of the number of premises,  having an existential negative [“particular negative” or “O-proposition”] as a premise can be completed using a direct deduction—assuming of course that no premises are redundant and that the conclusion is not among the premises. To be clear this means that for no such argument is it possible to deduce the conclusion from the premises without using reductio.

This result, called the EXISTENTIAL-NEGATIVE EXCLUSION [ENE], was circulated informally by Corcoran much earlier but it seem ... (read more)


JOHN CORCORAN AND HASSAN MASOUD, Three-logical-theories redux.

  The 1969 paper, “Three logical theories” [1], considers three logical systems all based on the same interpreted language and having the same semantics.

  The first, a logistic system LS, codifies tautologies (logical truths)—using tautological axioms and tautology-preserving rules that are not required to be consequence-preserving.

  The second, a consequence system CS, codifies valid premise-conclusion arguments—using tautological axioms and consequence-preserving rules that are not required to be cogency-preserving [2]. A rule is cogency-preserving if in every application the conclusion is known to follow from its premises if the premises are all known to follow from their premises.

  The third, a deductive system DS, codifies deductions, or cogent argumentations [2]—using cogency-preserving rules. The derivations in a DS represent deduction: the process by which conclusions are deduced from premises, i. e. the way knowl ... (read more)


JOHN CORCORAN, Two-method errors.

  Where there are two or more methods for the same thing, sometimes errors occur if two are mixed. Two-method errors, TMEs, occur in technical contexts but they occur more frequently in non-technical writing. Examples of both are cited.

  We can say “Abe knows whether Ben draws” in two other ways: ‘Abe knows whether or not Ben draws’ or ‘Abe knows whether Ben draws or not’. But a TME occurs in ‘Abe knows whether or not Ben draws or not’.

  We can say “Abe knows how Ben looks” using ‘Abe knows what Ben looks like’. But a TME occurs in ‘Abe knows what Ben looks’ and also in ‘Abe knows how Ben looks like’. Again, we can deny that Abe knows Ben by prefixing ‘It isn’t   that’ or by interpolating ‘doesn’t’. But a TME occurs in trying to deny that Abe knows Ben by using ‘It isn’t that Abe doesn’t know Ben’.

  There are two standard ways of defining truth for first-order languages: using finite sequences or infinite sequences. Quine’s discussion in the 1970 first ... (read more)


► JOHN CORCORAN AND WILLIAM FRANK, Cosmic Justice Hypotheses.

  This applied-logic lecture builds on [1] arguing that character traits fostered by logic serve clarity and understanding in ethics, confirming hopeful views of Alfred Tarski [2, Preface, and personal communication].

  Hypotheses in one strict usage are propositions not known to be true and not known to be false or—more loosely—propositions so considered for discussion purposes [1, p. 38].

   Logic studies hypotheses by determining their implications (propositions they imply) and their implicants (propositions that imply them). Logic also studies hypotheses by seeing how variations affect implications and implicants. People versed in logical methods are more inclined to enjoy working with hypotheses and less inclined to dismiss them or to accept them without sufficient evidence.

  Cosmic Justice Hypotheses (CJHs), such as “in the fullness of time every act will be rewarded or punished in exact proportion to its goodness or badness ... (read more)

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