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Philosophy of Mathematics

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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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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)

Hi Jack,

Nice paper!. However, if I may, I wasn't convinced by your response to objection five. The objection, I take it, is that the intuitions you are marshaling about incoherence derive from a non-moral standpoint, that is, they are intuitions that arise when one is doing metaethics and not when one is actually moralizing.  And it seems undeniable that Moore paradoxical sentences are straightforwardly bizarre when uttered by persons in the context of actual moralizing (just imagine actually having the relevant conversation). At the outset of your paper, you correctly note that expressivism is a theory about actual moralizing, so it seems like this is one objection to which you should be very sensitive.  You respond:

This is not really a rejection of C3, but a rejection of C1, since it admits that it is not always the case that affective or conative attitudes are expressed by moral assertions. If non-cognitive mental states are only sometimes expressed by moral assertions, then the clai ... (read more)

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Your article is very interesting.

In the same spirit I propose a more modal formalism to speak about "true announcements" and "learning" :

This representation allow to make the difference between a world before and after the learning act. Then it becomes easier to deal with expression about knowledge and learning.


Your work is technically very interesting neverthless I have some remarks.

1/ I don't always understand the claim that the Fitch Paradox threatens Anti-Realist philosophy.
If everybody accepts the Knowability Principle restricted to basic propositions,
it sounds more like a victory than a defeat for the knowability advocates.
It seems that what is threatened is more the capacity of modal logic to represent the knowability.

2/ In your intuitionistic frame, as you say in proposition 5.8, it is impossible to have 'A' and 'not K A' in the same world.
So the Fitch Paradox is avoided but the result is a very poor epistemic logic where you cannot express that some truths are unknown.

3/ more technically in the figure below the proposition 5.7
I don't understand what happens in the world y.
You have      y Rk y ;    y Rk z ;    y: p   z: not p
but you have not    'y: not K p' .
Does it stand that 'y: not not K p' ?
It is very counterintuitive.

Else I wrote a dissertation on these points.
I have published a - maybe insufi ... (read more)

notation : I use ! for 'not'

Perhaps you can avoid paradox but you have to admit this very strange proposition :
K !K x ->  !P K x
If you know that you ignore (x) it's impossible that you know (x)

I don't see how it could be compatible with the knowability principle :
x ->  P K x
else you can't have
(x) and (K !K x)

(excuse me if this message is out of place, I ignore the policy of tis forum,
excuse also my probable mistakes in english)
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This is awesome. Tractatus power!
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When we say that "P" is possible iff there is a possible world where "P" is true, we can continue and say that there is a possible world where "P" is true iff "A" is true, and "A" is not something about a non-actual possible world. Such an approach would allow us to use the vocabulary of possible worlds, while sustaining an agnostic or an anti-realist position about the the existence of non-actual possible worlds. Can such a reductive approach to possible worlds solve the problem of ontological commitmentt to the existence of non-actual possible worlds? And what do you think that "A" must be as non-actual possible worlds dont get implicated?

Cross-posted from


This week we discussed some unpublished material by Antony on ‘might’ counterfactuals. The handout is here, and the paper is here.

We thought a bit about cases in which ‘could’ and ‘might’ come apart. In the paper, Antony discussed sentences like

33b)  If we’d left the gate open, the dog could have got out; yet if we’d left the gate open, it isn’t the case that the dog might have got out.

The felicity of such sentences seems to show that at least some ‘might’ counterfactuals shouldn’t be analysed in terms of ‘could’, but instead should be given an epistemic reading. Antony isn’t averse to this idea – in fact, his final view is that ‘might’ is ambiguous in counterfactual contexts between the epistemic reading and the ability reading. However, this does invite the further question of what determines the appropriate reading for some given ‘might’ counterfactual.

Fron 33b we naturally conclude though the dog has the ability to get out, it is ... (read more)


Cross-posted from


The handout for this week is here, the original paper is here.

I found this a particularly interesting paper. I’m in firm agreement with the main gist of Williams’ view- that the notion of typicality is in principle better adapted to deal with chancy similarity than the notion of ‘non-remarkableness’. That said, we found plenty of potential pressure points.

- Firstly, I’m not sure that quantum mechanics really has as wide-reaching consequences as is assumed in the paper. Depending on your response to the measurement problem, it could be that outcomes such as plates flying off sideways are not genuine quantum possibilities after all, because the low-amplitude branches are in some way ‘lost in the noise’. Although I think this issue is worth further investigation, I don’t think it’s critical to the debate between Williams, Hawthorne, and Lewis. Their worries can be raised about considerably less unlikely events – in fact, we can restrict ... (read more)

Cross-posted from


You can find the handout for this week here. I thought this was a really good paper, and we didn’t find all that much to criticise in it. It was a bit frustrating not to hear more about Fitelson’s positive story, in particular about the bridge principle that he would endorse instead of the various versions of RTE that he criticises. He’s clearly saving the juicy stuff for his book.

In particular, I find it hard to see how he plans to steer a middle ground between the Carnap/Williamson-style ‘a priori priors’ version of objective bayesianism, and the subjective bayesian approach. My naive take on the matter is that you either think that there’s a unique correct set of priors or you don’t. Maybe these priors aren’t a priori knowable (contra the Carnap/Williamson approach), although it seems that a position like this would be committed to complete epistemic rationality being in principle unattainable.

I wasn’t sure how strongly Fitelson m ... (read more)

Cross-posted from

This week we discussed Cian Dorr’s ‘There are no abstract objects’, which isn’t currently available online, but is in ‘Contemporary Debates in Metaphysics’. Here’s the handout instead.

As we had Cian on the spot for this meeting, the discussion mostly took a question-and-answer format. So here are what I recorded of some questions and some answers, with a few that I didn’t get time to ask thrown in at the end. Apologies if my paraphrases of Cian's answers misrepresent him!

Q: What about people who would resist the paraphrase strategy (p.37) because they think that counterpossibles are all vacuously true (Williamson takes this line in The Philosophy of Philosophy).
A: Nominalism/anti-nominalism are both contingent theses. But even if you think that nominalism is necessary if true, there will be certain kinds of truths like ‘there are possibly some things with a number-like structure’ which can be used to ground the relevant counterfactua ... (read more)

Cross-posted from

Boris Kment - Counterfactuals and Explanation

A really interesting paper this week - it can be found here, and the presentation is here.

Kment's main proposal is that match of matters of particular fact should be relevant to closeness of two worlds for the purposes of evaluating counterfactuals if and only if the matters of fact have the same explanation in both worlds. Secondarily, he proposes that we should allow for laws to have exceptions, and hence that all worlds which share the same laws as ours should be closer to actuality than any world with different laws.

We quite liked the main proposal, but worried about the individuation of explanations it relies upon. What are the conditions for two events to have the same explanation? For example, consider the counterfactual 'if I had tossed the coin five minutes earlier, it would still have come up heads'. This seems false, but perhaps Kment can account for this falsity by sayi ... (read more)

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