Some philosophers aim to show how we can be justified in accepting certain basic logical truths by giving "rule circular" proofs of the soundness of these basic logical truths. They admit that most people will never have considered the proofs in question, and they admit that these people still count as justified in using logic. But, they say that they showing that such proofs can in principle be given makes sense of how we can be justified in believing the basic logical claims established by the proofs right now.
That idea seems prima face implausible. In general the fact that someone 100 years later will prove P from premises that I accept (like the ZF axioms) doesn't suffice to show that I am justified in believing that P now. So why should the case be any different for the proofs of logical principles?
[I would rather say that we are prima facie justified in believing these logical principles in a way that has nothing to do with the possibility of giving further argument; coming up with more or less circular ways of proving the soundness of our logical principles is (at best) a way of improving our justification]
Tuesday, March 29, 2011
Saturday, March 19, 2011
are stipulative definitions a source basic knowledge?
Random thought:
Whether or not its OK to make a certain stipulative definition can depend very messy questions - and not just mathematically messy questions like questions about harmony.
For example: it would seem that it's OK to stipulate that people are to count as "gleb" whereas bodies are not to count as "gleb" if and only if people are distinct from their bodies.
This suggests that knowledge by stipulative definition is not a source of basic knowledge. (basic knowledge= justified belief that doesn't depend on any other beliefs for justification) For, you can say 'of course people are gleb and bodies aren't, thats just what I mean by the term! remember when I stipulatively defined it...'. But (it would appear) the justificatory buck doesn't stop when you say this. If you are unjustified in thinking that bodies are distinct from people, this would seem to poison your justification for making and appealing to this stipulative definition.
However, perhaps we should say that only some stipulative definitions do have prima facie warrant, and the above stipulation about glep is just not one of the ones that does.
p.s. if we say that stiplative definitions aren't basic knowledge, we will probably want to say that analyticities aren't either.
Whether or not its OK to make a certain stipulative definition can depend very messy questions - and not just mathematically messy questions like questions about harmony.
For example: it would seem that it's OK to stipulate that people are to count as "gleb" whereas bodies are not to count as "gleb" if and only if people are distinct from their bodies.
This suggests that knowledge by stipulative definition is not a source of basic knowledge. (basic knowledge= justified belief that doesn't depend on any other beliefs for justification) For, you can say 'of course people are gleb and bodies aren't, thats just what I mean by the term! remember when I stipulatively defined it...'. But (it would appear) the justificatory buck doesn't stop when you say this. If you are unjustified in thinking that bodies are distinct from people, this would seem to poison your justification for making and appealing to this stipulative definition.
However, perhaps we should say that only some stipulative definitions do have prima facie warrant, and the above stipulation about glep is just not one of the ones that does.
p.s. if we say that stiplative definitions aren't basic knowledge, we will probably want to say that analyticities aren't either.
Sunday, February 6, 2011
Angst Over the Ordinals
Ohhh, which of these three options is correct? Given my focus on philosophy of math it's mildly embarrassing not to have a fixed position on this, but I keep going back and forth...
1. Just say the hierarchy of sets goes "all the way up"
2. Say the hierarchy of sets goes "all the way up" in the sense that it contains ordinals corresponding to every distinct combinatorially possible way for some objects to be well ordered *except for the one that it, itself is an instance of*. (this would be appealing but i think it may be impossible to spell out in a consistent way)
3. Say that the hierarchy of sets goes up at least far enough to satisfy the axiom of infinity+ the rest of ZF, and leaves it vague what there is beyond the inacessables - much as our concept mountain leaves it vague how many really tiny mountains there are given that there is such and such a bit of lumpy terrain.
1. Just say the hierarchy of sets goes "all the way up"
2. Say the hierarchy of sets goes "all the way up" in the sense that it contains ordinals corresponding to every distinct combinatorially possible way for some objects to be well ordered *except for the one that it, itself is an instance of*. (this would be appealing but i think it may be impossible to spell out in a consistent way)
3. Say that the hierarchy of sets goes up at least far enough to satisfy the axiom of infinity+ the rest of ZF, and leaves it vague what there is beyond the inacessables - much as our concept mountain leaves it vague how many really tiny mountains there are given that there is such and such a bit of lumpy terrain.
Five Reasons to be a Modal Carnapian
I currently believe mathematics is best understood in terms of combinatorial possibility plus Carnapian framework stipulations. One reason for thinking this is, of course, that understanding mathematical objects in this way lets you tell a nice story about access to abstract mathematical objects, like the one I tell in my dissertation! But here are 4 other reasons.
1. Thinking in terms of combinatoiral possibility lets you solve the bad company objection for neo-carnapianism about objects.
You can say: It is OK to stipulate that there are sets but not to make the stipulations for tonk, or for Boolos parities because...
One can safely extend one's language by introducing object stipulations S if and only if it would be combinatorially possible for the objects one's language currently acknowledges at each possible world to be supplimented by new objects in such a way as to satisfy S.
2. Thinking about things in terms of combinatorial possibility + carnapian framework stipulations explains why certain mathematical questions are substantive questions while others are not (substantive ones involve disagreement over combinatorial possibility as well as over which from among the many objects which it would be combinatorially possible to stipulate you actually do stipulate)
For example:
There is no substantive mathematical question about whether there are sets or categories or both, because everyone agrees that it would be combinatorially for there to be objects satisfy Carnapian framework stipulations phrased in terms of combinatorial possibility for the sets, and for the categories.
There is a substantive mathematical question about whether the Goldbach conjecture is true, because it is combinatorially impossible that the Fs and the Gs should satisfy the informal (non-first order logical!) framework stipulations for sets could yield different answers to Goldbach.
3. "Mathematical Hypothesis"-based alternatives like structuralism, fictionalism and plenetudinous platonism already need *some* powerful modal notion or a sense of possible and impossible which can take into account vocabulary that goes beyond the connectives of first order logic, if they want to capture intuitive claims about all questions in arithmetic having right answers.
Intuitively there are right answers to at least all questions about arithmetic. Each first order logical hypothesis about the numbers necessitates right answers to all questions about arithmetic. Therefore, any mathematical hypothesis that claims to cash out the truth conditions for our claim that phi in terms of the claim that if H then phi must crucially use non-first order vocabulary. Therefore if you want to make some hypothetical view of math work, you must either bite the bullet that some simple questions of arithmetic have no answer, or invoke a more mathematically powerful notion of possibilty/consequence
4. The notion of combinatorial possibility is very crisp and simple, indeed it maybe well the crispest and simplest notion that yields enough mathematical power to reconstruct intuitive verdics about truthvalues in arithmetic as per (3).
For example, if you accept Tarski's definition of logical truth in terms of facts about how it would be possible to reinterpret all properties and relations involved (where possible doesn't mean that a person could give a rule for how to do it, or somehow list the new extensions) then you pretty much already accept the powerful notion of how it would be in principle possible for an arbitrary n-place relation to apply to some objects which I have in mind when I talk about combinatorial possibility. So if you're not too much of a finitist/intuitionist to accept the very broad sense of in principle possible reassignment of extensions to predicates used in the Tarski definition, I think you should be OK with my notion of combinatorial possibility.
Secondly, combinatorial possibility faces none of the quandires that make people skeptical about the notion of metaphysical possibility e.g. it is possible for something to be both red all over and green all over, or for there to be zombies? We avoid these problem because facts about combinatorial possibility ignore all metaphysical facts about particular properties and relations involved. In some cases one might perhaps argue that some natural language sentences are vague with regard to what pattern of relationships between objects they assert. But once you specify that you are asking whether e.g. combiposs( Ex Redallover(x)&Greenallover(x)), it is perfectly clear that this is combinatorially possible, since it would be combinatorially possible to choose extensions for Redallover() and Greenallover() that make this true.
1. Thinking in terms of combinatoiral possibility lets you solve the bad company objection for neo-carnapianism about objects.
You can say: It is OK to stipulate that there are sets but not to make the stipulations for tonk, or for Boolos parities because...
One can safely extend one's language by introducing object stipulations S if and only if it would be combinatorially possible for the objects one's language currently acknowledges at each possible world to be supplimented by new objects in such a way as to satisfy S.
2. Thinking about things in terms of combinatorial possibility + carnapian framework stipulations explains why certain mathematical questions are substantive questions while others are not (substantive ones involve disagreement over combinatorial possibility as well as over which from among the many objects which it would be combinatorially possible to stipulate you actually do stipulate)
For example:
There is no substantive mathematical question about whether there are sets or categories or both, because everyone agrees that it would be combinatorially for there to be objects satisfy Carnapian framework stipulations phrased in terms of combinatorial possibility for the sets, and for the categories.
There is a substantive mathematical question about whether the Goldbach conjecture is true, because it is combinatorially impossible that the Fs and the Gs should satisfy the informal (non-first order logical!) framework stipulations for sets could yield different answers to Goldbach.
3. "Mathematical Hypothesis"-based alternatives like structuralism, fictionalism and plenetudinous platonism already need *some* powerful modal notion or a sense of possible and impossible which can take into account vocabulary that goes beyond the connectives of first order logic, if they want to capture intuitive claims about all questions in arithmetic having right answers.
Intuitively there are right answers to at least all questions about arithmetic. Each first order logical hypothesis about the numbers necessitates right answers to all questions about arithmetic. Therefore, any mathematical hypothesis that claims to cash out the truth conditions for our claim that phi in terms of the claim that if H then phi must crucially use non-first order vocabulary. Therefore if you want to make some hypothetical view of math work, you must either bite the bullet that some simple questions of arithmetic have no answer, or invoke a more mathematically powerful notion of possibilty/consequence
4. The notion of combinatorial possibility is very crisp and simple, indeed it maybe well the crispest and simplest notion that yields enough mathematical power to reconstruct intuitive verdics about truthvalues in arithmetic as per (3).
For example, if you accept Tarski's definition of logical truth in terms of facts about how it would be possible to reinterpret all properties and relations involved (where possible doesn't mean that a person could give a rule for how to do it, or somehow list the new extensions) then you pretty much already accept the powerful notion of how it would be in principle possible for an arbitrary n-place relation to apply to some objects which I have in mind when I talk about combinatorial possibility. So if you're not too much of a finitist/intuitionist to accept the very broad sense of in principle possible reassignment of extensions to predicates used in the Tarski definition, I think you should be OK with my notion of combinatorial possibility.
Secondly, combinatorial possibility faces none of the quandires that make people skeptical about the notion of metaphysical possibility e.g. it is possible for something to be both red all over and green all over, or for there to be zombies? We avoid these problem because facts about combinatorial possibility ignore all metaphysical facts about particular properties and relations involved. In some cases one might perhaps argue that some natural language sentences are vague with regard to what pattern of relationships between objects they assert. But once you specify that you are asking whether e.g. combiposs( Ex Redallover(x)&Greenallover(x)), it is perfectly clear that this is combinatorially possible, since it would be combinatorially possible to choose extensions for Redallover() and Greenallover() that make this true.
What is Combinatorial Possibility?
At the moment I think mathematics is best understood in terms of neo-carnapian/neo-logicist existence conditions for mathematical objects plus a kind of specifically mathematical modality (along the lines considered by Charles Parsons) which is looser than metaphysical possibility, and which I call "combinatorial possibility".
Here's a new way I thought of to explain what I mean by combinatorial possibility:
Combinatorial possibility is just like Tarski's notion of logical possibility, except a) we drop the assumption that you have to shanghai objects from the actual world to use in making a given claim true, allowing arbitrary choices of domain as well as arbitrary reassignments of extensions to properties and b) we allow more vocabulary to count as "logical vocabulary" in the sense that when reassigning extensions to predicates you can't change its meaning.
What more vocabulary?
In general anything like "finitely many" or "equinumerous" which functions like logical vocabulary in a sentence that its effect on the truthvalue of the whole sentence is systematically determined just by the domain and extension of relations in a set theoretic model is OK. Call these semi-logical expressions.
However I conjecture that we can cash out all of these all the semi-logical expressions, or at least all the ones that occur in modern mathematics and and its applications by merely using two things:
- an actuality opporator @F, so that you can say things about how it would be combinatorially possible for the things that are actually kittens, to be related by liking to the things that are actually baskets, e.g. there are enough different kittens and few enough baskets that it is combinatorially impossible that each kitten slept in a different basket last night.
- nesting, so that you can ask questions about the combinatorial possibility or impossibility of questions which are themselves described in terms of how it would be combinatorially possible to supplement them e.g. There are people at this party representing all combinatorially possible choices of whether to take sugar and/or milk in your tea = It would be combinatorially impossible to supplement the actual people with some additional person who differs from all actual people with respect to either whether they take coffee or whether they take tea = not combiposs(Ex person(x) & Ay [if @person(y) then ~x=y , and takessugar(x)iff~takessugar(y) and takesmilk(x) iff~takesmilk(y)) ] *
*yes, when you have multiple nesting of claims about combinatorial possibility, actuality opporators will need to be indexed to a particular instance "combiposs" so you will have combiposs_i and @F_i.
Here's a new way I thought of to explain what I mean by combinatorial possibility:
Combinatorial possibility is just like Tarski's notion of logical possibility, except a) we drop the assumption that you have to shanghai objects from the actual world to use in making a given claim true, allowing arbitrary choices of domain as well as arbitrary reassignments of extensions to properties and b) we allow more vocabulary to count as "logical vocabulary" in the sense that when reassigning extensions to predicates you can't change its meaning.
What more vocabulary?
In general anything like "finitely many" or "equinumerous" which functions like logical vocabulary in a sentence that its effect on the truthvalue of the whole sentence is systematically determined just by the domain and extension of relations in a set theoretic model is OK. Call these semi-logical expressions.
However I conjecture that we can cash out all of these all the semi-logical expressions, or at least all the ones that occur in modern mathematics and and its applications by merely using two things:
- an actuality opporator @F, so that you can say things about how it would be combinatorially possible for the things that are actually kittens, to be related by liking to the things that are actually baskets, e.g. there are enough different kittens and few enough baskets that it is combinatorially impossible that each kitten slept in a different basket last night.
- nesting, so that you can ask questions about the combinatorial possibility or impossibility of questions which are themselves described in terms of how it would be combinatorially possible to supplement them e.g. There are people at this party representing all combinatorially possible choices of whether to take sugar and/or milk in your tea = It would be combinatorially impossible to supplement the actual people with some additional person who differs from all actual people with respect to either whether they take coffee or whether they take tea = not combiposs(Ex person(x) & Ay [if @person(y) then ~x=y , and takessugar(x)iff~takessugar(y) and takesmilk(x) iff~takesmilk(y)) ] *
*yes, when you have multiple nesting of claims about combinatorial possibility, actuality opporators will need to be indexed to a particular instance "combiposs" so you will have combiposs_i and @F_i.
Saturday, December 25, 2010
Three Arguments for A Priori Knowledge of (Very) Contingent Facts
Can we have a priori knowledge of contingent facts? For example, consider the proposition below. Can we know truths like the following a priori? NOT PEA SOUP: 'It is not the case that everything outside of a 5 foot radius around me is made of pea soup, which stealthily forms up into suitable objects as I walk by' Here are three positive arguments (in ascending order of strength IMO) for the conclusion that we can know NOT PEA SOUP a priori.
1. Argument from Crude Reliablism
The belief-forming method of assuming that you aren't in a pea soup world is reliable. And even if we make things a little less crude by saying that good belief formation is belief formation that works via a chain of methods which are *individuated in a psychologically natural way* and are reliable, we will probably still get the same conclusion. For plausibly the most natural relevant psychological mechanism involved in generating that belief would be something like, 'believe not P when P is sufficiently gerrymandered'.
2. Argument from Probability and Conditionalization-Based Models of Good Inference
If you think that good reasoning is well modeled by the idea of assigning a certain probability measure to the space of possible worlds, and then ruling out worlds based on your observation, and asserting that P if and only if a sufficient fraction of the remaining probability is assigned to worlds in which P. There will be some propositions P that low enough prior probability to warrant asserting ~P before you have made any observations - and plausibly the pea soup hypothesis is one of them. Presumably in such cases your justification does not depend on experience. [I think Williamson has something like this in mind in one of his papers on skepticism, but his argument was more complicated]
3. Argument from Current Knowledge plus Inability to Cite Experiential Justification. The claim that NOT PEA SOUP is a priori follows from a claim about knowledge that only a skeptic would deny, plus a somewhat intuitive claim about the relationship between a priority and justification. The intuitive claim I have in mind is that if someone can count as knowing that P, without being able to point to any relevant experience (or memory of experience, or reason to believe that they had experience etc) as justification then they know that P a priori so P is a priori (i.e. a priori knowable). Everyone but the skeptic agrees that people know that they aren't in the pea-soup world. These people who know cannot point to any experience as justification. Hence, 'not-pea soup' must be knowable without appeal to experience for justification. You might try to defend the a posteriority of NOT PEA SOUP by saying that even if the man on the street can't make any argument from experience to NOT PEA SOUP, our intuition that people know that NOT PEA SOUP is based on the assumption that there exists some good argument from something about experience to NOT PEA SOUP, and philosophers just need to discover it. In this way, experience really is necessary to justify the belief that NOT PEA SOUP so the proposition is a posteriori.
However, this response threatens to generate the unattractive conclusion that people today do not know NOT PEA SOUP. For, in general, the mere existence of a good argument for some proposition that I believe does not suffice to make me justified in believing that proposition now, if I cannot (now) give that argument. If I believe some mathematical theorem T on a hunch or on the basis of tea leaf reading, the mere fact that there is a good argument for T on the basis of things that I accept, doesn't suffice to allow me to count as knowing that T. So even if there is some cunning philosophical argument yet to be discovered which justifies NOT PEA SOUP on the basis of experience, it would seem that this argument cannot suffice to justifies people now accepting that NOT PEA SOUP. If people now are justified that NOT PEA SOUP, and can give no argument from experience for this claim, it must be that the claim can be justifiably believed without appeal to experience.
1. Argument from Crude Reliablism
The belief-forming method of assuming that you aren't in a pea soup world is reliable. And even if we make things a little less crude by saying that good belief formation is belief formation that works via a chain of methods which are *individuated in a psychologically natural way* and are reliable, we will probably still get the same conclusion. For plausibly the most natural relevant psychological mechanism involved in generating that belief would be something like, 'believe not P when P is sufficiently gerrymandered'.
2. Argument from Probability and Conditionalization-Based Models of Good Inference
If you think that good reasoning is well modeled by the idea of assigning a certain probability measure to the space of possible worlds, and then ruling out worlds based on your observation, and asserting that P if and only if a sufficient fraction of the remaining probability is assigned to worlds in which P. There will be some propositions P that low enough prior probability to warrant asserting ~P before you have made any observations - and plausibly the pea soup hypothesis is one of them. Presumably in such cases your justification does not depend on experience. [I think Williamson has something like this in mind in one of his papers on skepticism, but his argument was more complicated]
3. Argument from Current Knowledge plus Inability to Cite Experiential Justification. The claim that NOT PEA SOUP is a priori follows from a claim about knowledge that only a skeptic would deny, plus a somewhat intuitive claim about the relationship between a priority and justification. The intuitive claim I have in mind is that if someone can count as knowing that P, without being able to point to any relevant experience (or memory of experience, or reason to believe that they had experience etc) as justification then they know that P a priori so P is a priori (i.e. a priori knowable). Everyone but the skeptic agrees that people know that they aren't in the pea-soup world. These people who know cannot point to any experience as justification. Hence, 'not-pea soup' must be knowable without appeal to experience for justification. You might try to defend the a posteriority of NOT PEA SOUP by saying that even if the man on the street can't make any argument from experience to NOT PEA SOUP, our intuition that people know that NOT PEA SOUP is based on the assumption that there exists some good argument from something about experience to NOT PEA SOUP, and philosophers just need to discover it. In this way, experience really is necessary to justify the belief that NOT PEA SOUP so the proposition is a posteriori.
However, this response threatens to generate the unattractive conclusion that people today do not know NOT PEA SOUP. For, in general, the mere existence of a good argument for some proposition that I believe does not suffice to make me justified in believing that proposition now, if I cannot (now) give that argument. If I believe some mathematical theorem T on a hunch or on the basis of tea leaf reading, the mere fact that there is a good argument for T on the basis of things that I accept, doesn't suffice to allow me to count as knowing that T. So even if there is some cunning philosophical argument yet to be discovered which justifies NOT PEA SOUP on the basis of experience, it would seem that this argument cannot suffice to justifies people now accepting that NOT PEA SOUP. If people now are justified that NOT PEA SOUP, and can give no argument from experience for this claim, it must be that the claim can be justifiably believed without appeal to experience.
Sunday, December 19, 2010
Reliablism and the Value of Justification: The Angel's Offer
A major objection to reliablism about justification is that it doesn't explain why we value having knowledge of a given proposition more than mere dogmatic true belief. For, believing a true proposition via a method that's reliable is just more likely to lead you to believe other true propositions; there's no obvious sense in which your relationship to true beliefs formed by reliable methods is thereby intrinsically any better or more valuable than you relation to mere true beliefs. If we don't like a particular cup of good expresso any better for it being the product of a machine that reliably makes good expresso, why should we like a particular state of believing a truth any better from the fact that it was produced by processes that reliably lead to believing the truth?
But maybe we DON'T value having the special relation we do to justified true beliefs over and above it's tendency to promote having stable true beliefs. Consider this thought experiment:
An angel convinces you that he knows the true laws of physics and maybe also that it can do super-tasks and thereby knows certain statements of number theory which cannot be proved from axioms which you currently accept. The angel offers to make it the case that you find these true principles feel obvious to you - the way that you now feel about 'I exist' or '2+2=4'. He will wipe your memory of this conversation so that you will not be able justify these feelings to yourself by appeal to the reliable way you got them - but of course you won't feel the need to justify them to yourself since they will just feel obvious and you will be inclined to immediately accept them. [Suppose also, if it matters, the angel will do the same to everyone in your community, that community members prefer to go along with whatever choice you make, that the angel is already going to blur your memories of not finding these claims obvious in the past etc.]
Would you accept the offer?
I personally would definitely take the offer. And I think many people would share this preference. If there were something intrinsically valuable about knowing verses merely dogmatically assuming a necessary truth, then this would be a strong reason not to take the angels offer. But if Plato is right (as thinking about the example tempts me to think that he is) to say that the only bad thing about dogmatically assuming truths rather than knowing them is that dogmatic assumptions don't stay tied down, then the angel's offer to make you and everyone else in your community find these truths indubitable fixes that problem - and you should take him up on his offer.
But maybe we DON'T value having the special relation we do to justified true beliefs over and above it's tendency to promote having stable true beliefs. Consider this thought experiment:
An angel convinces you that he knows the true laws of physics and maybe also that it can do super-tasks and thereby knows certain statements of number theory which cannot be proved from axioms which you currently accept. The angel offers to make it the case that you find these true principles feel obvious to you - the way that you now feel about 'I exist' or '2+2=4'. He will wipe your memory of this conversation so that you will not be able justify these feelings to yourself by appeal to the reliable way you got them - but of course you won't feel the need to justify them to yourself since they will just feel obvious and you will be inclined to immediately accept them. [Suppose also, if it matters, the angel will do the same to everyone in your community, that community members prefer to go along with whatever choice you make, that the angel is already going to blur your memories of not finding these claims obvious in the past etc.]
Would you accept the offer?
I personally would definitely take the offer. And I think many people would share this preference. If there were something intrinsically valuable about knowing verses merely dogmatically assuming a necessary truth, then this would be a strong reason not to take the angels offer. But if Plato is right (as thinking about the example tempts me to think that he is) to say that the only bad thing about dogmatically assuming truths rather than knowing them is that dogmatic assumptions don't stay tied down, then the angel's offer to make you and everyone else in your community find these truths indubitable fixes that problem - and you should take him up on his offer.
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