h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hi Bx4,

>>(i) Knowledge entails truth : KcP>P

and the Plantinga/Merricks definition of warrant (T):

(ii) K==T&W (where == ( stands for 'is equivalent to')<<
I prefer KcP==P&BcP&W(P). I'm not sure what your 'T' is here, could you clarify?

>>whence:

(iii) KcP == T&WcP

hence

(iv)T&WcP->P

is axiomatically true because (1) is axiomatically true. Since by the laws of logic, if the conjunction T&W is true then each of the individual propositions T and W are necessarily true. Hence:

(v) W(P)->P (P is warranted entails that P is true)

So Merricks conclusion 'Warrant entails truth' is part of premise (ii). <<
This must be equivalent to:

KcP == P&BcP&W(P)

Hence P&BcP&W(P)->P

but this is the same tautology. You have derived P via a premiss containing KcP. (your (iii)). But you are trying to show W(P)->P
which contains no such premiss. So it seems you have demonstrated your own circle again rather than shown circularity in Merricks' argument.

In summary, according to Merricks, if I know P, then it must be the case that P (axiomatically), that I believe P and that P is warranted. From this it follows tautologically that P is true. That isn't the issue though. The issue is whether you could have a scenario in which P is warranted but happens to be false. In that case, I can't know P and this seems to render your argument irrelevant.

I'll reply about the rest anon.

ttfn

hi psi

>>This must be equivalent to:

KcP == P&BcP&W(P)

Hence P&BcP&W(P)->P

but this is the same tautology<<

Well no since I specifically said before making the logical inference from premises to conclusion I said 'Leaving aside the additional complexity the tautological nature of 'mere (?) true belief (T) for the moment...'. I did this because to deconstruct Merricks premiss I needed to preserve the term warrant(W). The problem is that the the tautological nature of:

(i)BcP&P-> P***


(leaving aside the obvious sophistry of the use of the 'weasel' word "mere") as this seems to render the term warrant (whatever its definition redundant) since

(ii) KcP->P

hence

(iii)KcP==BcP&P->P

but the Merricks/Plantinga definition of warrant is that:

(iv) KcP=BcP&P&W(P)

however from (iii) since KcP==BcP&P then in (iv) W is redundant. That is to say that (iv) is only satisfied by the null (empty) subset {} of the set of definitions of warrant:

(v) KcP==BcP&P&{}

which would seem to be a defeater for warrant as that which distinguishes true belief from knowledge.

>>You have derived P via a premiss containing KcP. (your (iii)). <<

Surely Merricks 'S knows that P..., KSp, is the same as KcP, and hence forms part of his premiss?

'But you are trying to show W(P)->P which contains no such premiss. So it seems you have demonstrated your own circle again rather than shown circularity in Merricks' argument.<<

Rather, I am trying to show that the conclusion is //part// of the premiss:

'S knows that p[is true], therefore, if and only if S’s belief that p is warranted[is true] and p is true.'(1)

that is KSp->W[p]&p

So KsP can only be true if the logical conjunction is true and the logical conjunction Wp&P can only be true if both its constituent are true. Hence, the atomic proposition that P is warranted [W(p)] /cannot/ be false. Hence 'warrant entails truth' is part of the premiss a well as te conclusion.

>>In summary, according to Merricks, if I know P, then it must be the case that P (axiomatically), that I believe P and that P is warranted. From this it follows tautologically that P is true.>>

The problem here is while KcP->P is axiomatic it does not thereby follow KSp->W[p]&P is an axiom rather than an arbitrary postulate (premiss)

Another problem with your argument is that if we accept Merricks premiss, KSp->W[p]&p, then if Ksp is true and p is true then W[p] cannot be false.

>>That isn't the issue though. The issue is whether you could have a scenario in which P is warranted but happens to be false. In that case, I can't know P and this seems to render your argument irrelevant.>>

No, as I have shown above, Merricks premiss (1) means that by definition W[p] //cannot// be false. What is irrelevant is the rest of Merricks argument for infallibillism since it implicit in his definition (1).

Clearly, the paper is on in a long line of papers of the JTB+X variety attempting to preserve knowledge by providing a solution to the 'Gettier Problem'. However, as William Lycan has put it:

'Of course the Gettier Problem arises in the first instance only for those of us who are not skeptics.' ('On the Gettier Problem problem')

So it may be a problem for you but not for me.

bs


***(In order to avoid the 'scholastic' legerdemain of epistemological orthodoxy BcP&P is not a Gettier case)

Hi Bx4,

I'll stick to the logic strand for now and return to the analogy and Rorty issues in due course.

>>Well no since I specifically said before making the logical inference from premises to conclusion I said 'Leaving aside the additional complexity the tautological nature of 'mere (?) true belief (T) for the moment...'. I did this because to deconstruct Merricks premiss I needed to preserve the term warrant(W). The problem is that the the tautological nature of:

(i)BcP&P-> P*** <<
I don't understand this. The argument presented uses the model KcP == BcP&P&W(P) so that's what I'm interpreting. I don't see a problem with 'mere' or what your justification for the accusation of 'legerdemain' is if we are interpreting an argument within this frame of reference. You might well have valid reasons for rejecting this model, but I can't see how these are within the scope of the issue of whether there is circularity in Merricks' argument

>>(leaving aside the obvious sophistry of the use of the 'weasel' word "mere") as this seems to render the term warrant (whatever its definition redundant) since <<
It isn't obvious to me. If the epistemology used has as a premiss that true belief is insufficient for knowledge as it seems to, then the word 'mere' has clear meaning: true belief is not enough. It follows that if knowledge is possible, then something else is required and that is labelled 'warrant'. Your epistemology might be very different from this, but if so then I can't see how a debate about circularity in Merricks' argument is relevant, since rejection of the KcP == BcP&P&W(P) model is a different argument in my view.


>>(i)BcP&P-> P***


(leaving aside the obvious sophistry of the use of the 'weasel' word "mere") as this seems to render the term warrant (whatever its definition redundant) since

(ii) KcP->P

hence

(iii)KcP==BcP&P->P

but the Merricks/Plantinga definition of warrant is that:

(iv) KcP=BcP&P&W(P) <<
I'm sorry, this doesn't seem to relate to Merricks' argument apart from that your (iv) is the definition of warrant used in the paper. Your (i) is indeed tautologically true, (ii) is an axiom, but how did you get (iii)? Via just leaving out the warrant component?

>>however from (iii) since KcP==BcP&P then in (iv) W is redundant.<<
Well, yes but where did (iii) come from? It isn't part of Merricks' argument as far as I can tell.

>>Surely Merricks 'S knows that P..., KSp, is the same as KcP, and hence forms part of his premiss?<<
But substituting agent c for S, this is just KcP=BcP&P&W(P) which defines warrant. You have yet to show how (v) W(P)->(P), is contained therein. As far as I can tell, you keep assuming KcP but as I pointed out, the scenario that would be a counter example would be (vi) ¬P&W(P). Merricks' argument attempts to show that (vi) cannot be true. In other words, it cannot be the case that p is false and p is warranted. However,

(vii) ¬P&W(P)->¬Kc(P)

Since if ¬P then it cannot be the case that K knows P as this contradicts the axiom KcP->P. So any attempt to show circularity that takes KcP as a premiss is doomed.

>>Rather, I am trying to show that the conclusion is //part// of the premiss:

'S knows that p[is true], therefore, if and only if S’s belief that p is warranted[is true] and p is true.'(1)

that is KSp->W[p]&p <<
No, what we have is:

"S knows that p, therefore, if and only if S’s belief that p is warranted and
p is true."

and that is KSp <-> BSp&p&W(p). I'm unclear about putting [is true] in there, at best it is redundant.

>>So KsP can only be true if the logical conjunction is true and the logical conjunction Wp&P can only be true if both its constituent are true. Hence, the atomic proposition that P is warranted [W(p)] /cannot/ be false. Hence 'warrant entails truth' is part of the premiss a well as te conclusion. <<
I agree KsP can only be true if P is true and if W(P) is true. BsP must also be true, but even if we ignore that it does not follow that W(P)->P. That's because you haven't ruled out the possibility that ¬KsP&¬P&W(P).

>>The problem here is while KcP->P is axiomatic it does not thereby follow KSp->W[p]&P is an axiom rather than an arbitrary postulate (premiss) <<
Why is that a problem? Or rather why is using KsP==BsP&P&W(P) as a premiss a problem? That is part of the argument in the paper.

>>Another problem with your argument is that if we accept Merricks premiss, KSp->W[p]&p, then if Ksp is true and p is true then W[p] cannot be false. <<
Merricks' premiss is KSp<->W[p]&p&BSp. Now, consider what if KSp is false?

>>No, as I have shown above, Merricks premiss (1) means that by definition W[p] //cannot// be false. What is irrelevant is the rest of Merricks argument for infallibillism since it implicit in his definition (1). <<
This seems to me to be the same error. According to the definition W(p) cannot be false /given/ KSp is true. But if it could be the case that KSp is false and p is false and W(p) is true then we cannot assert that W(p)->p.

>>So it may be a problem for you but not for me.<<
I don't see how our respective attitudes to Gettier or our views on knowledge are relevant to the issue of circularity.

ttfn

Hi psi

Intermittent but frequent problem with home hub. Extended reply impractible.

Replacement tomorrow - hopefully

Bs

Hi Bx4,

While I wait for you to be up and running hubwise, I'll try to progress the analogy strand.

>>>>I don't think the analogy is like that. All that is needed for the analogy to work is that the solution set depends in some
way on the other elements.<<

I am rather unclear what you mean here - perhaps you would reduce the level of abstraction by giving a //concrete// example
of how such a mathematical analogue maps to a target involving truth, true belief, warrant and knowledge. <<
When I thought of the analogy, I was thinking of the role of something like the small positive number e in the following (I can't write the symbol for epsilon, I don't mean the famous other number e which is the base of natural logs):

Suppose we want to prove that if Sn and Tn are null sequences, then so is Sn + Tn. We might say that given e, we can find N1 such that:

-e < Sn < e for all n > N1

I won't give the full proof but the salient feature is that the role of e is to be any positive number which can be as small as you like. The proof does not rely on us considering what a specific value of e might be. Similarly, warrant has a specific role in Merricks' argument, but we don't need to consider a specific definition. It suffices to say warrant is whatever is added to true belief in order to produce knowledge, it can be any of the definitions you like. This analogy seems reasonable to me.

ttfn

hi psi

>>When I thought of the analogy, I was thinking of the role of something like the small positive number e in the following (I can't write the symbol for epsilon....<<
I think if one is considering sequences (ordered sets) the 'e' does not have to be small.

>>Suppose we want to prove that if Sn and Tn are null sequences, then so is Sn + Tn.<<

A consequence the general Sum Rule for convergent sequences.

'A sequence is usually defined as a function whose domain is a countable totally ordered set'

http://en.wikipedia.org/wiki/Sequence#Definition

'A sequence of real numbers that converges to 0 is sometimes called a null sequence. Null sequences are analogous to the number zero'

http://www.math.colostate.edu/~yzhou/course/math317_spring2012/KenNotes03.pdf

I am somewhat unclear why you are restricting your analogue to sequences real numbers that converge to zero.

>> We might say that given e, we can find N1 such that:

-e < Sn < e for all n > N1 <<

The argument of course presupposes that e is a member of the sequence determined by the rule(s) that specifies the sequence

>>I won't give the full proof but the salient feature is that the role of e is to be any positive number which can be as small as you like. The proof does not rely on us considering what a specific value of e might be.<<

However since the foundational definition of a sequence whether convergent, null or otherwise is:

'A sequence is usually defined as a function whose domain is a countable totally ordered set'

http://en.wikipedia.org/wiki/Sequence#Definition

Then a sequence requires a starting value and a rule which allows the generation of the other members of the sequence so we could, in principle, determine whether e is a member of that ordered set or not.


>>Similarly, warrant has a specific role in Merricks' argument, but we don't need to consider a specific definition. It suffices to say warrant is whatever is added to true belief in order to produce knowledge, it can be any of the definitions you like. This analogy seems reasonable to me. <<

Despite your 'similarly', I am not persuaded that your analogue of a sequence maps to your target because the ordered set that is a sequence needs must have a starting value and a rule that allows the other members of the ordered set to be generated and none of these features are to be found in your putative target.

I have already argued*** that the claim that 'it suffices to say warrant is whatever is added to true belief in order to produce knowledge' is problematic given that the tautologous nature of:

BcP&P->P (If c believes that P is true and P is true then P is true {strict implication}It suffices to say warrant is whatever is added to true belief in order to produce knowledge

that //nothing// needs to be added to true belief for it to be knowledge given the axiomatic definition of knowledge in epistemic logic:

KcP->P (if C knows that P is true then Pis true {strict implication}.

I await your refutation.

***For clarification this argument is distinct from the one in which I accept, arguendo, Merricks definition of warrant.

Hi Bx4,

>>I await your refutation.<<

I think you might have missed the post I gave before the last one? I did two in a row.
I'll reply to the analogy strand in full anon.

ttfn

Hi Bx4,

Sorry, no I didn't do two, I just can't see a reply to post #83 which seems to me to contain the refutation of what is essentially the same argument.

>>BcP&P->P (If c believes that P is true and P is true then P is true {strict implication}It suffices to say warrant is whatever is added to true belief in order to produce knowledge

that //nothing// needs to be added to true belief for it to be knowledge given the axiomatic definition of knowledge in epistemic logic:

KcP->P (if C knows that P is true then Pis true {strict implication}.<<
So, again yes BcP&P->P and KcP->P but it does not follow that warrant is superfluous, you'd have to show that simply believing something and it being true suffices to know it. The epistemology used by Merricks assumes the insufficiency of true belief for knowledge.

ttfn

hi psi

missed your #83. Will get round to a reply when I can.

>>It suffices to say warrant is whatever is added to true belief in order to produce knowledge<<

I am not sure it does since it /presupposes/ that true belief /requires/ such an addition. Merricks simply declares this by fiat.

>>So, again yes BcP&P->P and KcP->P but it does not follow that warrant is superfluous, you'd have to show that simply believing something and it being true suffices to know it.<<

Entia non sunt multiplicanda praeter necessitatem***. So rather you have to show that BcP&P==KcP is false.

>>The epistemology used by Merricks assumes the insufficiency of true belief for knowledge. <<

I am questioning that assumption.

*** I realise that the unnecessary multiplication of entities is the sine qua no of modern professionalised philosophy
bs

Hi Bx4,

I think the question of whether warrant is superfluous is a new topic.
We should probably try to settle the question of circularity in Merricks' argument before getting into that.
So far you have not shown that assuming by fiat that warrant has a role produces the circularity, or that there is circularity for any other reason in my view.

ttfn

hi psi

>><<

Well no since in my post 82 I had already said:

'... this seems to render the term warrant (whatever its definition) redundant... ' [corrected for typo]

>>So far you have not shown that assuming by fiat that warrant has a role produces the circularity, or that there is circularity for any other reason in my view.<<

Given that neither my post 89 nor your post 88, to which I was replying, mentioned circularity I'm somewhat at a loss as to why you should conclude that my skepticism about Merricks 'assuming warrant by fiat' has anything my views on the circularity of his argument. It is not.

My argument intended to show the circularity of the logical argument leading to the /conclusion/ 'warrant entails truth' is based on the extension of the logical law of conjunction to /any/ arbitrary three operands, r,s and t:

(a) r.s.t is true if and only if /all/ three operands, r,s and t are true

and the Plantinga/Merricks 'definition' of warrant (his /premise/)

(b) 'Warrantis “that...which makes the difference between knowledge and mere true belief.”'

which we can formalise as:

(c) c knows that P is true) if and only if (c believes that P is true) and that (P is warranted) and P is true

or symbolically:

(d) KcP iff BcP.W.P

but from (a) if follows that:

(e) BcP.W.P is true if and only if /all/ three operands, BcP,W and P are true. So the premiss requires that W is necessarily true for the conjunction to be true and by definition KcP is by definition the conjunction BcP.W.P or more abstractly, K==B.W.P so the proposition 'warrant entails truth' is implicit in the premiss as well as being the conclusion, aka 'circular reasoning'.

bs

Hi Bx4,

>>Well no since in my post 82 I had already said: <<
Fair point. What I meant was that I want to try to sort out your argument below which is about the circularity in Merricks' argument rather than deal with the issue of whether warrant is superfluous.

>>Given that neither my post 89 nor your post 88, to which I was replying, mentioned circularity I'm somewhat at a loss as to why you should conclude that my skepticism about Merricks 'assuming warrant by fiat' has anything my views on the circularity of his argument. It is not. <<
I was just trying to stick to the circularity issue and not conflate this with skepticism about warrant.

>>My argument intended to show the circularity of the logical argument leading to the /conclusion/ 'warrant entails truth' is based on the extension of the logical law of conjunction to /any/ arbitrary three operands, r,s and t:

(a) r.s.t is true if and only if /all/ three operands, r,s and t are true

and the Plantinga/Merricks 'definition' of warrant (his /premise/)

(b) 'Warrantis “that...which makes the difference between knowledge and mere true belief.”'

which we can formalise as:

(c) c knows that P is true) if and only if (c believes that P is true) and that (P is warranted) and P is true

or symbolically:

(d) KcP iff BcP.W.P <<
As I have said, I don't agree that this is how Merricks' argument works. I assume that the '.' operator in r.s.t is &, then I agree with your a), I agree that Merricks takes d) as a premiss and that this does follow from b) via c). I am still unclear how this demonstrates circularity. The truth of P is not contained in the premiss KcP iff BcP.W.P in such a way as to render the conclusion W->P circular. If you take KcP as a premiss, then trivially, BcP is true, W is true and P is true. As far as I can see, Merricks does not take this premiss. That means your r.s.t argument does not apply.

ttfn

hi psi

Sorry about the delayed reply. Somewhat busy for fun and profit in the world.


>>Fair point. What I meant was that I want to try to sort out your argument below which is about the circularity in Merricks' argument rather than deal with the issue of whether warrant is superfluous.<<

Fair enough but I was emphasising that while I found the Plantinga/Merricks attempt to bolt on their definition of 'warrant' to 'true belief' superfluous, I was accepting it, arguendo, since if I did not I could not make the case that their argument is a case of circular reasoning.


>>I was just trying to stick to the circularity issue and not conflate this with skepticism about warrant.<<

Yet your:

'So far you have not shown that assuming by fiat that warrant has a role produces the circularity...'[#90]

seems to do just that.

>>As I have said, I don't agree that this is how Merricks' argument works.<<

I presume you mean the argument that Merricks makes in support of infallibilism, etc., but I don't think counters my case for circularity:

"The circular argument uses its own conclusion as one of its stated or unstated premises. Instead of offering proof, it simply asserts the conclusion in another form, thereby inviting the listener to accept it as settled when, in fact, it has not been settled. Because the premise is no different from and therefore as questionable as its conclusion, a circular argument violates the criterion of acceptability."
(T. Edward Damer, Attacking Faulty Reasoning. Wadsworth, 2001)

The problem for Merricks argument is that its intermediate premisses that appeal to warrant necessarily contains, if only implicitly, the the original definitional premiss, albeit unstated, which necessarily means that such premisses also contribute to the argument's circularity.

>> I assume that the '.' operator in r.s.t is ...<<

Indeed the '.' is an alternative to the '&' notation for conjunction which I prefer largely for aesthetic reasons but also because it removes any confusion that might arise when I use 'and' in a grammatical sense rather than a strictly logical one.

>> ...then I agree with your a), I agree that Merricks takes d) as a premiss and that this does follow from b) via c). I am still unclear how this demonstrates circularity.<<

Merricks conclusion (the title of his paper) is

e) 'Warrant entails truth'

and since you accept the premiss:

d) KcP iff BcP.W.P

Then it follows that KcP can only be true iff W is is necessarily true. Hence given that:

'In logic, entailment (or logical implication) is a relation between sets of sentences and a sentence. Typically entailment is defined in //terms of necessary truth //some set T of sentences entails a sentence A if and only if it is necessary that A be true whenever each member of T is true.' [emphasis added].

http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Entailment.html

Hence since W is a necessary truth in terms of the sentence KcP it follows that W entails truth is implicit in the premiss d) as well as in the conclusion e) which is therefore a case of circular reasoning.

>>The truth of P is not contained in the premiss KcP iff BcP.W.P in such a way as to render the conclusion W->P circular<<

If 'W->P' is a symbolic representation of e) 'Warrant entails truth' as a //strict// implication (if W is true then P is necessarily true') then I have shown that the premiss d) also contains yhis strict implication so the circularity is maintained.

>> If you take KcP as a premiss, then trivially, BcP is true, W is true and P is true. As far as I can see, Merricks does not take this premiss. That means your r.s.t argument does not apply.<<

Surely, Merricks ordinary language definition of knowledge presupposes that KcP is true and includes 'mere true belief' as part of the definition and so necessarily assumes BcP is true for KcP to be true?

bs

Hi Bx4,

>>Sorry about the delayed reply. Somewhat busy for fun and profit in the world. <<
No problem.

>>Yet your:

'So far you have not shown that assuming by fiat that warrant has a role produces the circularity...'[#90]

seems to do just that. <<
No it doesn't. The full quote makes this clear, I am rejecting the two quite distinct cases that a) you thought the circularity is due to the fiat and b) any other reason or argument you have used. When I posted this I wasn't sure whether you were using a) as well as b) and that's why I said we ought not to conflate the two.

>>The problem for Merricks argument is that its intermediate premisses that appeal to warrant necessarily contains, if only implicitly, the the original definitional premiss, albeit unstated, which necessarily means that such premisses also contribute to the argument's circularity. <<
No the original definitional premiss does not produce a circularity, what you have done is overlook its conditional property. I have tried to explain where I think you are going wrong, I'll have one more go using the same context and then if we make no progress I'll see if I can think of a more intuitive scenario with the same logical structure.

>>e) 'Warrant entails truth'

and since you accept the premiss:

d) KcP iff BcP.W.P

Then it follows that KcP can only be true iff W is is necessarily true. <<
Modal scope fallacy alert. What is 'necessarily' doing here? It should be: Necessarily, KcP can only be true iff W is true. I agree. However, it does not follow that W -> P as I have explained, since you haven't ruled out the scenario that KcP is false and P is false and W is true. If you can preclude that possibility then you have W -> P. The important thing to realise is that you cannot assume KcP!

>>Hence since W is a necessary truth in terms of the sentence KcP it follows that W entails truth is implicit in the premiss d) as well as in the conclusion e) which is therefore a case of circular reasoning. <<
No, W is only true if you assume KcP which you are not entitled to do and Merricks doesn't. Merricks assumes that IF KcP is true THEN BcP.W.P is also true. That is the difference you haven't distinguished as far as I can tell.

>>If 'W->P' is a symbolic representation of e) 'Warrant entails truth' as a //strict// implication (if W is true then P is necessarily true') then I have shown that the premiss d) also contains yhis strict implication so the circularity is maintained. <<
See above.

>>Surely, Merricks ordinary language definition of knowledge presupposes that KcP is true and includes 'mere true belief' as part of the definition and so necessarily assumes BcP is true for KcP to be true? <<
This is the error you are making in a nutshell. Merricks' definition presupposes no such thing, any more than presupposing that 'if it is raining then it is wet' presupposes that it is raining.

ttfn

hi psi

Descending in to the Weihnachten 'social whirl' so I'm not sure when I will be able to reply.

bs

Hi Bx4,

I hope you have some fun

ttfn

hi psi

Apologies for delayed reply. Because we knew we wouldn't be here for most of last year we cancelled our internet link and it took longer to get it reinstated than anticipated

Things are finally quietening down here after extended Hogmanay and so I hope to manage a reply in the next couple of days.

bs

Hi Bx4,

Whilst I'm waiting for that I'd just like to mop something up from the analogy strand. Recall this example I gave:

We might say that given e, we can find N1 such that:

-e < Sn < e for all n > N1

You replied:

>>The argument of course presupposes that e is a member of the sequence determined by the rule(s) that specifies the sequence<<

This is a mistake I'm afraid, the argument presupposes no such thing and thus the rest of your critique of the analogy fails, proceeding as it does from this mistaken assumption.

ttfn

hi psi


Apologies for further delays but I have be suffering from some sort of flu-like virus infection for the the last few days though it has not yet emerged as a full blown flu or cold. The power of the hot toddy, perhaps?

However, I am sufficiently shivery, wheezy and generally out of sorts to be rendered incapable of putting together any kind of coherent argument on matters filosofickal.

bs

Hi Bx4,

Sorry to hear you are not well.

Get well soon.

ttfn