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

hi psi.

I finally recreated my reply offline in Notepad but when I paste it in I am now required to enter a CAPCHA before I can see the Preview which appears OK bu when I try to Post Message I get a blank!

bs

Hi psi

My hopefully recreated post. This proved to have a problem sometimes encountered in h2g2 which is that a document prepared offline in Notepad sometime (but curiously not always) inserts CR-LF characters when pasted into the Reply box. Hopefully I have edited all these out.

>>I am interpreting W(p) -> p as 'warrant entails truth', or more precisely, 'if a belief that p is warranted then p is true'. <<

I was confused by the use of the same symbol, ->, for entailment and material(?) implication. Your use would seem to imply that entailment and material(?) implication are equivalent but I am not persuaded this is the case as:

'The material conditional ‘A -> B ’ is true if as a matter of fact it is not the case that A is true and B is false, whereas for ‘A, therefore B’ to be a valid inference it must be impossible for B to be false when A is true. Since it is generally accepted that ‘A’ entails ‘B’ iff ‘A, therefore B’ is a valid inference, this means that to say that ‘A’ entails ‘B’ is to say that ‘A -> B ’ is not merely true, but is necessarily true.'

(Dictionary of Philosophy: Revised Second Edition, 1984)

Which would seem to point to an obvious difference between the two notions which may account for the fact that the symbol for entailment is either |= (semantic entailment) or |- (syntactic entailment). The latter is sometimes used in the sequent notation for inference schemes like modus ponens:

if p then q;p|-q

So given that saying 'that to say that ‘A’ entails ‘B’ is to say that ‘A -> B ’ is not merely true, but is necessarily true.' then then to say that p entails q, p|=q or p|- would seem to be closer to strict
implication p strictly implies q, p-|q (where -| stands for the non-printing Lewisian fishhook operator) than to material implication.

Moreover if you are interpreting W(p)->(p) as

(a)'if a belief that p is warranted then p is true'


then surely what you are actually saying is that

(b)'Necessarily, if a belief that p is warranted then p is non-accidentally true'


However, from Merricks 'Argument One' we have:

'(5) Necessarily, a warranted true belief is known. [definition of warranted belief]'


then surely it follows that saying (b)' is equivalent to saying:

(c)'Necessarily if p is known then p is non-accidentally true'

from which we can argue that

@(KSpP->P)=@(BSpP.W(P)->P)=@(BSP)->@p Where = stands for the equivalence symbol

So Merricks definition of warrant (his original premise/definition) contains an entailment that is an alternative expression of his conclusion that 'warrant entails truth' and hence his argument is circular.

>>Before you continue replying I would urge you to watch this video with a particular focus on what is meant by the 'conditional' and the 'converse', since your latest reply seems still to conflate these notions with the antecedent and consequent.<<

I'm not sure why you think I am am conflating the notions of the conditional and the converse with those of the antecedent and the consequent when I am seeking to differentiate them. Of course they are somewhat related through the rule of inference, biconditional elimination (in sequent notation):

(i) p<->q|-(p->q),(q->p)

http://en.wikipedia.org/wiki/Biconditional_elimination
http://logic.stanford.edu/intrologic/chapters/chapter_03.html

from which we can infer

(ii) p<->q|-(p->q).(q->p)

My point was simply that in the biconditional p is the conditional and q is the converse whereas in the the syntactic entailments (syntactic consequences) p is the antecedent and q the consequent in the first syntactic entailment and q is the antecedent and p the consequent in the second syntactic entailment so one can only talk of p and q as antecedent and consequence and vice versa with respect to the relevant syntactic entailment and in terms of the biconditional it is meaningless to refer to p or q in terms of either being an antecedent or a consequent or vice versa .

I hope this clarifies.

I am not clear you feel what I should get from the video since while the use of 'hypothesis' and 'conclusion'*** somewhat obscures the point that in both examples the conditional is presented as an
implication with an antecedent h ('the hypothesis) and a consequent c ('the conclusion'), that is h->c while in the converse the roles of h and c are reversed so that c becomes the antecedent and h the consequent, that is c->h)

This is formalised as the rule of inference biconditional introduction. In sequent notation

p->q,q->p|-p<->q

http://en.wikipedia.org/wiki/Biconditional_introduction
http://logic.stanford.edu/intrologic/chapters/chapter_03.html

from which we can infer

p->q.q->p|-p<->q

my intention in linking to the video was the comment at the end. viz:

'The conditional and its converse both have to be true'

which seems to run contrary to your argument that involved setting the conditional and the converse to be false. While no doubt this does yield a true biconditional it is not one in which 'the conditional and its converse both have to be true'. The truth table for a biconditional is:

http://www3.cs.stonybrook.edu/~skiena/113/lectures/lecture3/img74.gif

So while setting both conditional and converse to false yields a true biconditional it does not satisfy the additional constraint that for a biconditional to be a definition 'the conditional and its converse both have to be true'

It would seem to follow from this that Merricks definition (or 'purely formal characterisation') necessarily requires that both the conditional KSP an the converse BSP.W(P).P are necessarily true.

*** If one accept the usage of the terms 'hypothesis' and 'conclusion' as precise then this would suggest
that the author of the video is using an implicit implication elimination (modus ponens):

'Modus ponens allows one to eliminate a conditional statement from a logical proof or argument (the antecedents) and thereby not carry these antecedents forward in an ever-lengthening string of symbols; for this reason modus ponens is sometimes called the rule of detachment'

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

p->q;p|-q

if we apply this to the expanded conditional term of Merricks biconditional definition:

KSP->BSP.W(P).P

We get:

KSP->BSP.W(P).P;KSP|-BSP.W(P).P

and for the expanded converse term of Merricks biconditional definition:

BSP.W(P).P->KSP

we get:

BSP.W(P).P->KSP;BSP.W(P).P|-KSP

So the expansion of Merricks biconditional definition:

(KSP->BSP.W(P).P).(BSP.W(P).P->KSP)

would reduce to:

(BSP.W(P).P).KSP

when KSP is true and BSP.W(P).P is true as is required for the biconditional to be a definition.

bs

Hi psi The CAPCHA requirement seems to be down to a company called Cloudflare: https://www.cloudflare.com/features-security which the h2g2 site uses. Apparently, I'm not the first to encounter this nonsense. http://h2g2.com/feedback/A388325/conversation/view/F47996/T8302868 h2g2 seems to be getting flakier and flakier in addition to the irritating limitaion of its automated GuideML preprocessor. Apparently it might be possible to write entries using GuideML http://h2g2.com/entry/A5974374 Though I am to time poor to look into it. Might be preferable to find a more robust site but I don't know of one. Do you? bs

Hi psi

Got it slightly wrong GuideML is not a pre-processor but rather a local variant of the XML markup language

Playing around it appears that one can do bold and italic fairly easily:

hello and goodbye

In the Preview at least. Might be interesting to see if one can embed 'non-printing' logic symbols but that's for another day

bs

Hi Bx4,

>>I was confused by the use of the same symbol, ->, for entailment and material(?) implication. Your use would seem to imply that entailment and material(?) implication are equivalent but I am not persuaded this is the case as:<<

We have been through this before. Entailment can be used in a technical sense as distinct from implication, but I am not persuaded that Merricks was doing this and nor am I persuaded that it is relevant for the argument on circularity. In any case, if the material implication is not contained in the premise in a circular way, then a fortiori, the strict implication will not be either.

Even if we do introduce the modal element of necessity, there are problems here:

>>then surely what you are actually saying is that

(b)'Necessarily, if a belief that p is warranted then p is non-accidentally true' <<

Fine but:

>>However, from Merricks 'Argument One' we have:

'(5) Necessarily, a warranted true belief is known. [definition of warranted belief]' <<

Warranted true belief, rather than warranted belief. You can only collapse it to warranted belief if you already accept that warrant entails truth.

>>then surely it follows that saying (b)' is equivalent to saying:

(c)'Necessarily if p is known then p is non-accidentally true'<<

I don't see how, as (b) doesn't mention knowledge, whereas (c) links knowledge to truth.

The whole of the rest of your argument seems to rest on the misapprehension that I have set the conditional and the converse to F. I have not. Rather than go back to give examples, I'll quote from your latest:

>>My point was simply that in the biconditional p is the conditional and q is the converse<<

No, in the biconditional p <-> q, p -> q is the conditional and q -> p is the converse. I have set neither to be F and therefore the objection that they must be true for a definition does not bite.

Let's consider another example. Suppose we have the following:

(1) f <-> q

Where f = is a four sided figure, q = is a quadrilateral.

For this to be a definition of quadrilateral, what has to hold about the conditional and its converse? The conditional and converse have to be true.

The conditional is f -> q

That is, if this is a four sided figure then it is a quadrilateral. This is true if we consider a five sided figure because that isn't a quadrilateral. So f = F, q = F yet f -> q is T

A similar scenario holds for the converse which is q -> f, since if we have a non quadrilateral then is doesn't have four sides. So q -> f is true in this circumstance.

I can't see any prospect of further progress until you address the issue that in the biconditional p is not the conditional as you have stated above, p is the antecedent of the conditional.

ttfn

hi psi

Apologies. Been rather busy preparing for a symposium which I'm attending all this week so I'm unlikely to manage a reply until the weekend.

bs

Hi psi

Clearly this needs to be addressed before we can move on:



>>I can't see any prospect of further progress until you address the issue that in the biconditional p is not the conditional as you have stated above, p is the antecedent of the conditional.<<

However as I said in the conjunction:

(p->q).(q->p) (1)

from which a biconditional derives that p is also the consequent in the converse. So in the biconditional form of (1):

(p<->q) (2)

that p would be both an antecedent and a consequent which clearly causes confusion hence I preferred to avoid this by adopting a different nomenclature when referring to p and q in (2).

However, since a biconditional is a logical equivalence:

'If any two propositions are joined up by the phrase "if, and only if", the result is a compound proposition called an equivalence. The two propositions connected in this way are referred to as the left and right side of the equivalence. By asserting the equivalence of two propositions, it is intended to exclude the possibility that one is true and the other false; therefore, an equivalence is true if its left and right sides are either both true or both false, and otherwise the equivalence is false.'

http://www.personal.kent.edu/~rmuhamma/Philosophy/Logic/SymbolicLogic/3-equivalenence.htm

So perhaps we can cut the Gordian not which apparenly prevents us from proceeding by referring to p as 'left side' (LS) and q as 'right side' (RS)

bs

Oops! Typos:

'knot' not 'not'

'apparently' not 'apparenly'

bs

hi psi

Still time poor so trying to reply to parts of your #185 as and when I can.

>>We have been through this before. Entailment can be used in a technical sense as distinct from implication, but I am not persuaded that Merricks was doing this and nor am I persuaded that it is relevant for the argument on circularity<<


My point was simply that there are accepted notations that distinguish amongst material implication (-&gt, strict implication (-|) {as a printable alternative to Lewis' 'fish-hook' symbol} and entailment (|-) {without needing to worry about the technical distinction between syntactic and semantic entailment) and I was merely trying to determine whether you were using the same symbol for two distinct (non-equivalent) concepts.

There seems to be nothing in Merricks paper that would suggests that he thinks material implication and entailment are in some unspecified non-technical sense equivalent*** so I think you are doing him no favours by attributing such a view to him.

However, nowhere in Merricks paper can I see anything that would suggest that Merricks' is conflating the notion of entailment with the material conditional and I would be interested to see any argument that he is.

The relevance to my argument is that I am trying to establish what you or you as an 'interpreter' of Merricks means by 'entailment'. The reason I need to establish this is that I need to show that Merricks 'purely formal characterization':

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

implicitly contains his conclusion and to do that we need to establish that we have a common understanding of the meaning of entailment.


The OCP entry ('Entailment' 1995 (Paperback Edition,1995: p.237) says:

'A set of propositions (or statements or sentences entails a proposition (etc.) follows necessarily (logically, deductively) from the former, i.e. when an argument consisting of the former as premises and the latter as conclusion is a valid deduction.

However, it goes on to say

'The criterion of this is contentious The classical criterion identifies entailment with strict implication Where Set T strictly implies A' means it is impossible for for all members of T to be true without A being true.'

But goes on then goes on to note that:

'Some logicians search for a different criterion to escape the paradoxes (of strict implication) and more generally to respect the feeling that a set of propositions should have some 'relevance' to what in entails.'


However since the premises and conclusion would seem to satisfy Anderson and Belnap's 'variable sharing principle' (Anderson, A.R. and N.D. Belnap, Jr., 1975, Entailment: The Logic of Relevance and Necessity, Princeton, Princeton University Press) then there t no problem in Merricks entailment with the paradoxes of strict implication.


I am quite happy to proceed on any of these concepts of entailment but it would be helpful if you indicate which or, alternatively, indicate some other generally accepted concept that you prefer.

>> In any case, if the material implication is not contained in the premise in a circular way, then a fortiori, the strict implication will not be either.<<

Surely W(P)|-P is Merricks conclusion not his premise (unless you are conceding circularity.)


***{Merricks only uses the term conditional twice in a footnote (10) referring to Argument One Premise (2) where the conditional is preceded by the modal qualifier necessary suggesting a strict rather than a material conditional and again in a footnote (23) to premise (7) though here the position seems somewhat confused with ' the infallibilist thinks that(7) has a necessarily false antecedent, and therefore thinks the entire conditional is true.' the scope of the modal operator is restricted to the antecedent (a modal scope error?) and from this arsees that the third row of the truth table for the material conditional is met.}

bs

I have no idea why the symbol material implication -> came out as it did. Something to do with the bracketing perhaps? Let's see (-&gt

bs

Hi Bx4,

>>Clearly this needs to be addressed before we can move on: <<

I fear there may be a way to go yet.

>>that p would be both an antecedent and a consequent which clearly causes confusion hence I preferred to avoid this by adopting a different nomenclature when referring to p and q in (2). <<

This hasn't caused confusion /at all/. As far back as #106 I was pointing out that talk of consequent and antecedent was irrelevant as the biconditional can be read in either direction.

The confusion seems to be in your conflation between 'conditional/converse' and 'consequent/antecedent'. The evidence for this is your comment:

>>My point was simply that in the biconditional p is the conditional and q is the converse<<

P is not the conditional and nor is q the converse. It seems to me that it is this error that leads you to think that my assignment of truth values sets the conditional and consequent to be false. It doesn't.

This is the issue we need to resolve before progress can be made I think. To be absolutely clear, it does not matter whether p is both the antecedent and consequent, or whether we simply adopt LHS and RHS.

So do you accept that your "in the biconditional p is the conditional and q is the converse" is incorrect? If so, we can revisit the argument against circularity and hopefully you'll see it is valid.

ttfn

Hi Bx4,

>>I am quite happy to proceed on any of these concepts of entailment but it would be helpful if you indicate which or, alternatively, indicate some other generally accepted concept that you prefer.<<

I don't care, just pick one and assume that whatever symbol I'm using for implication means that. If anomalies arise as a result, just point to them and I'll fix them.

>>Surely W(P)|-P is Merricks conclusion not his premise (unless you are conceding circularity.<<

Yes, but what I'd said was:

"In any case, if the material implication is not contained in the premise in a circular way, then a fortiori, the strict implication will not be either."

By which I mean if the material implication which is Merricks' conclusion is not (implicitly or explicitly) contained in the premise, then there is no circularity. If it is not the case for material implication then it definitely won't be for strict implication.

I hope that clarifies.

ttfn

hi psi

hi psi

Apologies for delay in replying. Very time poor at moment - a combination of writing final reports on a number of projects before year end and (tedious but inevitable) seasonal social whirl so little time for things filosofikal. Hopefully back on piste in a few days.

Meanwhile have a good one.
bs

Hi Bx4,

You too!

ttfn

Hi psi

Sorry to have been of piste for so long but I made a 'resolution' to REALLY retire at the New Year (as opposed to my unsuccessful earlier attempt a few years ago) so I am a bit mire in the administrative details at the moment.

Hope to wrap these up soon at which point (after re-reading the latest posts) I will resume our 'conversation'.

bs

Hi Bx4,

Goodness, well I hope that goes smoothly. I'm sure you will have plenty to do if you actually manage to retire this time!

ttfn

Hi psi

Last time I 'retired' I made the mistake of agreeing to 'a few days consultancy a month'. The 'slippery slope' proving not to be a logical fallacy so this time a clean break!

Have been having some problems with h2g2. First getting asked to decode 'capchas' to make sure I', not a bot then a ongoing failure to have my id and password recognised but finally managed to contact 'the gurus'('No Researcher left behind') and all now appears OK.

Weather is quite mild for the time of year (cloudy, 3C, no snow) but not so mild as to encourage outdoor pursuits so I have spent the time reading and transferring my record/CD collection to my Brennan. Quite ejoyable since I've been playing stuff I havent listened to in years.

Anyhow back to matters filosofickal:

I have been thinking how to present my circularity contention in a simpler form.

A biconditional is true either when its left hand side and right hand side are both false or when the are both true.

However I gave two links which said that when a biconditional is functioning as a definition then the the second situation applies.

Before going any further I want to make sure you have no problem with this

bs

Hi Bx4,

Well it sounds like retirement is going smoothly so far anyway. Meanwhile, h2g2 seems as buggy as ever!

On the philosophy I think you have distilled the nub of it. This is the impasse. I'll try to state my disagreement clearly again.

>>I have been thinking how to present my circularity contention in a simpler form.<<

I don't think the complexity is the issue, but a simpler form might help us both to be looking at the same thing. When I point at 'conditional' and 'converse' we both need to be assured that the other has the same concept, as this is key here.

>>A biconditional is true either when its left hand side and right hand side are both false or when the are both true.<<

Here we agree.

>>However I gave two links which said that when a biconditional is functioning as a definition then the the second situation applies.<<

I haven't looked back at the links, but from memory what they said was that when a biconditional is fuctioning as a definition, both the conditional and its converse must be true. Most of my posts since then have been trying to make it clear that this does not mean that the only the second situation applies. The first situation, where LHS=RHS=FALSE also applies, since in this case, the conditional and its converse are TRUE.

>>Before going any further I want to make sure you have no problem with this<<

I think we need to address the problem above. Consider the biconditional:

p <-> q

This is true for p=q=T and for p=q=F.

You seem to think that in the latter case, where p and q are both false, the conditional and its converse are false. This is not so. Consider:

1) p -> q (conditional)
2) q -> p (converse)

Both 1) and 2) are TRUE for p=q=FALSE.

Nor is the situation changed in respect of this aspect of the argument if we change the type of implication. Whether material or strict, the above holds.

I hope that clarifies.

ttfn

Hi psi

>>smoothly<<

Indeed. I find I have more than enough to keep me busy/interested for the forseeable future.

>>buggy<<

I have again taken to writing my replies offline using Notepad and pasting them into h2g2 so even if problems occur I have the original.

>>You seem to think that in the latter case, where p and q are both false, the conditional and its converse are false. This is not so.<<

I have not so claimed for the biconditional form:

I accept that for any any arbitrary biconditional p <-> q four possible truth conditions exist

(a) p = T; q = F : p <-> q = F

(b) p = F; q = T : p <-> q = F

(c) p = F; q = F : p <-> q = T

(d) p = T; q = FF ; p <-> q = T

http://2.bp.blogspot.com/_YgQSxV-QRS4/SioXsxBqjMI/AAAAAAAAAA0/8jpNSBkAvQQ/s320/Biconditional.JPG

So the biconditional is true when both the left hand side and the right hand side hand side are true and is also true when both the left hand side are both false.

My point however is when a definition is presented as a biconditional then both its left hand side and its right had side are necessarily true whether one is using the unexpanded [p<->q] or expanded [(p->q).(q->p)] form of the biconditional:

https://docs.google.com/presentation/d/1JVSuRiX06kpJpdtakIkHjyy0Tj4uGdif-QHglEGv8Iw/edit?pli=1#slide=id.i57

http://www.regentsprep.org/regents/math/geometry/gp1/bicon.htm

>>Consider:

1) p -> q (conditional)
2) q -> p (converse)

Both 1) and 2) are TRUE for p=q=FALSE.<<

I don't see the problem here setting p and q to be false is simple equivalent asserting their negations:

'A figure is NOT and quadrilateral iff the figure does not have four sides.'

or in the expanded form:

'If a figure is NOT a quadrilateral then the figure does NOT have four sides AND if a figure does NOT have four sides then it is NOT a quadrilateral

bs

Hi psi

An afterthought to my last.

I think I have understood why we are at cross purposes on the truth values of the expanded biconditional:

(p->q).(q->p)

Which is simply that you were considering the consequences of the falsity of the component propositions, p and q, of the conditional and converse conditional whereas I was looking at the falsity of the complete conditional and converse conditional expressions.

It seems to me that in setting p and q to false (that is setting ¬p and ¬q you are correctly asserting that you will get a true conditional and a true converse conditional. However this gives you a true expanded biconditional but one which is different (that is has a different semantic meaning) from that which one gets by setting p and q to true; for example:


(a) IF it is TRUE that a figure is a quadrilateral THEN it is TRUE that the figure has four straight sides.

(b) IF it is TRUE that a figure has four straight sides THEN it is TRUE that the figure is a quadrilateral

(c) IF it is FALSE that a figure is a quadrilateral THEN it is FALSE that the figure has four straight sides.

(d) IF it is FALSE that a figure has four straight sides THEN it is FALSE that the figure is a quadrilateral


from which we can eliminate the embedded truth values by the transformations:

(a1) IF (a figure is a quadrilateral {p}) THEN (the figure has four straight sides {q}).

(b1) IF (a figure has four straight sides {q}) THEN (the figure is a quadrilateral {p}).

(c1) IF (a figure is NOT a quadrilateral {¬p}) THEN (the figure does NOT have four straight sides {¬q}).

(d1) IF (a figure does NOT have four straight sides {¬q}) THEN (the figure is a NOT quadrilateral {¬p}).

So now we have two expanded //true// biconditionals:

(i) (p->q).(q->p)

and

(ii) (¬p->¬q).(¬q->p)

which are true by virtue having //true// conditionals and converse conditionals and thus meet the criterion that when a biconditional is a definition both the conditional and the converse conditional are true. However they do not have the same semantic content; (i) is a definition of the set of all straight sided figures which are quadrilaterals whereas (ii) is the (mutually exclusive) set of all straight side figures which are not quadrilaterals.

Therefore reduce unnecessary complexity I am quite happy to limit the starting point of my argument by using only the true unexpanded biconditional (as definition) to the case where the conditional and converse conditional are true.

bs