Archive for September, 2009

Consider this set of propositions:

P1. The sentence “The current king of France is bald.” expresses a proposition.

P2. The sentence “The current king of France is not bald.” expresses a proposition.

P3. The propositions expressed by the sentences “The current king of France is bald.”and “The current king of France is not bald.” are contradictory.

P4. For all p and for all q, if p is a contradictory of q, then (p is true and q is false, or q is true and p is false).1

P5. Iff the sentence “The current king of France is bald.” expresses a proposition, then the sentence “The current king of France is not bald.” expresses a proposition.

P6. If the sentences “The current king of France is bald.” and “The current king of France is not bald.” express propositions, then the propositions are both false.

These propositions form an inconsistent set. When a person is confronted with that he believes is inconsistent, then that person will almost always stop believing at least one of the things he believes. However in this case people usually stop believing both (P1) and (P2), and that is because of (P5). To stop believing one of them only would be futile as the set would still be inconsistent. I suppose that the same people find the other propositions in the set much more justified than (P1) and (P2), and that is the reason why they are rejected.

About (P3)

But why is it that people believe (P3)? It seems to me to be a interpretation failure. Perhaps they interpret the two sentences:

S1. The current king of France is bald.

S2. The current king of France is not bald.

As:

P7. (∃x)(Kx∧Bx)

P8. ¬(∃x)(Kx∧Bx)

(P7) and (P8) may be read as:

P7′. There exists at least one x such that he is the current king of France and he is bald.

P8′. It is not the case that there exists at least one x such that he is the current king of France and he is bald.

(P7) and (P8) are rightly seen as contradictory, but they are not equivalent to the propositions expressed by (S1) and (S2). To see this notice where the negation in placed in (S2). It is not placed in the front of the sentence as it is in (P8), and neither is it placed in the first sentence part.2 It is placed in the second sentence part, and it is that part that is negated, not the entire sentence. The correct interpretation of the propositions expressed by (S1) and (S2) is therefore:

P9. (∃x)(Kx∧Bx)

P10. (∃x)(Kx∧¬Bx)

They may be read as:

P9′. There exists at least one x such that he is the current king of France and he is bald.

P10′. There exists at least one x such that he is the current king of France and he is not bald.

Clearly, now we can see that the propositions expressed by (S1) and (S2) are not contradictory but they are merely contrary, that is, they cannot both be true but they can both be false. This implies that (P3) is false, and so the threat to the set is removed.3

Why (P6) is true

Additionally we can now see why (P6) is true. Before we just thought (P6) evidenced by our intuitions. But since both (P9) and (P10) imply (by simplification) that:

P11. (∃x)(Kx)

P11′. There exists at least one current king of France.

And, as we know there is no current king of France, therefore, (P11) is false. If (P11) is false, then so are (P9) and (P10). If they are both false, then (P6) is true.

We can also generalize a bit from this. What seems to confuse people about (S1) and (S2) is that the subject, that is, the current king of France, is missing, that is, he does not exist.

Appendix

Here is a predicate logic version of the original set.

Translation key

Ex ≡ x expresses a proposition.

C(AB) ≡ A and B are contradictories.

a ≡ (The sentence) “The current king of France is bald.”

b ≡ (The sentence) “The current king of France is not bald.”

A ≡ (The proposition) “The current king of France is bald.”

B ≡ (The proposition) “The current king of France is not bald.”

The set

P1′. Ea.

P2′. Eb.

P3′. (Ea∧Eb)→C(AB)

P4′. (∀P)(∀Q)(C[PQ]→[(P∧¬Q)∨(Q∧¬P)])

P5′. Ea↔Eb

P6′. (Ea∧Eb)→(¬A∧¬B)

Notes

1I complicate matters a bit with bivalance because it is difficult to formulate things without bivalance.

2More on this in a forthcoming essay.

3The set may still be inconsistent for other reasons, though it doesn’t seem inconsistent to me.


“[W]hat follows from a true premiss must be true” (The Problems of Philosophy, p. 60, link)

Wrote Russell as an example of a principle of logic that is more self-evident than the inductive principle. If we were to formalize this we would perhaps write it like this:

E1. □[([∀P][Q∧Q⇒P])→P]1

Or perhaps just just in propositional logic:

E2. □(P→Q) where “P” means P is true and P implies Q.

As the reader of my blog should know by now, the modal fallacy consists of trusting language and placing the modal operator of necessity in the consequence instead of before the conditional:

E3. P→□Q

However we could also put the modal operator somewhere else in our formalization:

E5. P□→Q

Operators solely ‘work on’ whatever is to the right of them.2 Thus the modal operator in (E5) works on the material conditional and not the proposition to the left of it. Similarly in (E2) the modal operator works on the parentheses-set which is treated as a single entity.

(E5) is closer to normal english (and danish) than (E2) which we can express in normal english like this:

E4. Necessarily, if P follows from Q and Q is true, then P is true.

Consider the sentence:

E5. If P follows from Q and Q is true, then P must be true.

(E5) is a reformulation of (E2). (E5) is worded like it would be by a normal english speaking person. In (E5) it may seem as though the modal operator is intended to work on the consequent. Indeed some people think this and commit the modal fallacy.

However the modal operator may also be thought of as working on the second part of the “if, then” clause, that is, the “then” part. The only problem with this interpretation that I know of, is that it makes the operator work on something that is to the left of it instead of to the right of it: Because “then” is to the left of “must” in (E5).

Notes

1Ignoring the complexities of bivalance.

2Monadic operators. Not dyadic operators like implication, consistency etc.

In Longman’s texts Talking about the present, and Talking about the Past it is claimed that some verbs cannot be used in the present and past progressive. But they can. Here is the list of verbs it is calimed that cannot be used: (to) be, have, see, believe, like, agree, know, love, disagree, recognize, hate, mean, remember, prefer, need, understand, want, deserve, wish, belong.

Now consider these commonly used sentences:

E1. She is seeing another man.

E2. She saw another man.

Clearly then the verb “see” can be used both in the present and the past tense.

Consider these sentences:

E3. I am standing here and agreeing with you, but then you give me that. That was uncalled for.

E4. I was standing here and agreeing with you, but then you give me that. That was uncalled for.

It seems to me that there is nothing wrong with these sentences. So the verb “agree” can be used both in the past and in the present.

Consider these sentences:

E5. I am being very nice to you.

E6. I was being very nice to you.

Again nothing seems to be wrong with these sentences. Sentences like (E5) and (E6) seem to be used to stress the act of being nice to someone. Perhaps someone who denies it is the case.

I’m undecided as to which of the other verbs mentioned that cannot be used in the progressive form.

What is the present progressive? It is a sentence form. Longman Dictionary of Comtemporary English explains it like this:

“You make the present progressive by using a form of the verb be in the present tense,

followed by the main verb with an  ing ending, for example l am waiting, she is coming.”

Later in the same text, Talking about the present, Longman notes that:

“You use the present progressive to say that something is happening now, but will only continue for a limited period of time. Compare these pairs of sentences:

We live in France. (=”France” is our permanent home)
We’re living in France. (=”we” are living there for a limited period of time)

He cooks his own meals. (=”he” always does it)
He’s cooking his own meals. (=”he” does not usually do it)”

I take it that “He cooks his own meals.” means that he has always and will always cook his own meals. So the “always” stretches both back in time and forward in time.1

If so, then I disagree with it. I don’t think think there is any change in meaning by using the present progressive instead of the simple present. I have created an example that seems to show that there is no implication about an eternity by using the present progressive. Consider:

E1. He cooks his own meals now.

I have inserted the word “now” in the end of Longman’s example. When “now” is used like this it implies that there has been some change. If there has been some change, then the past has not always been the same. If the past has not always been the same, then the proposition (E1) does not imply that there has been change in the past. Thus, the proposition (E1) and, by extension, the propositions expressed by the sentences of the form that (E1) has does not imply there has never been change in the past.

Moreover, if the proposition (E1) did imply that there has been no change in the past, then the proposition (E1) would be a contradiction, but it isn’t. Thus, the proposition (E1) and, by extension, the propositions expressed by the sentences of the form that (E1) has does not imply there has never been change in the past.

Similar considerations seems to show that the proposition (E2) does not imply that there is something non-permanent about the situation. Consider this sentence:

E2. It has always been and will always be the case that he’s cooking his own meals.

If propositions expressed by sentences that has the present progressive form did imply that there is something non-permanent about the situation, then the proposition (E2) would be a contradiction, but it isn’t. Thus, propositions expressed by sentences that has the present progressive form does not imply that there is something non-permanent about the situation.

1That a proposition expressed by a sentence that contains “always” does not imply that the individuals mentioned in the proposition will continue to exist for an eternity, of course.

Wikipedia on Jacqui Dean.

“Jacqueline Isobel (Jacqui) Dean (born 13 May 1957 in Palmerston North) is a New Zealand politician and the current Member of Parliament for the Waitaki electorate.”

(skip)

“Party pills

Jacqui Dean campaigned for the banning of the sale of “party pills”, namely Benzylpiperazine (BZP), over which Associate Health Minister Jim Anderton (Progressive party) has accused her of indulging in political grandstanding, saying – “Perhaps Mrs Dean doesn’t subscribe to the idea that any Government must balance the need to act promptly with its responsibilities to act fairly and follow due process, particularly where its actions affect those who are currently acting within existing legal constraints.”[4] Dean’s press releases refer to BZP as either “cattle drench” or a “worming agent”[5][6]. BZP was developed for this use, but has never been commercially used as a wormer or drench.[7][1] Evidence that Dean has used to promote the BZP ban (such as the MRINZ report on BZP) has been criticized as consisting of flawed research which does not meet peer review requirements.[8]

Salvia divinorum

In November 2007 Jacqui Dean called for the government to take action against Salvia divinorum, saying – “Salvia divinorum is a hallucinogenic drug, which has been banned in Australia, and yet here in New Zealand it continues to be sold freely.” and “We’re dealing with a dangerous drug here, with the minister’s wait and see approach like playing Russian Roulette with young people’s lives.”[9] In March 2008 she was reportedly pleased on hearing about plans for action against salvia, but saying she was not hopeful it would be fast, given that it had taken the Government two and a-half years to move on BZP. Her concern about salvia was that people were self-medicating with it and combining it with other drugs including alcohol. “I don’t think we understand the long-term effects of Salvia divinorum.” she said.[10]

Opponents of prohibitive Salvia restrictions argue that such reactions are largely due to an inherent prejudice and a particular cultural bias rather than any actual balance of evidence, pointing out inconsistencies in attitudes toward other more toxic and addictive drugs such as alcohol and nicotine.[i][11] While not objecting to some form of regulatory legal control, in particular with regard to the sale to minors or sale of enhanced high-strength extracts, most Salvia proponents otherwise argue against stricter legislation.[ii][12]

Alcohol and tobacco

When questioned by Maori Party MP Tariana Turia, on why she was unwilling to take the same prohibitory line on smoking cigarettes and drinking alcohol as she took on BZP. Ms Dean said – “Alcohol and tobacco have been with our society for many, many years.”[13] It is estimated that alcohol-related conditions account for 3.1% of all male deaths and 1.41% of all female deaths in New Zealand.[14]

Dean’s Otago electorate is also home to approximately 5% of New Zealand’s wine production, described by the New Zealand Wine Growers Association as a new but aggressively expanding wine area, which is now New Zealand’s seventh largest wine region.[15]

Water

In August 2007, as a result of emails from ACT on Campus members based loosely around the well known Dihydrogen monoxide hoax, she sent a letter to Associate Health Minister Jim Anderton, asking if there were any plans to ban “Dihydrogen Monoxide”, apparently not realizing that this is water.[16][17]

In September 2007, the Social Tonics Association of New Zealand (STANZ) called for Jacqui Dean to step down from speaking on drug issues after she demonstrated – “a lack of credibility in calling for the ban of dihydrogen monoxide (water.)” STANZ Chairman Matt Bowden said – “The DHMO hoax played on the member this week is not a joke, it highlights a serious issue at the heart of drug policy making. Ms Dean demonstrated a ‘ban anything moderately harmful’ reflex. This approach is just downright dangerous.” – “Jacqui Dean has clearly demonstrated a lack of credibility in her requests to the Minister to consider banning water; She has also seriously embarrassed her National Party colleagues who can no longer have confidence in her petitions to ban BZP or anything else.”[18]

When interviewed on the radio by Marcus Lush on 14 September 2007, she referred to the members of ACT on Campus as “left wingers”. She also suggested that there were no lessons to be learned from her attempts to call for a ban on water.[19]”

This has to be one of the dumbest fucking politicians that I’ve ever seen.

Wikipedia om bogen.

“This seems plainly absurd; but whoever wishes to become a philosopher must learn not to be frightened by absurdities.” p. 10.

Download: Bertrand Russell – The Problems of Philosophy

I have encountered the following problem a couple of times. This problem is this: When formalizing something in predicate logic, the predicate uses propositions as variables.1 We may refer to this as the predicate acting upon the variable. The predicate is a function similar to functions in mathematics like “F(x) = x4”. Predicates were also written like this (with parentheses) to begin with. Variables are written in the english lower-case letters beginning from x and i.e. {x, y, z, etc.}. For variables that have to do with time ‘t’ is often used. Propositions are written in upper-case english letters beginning with P i.e. {P, Q, R, S, etc.}. What ought one to write, then, when a predicate acts upon propositions? Ought one to switch to lower-case but keep the same letters i.e. {p, q, r, s, t}? We ought to use the way which is the least confusing and which is powerful enough to express whatever meaning clearly we might want to express. Here are some possible ways to formalize it:

  1. Upper case beginning from P.
  2. Lower case beginning from p.
  3. Upper case beginning from X.
  4. Lower case beginning from x.
  5. Upper case beginning from P in parentheses.
  6. Upper case beginning from P in square brackets.
  7. Upper case beginning from P in curly brackets.
  8. Upper case beginning from P in sub-script.
  9. Upper case beginning from P in sup-script.

Some possible causes of confusion

Upper case beginning from P

Example: s knows that P is formalized as KsP.

Normally only the predicate is written in upper case e.g. “Fx” and all variables are written in lower case e.g. Rxyz.

Lower case beginning from p

Example: s knows that P is formalized as Ksp.

Normally propositions are written in the upper case. The confusion may be enhanced when the proposition used in some predicate is also used elsewhere in the analysis without being acted upon by a predicate e.g. Kxp⇒P where “Kxp” means x knows p.

Upper case beginning from X

Example: s knows that P is formalized as KsX.

Normally propositions are not referred to as “X”. “X” is usually reserved for variables in predicates. So this may cause confusion.

Lower case beginning from x

Example: s knows that P is formalized as Ksx.

For the same reason as above and that propositions are normally written in the upper case.

Upper case beginning from P in parentheses.

Example: s knows that P is formalized as Ks(P).

This is sometimes used. Parentheses are normally used for other purposes, so that may cause confusion. The good thing about using some sort of surrounding formalization is that it makes it much less confusing to have the propositional variable be complex e.g. an agent might know that someone knows a proposition logically implies that that proposition is true. Formalized as Ka(Kx(P)⇒P).

Upper case beginning from P in square brackets

Example: s knows that P is formalized as Ks[P].

I have used this from time to time. I have not seen anyone else use it. It suffers from the same problem as above although to a lesser degree since square brackets are not as commonly used as parentheses. When square brackets are used, then they are usually used as alternate parenthesis. When some wff becomes so complicated that multiple sets of parentheses are needed, then one alternates between regular parentheses and square brackets e.g. (P→[Q→(R↔T)]). It makes it easier to keep track of the parenthesis sets. I sometimes use square brackets to mean that some wff is a formalization of something that was written in normal language. One might combine square brackets with parentheses so one can alternate surrounding characters even within some predicate e.g. an agent might know that someone knows a proposition logically implies that that proposition is true. Formalized as Kx[Kx(P)⇒P].

Upper case beginning from P in curly brackets

Example: s knows that P is formalized as Ks{P}.

I have never seen this used and I don’t know of any other uses of curly brackets that may be confused with this.

Upper case beginning from P in sub-script

Example: s knows that P is formalized as KsP.

Sub-script is sometimes used to number propositions e.g. (P1, P2, P3 … Pn.) to mean all P propositions. I suppose one could formalize this in second order predicate logic as (∀P)(P). It is usually employed for infinite sets or ‘changing’ sets e.g. a set of premises of an argument.2 It is also sometimes used to symbolize an alternate meaning of a word e.g. atheist2 but words are not used in wff’s, so that should not be a source of confusion. Another thing is that it may be hard to read since the letters are so small. It’s hard to even see that the “P” is in upper case.

Upper case beginning from P in sup-script

Example: s knows that P is formalized as KsP.

I don’t know of any usage of sup-scripts in formalization at all. So I don’t know of any potential sources of confusion. The problem with readability also applies here.

Summary and conclusion

Given the 9 suggestions above I think one should go with the method of surrounding the acted upon proposition with alternatingly parentheses and square brackets. The reason for choosing this method is that it introduces no new characters, so it is characteristic parsimonious which is important. The less different characters used the better all other things being equal. This is a general fact of language. The curly bracket solution introduces two new characters, “{“ and “}”, but doesn’t have any large advantages as far as I can see over the chosen method. The other methods are either confusing or too hard to read. To make it very simple: The chosen method is both parsimonious and powerful.

1Predicates with more than one variable are sometimes called relations. I shall just refer to them all as predicates in this essay.

2That set is ‘changing’ in the sense that the number of premises vary from argument to argument.

It is clear that when we use the phrase “It is possible that…” it is not in all cases used to express mere alethic possibility, that is, “It is logically possible that p.” [◊P] Other times it is used to express what is called epistemic possibility, that is, “For all we (or I) know p might be true.”. It preliminarily seems like a good idea to explicate this as “It is compatible with everything we know that p is true and that p is false.”.1 But this is an improper explication as pointed out in Possible Worlds.2

Consider the example of Goldbach’s Conjecture (GC), that is, that every even number greater than 2 is the sum of two prime numbers.3 A mathematician might say that it is possible that (GC) is true. If we explicate that as suggested above, then we get that (GC) and not-(GC) is consistent with everything we know. We may formalize this explication as:

(∀P)(EP↔◊[P∧(∀n)Q1∧Q2∧Q3∧...∧Qn∧([∀Q][KQ])]) where “EP” means “P is epistemically possible”, “KQ” means “Q is known”.4

However, since (GC) is a mathematical proposition, then it is either necessarily true, or necessarily false. If it is necessarily true, then it’s negation is not consistent with everything we know. All necessary falsehoods are inconsistent with any proposition.5 If (GC) is false, then (GC) is necessarily false, and, thus it is not consistent with everything we know. If (GC) is true, then it is necessarily true, but then the claim that it is false is necessarily false and thus not consistent with everything we know. I note that this objection applies when one deals with non-contingent propositions.

The authors of Possible Worlds suggest instead that epistemic possibility should be explicated without alethic terms at all. They suggest the plain explication of: We (or I) do not know that (GC), and we do not know that not-(GC).

1Simplifying here. It is possible to formulate it without assuming bivalence.

2N. Swartz, R. Bradley, 1979, pp. 229-230.

3Some examples: 4 is the sum of 1 and 3. 6 is the sum of 3 and 3. 8 is the sum of 5 and 3. Etc. en.wikipedia.org/wiki/Goldbach%27s_Conjecture

4This is a bit complicated because it uses propositions as variables and propositions are written in the upper case in formalizations. It is to be read as: For all propositions, that P is epistemically possible is logically equivalent with that it is logically possible that (P and Q1 and Q2 and Q3 etc, and that for all Q’s, Q is known.

5To say that two propositions are consistent is to say that they are both true in some possible world, but a necessary falsehood is not true in any possible world, thus, it is not true together with any other proposition in any possible world. Hence, it is not consistent with any proposition. More about this in chapter 1 of Possible Worlds.

I could write a long detailed review but it is entirely unneeded. This book is without doubt the most enlightening book that I have ever read about logic, and it doesn’t even cover predicate logic! So that says a lot. It is recommended for anyone who wonders if talk of possible worlds is really worth it, who wants a systematic introduction to propositional logic and modal propositional logic, and who is not afraid of symbols.

Rapidshare link (updated as of 21. September 2009)

Follow the link and download these books

ScreenShot_01

I will update this post if I can figure out a way to avoid having to write off all the file names or copy them individually. If someone has more books in PDF format, just email me them and I’ll update this post at some time with the newly included books.