Tuesday, August 30, 2011

More on predicate logic and direct reference

David Brightly asks here whether modern predicate logic (MPL) in fact rests on ‘weak’ (relativistic) reference, not strong (direct) reference. He appeals to the way that singular sentences in MPL seem to have a meaning even when the singular term is empty.

There are a number of connected reasons why this won’t work. Here are the main characteristics of the relativistic logic (RL) I have defended here.

  • In RL there are two forms of negation for singular sentences. There is a narrow form ‘nonF(a)’, which is true when a exists and it is not the case that Fa, i.e. a is a nonF; and a wide form, which is true when either nonF(a), or when a does not exist. Excluded middle applies only to the wide form, naturally. By contrast, MPL has no such feature.
  • In RL, a sentence of the form ‘An S is P’ has a narrow existential sense: it is convertible with ‘An S-P exists’. There is no distinction between a wide sense conveyed by ‘some’, and the narrow sense conveyed by ‘exist’. RL is not Meinongian. By contrast, it is open to MPL to invent an existence predicate ‘E()’, which may satisfied by some members of the domain, but not by others.
  • As a direct consequence, in RL, the wide negation ‘it is not the case that Fa’ never implies the existential form ‘some x is non-F’. By contrast, in MPL a singular sentence is existential, at least in the wide sense: ‘~Fa’ implies ‘Ex ~Fx’.
  • A further difference, though probably not relevant here, is that some relational statements in RL are not existential. ‘aRb’ does not always imply ‘Ex aRx’, namely in the case where ‘R’ is not logically transitive. This is how RL avoids the problem of intentionality without invoking Meinongian non-existent objects.
The net result is that if a singular sentence in MPL cannot be empty in the sense required by RL. Let ‘v’ be a term that purports to denote Vulcan (a non-existent planet). Then either Vulcan lies within the orbit of Mercury - OM(v), or not - ~OM(v). The problem is the negation. If ~OM(v), standard predicate logic implies Ex ~Fx. But that leads to Meinongianism, i.e. positing objects that do not exist, and this, as I have argued, is a form of direct reference. Direct reference is the thesis that if ‘Fa’ is meaningful, then ‘a’ refers to something - possibly a non-existent something.

Another argument: how can we even say in MPL that Vulcan does not exist? The sentence ‘~Ex x=v’ will not do, for it asserts of something in the domain that it is not in the domain; ‘~E(v)’ will do, where ‘E’ means ‘exists’, but this comes at the price of Meinong’s junkyard. For ‘~E(v)’ implies ‘Ex ~E(x)’, which is precisely Meinongianism. Or it can be denied that ‘OM(v)’ means anything at all, which is strong Direct Reference of the familiar variety. There is no escape. Either we adopt a radically different semantics and inference schemata, on the lines of RL above, or we are left with Direct Reference.

Labels: ,

Monday, August 29, 2011

Google takes up Latin

I was excited when I saw that the Google translator now includes Latin, and immediately gave it an untranslated section of the Logic Museum to look at.  Results below.  Not promising, sadly, so I am stuck with the tedious manual method for the moment.

Logic MuseumGoogle
Fit autem fallacia consequentis vel arguendo ex propositionibus quarum una sequitur ad aliam et non e converso; vel arguendo ex una condicionali ad aliam condicionalem in qua oppositum antecedentis primae condicionalis ponitur antecedens et oppositum consequentis primae condicionalis ponitur loco consequentis.Or by arguing the fallacy of the consequent becomes one of which follows from propositions to another and not vice versa, or by arguing on the one in which the opposite of the antecedent of the first conditional to another conditional is placed opposite of the consequent of the first antecedent and the consequent CONDITIONAL place is placed.

Labels: ,

Wednesday, August 24, 2011

Is God fair to scientists?

Is God fair? We have to assume that God will punish, harshly, those who do not believe his word. In which case, why did God design the universe in such a way that those of an intelligent and inquisitive disposition would conclude that the universe was not designed by God in the way he said he designed it?

Take the case of epicycles, of which ‘Belette’ comments.

“The massive difference, now, over say Copernican days is that the observations and calculating ability we have are so much better than before. It is no longer possible tlo believe in epicycles, because observations demonstrate clearly that they don't work, except in the trivial sense that any path can be fit by enough epicycles to an arbitrary degree of precision.”
I agree. One of Galileo’s motivations for his heliocentric theory was his observation of the phases of Venus. These cannot be observed properly without a telescope. Venus is lit by the sun from angles that are not consistent with the Ptolemaic system. There is a good explanation of this in the Wikipedia article on the Ptolemaic system. Why did God design the geocentric system in such a complex way that those of an inquisitive disposition, after inventing telescopes, would observe such apparent inconsistencies? In 1838, Friedrich Wilhelm Bessel successfully measured the parallax of the star 61 Cygni, which is also difficult to explain in the geocentric system. Why did God go to the trouble of designing the geocentric universe in such a complicated way that an detailed observations such those of Bessel appeared to contradict the way God said (in the Bible) that he had in fact designed things?

Note, I am not arguing that the geocentric model is wrong. What I mean is that, if God says it is correct, and if, as seems to be the case, detailed observation and deduction (such as by Galileo and Bessel) suggests that it is not correct, and if God punishes harshly those who disbelieve what he says, then it seems as though God is not being fair. He has designed the universe in such a way that the way it appears to be designed is not the way he has said it was designed, and thus designed it in a way that invites punishment of the inquisitive and intelligent.

What reply can we make to this? Well, we could avoid fundamentalism by distinguish between literal and analogical truths. When it says in Psalm 93 that “the world also is established, that it cannot be moved”, perhaps it does not mean that the earth is literally immoveable. Perhaps it is immoveable in a spiritual sense. Or that it cannot be moved from its orbit around the sun, all things being equal. Perhaps it is a statement about the constancy of physical constants, such as gravitational acceleration etc. Yes, that is a reasonable objection, but then we can turn the whole question around. Why did God make statements that are so easily open to a literalist misinterpretation? If so, then it seems God is discriminating against fundamentalists. Is that fair? Surely not.

Or it could be argued that the inquisitive and intelligent are simply wrong. A careful examination of the matter (see the websites linked to in earlier posts) shows that the geocentric theory is the correct one. I reply: whichever theory is correct, it remains that God’s design has misled an extraordinary large number of people. Either he is being unfair to astronomers and scientists, or he is being unfair to fundamentalists. We have to choose, and either way it seems that he is not being entirely fair.

Or it could be argued that the inquisitive and intelligent have violated a version of the charity principle. They have followed a line of observation and reasoning that leads them to conclude that God has not told the truth. But logic, as well as the principle, should tell them that God would not have designed the universe in such a way as to contradict what he says. If any theory or reasoning of yours leads to a contradiction with sacred scripture or the teaching of the church, there is something wrong with your reasoning. Indeed, as Ockham argues – as discussed in my last post), we should even make exceptions to the laws of logic when this happens*.

So, is God not being fair to someone?  And is that not inconsistent with an essence into which goodness, justice, fairness etc. are built into as though by definition?

* Yes, I realise that Ockham’s point is more subtle than that. But this is internet land, where we lay things on with trowels.

Labels: , ,

Ockham’s solution to the Trinity problem

In the fifth chapter of the third book of his monumental Summa Logicae, Ockham presents an interesting solution to the problem of the Trinity. He says that the following syllogism is not valid:

Every divine essence is the Father
The Son is a divine essence
Therefore, the Son is the Father
Why not? Surely it is valid by the rule of dici de omni that Ockham frequently applies to arguments. According to this rule, a universal proposition denotes that the predicate is truly affirmed or denied of anything of which the subject is predicated. Thus “every divine essence is the Father” denotes that of whatever ‘divine essence’ is predicated, so also ‘the Father’ is predicated. Since the minor premisses states that ‘divine essence’ is predicated of the son, it follows (by this rule) that ‘the Father’ is also predicated of the Son. So the conclusion “the Son is the Father” apparently follows. Why, then, does he now say that it does not follow?

Ockham answers that the proposition “every divine essence is the Father” does not denote that the predicate is truly affirmed or denied of anything of which the subject is predicated. And so the argument above is not governed by dici de omni, and so it does not have to be valid.

We can only know when such an argument holds, or when not, from sacred scripture, or from the determination of the church. If scripture, or the church, states that the premisses of such an argument are true and the conclusion false, we can be sure that the argument is not governed by ordinary logical laws. He notes also that such an argument is always valid in the physical or ‘created’ world. For “in the world of created things it is impossible that something one in number, simple and singular, really be several things, really distinct” (inter creaturas impossibile est quod una res numero, simplex et singularis, sit realiter plures res distinctae realiter).

Labels: ,

Tuesday, August 23, 2011

More fixed earth

There is more information about the geocentric theory here. This suggests a problem. The original motivation for the geocentric theory was its apparent simplicity, given that the heliocentric theory requires an extra assumption – a tilted earth – to explain the seasons. But it turns out the geocentric theory also needs an extra assumption to explain the seasons, and the apparent motion of the sun that corresponds to this.

A good way to imagine this is to look at a potters wheel as it turns . . . it turns once every 24 hours . . . at the center is the Earth . . . place your finger in the clay, you make a circle, this is the sun . . . now slowly move your finger out 3% of the radius of your first circle . . . now move your finger inward 3% . . . that is the motion of the sun albeit the sun would also move in a spiral helix up and down while it is moving out and in by 3% of the radius . . . there are two motions here, one is up then down the other is in then out again. So there is the spiral up and down motion to account for the seasons but the size of the spiral is constantly changing to account for changes in the distance over the course of a year.
Yes but why does the sun move that way? The assumption of a tilted earth plus gravity seems a simpler explanation and thus (pace Dr Connolley) should be preferred. Obviously the underlying rationale for neo-geocentricism (‘neo-geo’) is the Bible and God, but that raises another question: is God fair? It seems unfair to create a universe whose nature seems almost designed to be misunderstood, once observed carefully. Observations by Galileo and other astronomers, phenomena like Foucault’s pendulum, and many other observable phenomena, suggest a reality that is quite different from the literal truth of the Bible. Geological observations suggest the Earth is much older than 6,000 years. Why did God create the Earth in a way that seems almost designed to mislead? Why did God create dinosaur bones, without mentioned dinosaurs in the Bible? Was that fair of God?

It could be replied that, correctly understood (where correct understanding means theories like ‘neo-geo’), these phenomena do not contradict the literal truth of the Bible. But then, clearly, many millions or billions of people, including all eminent scientists since Galileo’s time, have not correctly understood the evidence. Was it fair of God to create a world whose true nature misled so many brilliant people?

Labels: ,

Thinking about Frodo

Dr Vallicella has a post about intentionality here which reminds me I haven’t addressed the question of when two people are having thoughts about the same individual.

I characterised intentionality in this series of posts and particularly this. I have argued that it is merely a semantic phenomenon: certain verbs are ‘logically intransitive’ – they have a grammatical accusative, but no logical one. ‘Tom wants a cigarette’ has the grammatical subject ‘Tom’ and the grammatical accusative ‘a cigarette’. But, unlike ‘Tom is smoking a cigarette’, the sentence is consistent with ‘nothing is a cigarette’. Thus the verb ‘wants’ is logically intransitive. I also argued that we can parse a sentence like ‘Tom has a thought about a hobbit’ in two ways. First, as

Tom / has a thought about / a hobbit

where the verb phrase ‘has a thought about’ is logically intransitive (it appears to relate two objects: Tom and a hobbit, but clearly it can’t, as there are no hobbits). The other way of parsing it is

Tom / has / a thought about / a hobbit

where the verb ‘has’ relates Tom and a thought. This is logically transitive, given that there are such things as thoughts, and given that Tom can be related to his thoughts by thinking them. That suggests we can analyse any sentence containing a logically intransitive verb phrase into a sentence containing a logically transitive one. If possible, that would explain intentionality completely. But what about

(*) Tom has a thought about Frodo ?

It seems easy to explain a thought ‘about hobbits’. It is a thought we would express using the word ‘hobbit’, or which would involve the attributes we commonly attribute to hobbits (being short, having furry feet, being prone to finding magic rings etc). But Frodo? How do we explain the possibility of two different people having a thought about the same individual fictional character, without being drawn into Meinong’s junkyard of non-existing objects?

To resolve this, I shall invoke two principles. The first is the ancient view that thought is a form of silent speech which, if expressed, would signify to the thought to another. Since signification is in some sense public – many people can grasp the meaning of speech, which is indeed the whole point of language - this resolves the apparent problem of thoughts being unobservable or inaccessible. The second is that the term ‘thought of X’ is true of any thought which, if expressed, would contain a term synonymous with ‘X’. I.e. ‘Tom has a thought about Frodo’ is true iff Tom has a thought which, if expressed, would contain a term synonymous with the name ‘Frodo’ as it occurs in ‘Tom has a thought about Frodo’.

Thus the apparently intractable problem of explaining when two thoughts – scientifically unobservable events – can be reduced to the simpler problem of explaining when two proper names have the same meaning. More tomorrow.


How the left conquered Wikipedia

Here. Correct in all the main points (except for its faint suggestion that we might even begin to take Conservapedia seriously).

When I worked on Wikipedia, I was continually irritated by the politically correct view that, when it comes to philosophy, all cultures and all traditions had to be treated absolutely equally. So that, for instance, ‘Eastern philosophy’ had to receive equal coverage. There were two problems with this. First, the average Wikipedia editor knows even less about Eastern philosophy than about ‘Western’ philosophy (which is not very much). Second, there is no way that these subjects can be given the same sort of treatment. Eastern philosophy is so absolutely different to the Western variety that they are effectively different subjects, which happen to have the same name.

Labels: ,

Monday, August 22, 2011

Wittgenstein on relativity

William has given us an interesting link here.

Meeting a friend in the corridor, Ludwig Wittgenstein (1889-1951) said: "Tell me, why do people always say that it was natural for men to assume that the sun went around the earth rather than the earth was rotating?"

His friend said: "Well, obviously, because it just looks as if the sun is going around the earth."

To which the philosopher replied: "Well, what would it look like if it had looked as if the earth were rotating?"
How clever of Wittgenstein. He is not asking what it would have looked like if the earth were rotating, but what it would have looked like if it had looked as if the earth were rotating. Implying that to look as though the observer is moving with respect to their environment is the same as looking as though the environment is moving with respect to the observer.

But is that true? As a postgraduate I house-shared with a research assistant who was working on the neurophysiology of perception. Some of his work showed that the world actually looks different dependent on whether the observer is moving with respect to a stationary environment, or the other way round. There are a number of kinaesthetic sensors in the body which respond to bodily motion or rotation, and these interact with the visual sense in various ways. You can fool these sensors in all sorts of ways – for example you can create the illusion of acceleration by tilting their seat backwards, which is how flight simulators like this work. The answer to Wittgenstein’s question could well be “well, it would look exactly like that” – namely, looking as if the earth were moving, rather than the sun.

Labels: , ,

Translating ordinary language into predicate logic.

Vallicella has now apparently accepted my arguments against grammatically proper names being translatable into the ‘logical constants’ of modern predicate logic (MPL). But this leaves him with the problem that there are also strong arguments against translating them into the predicates of MPL.  I summarised these arguments earlier in June, as follows.

Argument 1 was that a proper name does not signify something that is repeatable, therefore does not signify a property. Therefore it signifies an object. A reply is here.

Argument 2 was that a name cannot be significant or intelligible to another unless the idea of what the name applies to is in the other person’s mind. But we can only have the idea of a particular thing by being acquainted with that thing, which is only possible if that thing actually exists. A reply is here.

Argument 3 was that definition proceeds by genus and specific difference. Therefore a proper name cannot be defined, for they name individuals, and individuals are not species. They have no specific difference, and can only be distinguished by the proper name itself.  A reply is here.

Argument 4 was that truth-conditional semantics rests on the assumption that the conditions for the truth of a sentence give the sentence’s meaning or significance. But there is no truth evaluable content when reference failure occurs. If there are no truth conditions, then there is no meaning or significance. I have not replied to this argument yet.

If Vallicella accepts these arguments as well as those he summarises in his post, he is apparently left with the problem that ordinary language cannot be translated into MPL at all. Is that a problem?

Labels: ,

Sunday, August 21, 2011

Does the earth actually rotate?

Perhaps not.  Listen to these curious and entertaining broadcasts, particularly these two sessions, about the 'nine assumptions' of Copernican science. The argument is broadly this: if you start with the first assumption that the earth rotates (as opposed to what your senses tell you, namely that the earth is fixed, and that the sun and moon revolve around it), then you need eight further assumptions in order to make the first one consistent with what we observe.  This, as the broadcasters say, is not science.  And they have a point, no?  We tend to reject any theory whose basic assumption we have to save by a series of further assumptions. Isn't the simplest theory of the solar system the Aristotelian and biblical one, which is evidently supported by our senses?

Labels: ,

Saturday, August 20, 2011


About half the hits to Beyond Necessity come from Google. The keywords are mostly self-explanatory, e.g. "existential import", "Augustine", and so on.  But one of the most frequent is "ant".  I have no explanation for this, unless it has something to do with my post about ants here.


Augustine on lust

Someone said to me recently that Augustine had some very strange views on sex.  So I had a look at what he says.  And yes, some of it does seems strange to us moderns.  For example:
Although, therefore, lust may have many objects, yet when no object is specified, the word lust usually suggests to the mind the lustful excitement of the organs of generation. And this lust not only takes possession of the whole body and outward members, but also makes itself felt within, and moves the whole man with a passion in which mental emotion is mingled with bodily appetite, so that the pleasure which results is the greatest of all bodily pleasures. So possessing indeed is this pleasure, that at the moment of time in which it is consummated, all mental activity is suspended. What friend of wisdom and holy joys, who, being married, but knowing, as the apostle says, "how to possess his vessel in santification and honor, not in the disease of desire, as the Gentiles who know not God," would not prefer, if this were possi ble, to beget children without this lust, so that in this function of begetting offspring the members created for this purpose should not be stimulated by the heat of lust, but should be actuated by his volition, in the same way as his other members serve him for their respective ends? But even those who delight in this pleasure are not moved to it at their own will, whether they confine themselves to lawful or transgress to unlawful pleasures; but sometimes this lust importunes them in spite of themselves, and sometimes fails them when they desire to feel it, so that though lust rages in the mind, it stirs not in the body. Thus, strangely enough, this emotion not only fails to obey the legitimate desire to beget offspring, but also refuses to serve lascivious lust; and though it often opposes its whole combined energy to the soul that resists it, sometimes also it is divided against itself, and while it moves the soul, leaves the body unmoved. (City of God, Book 14 c. 16)
On the other hand, the logic is clear.  Lust is beyond our volition, out of our control.  So there is something wrong with it.  (But then so is hunger, of course).

Labels: ,

Friday, August 19, 2011

How often does the moon rotate around the earth?

Connolley makes a curious objection here to my claim that the moon rotates around the earth once a day. What is wrong with that? Doesn’t it? From the beach on holiday this effect was clearly visible. I concede that it does not do this exactly once a day. But simplification is a virtue, as Connolley himself shows when he refers to ‘two tides a day’ (actually there are two tides every 24h and 50 minutes).  So what, broadly speaking, is wrong with my claim?

Labels: , ,

Referential and attributive use of descriptions

Vallicella mentions Keith Donnellan’s distinction between the ‘attributive’ and ‘referential’ use of definite descriptions. This prompted me to look at Donnellan’s paper again after some years*. 

According to Donnellan, a speaker who uses the description 'the F' attributively means to state something about whoever or whatever is F.  A speaker who uses the description 'the F' referentially uses it to pick out who or what he is talking about. By implication, then, the attributive use does not pick out who or what we are talking.  Probably the most well-known claim of the paper is that we can use a description referentially even when nothing fits the description.  Donnellan gives the example of 'the man with the Martini', which could be used to successfully pick out someone who (in fact) is drinking water, not Martini, and to say something (according to Donnellan) which is true.  For example, we could truly say 'the man with the Martini has glasses' in the event that the man drinking water has glasses.  I noted with interest that Donnellan attributes this idea to Leonard Linsky - we can say of a spinster that 'her husband is kind to her', referring to her lodger.

I don't understand the distinction between referential and attributive at all.  In the sense that 'referential' means that the semantics of the referring term is dependent on the existence of an object referred to, I hold that no terms are referential. Meaning is independent of external reality.  In the sense that 'referential' means picking out or signifying which individual we are talking about, all meaningful singular terms do this, as I argued here and particularly here.  Even 'the King of France' - Donnellan's favourite example of an attributive use - tells us which king we are talking about.  Not the King of England, nor of Ireland, nor of Spain, but the one who rules France.  Of course, there is no such person as the King of France, but even as I say this, I have just told you which king there is no such king as.

So Donnellan's distinction is artificial.  And certainly it cannot be invoked to explain why the following argument is valid in one sense, invalid in another.
The prime minister of the UK is a man
The prime minister of the UK used to be a woman
Some man used to be a woman
Vallicella means (I think) that the argument is valid if the description 'the prime minister of the UK' is used referentially, i.e. to refer to Cameron, invalid if not.  By contrast, I say that the argument is valid when the description has the same sense in both premisses (meaning 'the UK Prime Minister in 2011') but invalid when it has different senses.  If it means 'the UK Prime Minister in 2011' in the first premiss, but 'the UK Prime Minister from 1979-1990' in the second, then it is invalid, just as any argument is invalid when the middle term is used equivocally.

*”Reference and Definite Descriptions”, Philosophical Review, 75 (1966), pp. 281-304.

Labels: ,

Thursday, August 18, 2011

Rigid designation

Is the following argument valid?
The UK Prime Minister is a man
The UK Prime Minister used to be a woman
Some man used to be a woman
You will probably say not.  The first is true because David Cameron (who at the time of writing is Prime Minister) is a man.  The second is true because Margaret Thatcher (who used to be the Prime Minister until the 1990s) is a woman, and always has been.  But the implication is clearly invalid, since the premisses could be true and the conclusion false if no man had ever undergone a sex change.  

Kripke's take on this is famous: definite descriptions such as 'The UK Prime Minister' are not rigid. They have different referents in different possible worlds, and so the inference is not valid, because of the non-ridigity of reference.  By contrast, proper names are rigid designators.  Understood in the same sense, they always refer to the same individual.  Thus, if you substitute 'David Cameron' into the inference above, you get something that is valid (although not sound, of course, for the second premiss is false - as far as we know, David Cameron never used to be a woman).

But how about this?
The leader of Cambridge City Council is a woman
The leader of Cambridge City Council used to be a man
Some woman used to be a man
As it happens, the (current) leader of Cambridge City Council is Jenny Bailey, who underwent a sex change operation to become a woman. So Jenny Bailey is now a woman, but used to be a man, and so some woman used to be a man.  The premisses can't be true (assuming "The leader of Cambridge City Council" has the same referent in both premisses) and the conclusion false, so the inference is valid.

How do we explain this?  Are definite descriptions rigid or not?

Ockham has a quite different take on this, distinguishing between 'sense of composition' and 'sense of division'.  He explains it in Book III of his great Summa Logicae here.  In the sense of composition it is signified that a mode (such as 'is possible' or 'was the case that') is predicated of the whole proposition. In the sense of division it is denoted that the predicate is predicated of the subject by a verb determined by such a mode. Thus we must distinguish between 'it was the case that the (then) PM was a woman' and 'it is the case that the (current) PM was a woman'.  Clearly, read in the sense of composition, the subject term may have a different sense in the different premisses.  In 'The UK Prime Minister used to be a woman', the subject term means Margaret Thatcher. In 'The UK Prime Minister is a man' it means David Cameron.  Hence (read in the sense of composition) the inference is not valid because of equivocation, just as any syllogism is invalid because of equivocation.  But read in the sense of division it is valid.  If by 'The UK Prime Minister used to be a woman' I mean that David Cameron used to be a woman, the first inference is valid, just as the second is (where 'The leader of Cambridge City Council' refers in both premisses to Jenny Bailey).

Hence we do not have to accept any fundamental distinction between definite descriptions and proper names.  Rather, it is easier to read definite descriptions in an ambiguous way.  When we say 'The UK Prime Minister used to be a woman', we can read it either as saying that the present PM used to be a woman, or that some former PM used to be a woman.  And this ambiguity is not fundamentally different from the one we find here.

William Pitt was the youngest Prime Minister
William Pitt was not the youngest Prime Minister
the youngest Prime Minister was not the youngest Prime Minister

For there was more than one person called 'William Pitt', just as there was more than one person called 'Prime Minister of the UK'.

Labels: ,

Wednesday, August 17, 2011

Sentence negation, predicate negation and direct reference

There is a good post here by Vallicella, about a comment I made to an earlier post of his. My claim, in essence, is that if we cannot distinguish between sentence negation and predicate negation of a singular sentence, then we have to accept ‘direct reference’. Let me explain. Predicate negation – e.g. ‘a is non-F’ is such that its truth requires that the object denoted by ‘a’ exist, and that this object is not F. Sentence negation, by contrast, is when we simply deny whatever is asserted by ‘a is F’.

In modern predicate logic, we do not distinguish between these two forms of negation, at least for singular sentences. The negation ‘~Fa’ is true whenever some thing is referred to by the singular term ‘a’, and when that thing is not F. Thus it is predicate negation. But it is also the contradictory of ‘Fa’. If it is not true that ~Fa, then it is true that Fa, and conversely. Thus it is sentence negation.

By contrast, in traditional logic (or at least in standard forms of it) we do distinguish between these forms of negation. ‘Socrates is a non-runner’ is true when there is some such person as Socrates, and that person is not running. Thus ‘Socrates runs’ and ‘Socrates is a non-runner’ are contraries. Both can be false when Socrates does not exist.  This is predicate negation (or ‘indefinite negation’ as the scholastics called it). This was considered distinct from propositional or ‘extinctive’ or ‘destructive’ negation, where everything asserted by ‘Socrates runs’ is denied, and which can be true even when there is no such person as Socrates.  Thus ‘Socrates runs’ and the destructive negation ‘It is not the case that Socrates runs’ are contradictories. Both cannot be false, for one denies everything the other asserts.

Now I claim that in systems where there is no distinction between predicate and sentence negation, we have ‘direct reference’. This is easily shown. Direct reference in a singular sentence is when the sentence is meaningless when the singular subject fails to refer. Assume that ‘a is F’ is not meaningless. If it is true, then there is a referent for ‘a’. If it is not true, the sentential negation ‘It is not the case that a is F’ is true. If sentential negation is equivalent to predicate negation, it follows that ‘a is non-F’ is true, and so a exists, and so, there is a referent for ‘a’. But (by excluded middle) either ‘a is F’ is true, or its contradictory (the sentential negation) is true. In either case, ‘a’ has a referent. Thus if ‘a is F’ is not meaningless, ‘a’ has a referent. Conversely if ‘a’ does not have a referent, ‘a is F’ is meaningless. But that is Direct Reference, as I have defined it.

In systems where we can make a distinction between the two forms of negation, we do not have to accept Direct Reference. If ‘a is F’ is meaningful but false, this could either be because the predicate negation is true, or because there is no referent for ‘a’.

Both Peter Lupu and David Brightly challenged this idea in their comments to the post.

Peter argued that we do not have to accept Direct Reference if we accept the distinction between meaning and reference (or ‘sense’ and reference or whatever you call it). I reply: in accepting this distinction we have (tacitly) accepted the distinction between the two forms of negation. If we are happy that ‘a is F’ may not be true in the case that ‘a’ has a sense but not a reference, and assuming that (in this system) we can say that this is so, i.e. state that a does not exist, then we can state the disjunction ‘a does not exist or a exists but is non-F’. But that disjunction is equivalent to the sentential negation of ‘a is F’, and is therefore distinct from the predicate negation ‘a does exist and is non-F’.

David argued that we can suppose a singular term has a referent, and prove a contradiction in order to show that it does not have a referent. For example, let ‘a’ denote the largest ordinal number. That supposition leads to a contradiction, therefore ‘a’ cannot denote that. Yet (according to David) it is meaningful to make assertions using the singular term ‘a’. I reply: what is meant by ‘denotes’ here? If it means ‘signifies’ or ‘means’, then the supposition that it has a meaning leads to a contradiction, and so it does not have a meaning. This is Direct Reference. On the other hand, if it does not mean ‘signifies’, but rather that a exists, then my reply is the same as to Peter. To accept the possibility that ‘a’ is meaningful but fails to correspond to any existing number, is tacitly to accept the distinction between the two forms of negation.


Tuesday, August 16, 2011

On simple explanations

Clever Connolley caught me out with a comment here on simple explanations. Mencken says that the reason for the 'inferior man' hating knowledge is because it is complex. All superstition is a short cut to make the unintelligible simple. Connolley, apparently agreeing with Mencken, chides me for wanting a simple explanation for global warming.

But is Mencken right? The appeal of true scientific explanation generally does lie in its simplicity. There are obvious exceptions - the proof of the four colour theorem for example. But consider the explanation of an eclipse. That is pretty simple. The moon goes round the earth, the earth goes round the sun. The sun lights up the earth. Occasionally the moon gets in the way and casts a shadow. How simpler could it get? The theory that the eclipse is caused by a dragon crossing the sun, by contrast, requires a theory of dragons, and no theory of dragons - at least not one that gives a comprehensive treatment of them, including their metabolism, genetic structure etc - could be simple at all.
Or consider Augustine's explanation of why evil exists:
That the whole human race has been condemned in its first origin, this life itself, if life it is to be called, bears witness by the host of cruel ills with which it is filled. Is not this proved by the profound and dreadful ignorance which produces all the errors that enfold the children of Adam, and from which no man can be delivered without toil, pain, and fear? Is it not proved by his love of so many vain and hurtful things, which produces gnawing cares, disquiet, griefs, fears, wild joys, quarrels, lawsuits, wars, treasons, angers, hatreds, deceit, flattery, fraud, theft, robbery, perfidy, pride, ambition, envy, murders, parricides, cruelty, ferocity, wickedness, luxury, insolence, impudence, shamelessness, fornications, adulteries, incests, and the numberless uncleannesses and unnatural acts of both sexes, which it is shameful so much as to mention; sacrileges, heresies, blasphemies, perjuries, oppression of the innocent, calumnies, plots, falsehoods, false witnessings, unrighteous judgments, violent deeds, plunderings, and whatever similar wickedness has found its way into the lives of men, though it cannot find its way into the conception of pure minds? (City of God, Book 22 chapter 22)
The explanation - that Adam and Eve offended God, and that these evils are a punishment - appears simple at first sight, just like the dragon explanation.  But it is not, for it requires a theory of God, and also a theory of Paradise, which is problematic.  Sociobiology could probably provide a simpler one (although I'm not sure it has, yet).

Superstition is not necessarily a simpler theory. So, what distinguishes superstition from science?

Labels: ,

Hello again

Apologies for the absence of posts recently.  The manuscript of the Scotus translation (plus extensive commentary) is to be submitted to the publishers soon and work on that takes precedence.  I shall be posting sporadically, however.