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[Londoner]

Fafner

The analytic truth "bachelors are unmarried" is true because we happen to use the term 'bachelor' to signify unmarried people, but this is an arbitrary convention about language (we may not had the term 'bachelor' for example), hence it's not an interesting fact about reality that we happened to have a special term for unmarried men.

It is true because both 'bachelor' and 'unmarried men' signify the same thing.

On the other hand, a synthetic proposition like "dolphins are mammals" isn't true by virtue of linguistic convention but by the way the world is, we can't know that just by analyzing the meaning of the words, we have to see how the world is like to know it.

Also, 'dolphin' and 'mammal' do not signify the same thing.

If anybody can make up their own word then how it isn't arbitrary?

Because we use words to communicate, which means we have to agree on what they mean. A word where we didn't have such an agreement wouldn't be a word; it would just be a sound.

Actual scientists don't really care about philosophy of science.

I think you are very wrong.

For example the conditional "if it is raining, then I will bring an umbrella" can nevertheless be true even if it isn't raining (and hence I didn't bring an umbrella). This is just how the truth table of logical implication works, you can check any logic textbook and see.

To do logic you have to look carefully at what you say, and only what you say. Are you saying that rain causes you and an umbrella to 'be', to spring into existence?

If that wasn't what you meant, then what also needs to be true is that you exist and your umbrella exists. If either don't exist, then it will be impossible for you to 'bring your umbrella', whether it is raining or not. That these things exist is implied in ordinary speach when we name them, but not in logic. In logic their existence must be marked as a distinct assumption.

Further, that you 'will' bring your umbrella is you telling us of your intentions; the proposition is not about the rain (or the umbrella) but about you. What makes it true or false is whether it is an accurate prediction of your behaviour. Again, in logic, that this was an assumption - something that might be true or false - would have to be acknowledged.

So:

'You' must be true (1) and 'Your umbrella' must be true (2) and 'You carry an umbrella when it rains' must be true (3). Only then, if 'it rains' is true (4), would your proposition be true - and your proof would have to be annotated (1,2,3,4) to show that it rested on all those assumptions.

[Fafner88]

To do logic you have to look carefully at what you say, and only what you say. Are you saying that rain causes you and an umbrella to 'be', to spring into existence?

If that wasn't what you meant, then what also needs to be true is that you exist and your umbrella exists. If either don't exist, then it will be impossible for you to 'bring your umbrella', whether it is raining or not. That these things exist is implied in ordinary speach when we name them, but not in logic. In logic their existence must be marked as a distinct assumption.

Further, that you 'will' bring your umbrella is you telling us of your intentions; the proposition is not about the rain (or the umbrella) but about you. What makes it true or false is whether it is an accurate prediction of your behaviour. Again, in logic, that this was an assumption - something that might be true or false - would have to be acknowledged.

So:

'You' must be true (1) and 'Your umbrella' must be true (2) and 'You carry an umbrella when it rains' must be true (3). Only then, if 'it rains' is true (4), would your proposition be true - and your proof would have to be annotated (1,2,3,4) to show that it rested on all those assumptions.

It seems to me that you've missed the point of the example. I was arguing that you can have a true conditional of the form "if P then Q" even if P, or both P and Q, are false. In the previous example -

"if it is raining, then I will bring an umbrella"

P= It is raining Q= I will bring an umbrella

We can imagine a story in which P and Q are false, namely It's not raining and and therefore I don't bring an umbrella, while it's nevertheless true that If it was raining then I would bring an umbrella (i.e. the conditional as a whole).

Now, to go back to the original point: It's is true that a 6 meter tall man must be taller then a 5 meter tall man, even though there are no such people, and this is no objection to the truth of the sentence because it's a conditional, and as I illustrated in the previous example, a conditional can be true while both its antecedent and consequent are false.

A second point is that the only way to show that a conditional "P->Q" is false is to show that P could obtain while Q is false. In the context of my example that means that you have to show that it's possible for such two people to exist while the 6 meter tall person would not be taller then the 5 meter tall person in order to show that the proposition is false, otherwise the proposition must be true (because if its impossible for a 6 meter tall person to be shorter then (or the same as) a 5 meter tall person then it's true that any 6 meter tall man must be taller then any 5 meter tall man, which is exactly what the proposition says).

[Belinda]

Fafner, we attribute mammal to dolphins. 'Mammal' is thus an attribute of dolphin , in the same relationship as 'good swimmer' or 'fish eating' is an attribute of dolphin.

' Mammal' may also be attributed to cat, or cow, even although those are not dolphins.Likewise good swimmer and fish-eating are attributes of other than dolphins. Attributes are transferrable.

But bachelorhood is not an attribute of unmarried men, because the- state- of- being- a- bachelor is synonymous with the-state-of-being-an-unmarried-man. Thus, there is nothing that is an unmarried man which is not also a bachelor, and vice versa. Attributes are not synonyms and synonyms are not attributes.

'Unmarried man' is an attribute of my friend Richard, and if this is indeed the case my friend Richard is a bachelor. If my friend Richard gets married he will inevitably relinquish the state of bachelorhood . Richard may well retain the attributes of his bachelor days such as drinker of beer at breakfasttime.

To know tautological terms such as bachelor and unmarried man is not a priori knowledge. Neither is knowing the uses of zero, or any other mathematical functions, a priori knowledge

[Londoner]

We can imagine a story in which P and Q are false, namely It's not raining and and therefore I don't bring an umbrella, while it's nevertheless true that If it was raining then I would bring an umbrella (i.e. the conditional as a whole).

OK. Then taken on its own, 'If it was raining then I would bring an umbrella' is a statement about your behaviour. It might be true or false. Certainly, it isn't conditional on the rain; it has nothing to do with the weather. 'Saying what my behaviour would be if it rains, doesn't depend on whether that condition is currently occurring.

You tack on a superfluous 'If it is raining' or 'If it isn't raining' to make it look as if it has a connection with a 'if P then Q' type statement in order to make an argument something to do with logic and being conditional. It isn't; it has no antecedent and consequent. Although it contains the words 'if' and 'then' it is just a simple assertion about you; 'If it was raining then I would bring an umbrella' has the form 'P'

[Fafner88]

To know tautological terms such as bachelor and unmarried man is not a priori knowledge. Neither is knowing the uses of zero, or any other mathematical functions, a priori knowledge

You are wrong, it is a priori.

OK. Then taken on its own, 'If it was raining then I would bring an umbrella' is a statement about your behaviour. It might be true or false. Certainly, it isn't conditional on the rain; it has nothing to do with the weather. 'Saying what my behaviour would be if it rains, doesn't depend on whether that condition is currently occurring.

You tack on a superfluous 'If it is raining' or 'If it isn't raining' to make it look as if it has a connection with a 'if P then Q' type statement in order to make an argument something to do with logic and being conditional. It isn't; it has no antecedent and consequent. Although it contains the words 'if' and 'then' it is just a simple assertion about you; 'If it was raining then I would bring an umbrella' has the form 'P'

Sorry, I don't understand what is the objection here or how does it bear on my main argument. It isn't clear to me if you accept or not the claim that "P->Q" can be true while P&Q are false? (and what makes "P->Q" true is a different question, let's settle the logical issue first).

[Londoner]

Fafner

Sorry, I don't understand what is the objection here or how does it bear on my main argument. It isn't clear to me if you accept or not the claim that "P->Q" can be true while P&Q are false? (and what makes "P->Q" true is a different question, let's settle the logical issue first).

It isn't a different question; both embody the same confusion. I'll try and explain one more time.

Of your example you say:

I was arguing that you can have a true conditional of the form "if P then Q" even if P, or both P and Q, are false. In the previous example -

"if it is raining, then I will bring an umbrella"

P= It is raining Q= I will bring an umbrella

What you call the 'true condional' is the connection between P and Q. In it's symbolic form, this is just the word 'then' or an arrow. A 'then' isn't a thing; it isn't true or false on its own.

(You can see this if you forget what your P and Q are supposed to stand for; consider instead the sum; 'A + B = C'. Is that true, or false? It is a silly question! Until it has some values it is neither. That we can observe it contains a 'plus' sign does not change that.)

But when you write your own example in the long form you confuse that by writing it as 'then I will'. This does introduce a thing; 'I'. Either that 'I' is superfluous, or it is part of a further condition, something like 'I have the will and ability to keep dry using an umbrella'.

You skip between the two: You claim it asserts something (because it contains 'then I will') but also that it cannot be shown false (because it is only a 'then'.)

Perhaps you are also confused by the fact that in expressing the 'then' connection we name those things that connection relates to; 'P' and 'Q'. But that does not mean the same line that asserts the connection also asserts that P and Q exist/are true. That is why, in a formal proof, we have to assume the P separately.

[Belinda]

Fafner,

I wrote, to which you objected:

To know tautological terms such as bachelor and unmarried man is not a priori knowledge. Neither is knowing the uses of zero, or any other mathematical functions, a priori knowledge

I admit that this that I wrote does look odd. What I mean is that learning the lexicon of any language is not learning the difference between tautologies and synthetic propositions. To know this difference we need to engage in language about language. Within the practise of metalanguage we are, for the purpose of this discussion, not talking about how terms are used in everyday language but are talking about what knowledge if any is embedded in the matrix of everyday language.

My claim is that a priori knowledge therefore is impossible except within the strict confines of some arbitrary lexicons such as those of mathematics and formal logic.

[Consul]

Remark:
To say that a priori truths such as "Bachelors are unmarried" are knowable experience-independently is not to say that the meaning of this sentence and the meanings of the words it contains are knowable experience-independently. If you haven't learned the English language, you certainly cannot understand the sentence in question. And if you cannot understand it, you cannot know its truth-value.
So, to say that a priori truths are knowable experience-independently is to say that once you know the sentence's meaning and thus understand it, no additional experiences are needed in order for you to be able to know its truth-value.

"I choose to follow Kant and the overall tradition by stipulating that a proposition will count as being justified a priori as long as no appeal to experience is needed for the proposition to be justified once it is understood, where it is allowed that experience may have been needed to achieve such an understanding."
(p. 10)

"I propose to count a proposition P as being justified a priori (for a particular person, at a particular time) if and only if that person has a reason for thinking P to be true that does not depend on any positive appeal to experience or other causally mediated, quasi-perceptual contact with contingent features of the world, but only on pure thought or reason, even if the person's ability to understand P in question derives, in whole or in part, from experience."
(p. 11)

(BonJour, Laurence. In Defense of Pure Reason: A Rationalist Account of A Priori Justification. Cambridge: Cambridge University Press, 1998.)

[Fafner88]

What you call the 'true condional' is the connection between P and Q. In it's symbolic form, this is just the word 'then' or an arrow. A 'then' isn't a thing; it isn't true or false on its own.

(You can see this if you forget what your P and Q are supposed to stand for; consider instead the sum; 'A + B = C'. Is that true, or false? It is a silly question! Until it has some values it is neither. That we can observe it contains a 'plus' sign does not change that.)

Of course "->" by itself can't be true or false, as I said already only propositions can be true or false, and "if" statements are propositions therefore they can have truth value, so I don't understand your point here.

But when you write your own example in the long form you confuse that by writing it as 'then I will'. This does introduce a thing; 'I'. Either that 'I' is superfluous, or it is part of a further condition, something like 'I have the will and ability to keep dry using an umbrella'.

Nothing is superfluous in P->Q, and I have a right to put as "p" and "q" anything I want. And no you can't paraphrase the sentence as meaning "I have the will and ability to keep dry using an umbrella" because it's a completely different proposition that doesn't say the same thing.

You skip between the two: You claim it asserts something (because it contains 'then I will') but also that it cannot be shown false (because it is only a 'then'.)

Again, I don't understand you. A conditional of the form "P->Q" has three components: P, Q and a logical relation between them, you can't get rid of or replace any of them without changing the meaning of the sentence as a whole. What part of this you don't agree with?

Perhaps you are also confused by the fact that in expressing the 'then' connection we name those things that connection relates to; 'P' and 'Q'. But that does not mean the same line that asserts the connection also asserts that P and Q exist/are true. That is why, in a formal proof, we have to assume the P separately.

I said already that the truth of "P->Q" doesn't imply the truth of P or Q, it's precisely my point. It's supposed to be a counterexample to the claims you made earlier- there are true propositions about non existing objects, therefore not all the facts in the world are facts about actual objects.

[Londoner]

Again, I don't understand you. A conditional of the form "P->Q" has three components: P, Q and a logical relation between them...

No; I know it looks like that but they are all one thing. The P and Q are there as part of the description of the relationship, not as seperate assertions.

Are you forgetting the conditional is, indeed, conditional? The conditional is an assumption; it starts with 'if'. 'If it is true that a P would result in a Q'.

You agree (quote below) that the inclusion of P and Q don't imply they are 'true', yet you still say they are included as 'components'. I do not see how you could use something as a component while accepting that it may not exist. You can only square this if P,Q and the sign are not seperate components, but all one thing.

I said already that the truth of "P->Q" doesn't imply the truth of P or Q, it's precisely my point. It's supposed to be a counterexample to the claims you made earlier- there are true propositions about non existing objects, therefore not all the facts in the world are facts about actual objects.

But the line on its own doesn't propose anything! Look at it: "P->Q". What fact does that tell us? Suppose I write "A->B". Does that contradict your fact? I have no idea! So in effect your fact can be expressed: "? -> ?"

It only conveys information if P and Q stand for something. To have meaning you have to have that separate line in your proof where you assume/assert them.

But let us move onto the much more interesting notion of facts about non existing objects, ('objects' being an important word) as this threatens to raise the ghost of Wittgenstein! Can you give an example of what you have in mind?

[Fafner88]

Again, I don't understand you. A conditional of the form "P->Q" has three components: P, Q and a logical relation between them...

No; I know it looks like that but they are all one thing. The P and Q are there as part of the description of the relationship, not as seperate assertions.

Did I say they are separate assertions?

Are you forgetting the conditional is, indeed, conditional? The conditional is an assumption; it starts with 'if'. 'If it is true that a P would result in a Q'.

So what's the point?

You agree (quote below) that the inclusion of P and Q don't imply they are 'true', yet you still say they are included as 'components'. I do not see how you could use something as a component while accepting that it may not exist. You can only square this if P,Q and the sign are not seperate components, but all one thing.

They are components in the sense that you can arrange them in different sentences (e.g. P&Q, P<->Q, ~Q->P etc.). It seems to me pretty obvious that sentences are made of components, you can have the same words but put them in different ways etc.

About non existing components, yeah that's how language works, and you must live with that. When I say "the first person to be born in the year 2017 will be ..." it's a perfectly intelligible sentence even though it doesn't refer to any actual object. So if you think that this sentence is meaningless, then there's a heavy burden of proof on you to show why.

But the line on its own doesn't propose anything! Look at it: "P->Q". What fact does that tell us? Suppose I write "A->B". Does that contradict your fact? I have no idea! So in effect your fact can be expressed: "? -> ?"

Well yes "P" and "Q" don't stand for anything because they are variables, but I thought it was understood that I meant to schematically represent actual propositions of this form that you can fill in with content, like the one I gave about me taking an umbrella.

It only conveys information if P and Q stand for something. To have meaning you have to have that separate line in your proof where you assume/assert them.

Which proof?

But let us move onto the much more interesting notion of facts about non existing objects, ('objects' being an important word) as this threatens to raise the ghost of Wittgenstein! Can you give an example of what you have in mind?

It's the example I already gave- "a 6 meter man is taller then a 5 meter man".

I should qualify this statement though, of course I don't mean by "a fact about a non existent object" to say that there are objects such that they are non-existent, or something like that (a-la Meinong), this is of course obvious nonsense. What I mean is that sentences about objects which don't exist have nevertheless a meaning and can be true or false. So to take the famous example of Russell, "the present kind of France is bald", which is supposed to be a false statement on his account, but according to Russell, it doesn't mean that there's actually an object which is the present king of France who isn't bald, but merely to say that there's no such object that is both bald and the present king of France (the negation is ambiguous in this context). But this is not to say that "the present king of France is bald" is not a sentence that isn't about an object, because it falsely asserts the existence of such and such object.

[Londoner]

Fafner

They are components in the sense that you can arrange them in different sentences (e.g. P&Q, P<->Q, ~Q->P etc.). It seems to me pretty obvious that sentences are made of components, you can have the same words but put them in different ways etc.

This may be the key to your misunderstanding. The lines of a proof are not like sentences. Logic is like maths; a 'formal science'.

With sentences, the meaning of the words is important. But 'all women have beards' is just as valid in logic as 'grass is green'. Both will be symbolised by a letter, so both can create 'valid' proofs ('valid' also having a special meaning). It is like 2 + 3 = 5; it is not saying the same as 'I have two apples then three more apples, now I have five'. The first is necessarily correct; the second may be false.

This means the symbol; '->' etc. is not really the equivalent of a word like 'then'. If we say 'If it rains then we get wet' then this doesn't imply that getting wet can only be the result of rain, nor does it imply that it can't be negated (by having an umbrella). But in the logical P -> Q, which has nothing to do with empirical facts, the relationship is much more strict. It is saying that wetness must follow rain; and does not follow from anything else. But even the 'follow' in my attempt to use words to explain it is misleading, since no cause-effect relationship is implied. The relationship is more like 'If you are a bachelor then you will be unmarried' - it is saying the conditions that would make P true are identical to the conditions that would make Q true.

So how can you substitute words for letters in P->Q? If those words denoted different things, then P -> Q isn't true. (If Q denoted something different to P, then the line should be P (+ something else) -> Q).

Like maths, superficially logic seems to match up with 'common sense' and the practical ways in which we deal with the world of objects, but this is misleading. Logic is not about 'things' and the words which denote them - and this becomes clear when you examine it closely.

About non existing components, yeah that's how language works, and you must live with that. When I say "the first person to be born in the year 2017 will be ..." it's a perfectly intelligible sentence even though it doesn't refer to any actual object. So if you think that this sentence is meaningless, then there's a heavy burden of proof on you to show why.

No; it isn't intelligible as it stands. We can only understand it if we know what sort of a proposition it is; the meaning of the words, what conditions would make it true or false.

For example, what is the meaning of your word 'person'? In normal speech, that word denotes something that exists now, so we need to find out what you might mean by it.

And is this sentence to be understood as an empirical claim? (...will be a girl) Or a tautology? (...will be human). Since you don't finish it, we cannot guess.

Although you disguise it with verbs like 'born', your sentence consists of a subject...and nothing else. It can be condensed to; "Person", (plus the hint that 'person' may not have the normal meaning). That isn't intelligible, it doesn't say anything.

Which proof?

1. P (A) (A meaning we assume)

2. P -> Q (A)

3. Q Modus ponens (1,2) (meaning it rests on the assumptions of lines 1 and 2)

Note; you can't leave out line 1, even though P also appears in line 2. On its own, P -> Q cannot be used to prove Q.

Me: But let us move onto the much more interesting notion of facts about non existing objects, ('objects' being an important word) as this threatens to raise the ghost of Wittgenstein! Can you give an example of what you have in mind?

It's the example I already gave- "a 6 meter man is taller then a 5 meter man".

'A 6 meter man' is an object.

'6 meters' is not an object, but nor is it a fact. A fact can be true or false; '6 meters' isn't either.

You want to borrow the object-ness of 'man' to turn that '6 meters' into a fact, but then deny the sentence concerns objects.

What I mean is that sentences about objects which don't exist have nevertheless a meaning and can be true or false. So to take the famous example of Russell, "the present kind of France is bald", which is supposed to be a false statement on his account, but according to Russell, it doesn't mean that there's actually an object which is the present king of France who isn't bald, but merely to say that there's no such object that is both bald and the present king of France (the negation is ambiguous in this context). But this is not to say that "the present king of France is bald" is not a sentence that isn't about an object, because it falsely asserts the existence of such and such object.

That is not what it is about. Russell is about logic. The problem here is the logic of the 'excluded middle'; it appears logically necessary that the statement 'the present king of France is bald' must either be true or false. But both seem to involve the necessity for there to be a King of France. But (crudely) Russell says the logical meaning of 'is' when talking of identity, prediction and existence are different. By changing the focus of our negation we can say 'It is not the case that there exists a King of France (who is bald)'.

Which, of course, seems obvious. But it seems obvious to us because it corresponds to an empirical fact, but again I make the point that the rules of logic are not about empirical facts. That this negation is true is irrelevant. Russell's object is to prove that such a negation is 'valid'.

(Another way of looking at it is what terms like 'the present king of France' describe. Because if we substitute a proper name then the logic works differently. The 'King of France..' stuff is discussed as an aspect of Russell's 'Theory of Definite Descriptions'.)

[Consul]

This means the symbol; '->' etc. is not really the equivalent of a word like 'then'.

Does the classical logical semantics of "–>" correspond to the natural-language semantics of "if…then"? Arguably, it doesn't, because it is irrelevant to the truth of the material conditional of classical logic whether or not there is a real relation or connection between the antecedent and the consequent.

Relevance Logic: http://plato.stanford.edu/entries/logic-relevance/

So how can you substitute words for letters in P->Q? If those words denoted different things, then P -> Q isn't true. (If Q denoted something different to P, then the line should be P (+ something else) -> Q).

If "p" and "q" stand for different propositions, then "p –> q" is certainly not an a priori knowable logical truth.

'A 6 meter man' is an object.

It's a linguistic object called an indefinite noun phrase. For the logic of indefinite descriptions, see: http://plato.stanford.edu/entries/descriptions/#IndDes

'6 meters' is not an object, but nor is it a fact. A fact can be true or false; '6 meters' isn't either.

A fact is either a true proposition or an actual state of affairs. In the former case, "false fact" is a contradiction in terms; and in the latter case, "false fact" is a category mistake, because states of affairs are not the kind of entities which can be bearers of truth-values.

[Rodion]

2+2=4

analytic (you only need to know what the symbols 2, +, = and 4 mean.) a priori (no a posteriori reasons necessary)

[Londoner]

A fact is either a true proposition or an actual state of affairs. In the former case, "false fact" is a contradiction in terms; and in the latter case, "false fact" is a category mistake, because states of affairs are not the kind of entities which can be bearers of truth-values.

Quite right. It was badly expressed. But this happens if you keep switching between technical language and ordinary speech.

-- Updated May 12th, 2014, 12:07 pm to add the following --

It's a linguistic object called an indefinite noun phrase. For the logic of indefinite descriptions, see: http://plato.stanford.edu/entries/descriptions/#IndDes

For anyone who is interested, the 'theory of descriptions' page in a well-known on-line encyclopedia (this site won't allow us to link to) may be more comprehensible! You can see how it connects with Russell, which I refer to later in the same post.

But I'm not so sure it applies to the '6 meter man'. I don't think we get enough information to tell what sort of an object he is in that sense.