kennethamy wrote:
That all dogs are dogs is a necessary truth, since it can be shown on a truth table that it is a tautology for it is a substitution instance of the propositional form, all X is X. Consult any elementary logic book for how to operate truth tables. All dogs are dogs is a logical truth, and all logical truths are necessary truths for their negations are logically impossible.
1. All dogs are dogs is a logical true
2. All logical truths are necessary truths.
Therefore, 3, all dogs are dogs is a necessary truth. QED.
The second premise clearly needs further treatment before it can be accepted as sound. For me, something more substantive than, 'it is impossible for a logical truth to be false', is required. An appeal to possible worlds does, after all, need to be grounded in something. Conceivability is appealed to quite a lot, but might we not have had different concepts? Concepts are a very tricky subject, but its a more than plausible claim that our concepts could have been different if certain biological, socialogical, or natural facts had been different. A presumable result of this may be that certain things we hold to be impossible might be perfectly conceivable. Pv~P is a necessary truth, but what if we only had paraconsistent logics?...
There are other reasons to reject conceivability and possibility as being co-extensive. The necessarily existing God used in the ontological argument seems to be conceivable using possible world semantics: a necessarily existing God simply exists in all possible worlds. However, the notion of a necessarily existing being seems impossible to most.
mickalos wrote:any proposition of the form, A or not-A is not only true, but it must be true, since for all substitutions of A, the proposition is true.what if we only had paraconsistent logics?
kennethamy wrote:Isn't Mickalos' point that something being true in a particular class of logics, doesn't make it true in all logics? Thus the idea of necessary truth isn't primary and cant be appealed to for primary distinctions.mickalos wrote:any proposition of the form, A or not-A is not only true, but it must be true, since for all substitutions of A, the proposition is true.what if we only had paraconsistent logics?
ughaibu wrote:I see no reason to deny the correctness of standard logic just because there are peculiar logics floating about.Isn't Mickalos' point that something being true in a particular class of logics, doesn't make it true in all logics? Thus the idea of necessary truth isn't primary and cant be appealed to for primary distinctions.
When pure mathematics is applied to "the real world", funny things begin to happen. That is also true when theoretical logic is applied to "the real world".
For me, something more substantive than, 'it is impossible for a logical truth to be false', is required.
I see. You don't know what a logical truth is. It is a technical term in logic. It means that if when a truth is subjected to a truth table test, it tests out to a tautology. Now, a tautology is a proposition which is true under all interpretations of its variables. Thus, for instance, any proposition of the form, A or not-A is not only true, but it must be true, since for all substitutions of A, the proposition is true. Thus, it is impossible for any proposition of that form to be false, and so, it is not only true, but it is a necessary truth. You may need to learn a little logic. We can put this into the language of possible worlds too (although I find that language cumbersome). But we can say that a logical truth is true in all possible worlds. Which is equivalent to saying that it is true under all possible interpretations of its variables.
The notion of "necessary truth" has been extended somewhat outside of the strict notion of logical truth. So that to call a proposition a necessary truth is to say that it is either a logical truth or it is reducible to a logical truth. Issues sometime arise about whether or not a proposition is reducible to a logical truth as famously discussed by Quine. But there is no issue that some truths are clearly logical truths, and that all logical truths are necessary truths. The only problem that arises is whether a truth is reducible to a logical truth and is, therefore, a necessary truth.
I see no reason to deny the correctness of standard logic just because there are peculiar logics floating about.
Mathematics is not a science.
You have missed my point. Does a certain model/interpretation force a truth value? Certainly, we feel a psychological compulsion to say Pv~P is an eternal truth (I'm not trying to be technical), but I dispute that it is any more than that, which is what my point about paraconsistent logics was about. With certain sociological and historical developments or changes, and perhaps with certain things we learn about the world, we often break free of psychological compulsions. History is full of examples of primitively compelling claims and arguments that we now regard as false concerning God, politics, reality, and even, yes, patterns of reasoning. Is there any fact of the matter that distinguishes "logical compulsion" from mere dogma (this, I think, is one of the things that makes Quine's arguments against the analytic synthetic distinction so compelling)? I think it can only be a naturalistic/anthropocentric fact of the matter, e.g. "People do not allow ~(Pv~P) in classical logic", but this is a very different fact of the matter than most realists about necessity and logical truth think there is.
You subscribe to a correspondence theory of truth, and I take it "correctness" is a notion concerned with truth. As the correctness of logics is independent of reality and what is correct in any particular logic is a matter of how the formalism is defined, I dont see how you can talk about the correctness of logics in any interestingly non-circular manner. So, what do you mean by "correctness"?
mickalos wrote:You have missed my point. Does a certain model/interpretation force a truth value? Certainly, we feel a psychological compulsion to say Pv~P is an eternal truth (I'm not trying to be technical), but I dispute that it is any more than that, which is what my point about paraconsistent logics was about. With certain sociological and historical developments or changes, and perhaps with certain things we learn about the world, we often break free of psychological compulsions. History is full of examples of primitively compelling claims and arguments that we now regard as false concerning God, politics, reality, and even, yes, patterns of reasoning. Is there any fact of the matter that distinguishes "logical compulsion" from mere dogma (this, I think, is one of the things that makes Quine's arguments against the analytic synthetic distinction so compelling)? I think it can only be a naturalistic/anthropocentric fact of the matter, e.g. "People do not allow ~(Pv~P) in classical logic", but this is a very different fact of the matter than most realists about necessity and logical truth think there is.
But following this, where do you draw the line? When do we consider something an actual matter of fact, instead of *only* a matter of fact for X discipline or position?
May I ask what position on truth you take?
ughaibu wrote:You subscribe to a correspondence theory of truth, and I take it "correctness" is a notion concerned with truth. As the correctness of logics is independent of reality and what is correct in any particular logic is a matter of how the formalism is defined, I dont see how you can talk about the correctness of logics in any interestingly non-circular manner. So, what do you mean by "correctness"?
Echoing my post above, where do you draw the line? What has led you to conclude that the correctness of a particular logic is independent of reality? Because"correct" is defined by the particular formalism in question, doesn't mean that "correctness" cannot be part of a proposition (describing reality), does it? If I told you that the correct way to move a pawn in the game of chess is to advance it a single square, except during the first time the pawn is moved where it can advance two squares, am I not expressing something about reality?
ughaibu wrote:Echoing my post above, where do you draw the line?You subscribe to a correspondence theory of truth, and I take it "correctness" is a notion concerned with truth. As the correctness of logics is independent of reality and what is correct in any particular logic is a matter of how the formalism is defined, I dont see how you can talk about the correctness of logics in any interestingly non-circular manner.
What has led you to conclude that the correctness of a particular logic is independent of reality?
Because"correct" is defined by the particular formalism in question, doesn't mean that "correctness" cannot be part of a proposition (describing reality), does it?
If I told you that the correct way to move a pawn in the game of chess is to advance it a single square, except during the first time the pawn is moved where it can advance two squares, am I not expressing something about reality?
May I ask what position on truth you take?
TuringEquivalent wrote:Mathematics is not a science.
Obviously. But would it be more accurately classified as a Philosophy or a Language?
There is NO sector of mathematical thought that can be supplied with a set of axioms sufficient for developing systematically the endless totality of true propositions about that sector. That is the essence of the great Gödel's incompleteness theorem.
.... It might seem at first that the set-theoretical paradoxes would doom to failure such an undertaking, but closer examination shows that they cause no trouble at all. They are a very serious problem, not for mathematics, however, but rather for logic and epistemology. As far as sets occur in mathematics (at least in the mathematics of today, including all of Cantor's set theory), they are sets of integers, or of rational numbers, (i.e., of pairs of integers), or of real numbers (i.e., sets of rational numbers), or of functions of real numbers (i.e., of sets of pairs of real numbers), etc. When theorems about all sets (or the existence of sets in general) are asserted, they can always be interpreted without any difficulty to mean that they hold for sets of integers as well as for sets of sets of integers, etc. (respectively, that there either exist sets of integers, or sets of sets of integers, or... etc., which have the asserted property). This concept of set, however, according to which a set is something obtainable from the integers (or some other well-defined objects) by iterated application of the operation 'set of', not something obtained by dividing the totality of all existing things into two categories, has never led to any antinomy whatsoever; that is, the perfectly 'naive' and uncritical working with this concept of set has so far proved completely self-consistent
First of all, I wouldn't characterise my view as being that 'Pv~P is a fact only in classical logic'. Certainly, it is a theorem of classical logic, and not a theorem of other calculi, but that is uncontroversial. My view is a naturalistic scepticism about rule following and necessity. There is no fact of the matter that makes Pv~P follow from the rules of logical inference apart from the fact that we all agree that it does, i.e. no expression or interpretation of those rules forces a particular application on me; logic does not "grab you by the throat". We might have agreed on something different.
I don't understand what you mean when you distinguish between an "actual fact of the matter" and other facts.
Certainly, I think the idea that a sentence is true because that's the way the world is, seems to me to be a metaphysical tip of the hat that ought not be taken too literally
Only if you take a platonic realist approach to rules. What is a rule? Well, the rules of chess establish criteria of correctness, but how? Are they "abstract entities" that sort correct action from incorrect action all by themselves? Seems a bit far fetched for my liking.
Logics are linguistic systems which preserve truth values according to defined rules, that is to say that they are human inventions, and I hold that reality is not a subset of human invention.
Humans invented my air conditioner. The sentence, "There is an air conditioner in my room" expresses a proposition (that is, it has relation to reality), does it not?
The line I'm asking to be drawn is between where you consider something to be a matter of fact, and where you do not.
Is it all that is mind-dependent that you do not believe is part of reality? That is, you would deny that things like ideas are part of reality?
Yes, now will you please spell out how this is meant to be relevant.
I didn't write that either logics or mathematics aren't part of reality, I wrote that they're independent of reality
I take a naive view that reality is that which human beings have in common...
ughaibu wrote:Yes, now will you please spell out how this is meant to be relevant.
It's meant to be relevant because propositions, as far as I know, are a reflection of reality. They are not independent of reality.
ughaibu wrote:I didn't write that either logics or mathematics aren't part of reality, I wrote that they're independent of reality
So, logics and mathematics are both part of reality, but also independent of reality? What does that mean?
ughaibu wrote:I take a naive view that reality is that which human beings have in common...
That may explain things. I don't agree with your view here, as I don't believe reality is defined by human intersubjectivity. The moon, for instance, was around for millions of years before any human concurrence, and the moon, of course, is part of reality.
mickalos wrote:First of all, I wouldn't characterise my view as being that 'Pv~P is a fact only in classical logic'. Certainly, it is a theorem of classical logic, and not a theorem of other calculi, but that is uncontroversial. My view is a naturalistic scepticism about rule following and necessity. There is no fact of the matter that makes Pv~P follow from the rules of logical inference apart from the fact that we all agree that it does, i.e. no expression or interpretation of those rules forces a particular application on me; logic does not "grab you by the throat". We might have agreed on something different.
Can you give me an example of something you consider to be a matter of fact, and then detail how, or why, this differs from a rule in logic? Is it because logic, as ughaibu proclaimed, is an invention of man?
mickalos wrote:I don't understand what you mean when you distinguish between an "actual fact of the matter" and other facts.
You said, "...this is a very different fact of the matter...", and I thought that meant you believed there to be different sorts of facts of the matter. That is why I asked.
mickalos wrote:Certainly, I think the idea that a sentence is true because that's the way the world is, seems to me to be a metaphysical tip of the hat that ought not be taken too literally
If my mother is in the kitchen, wouldn't we say that it is literally true my mother is in the kitchen? A metaphysical tip of the hat? I'm not sure where you got that idea.
mickalos wrote:Only if you take a platonic realist approach to rules. What is a rule? Well, the rules of chess establish criteria of correctness, but how? Are they "abstract entities" that sort correct action from incorrect action all by themselves? Seems a bit far fetched for my liking.
I simply wish to know how you differentiate matter of facts, from matters which are not factual. I don't see how a rule is "independent of reality" as ughaibu states. I see no reason why a sentence, concerning rules, cannot express a proposition (that is, have relation to reality).
It's meant to be relevant because propositions, as far as I know, are a reflection of reality. They are not independent of reality.
So, logics and mathematics are both part of reality, but also independent of reality? What does that mean?
I don't agree with your view here, as I don't believe reality is defined by human intersubjectivity. The moon, for instance, was around for millions of years before any human concurrence, and the moon, of course, is part of reality.
