Predicate logic
Predicate logic
Let's discuss interpretations (yours and mine) of predicate logic, aka quantificational logic. It is the familiar logical language which includes
quantifiers (for all x, there exists an x such that  i.e. (x) and E(x)); the logical connectives of sentence logic (not, and, if..then, or); and then the atomic sentences of the form Fa where F is a predicate and a is a constant.
That's it, that's all predicate logic consists of. But here's the interesting part. It cannot function without the following metalogical apparatus to support it. It is assumed in predicate logic, that there exists a domain of objects  an ontology in philosophical terms, a domain in mathematical terms. Each object has a unique name. These are usually represented by lower case letters from the beginning of the alphabet  a,b,c, d. Of course we can generate as many as we want by means of something like a', b', c' ... and then a'', b'', c'''... etc..
Each unique name (denoting an individual object of the domain) then can be joined with a predicate symbol, represented by upper case letters F, G, H, etc. to form atomic sentences of the subjectpredicate form, Fa. The predicates, as with the objects and names of the domain, are given  that is, they are assumed and stipulated to at the beginning, as part of the formal language.
I am asking, in this thread (if anyone has read this far), for an interpretation of this logical apparatus. For instance, are the objects of the domain predicateless simples that exist prior to the formation of atomic sentences? How are the predicates of a particular system chosen? Aren't they chosen specifically to apply to a preconceived domain of predicated objects? In which case, isn't this whole apparatus just a bit of contrived smoke and mirrors?
quantifiers (for all x, there exists an x such that  i.e. (x) and E(x)); the logical connectives of sentence logic (not, and, if..then, or); and then the atomic sentences of the form Fa where F is a predicate and a is a constant.
That's it, that's all predicate logic consists of. But here's the interesting part. It cannot function without the following metalogical apparatus to support it. It is assumed in predicate logic, that there exists a domain of objects  an ontology in philosophical terms, a domain in mathematical terms. Each object has a unique name. These are usually represented by lower case letters from the beginning of the alphabet  a,b,c, d. Of course we can generate as many as we want by means of something like a', b', c' ... and then a'', b'', c'''... etc..
Each unique name (denoting an individual object of the domain) then can be joined with a predicate symbol, represented by upper case letters F, G, H, etc. to form atomic sentences of the subjectpredicate form, Fa. The predicates, as with the objects and names of the domain, are given  that is, they are assumed and stipulated to at the beginning, as part of the formal language.
I am asking, in this thread (if anyone has read this far), for an interpretation of this logical apparatus. For instance, are the objects of the domain predicateless simples that exist prior to the formation of atomic sentences? How are the predicates of a particular system chosen? Aren't they chosen specifically to apply to a preconceived domain of predicated objects? In which case, isn't this whole apparatus just a bit of contrived smoke and mirrors?
Re: Predicate logic
I'm no expert on logic but my understanding is that we have syntax, strings of symbols constructed according to rules. And we have semantics, in which we assign elements of some domain to the symbols. It's only when you specify a model that you can talk about truth.Wyman wrote: ↑Mon Jul 24, 2017 2:25 am
I am asking, in this thread (if anyone has read this far), for an interpretation of this logical apparatus. For instance, are the objects of the domain predicateless simples that exist prior to the formation of atomic sentences? How are the predicates of a particular system chosen? Aren't they chosen specifically to apply to a preconceived domain of predicated objects? In which case, isn't this whole apparatus just a bit of contrived smoke and mirrors?
For example consider the legal sentence
∃x (2x = 3)
For sake of my example I'm adding multiplication and equality to the basic language, I hope this doesn't make any difference.
My point is that this sentence is neither true nor false. The best we can say is that it's syntactically valid.
Now if I tell you that the domain, or intended interpretation, is the integers, then the statement is false. But if I tell you it's the rationals, then it's true.
I am not sure if my response is on topic to your question. Am I in the ballpark? Point is that we have strings of symbols, and we have interpretations or models in which we interpret the strings so that we can determine their truth values in that model. And the truth of a given string is always relative to the chosen model.
As far as whether it's all smoke and mirrors, perhaps it is. Is logic an aspect of the universe? Or only something people project onto a random universe, like seeing constellations in the stars?
If you're asking which came first, syntax or semantics, that's a good question.
Last edited by wtf on Mon Jul 24, 2017 3:14 am, edited 2 times in total.
Re: Predicate logic
That's an excellent example. If the logical language includes the symbols 2,3,4, etc. and the interpretation given is that the domain consists of integers, then those symbols are names of the integers. And the predicates are things like 2 is even or 3 is prime and then all the relations.wtf wrote: ↑Mon Jul 24, 2017 3:04 amI'm no expert on logic but my understanding is that we have syntax, strings of symbols constructed according to rules. And we have semantics, in which we assign elements of some domain to the symbols. It's only when you specify a model that you can talk about truth.Wyman wrote: ↑Mon Jul 24, 2017 2:25 am
I am asking, in this thread (if anyone has read this far), for an interpretation of this logical apparatus. For instance, are the objects of the domain predicateless simples that exist prior to the formation of atomic sentences? How are the predicates of a particular system chosen? Aren't they chosen specifically to apply to a preconceived domain of predicated objects? In which case, isn't this whole apparatus just a bit of contrived smoke and mirrors?
For example consider the legal sentence
∃x (2x = 3)
For sake of my example I'm adding multiplication and equality to the basic language, I hope this doesn't make any difference.
My point is that this sentence is neither true nor false. The best we can say is that it's syntactically valid.
Now if I tell you that the domain, or intended interpretation, is the integers, then the statement is false. But if I tell you it's the rationals, then it's true.
I am not sure if my response is on topic to your question. Am I in the ballpark? Point is that we have strings of symbols, and we have interpretations or models in which we interpret the strings so that we can determine their truth values in that model. And the truth of a given string is always relative to the chosen model.
As far as whether it's all smoke and mirrors, perhaps it is. Is logic an aspect of the universe? Or only something people project onto a random universe, like seeing constellations in the stars?
If you're asking which came first, syntax or semantics, that's a good question.
Logicians imagine that the syntax and logical language are independent of the interpretations, but common sense tells us this is not the case. For example, if instead of naming the elements of the domain 2,3,4 (which obviously shows what the intended interpretation is to be) we name them a,b,c, etc. and assigned 'neutral' predicates F,G,H,I etc.  how are we to develop a syntax?
Syntax is not created at random (I suppose it could be, but you'd just get a jumble of symbols). We assign a predicate (let's use 'P') to certain elements of the domain  let's say a,b,c,e,g...  only because we have in mind that 1,2,3,5,7 are the first five prime numbers. We cannot develop a syntax in the dark, we need an intended interpretation to guide us. Another example is that of a claim Hilbert made, that in doing geometry, he could just as well call points, lines and planes 'beer mugs,' 'tables' and 'chairs.' That is true only after the original system, with its intended interpretation, is developed. For what reason would one have to use as an axiom 'Two beer mugs determine a unique table' (two points determine a unique straight line)  you'd have no reason whatsoever, other than the understanding that your terms are meant to be interpreted in a certain way from the start.
So why do we claim that there is any such thing as an uninterpreted formal system?
Re: Predicate logic
Yes of course. There is always an intended interpretation. That's why we have "nonstandard" models of some of our axiom systems in math. There is an intended interpretation, and the unintended ones.
In your example of the natural numbers, of course we are "secretly" thinking of the familiar counting numbers 1, 2, 3, ... when we formalize them with the Peano axioms.
That makes the formalization useful for something, namely studying the natural numbers.
But of course there are arbitrary formal systems. You could write down some random rules, and perhaps prove that they don't lead to a contradiction, but they're not really "about" anything and for that reason they are not very interesting.
Or consider the rules of games. Chess is a formal system but it does not refer to anything in the real world. We can think of a win in chess as a proposition to be proven or a particular state to be achieved. The best player is the one who can get to winning positions more often than the other players. Almost like a mathematician who can prove hard theorems better than others.
But there is no reason a formal system must be about something. It's just that the formal systems that aren't about anything are generally not interesting to anyone, unless working out the rules is regarded as fun, in which case it's a game.
Consider the formal systems kids dream up when the make up games to play. They make up the rules as they go, and have fun following the arbitrary rules. Games in general are formal systems that are not about anything but themselves.
Re: Predicate logic
You've clarified things nicely.wtf wrote: ↑Wed Jul 26, 2017 3:10 amYes of course. There is always an intended interpretation. That's why we have "nonstandard" models of some of our axiom systems in math. There is an intended interpretation, and the unintended ones.
In your example of the natural numbers, of course we are "secretly" thinking of the familiar counting numbers 1, 2, 3, ... when we formalize them with the Peano axioms.
That makes the formalization useful for something, namely studying the natural numbers.
But of course there are arbitrary formal systems. You could write down some random rules, and perhaps prove that they don't lead to a contradiction, but they're not really "about" anything and for that reason they are not very interesting.
Or consider the rules of games. Chess is a formal system but it does not refer to anything in the real world. We can think of a win in chess as a proposition to be proven or a particular state to be achieved. The best player is the one who can get to winning positions more often than the other players. Almost like a mathematician who can prove hard theorems better than others.
But there is no reason a formal system must be about something. It's just that the formal systems that aren't about anything are generally not interesting to anyone, unless working out the rules is regarded as fun, in which case it's a game.
Consider the formal systems kids dream up when the make up games to play. They make up the rules as they go, and have fun following the arbitrary rules. Games in general are formal systems that are not about anything but themselves.
Suppose Chess is a formal system. The pieces are its domain. What is the status of verbal reports about movements in the game, such as 'he moved his queen to square R3'? Are descriptions of this sort (potentially) part of a formal system having Chess as its intended interpretation? Or are descriptions always metasystematic?
The same questions apply to statements of the rules of the game.
Suppose that the moves in the game are true propositions (theorems) of the system. Suppose there are an infinite number of possible moves in any given game. How does Godel's incompleteness theorem apply if at all? I suppose the natural numbers within the system is a prerequisite to application of Godel's theorem and I don't know if Q1, Q2, Q3, etc. count  but what do you think? Is there always a possible move in Chess that we cannot determine to be either within the rules or not (true or false)?
Re: Predicate logic
Status of verbal reports. That sounds like more philosophy than I know. What's the status of verbal reports that the sun rose in the east today? Can you put that question in context for me so I understand what you mean?
If I'm understanding, suppose I make the following inference in logic.
Premise 1: X => Y
Premise 2: X
Inference rule: We may conclude Y.
Metasyntactic comment: "We call this rule modus ponens."
Are you asking what is the philosophical nature of applying a name to an inference rule? Way outside my modest sphere of knowledge. But at least tell me if I'm understanding the question.
I can't suppose that since it's false in chess. If you are using an alternate system of rules please specify.
One of the most interesting (and not widely known) rules in chess is that if the players repeat 50 moves in a row then the game is a draw. (I misstated that slightly, the official rule is here). This rule is designed so that players can't chase each other around the board forever with their kings. It also makes endgame technique important. For example king/knight/bishop can checkmate a lone king, but it's not easy. The 50 move rule makes sure the k/n/b side checkmates the opponent within 50 moves or it's a draw. Otherwise an inept player could chase the opposing king around forever without figuring out how to achieve the mate.
So like I say, it's on you to say exactly what variant of the rules of chess you are proposing.
But it's clear that even if you dropped the 50 move rule, chess is still finite. That's because there are only finitely many ways the finite number of pieces can be assigned to the finite set of squares. Even if you play infinitely long games, states must repeat because there are only finitely many states.
Doesn't, as I understand it, because chess can't model the Peano axioms. Incompleteness only applies to formal systems that can model the arithmetic of the natural numbers, including induction.
Yes, that's my understanding too.
Couldn't be. In principle we work out every possible game (the move tree is finite, because at some point you're only reaching positions that you've seen before. If you stop branching at those points, you have a finite tree of every possible game. Given a position, you just do a search on the finitelylong list of legal positions. If your position is in there, it's legal, else not.
But remember the key point was your original question. Chess is an example of an arbitrary formal system that does not refer to anything in the world, either physical or abstract. It's not about trees (green ones I mean) and it's not about topological spaces. It's not a tool for modeling economies or voting strategies. The only purpose of the formal system chess is for people to amuse themselves by being really good at using the rules to achieve certain positions.
So formal systems can be completely arbitrary (why does the knight move funny?) but still of interest to people.

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Re: Predicate logic
That's got to be one of the funniest skits ever. I also like the dead parrot and Department of Verbal Abuse (I paraphrase).Impenitent wrote: ↑Thu Jul 27, 2017 12:58 amwhy the knight moves funny...
https://www.youtube.com/watch?v=zKhEw7nD9C4
Imp
Re: Predicate logic
Let me try to clarify my thought process. Suppose geometry is based upon drawings of points and lines on a flat surface. Drawing points and (straight)lines is like the game of chess  moving pieces/objects according to rules. The mathematical discipline of geometry may have started as mere descriptions of the rules  i.e. what can be drawn with a straight edge and pencil (I understand the historical development of geometry is a tad more complicated, I'm simplifying). Euclid's language was very much like ordinary language description  e.g. Given any two points there may be only one straight line passing through both. This sounds a lot like ordinary language descriptions of the rules of chess  e.g. Any pawn may move only one square forward on a given turn, or one square diagonally if doing so takes an opponent's piece.wtf wrote: ↑Thu Jul 27, 2017 12:48 am
Status of verbal reports. That sounds like more philosophy than I know. What's the status of verbal reports that the sun rose in the east today? Can you put that question in context for me so I understand what you mean?
If I'm understanding, suppose I make the following inference in logic.
Premise 1: X => Y
Premise 2: X
Inference rule: We may conclude Y.
Metasyntactic comment: "We call this rule modus ponens."
Are you asking what is the philosophical nature of applying a name to an inference rule? Way outside my modest sphere of knowledge. But at least tell me if I'm understanding the question.
But remember the key point was your original question. Chess is an example of an arbitrary formal system that does not refer to anything in the world, either physical or abstract. It's not about trees (green ones I mean) and it's not about topological spaces. It's not a tool for modeling economies or voting strategies. The only purpose of the formal system chess is for people to amuse themselves by being really good at using the rules to achieve certain positions.
So formal systems can be completely arbitrary (why does the knight move funny?) but still of interest to people.
Eventually, geometry was formalized. A formal language was created with rules of syntax, etc.(Forget for now analytical geometry). Now, the formal system of geometry can be expressed via predicate logic. Is this formal system now a description of the intended interpretation  or is there a fundamental difference between formal systems and language used to describe things in the world? Or is the question just nonsense because I have misunderstood the nature of formal systems?
I am sympathetic to your point that formal systems need not describe anything. But logicians do call at least some of them 'formal languages' and much of logic is concerned with descriptions  such as Russell's theory of definite descriptions. I guess my thesis (which you may agree with given your comments) is that the formal language of predicate logic  which was always designed to make (verbal, ordinary language) arguments more precise  is not a language at all. I think that it is more like chess  a manipulation of symbols according to rules.
That'll have to do for now, have to go .
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