Predicates and Models ... Goldrei

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In summary, Goldrei's language for representing the predicate "less than" involves the symbols "*", "e", and "|", which are interpreted as addition, 0, and equality respectively. In a more standard mathematical notation, the predicate can be represented as ##(\exists z (x * z) = y \wedge \neg x = y)## or ##(\exists z (x * z) = y \wedge \neg z = e)##. However, it is worth noting that any variables other than "x" and "y" must be bound in order for the formula to accurately represent the predicate. Additionally, the formulas may not work for certain domains of quantification such as ##\mathbb{R}
  • #1

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I am reading Goldrei's book: "Propositional and Predicate Calculus: A Model of Argument", Chapter 4: Predicates and Models.

I need help in understanding Goldrei's answer to Exercise 4.5 (a) ...

Exercise 4.5 together with Goldrei's solution reads as follows:
Goldrei page 138.png


Can someone please explain exactly how x < y can be represented by

## ( \exists z \ (x * z) = y \land \neg x = y ) ##
Help will be much appreciated.Goldrei mentions the language given above ... that reads as follows:
Goldrei page 137.png

Peter
 
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  • #2
Your scanned or photographed images are difficult to read, especially because some symbols don't show up.
In the image for ex. 4.5, it says "(involving *, e and , interpreted respectively by 1 (?), 0, and equality)." It looks like whatever was written after "and" doesn't show up. Possibly it was in a different color. Also, is the first symbol after "respectively by" the digit 1 or the symbol |? It looks like the latter, but I don't know what that means in this context.

Math Amateur said:
Can someone please explain exactly how x < y can be represented by
##(\exists z (x * z) = y \wedge \neg x = y)##
The text in the second image is smaller, so more difficult to read. Several symbols in the explanation in this image also don't show up, so I still don't understand what is meant by *.
 
  • #3
* represents addition (I agree it's very tough to read). In more standard mathematical language:
$$( \exists z \ (x * z) = y \land \neg x = y )$$
says that there's a ##z## such that ##x+z=y## and ##x\neq y##. Alternatively, the other explanation
$$( \exists z \ (x * z) = y \land \neg z = \mathbf{e} )$$
says that there's a ##z## such that ##x+z=y## and ##z\neq 0##.
 
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  • #4
TeethWhitener said:
Alternatively, the other explanation
$$( \exists z \ (x * z) = y \land \neg z = \mathbf{e} )$$
says that there's a ##z## such that ##x+z=y## and ##z\neq 0##.
For clarity of notation, I think it is probably better to have the ##z## variable to be outside all of the brackets.

===========One thing to note about these formulas is that since the predicate is of the form ##less(x,y)##, any variables other than ##x## and ##y## that are used must be bound. This is true in general regardless of the specific predicate so it is worth a mention [since checking for it can sometimes help identify a mistake in more complicated formulas].

Also, I agree that it is not difficult to understand the solution if you write it with more familiar symbols.
(1) ##\exists z [x+z=y] \land (x\neq y)##
(2) ##\exists z [(x+z=y) \land (z\neq 0)]##

In the first formula if you look separately at:
##\exists z [x+z=y] ##
then this is just the representation of predicate ##x \leq y##. So the first formula is just saying that ##x<y## iff we have (i) ##x \leq y## and (ii) ##x \neq y##.
 
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  • #5
SSequence said:
For clarity of notation, I think it is probably better to have the z variable to be outside all of the brackets.
I agree, but that’s how it was written in the book. Not sure why.
 
  • #6
TeethWhitener said:
I agree, but that’s how it was written in the book. Not sure why.
Given the first image posted in OP, the book does put ##\exists z## outside all of the brackets in the second formula!

Anyway, it also occurred to me that formula-(1) in post#4 also works [as in correctly describes "less than" predicate] even if the domain of quantification is ##\mathbb{R}## or ##\mathbb{Q}##. However, while formula-(2) works for ##\mathbb{N}##, it doesn't work for ##\mathbb{R}## or ##\mathbb{Q}##.
 
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  • #7
SSequence said:
Anyway, it also occurred to me that formula-(1) in post#4 also works [as in correctly describes "less than" predicate] even if the domain of quantification is ##\mathbb{R}## or ##\mathbb{Q}##.
Sorry for the mistake. Formula-(1) in post#4 doesn't represent the "less than" predicate either [if the domain of quantification is ##\mathbb{R}## or ##\mathbb{Q}##]. It represents the "not equal" predicate in that case.

In fact, if we look at it:
(1) ##\exists z [x+z=y] \land (x\neq y)##
Then basically it is just equivalent to the predicate ##x\neq y## if the domain of quantification is ##\mathbb{R}##, ##\mathbb{Q}## or ##\mathbb{Z}##.And while we are at it, we might also look at the second formula:
(2) ##\exists z [(x+z=y) \land (z\neq 0)]##
This formula is also equivalent to the predicate ##x\neq y## if the domain of quantification is ##\mathbb{R}##, ##\mathbb{Q}## or ##\mathbb{Z}##.
 
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1. What is a predicate in mathematical logic?

A predicate is a statement that contains one or more variables and becomes a proposition when specific values are substituted for the variables. It is used to describe properties or relations between objects or concepts.

2. What is a model in mathematical logic?

A model is a mathematical structure that satisfies a given set of axioms or statements. It is used to interpret the logical symbols and determine the truth value of propositions.

3. How are predicates and models related?

Predicates are used to define the properties or relations in a model. Models, in turn, are used to interpret the truth value of predicates. A model can be seen as a concrete instance of a logical system, with predicates representing the rules and properties of that system.

4. What is the significance of Goldrei's book on predicates and models?

Goldrei's book is a comprehensive introduction to mathematical logic, with a focus on the concepts of predicates and models. It covers the fundamental principles and techniques used in the study of these topics, making it a valuable resource for students and researchers in the field.

5. Are there any real-world applications of predicates and models?

Yes, predicates and models are widely used in computer science, artificial intelligence, and linguistics. They are used to represent and reason about complex systems and to understand the structure and behavior of natural languages. They also have applications in fields such as philosophy and mathematics.

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