# Questions tagged [predicate-logic]

Questions concerning predicate calculus, i.e. the logic of quantifiers.

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### Creating a predicate checking if a string is a subsequence of another string

I've been trying to come up with a predicate that takes in two strings and checks whether the first string is a subsequence of the second string. Subsequence(a, b): predicate to check is 'a' is a ...
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### Is there an error in this proof the the “strong induction” theorem? Is this Escherian logic?

By set predicates, I mean the "tests for inclusion" used in defining sets. That may be more of a computer scientific than mathematical use of the the term predicate. I included predicate logic in ...
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### How to transform a sequent notation to rule form?

I can write this proposition in sequent notation: $$(P\rightarrow Q)\rightarrow (\neg P \lor Q)$$ as this one in rule form (see here): $$\frac{(P\rightarrow Q)}{(\neg P \lor Q)}$$ But how can I ...
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### Logic of predicates over predicates?

Is there logic, that allow to form predicates over predicates (or even formulas), I.e. derive is-interesting-statment(is(I, liar)). Of course, the model operators in the modal logic can take ...
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### Syntactical proof of universal instantiation rule

First: I am not mathematician but philosopher. I understand why the universal instantiation rule is working. $\frac{\vdash\forall xA}{\vdash A^x_t}$ But is there actually a serious proof in a ...
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### Is there a definition for free and bound variables in logic?

That makes me crazy to think about it because my book and other pages on the web talks about free and bound variables without any definition. I think everything in mathematics has a definition so ...
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### First-order logic task with limited clauses

I have the following axioms The sum of a natural number x and 0 is equal to x. The sum of a natural number x and the successor of a natural number y is equal to the successor of the sum of x and y. ...
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### Help to prove this formula (Sequent calculus for predicate logic)

I have a formula: $$¬(\exists x)ϕ(x) ⇒ (∀x)¬ϕ(x)$$ to prove. If the "$(∃x)ϕ(x)$" was in brackets like this $¬((∃x)ϕ(x))$, I could easily prove this formula, but without it I'm stucked. Can you ...
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### Logic: “the cubic root of a rational number is also a rational number”

I was attempting an online logical and mathematical statements self-test from the University of Toronto and came across the following statement in question 1: The cubic root of a rational number is ...
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### What are the features that make this proof constructive?

Below is an inductive proof of an integer square root theorem from a previous posting. $\forall x: \mathbb{N}, \exists y : \mathbb{N}((y^2 \leq x) \land (x < (y+1)^2))$ The proof was done using ...
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### Notation: $P(x)$ iff $x$ has property $P$

In set theory (for example), people write $P(x)$ to indicate that $x$ has property $P$. What is the meaning of this "expression" formally? Is $P$ a predicate (a Boolean-valued function on some set [...
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### Need help with relation properties using logical operators

I was wondering how should I proceed to determine what will be in the relation and what will not given these properties. Operating with integers: $R: \{(a, b)|(a= 0∧b= 0)∨ GCD(a, b) = 5\}$