Articles of predicate logic

How to translate set propositions involving power sets and cartesian products, into first-order logic statements?

As seen from an earlier question of mine one can translate between set algebra and logic, as long as they speak about elements (a named set A is the same as {x ∣ x ∈ A}). However I’ve stumbled upon propositions that involve cartesian products and power sets and I’m not sure how to translate […]

Logical formula of definition of linearly dependent

A subset $S$ of a vector space $V$ is said to be linearly dependent if there exist a finite number of distinct vectors $x_1, \ldots , x_n$ in $S$ and scalars $a_1 , \ldots ,a_n$ not all zero, such that $$ a_1 x_1 + \cdots + a_n x_n =0 $$ I want to translate this […]

Can finite theory have only infinite models?

I always thought, that when creating a theory (set of formulas of predicate logic of first order in some language) and when you want to have only infinite models, you must use infinite number of axioms. That’s how Peano arithmetic or ZF are made. But when I look at Robinson arithmetic, it is finite. Am […]

Qualification of a Universal Quantification

Let us say I have a predicate, $P(n)$, and I want to say that it holds for every integer greater than $2$ (an example would be $P(n) = 2n>2+n$). Let us furthermore say that the UOD (universe of discourse) is larger than $\mathbb{N}$, and that the order relation $>$ is not defined throughout the UOD […]

When do free variables occur? Why allow them? What is the intuition behind them?

In the formula $\forall y P(x,y)$, $x$ is free and $y$ is bound. Why would one write such a formula? Why are free variables allowed? What is the intuition for allowing free variables?

How to express the statement “not all rainy days are cold” using predicate logic?

I am trying to figure out how to express the sentence “not all rainy days are cold” using predicate logic. This is actually a multiple-choice exercise where the choices are as follows: (A) $\forall d(\mathrm{Rainy}(d)\land \neg\mathrm{Cold}(d))$ (B) $\forall d(\neg\mathrm{Rainy}(d)\to \mathrm{Cold}(d))$ (C) $\exists d(\neg\mathrm{Rainy}(d)\to\mathrm{Cold}(d))$ (D) $\exists d(\mathrm{Rainy}(d)\land \neg\mathrm{Cold}(d))$ I am really having a hard time understanding […]

Correct notation for “for all positive real $c$”

I am wondering how to notate “for all positive real value $c$” Is there a correct notation among the following? $$ \forall c \in \mathbb{R} > 0\\ \forall c \left( \in \mathbb{R} \right) > 0\\ \forall c > 0 \in \mathbb{R}\\ \forall c > 0 \left( \in \mathbb{R} \right)\\ $$ My ultimate goal is notating […]

Predicate logic for statements about functions?

Given the relation $R$ defined on the cartesian map $A\times A$ where $|A|=n$. How to use the predicate logic to express the statements about functions? Examples. The relation R corresponds to a function from A to A. The relation R corresponds to an injective function from A to A. The relation R corresponds to and […]

How to write “there exists an infinite number of”?

We all know that ∃ means “there exists” and ∃! means “there exists exactly one”. Is there a similar notation for existence of an infinite number of examples?

How to logically analyze the statement: “Nobody in the calculus class is smarter than everybody in the discrete math class.”

I am self-studying Daniel Velleman’s “How to Prove It.” In the exercises for section 2.1, for question # 1b, I got a different answer than he did (his answer is in the back of the book). I think my answer is equivalent to his, and I also think I see yet another equivalent answer. But […]