Articles of logic

Is $(P\implies Q)\implies (\lnot Q\implies \lnot P)$ always true?

I’ve discovered that many theorems in mathematics are often in forms: $$P\implies Q$$ I’ve also discovered that usually $\lnot Q\implies \lnot P$, if $P \implies Q$ is a theorem and I’m interested if this is always the case. So in language of mathematical logic, is the following statement always true: $$(P\implies Q)\implies (\lnot Q\implies \lnot […]

Understanding common knowledge in logic and game theory

For $k = 2$, it is merely “first-order” knowledge. Each blue-eyed person knows that there is someone with blue eyes, but each blue eyed person does ”not” know that the other blue-eyed person has this same knowledge. (from I am not getting this. If there are more than one people that have blue eyes, […]

Rule C (Introduction to mathematical logic by Mendelson fifth edition)

By Existential Rule E4 $\mathscr B(t, t)\vdash (\exists x) \mathscr B(x, t)$. But how can we get back? How can we formalize $(\exists x) \mathscr B(x)\vdash \mathscr B(t)$? It is shown on page 74 and it is called “rule C (“C” for “choice”)”. But two questions: Designation. “… rule C deduction in a first-order theory […]

Does weak completeness (“If $\vDash\phi$, then $\vdash\phi$”) imply strong completeness (“If $\Gamma\vDash\phi$, then $\Gamma\vdash\phi$”)?

Suppose we have a proof system for classical first-order logic, where $\vDash$ denotes model-theoretic consequence and $\vdash$ denotes proof-theoretic consequence. We can distinguish two forms of completeness of the proof system. Call them weak and strong, respectively*: $$\text{for all sentences $\phi$, if $\vDash\phi$, then $\vdash\phi$}\tag{weak}$$ $$\text{for all sets of sentences $\Gamma$ and sentences $\phi$, if […]

If $\phi$ is a tautology then dual $\phi$ is a contradiction.

If $\phi$ is a statement form, Prove that: $\vDash \phi$ iff $\phi^{d}$ is a contradiction. where $\phi^{d}$ is the dual of $\phi .$ $\phi^{d}$ is the dual of $\phi $ is defined by: (i) $A^{d} = A$ for any statement letter A. (ii)$(\lnot \phi)^{d} = \lnot (\phi^{d})$ (iii)$(\phi \land \psi )^{d} = \phi^{d}\lor \psi^{d}.$ and […]

Gluing together mathematical structures, how?

By structure, I mean that which is defined here: What I’m looking for is a way of gluing together structures so that each structure used is embedded within the whole glued-together object. (Each–meaning not just “most”) I don’t need this embedding to be elementary; just something that “preserves” function, relation, and constant symbols. I […]

a convex function on a 2 dimensional closed convex set

Let us say I have a closed compact convex set $\mathbb{S}$ on the 2-D plane (eg: a circle). Let any point $p$ in the 2-D plane be represented by $p=(x,y)$. I define the max function over 2-D plane \begin{align} f(p)&=\max{(x,y)} \\ &= \begin{cases}x & \text{if $x\geq y$} \\ y & \text{if $x<y$}\end{cases} \end{align} Eg: At […]

Do the proofes in set theory rely on the semantics of the formulas used in the axioms?

Motivation: The Axiom of separation $$\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \phi(x, w_1, \ldots, w_n, A) ] )$$ is used to guaranty the existence of subsets by constructing formulas $\phi$ which, I assume, say true or false for every […]

In NBG set theory how could you state the axiom of limitation of size in first-order logic?

Limitation of size: “For any class $C$, a set $x$ such that $x=C$ exists if and only if there is no bijection between $C$ and the class $V$ of all sets.” In Von Neumann–Bernays–Gödel set theory how could you state the axiom of limitation of size in first-order logic similarly to the axioms stated in […]

difference between some terminologies in logics

$$1) \alpha_1,\alpha_2,\alpha_3…….\alpha_{k-2}, \alpha_{k-1}, \alpha_k\vdash\alpha $$ Is a valid sequesnt. $$2) \alpha_1,\alpha_2,\alpha_3…….\alpha_{k-2}, \alpha_{k-1}, \alpha_k\models\alpha$$ $$3) \alpha_1\land\alpha_2\land\alpha_3…….\alpha_{k-2}\land \alpha_{k-1}\land \alpha_k\implies\alpha$$ From what I read in book . I am not being able to differentiate beteen these three arguments.Do they really mean same thing? If not,then when do they mean same thing?