Let $v_0$ be the valuation that assigns true ($T$) to every propositional variable. I’m trying to show that any formula $\phi$ is logically equivalent to one with only propositional variables and the binary connectives $\wedge$ and $\to$ if and only if the natural extension of $v_0$, $v$ say, assigns the value $T$ to $\phi$. If […]

I’m self-taught in logic, started with programming. Texts stress the importance of the implication operator but I fail to see operator’s importance. It was always intuitively clear that, $$ p \rightarrow q \equiv \neg \ p \lor q $$ Keeping that in mind, my questions are: Does implication allow for additional rewrite rules or higher […]

It may be a silly doubt, but let me ask this. What is the difference between a proposition and an assertion? I know there’s a very thin line between the two terminologies, but I’m unable to get that.

I’m trying to show that the following is a tautology: $(p \vee q) \wedge (\neg p \vee r) \Rightarrow (q \vee r)$ Can anyone help, as far as I can get is to the following: $[(\neg p \wedge q) \vee (p \wedge \neg r)] \vee (q \vee r)$

Most of what I am asking is based off this (fairly popular) article I’ve read here : https://bobobobo.wordpress.com/2008/01/20/how-to-do-epsilon-delta-proofs-1st-year-calculus/, but most lecturers, use this same process to tackle epsilon-delta proofs, so what I am asking should be pretty universal to epsilon delta proofs. Epsilon-Delta Definition of a Limit I’ve just referenced this here, for some added […]

Suppose I have a finite list of logical statements (would these be called axioms?) and for the sake of discussion say that there are 6 such statements. All statements are in the form of propositional logic. From these statements I can prove other statements true. However, I wish to show that my list of logical […]

Consider the statement, $1.$ “If it is Tuesday, then it is raining”. In propositional logic, 1 would read as, “$p \implies q$.” Now, in accordance with the rules and definitions prescribed in logic, we have a plethora of logical equivalences. We can rewrite 1 as $ \neg p \vee q$, and in English, $2.$ “It […]

Implicational propositional calculus is a system of propositional calculus in which implication is the only logical connective, and all other connectives are defined with respect implication and a single false statement. Consider the system of implicational propositional calculus with the following two rules of inference: the Deduction Theorem, which states that if by assuming P […]

I have some trouble with translating certain sentences into a statement of propositional logic. It is homework, so I will also be happy with some hints. Please keep in mind that I translated these sentences from dutch into english, so there can be mistakes. But the keywords are the same. The sentences are: a. To […]

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 […]

Intereting Posts

Degree of a function
Countable basis of function spaces
Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.
Non-concrete categories constructed from concrete categories
Economically computing $d\beta$
Limit of $L^p$ norm when $p\to0$
Properties of Weak Convergence of Probability Measures on Product Spaces
Expressing in the form $A \sin(x + c)$
Why are the differences between consecutive squares equal to the sequence of odd numbers?
In any Pythagorean triplet at least one of them is divisible by $2$, $3$ and $5$.
Probability that A meets B in a specific time frame
Approximating $\arctan x$ for large $|x|$
Minimum number of out-shuffles required to get back to the start in a pack of $2n$ cards?
Questions about Serre duality
Finding the sum of this alternating series with factorial denominator.