Articles of propositional calculus

A formula $\phi$ is logically equivalent to a another formula which contains only propositional variables and the connectives $\wedge$ and $\to$

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 […]

importance of implication vs its tautology

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 […]

Difference between a proposition and an assertion

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.

Diffucult Tautology to Prove

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: $[(\neg p \wedge q) \vee (p \wedge \neg r)] \vee (q \vee r)$

Logic Behind Epsilon-Delta Proofs (Single-Variable Calculus)

Most of what I am asking is based off this (fairly popular) article I’ve read here :, 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 […]

Proving and Modeling Logical Consistence

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 […]

Does the logical equivalence of 2 statements imply their semantic equivalence in everyday language?

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 […]

Deduction Theorem + Modus Ponens + What = Implicational Propositional Calculus?

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 […]

Translating sentences into propositional logic formulas.

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 […]

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 […]