# Tag Info

### Prove that [(p ∨ q) ∧ (p → s) ∧ (q → t)] → (s ∨ t) without using truth tables. What is the proof technique used?

This is the propositional form of a derived inference rule which is known as constructive dilemma or separation of cases. It is actually a disjunction of two modus ponens statements (hence, the name '...
• 2,561
1 vote

### Simplify, equivalent for $p \land(p\implies q)\land(p\implies\lnot q)$

Make a truth table. p q (p → q) p ∧ (p → q) (p → ¬q) p ∧ (p → q) ∧ (p → ¬q) T T T T F F T F F F T F F T T F T F F F T F T F So your expression is false all the time. Answer is $\boxed{F}$.
• 31

### An example of a maximal consistent set?

You can build a maximally consistent set of formulas as $\Sigma = \{ \psi \mid v(\psi) = T\}$ where $v$ is a truth assignment and $\psi$ is a formula of propositional logic. $\Sigma$ is satisfiable ...
• 163
Accepted

• 131k
1 vote

### The prenex form doesn't seem equivalent to the original sentence

$∀x\Big(∀y f(y) → g(x)\Big)\tag{A1}$ for all x, if for all y, f(y) is true, then g(x) is true $∀x∃y\Big(f(y) → g(x)\Big)\tag{A2}$ ...
• 40.6k

### Do De Morgan's laws hold in propositional intuitionistic logic?

This is basically the counterexample of that excellent answer, put a bit more abstractly. Choose a Boolean algebra $B$ that has elements besides $0$ and $1$. Define $A=B\cup\{\star\}$ with $\star$ ...
• 6,645

• 13.3k

### Simplifying $(p\vee\neg q)\wedge(q\vee\neg r)\wedge(r\vee\neg p)$

When I look at the propositions as Venn diagrams, it looks like you're saying that you need the inside of a Venn diagram OR the outside of another one, and this for all three Venn diagrams. The result ...
• 2,868
1 vote

### Notation for equivalent equations

What is the notation for showing that equations are equivalent after rearranging terms? For example, $$s=r\theta\implies r=\frac{s}{\theta}.\tag{✘ 1}$$ Is this the correct way to write it? Consider ...
• 40.6k

### Proving $\vdash \neg \neg P \to P$ (double negation elimination) in first order logic, preferrably without deduction theorem

Minimal proof (17 steps) You're asking for *2.14 from Metamath's pmproofs.txt database, which has a condensed detachment proof DD2DD2D13DD2D1311 in D-notation (that ...
• 282

### What does 'imply' mean in maths?

I'm not sure we need another answer here. But most of the other answers follow classical logic closely. I think the constructive logic meaning may be more intuitive. It's simply: We say that $A$ ...

### What does 'imply' mean in maths?

Adding to the other answers (which have pointed out that $\boldsymbol P$ implies $\boldsymbol Q$ means that $P$ being true necessitates that $Q$ be true): When we know that $\boldsymbol P$ is false ...
• 40.6k

### What does 'imply' mean in maths?

It means the same as it means in regular English. If we say “A implies B”, it means that the truth of B is a consequence of the truth of A. There are many different ways of expressing this. For ...
• 43.8k

### Substitution principle for hypothetical judgements

I shall provide an interim, but I hope, workable enough answer, later to be replaced with a permanent one. Discharging a hypothesis is essentially to relieve a formula of its function as a hypothesis ...
• 2,561

### Is "Alice loves candies" actually necessary for "Alice loves all sweet foods"?

The English-language statement "Alice loves candies" may either be taken to imply that Alice loves all candies, that Alice loves at least some candies, or most typically that Alice loves ...
• 563

### Is "Alice loves candies" actually necessary for "Alice loves all sweet foods"?

It seems like the analysis is made more complicated by thinking about negations. More simply, consider the implication $B \implies A$; that is, "If Alice loves all sweet foods, then Alice loves ...
• 8,668
Accepted

### Is "Alice loves candies" actually necessary for "Alice loves all sweet foods"?

Your mistake is that for $A$ to be a necessary condition for $B$ it must hold that $\neg A\Rightarrow\neg B$ (if the necessary condition is not met, then the condition for which it is necessary is ...
• 14.5k
1 vote

### Is "Alice loves candies" actually necessary for "Alice loves all sweet foods"?

A:Alice loves candies. B:Alice loves all sweet foods. I am trying to identify if A is a logically necessary condition (LNC) for B. the answer key says that A is a ...
• 40.6k

### How to make a formal proof with A → (B ∨ C) ⊢ (A → B) ∨ (A → C)

what about to use a truth table?
1 vote

• 6,679
1 vote

### Did Jim Carrey get away with lying in 'Liar Liar'?

Since imperatives like "write it" and "write a poem" don't have truth values (regardless of whether they are obeyed, and of the ambiguity of "it"), they are not ...
• 40.6k

Top 50 recent answers are included