# Tag Info

### What is meant by "logically incorrect"?

Rather than assume what the author means, I consulted the textbook (p. $155$) and examined the excerpt in context... "Logic is the process of deducing information correctly -- it is not the ...
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### What exactly does $X - (Y ∪ Z)$ mean?

See the image of your expression in a Venn diagram:
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### Deriving A, ¬A ⊢ B in a weak Hilbert proof system

Mayhap you missed an additional axiom, or whoever assigned this problem tried to pull a fast one on you? Either way, you cannot derive $A \rightarrow \neg A \rightarrow B$ in the Hilbert system given ...
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### Tautologies in classical logic

In Classical Logic it is always the case that either $P$, $\neg P$, or both. You will always be dealing with one of those three options, even if the third never happens. It is also always true for ...
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### Can an implication and its converse both be false?

Put $Px$ for $x$ is prime, and $Gx$ for $x > 100$. Now distinguish $\forall x(Px \to Gx) \lor \forall x(Gx \to Px)$ from $\forall x((Px \to Gx) \lor (Gx \to Px))$ The first I think properly ...
• 54.5k
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### Seeking a More Efficient Solution for Knights and Knaves Logic Puzzle

Whenever anyone in a knight-or-knave problem says a statement $P$, we can read it as the guaranteed to be true statement "Either I am a knight and $P$ is true, or I am a knave and $P$ is false.&...
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### Is "yesterday it was sunny" a proposition?

As far as mathematical logic is concerned, a proposition is simply whatever you can assign a truth value to. But it doesn’t matter how you do it, what the sentence means and whether your assignment ...
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### Asserting that when (P→Q) and (Q→R) are true, then so is (P→R)

You ask: But there are also instances where $P\rightarrow R$ is true and $P\rightarrow Q$ and $Q\rightarrow R$ is false. Is this indeed correct or did I do something wrong with the truth table? I ...
• 99.6k
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### In a direct proof, do your chain of deductions have to involve the antecedent in any way in order for this to be considered a "direct proof"?

A direct proof is simply a proof whose assumptions are drawn from just the set of axioms, established theorems and other givens. This is not to say that a direct proof (actively) assumes any ...
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### Should I completely forget about the meaning of if-then sentences as used in ordinary language and assign them a new meaning?

We go to Jack and Jill's party in April. Later we can't remember whose birthday was being celebrated. But I tell you that I think Jill has an autumn birthday. So you say "Ah, if Jill has an ...
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### On the tautology $(P \implies Q) \vee (Q \implies P)$

Yes, it is always true. I guess you are thinking about real life examples and that is confusing you because of the time dimension added to the situation. For instance: Let $Q$ be "I eat" and ...
Accepted

### Tautologies in classical logic

"According to its truth table, $P \lor \lnot P$ is a tautology, i.e. it is true for all truth values of its constituent propositions." OP is missing the Point that the constituent ...
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### Context on the theory of category theory

As Alex says in the comments, you can use two-sorted first order logic for this, with a sort for objects and a sort for morphisms. But actually this is technically unnecessary, and (small) categories ...
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### Why is $((p \land q) \Rightarrow z) \Rightarrow (p \Rightarrow z) \lor (q \Rightarrow z)$ true?

The sentences \begin{align}\Big((P \land Q) → Z\Big) → \Big((P → Z) \lor (Q→ Z)\Big)\tag{✔️1}\end{align} and \begin{align}∀\color{green}n\bigg(\quad\Big(\big(P\color{green}n ∧ Q\color{green}n\big) → Z\...
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### How to prove that if $P, \mathord{\sim}Q\vdash\bot$, then $P\vdash Q$

Strictly speaking, ripped out of context, $P \vdash Q$ doesn't have a determinate meaning. It is like “Jill is before Jack". In what way? On the class list? In who gets to bowl first? In order of ...
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### If $P \implies Q \implies R$ holds, does it follow that $\lnot R \implies \lnot Q \implies \lnot P$ is also true?

You want to be a little careful in not confusing logical notation with mathematical notation. In particular, the logical operator called the material implication (more often written using $\to$ rather ...
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### If P and Q can never be true, are they still "necessary" and "sufficient" for each other?

In mathematics, "P is necessary for Q" is defined as meaning "it is impossible for P to be false and Q to be true." If both P and Q are totally impossible, then it's certainly ...
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### Does $\Gamma \vDash \alpha \iff \Gamma \vDash \beta$ implies $\alpha \iff \beta$?

No it doesn't. Example: whenever p is true then p or q is true. Whenever p is true p is true. But it is not true that p iff p or q.
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### What is the relationship between ultrafilters and propositional theories?

The relationship is Boolean algebras A filter in a boolean algebra corresponds $1:1$ with surjective homomorphisms from that algebra to a some other algebra. An ultrafilter in a boolean algebra ...
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### What is meant by "logically incorrect"?

And If I said "Socrates is a Martian and Martians live on Pluto, therefore 2 + 2 = 4" then what I said was logically incorrect. I agree with your observation that this implication is (...
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### Are the derivations of a consistent set a consistent set?

$\Omega$ is indeed a consistent set of formulas. In other words, it is indeed possible for all formulas in $\Omega$ to be simultaneously true. This is because every formula $\varphi \in \Omega$ is a ...
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### Short proof of the principle of explosion in propositional calculus

To save us from parenthesis hell, let's follow the proof-theory convention of writing $P \rightarrow (Q \rightarrow R)$ as $P \rightarrow Q \rightarrow R$. We can write down a short, seven-line proof ...
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