# Tagged Questions

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### Proposition into spoken language

Given: $\sim( p \leftrightarrow (q \vee r) )$ $p:$ It's raining $q:$ The sun is shining $r:$ There are clouds in the sky. Translate the proposition into spoken language. ...
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### Is $'' \sum_{n = 1}^{\infty} (-1)^n \; \text{is a real number}''$ an invalid statement or a false proposition?

So we're beginning an introductory logic course and my professor is giving examples for valid statements/ propositions - meaningful statements that are either true or false but not both. So he puts ...
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### Laws of equivalence needed to prove $\;q \leftrightarrow (¬p ∨ ¬q) ≡ (¬p ∧ q)\;?$

I'm not sure which laws should be applied and how I can tell for myself how to discern which laws I should use - any and all help is appreciated.
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### Logical argument [on hold]

Let A, B, and C be propositions. Let ∧ denote logical AND, let ∨ denote logical OR, and let ¬ denote logical NOT. Argue that if (𝐴∨𝐵) ∧(¬𝐵∨𝐶) is true, then (𝐴 ∨𝐶) must be true as well.
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### Why is removing the negation worse than adding it?

Natural Deduction Rule (¬I): Natural Deduction Rule (RAA): My book [Ian Chiswell & Wilfrid Hodges, Mathematical Logic (2007)]presents these two rules and then adds: The use of (RAA) can ...
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### A simpler derivation of ($\phi \lor (\neg \phi)$)

In Chiswell&Hodges [Ian Chiswell & Wilfrid Hodges, Mathematical Logic (2007)] they use this derivation to prove ($\phi \lor (\neg \phi)$): A page earlier they used a simpler derivation that ...
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### Is there a proof of this statement about deductions?

Is there a proof of the following statement: you cannot prove with natural deduction theorems that are unprovable in a Hilbert-style proof system? The logic in discussion is either propositional logic ...
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### Prove that the disjunctions of all conjucts is a disjunctive normal form

Question: I am attempting the following exercise from An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof by Peter B. Andrews: X1408. Prove that if $\mathbf{A}$ is a wff ...
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### alternative Compactness theorem proof

I'm attempting a problem which requires me to prove the compactness theorem for propositional logic ![enter image description here][1]in a slightly different way to normal. I'm struggling to ...
I have been stuck on this proof for a while. Here's where I'm at: Goal $(\neg B \to \neg A) \leftrightarrow (A \to B)$ l 1. $A \to B$ ll 2. $\neg B$ lll 3. $A$ lll 4. $B$ Elim 1,3 ...