# Tagged Questions

Questions about truth tables, conjunctive and disjunctive normal forms, negation, and implication of unquantified propositions would fit very nicely under this tag. Questions about other kind of logics should be tagged with [logic] instead.

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### Proving that the theorems of one logistic system are also theorems of another logistic system

Question: I am developing the proof for the following exercise from An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof by Peter B. Andrews: X1102. Let $\mathscr{M}$ be ...
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### Help to understand material implication

This question comes from from my algebra paper: $(p \rightarrow q)$ is logically equivalent to ... (then four options are given). The module states that the correct option is $(\sim p \lor q)$. ...
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### How do I Prove the Theorems Needed for “The Deduction Meta-Theorem” from CδCpqCpδq?

I use Polish notation. δ stands for a variable function (or functor) of one argument. A section of Lukasiewicz's book on Aristotle's syllogistic from the modern point of view reads: "I should like ...
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### How to prove that $P \rightarrow Q$ is equivalent with $\neg P \lor Q$?

In my book about Logic, which is called 'Language, Proof and Logic', by the way, there is explained that the conditional $P \rightarrow Q$ is equivalent with $\neg P \lor Q$. There is another ...
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### Need Hints Prove “$((\neg \alpha \to \alpha) \to \alpha)$” Using Axiom 1,2,3 and MP and deduction theorem

$((\neg \alpha \to \alpha) \to \alpha)$ Hi, I am trying to prove this. Can someone gives me some hints to start the question... My friend told me I might need to use deduction theorem here, but I ...
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### Does $\neg(x > y)$ imply that $y \geq x$?

Given any arbitrary binary relation $\geq$ defined on some set $S$, we define a new binary relation $>$ on $S$ by: $$x > y \quad\text{iff}\quad (x \geq y) \wedge \neg(y \geq x)$$ In accordance ...
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### Distinguishing between valid and fallacious arguments (propositional calculus)

I am having some difficulties understanding logical arguments. I was taught that the notion of a valid argument is formalized as follows: "An argument $P_1, P_2,\cdots , P_n ⊢ Q$ is said to be ...
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### Prove $\;\big((p\rightarrow q) \lor (p \rightarrow r)\big) \rightarrow (q\lor r)\equiv p \lor q \lor r$ without use of a truth table.

Without using the truth table, I need to prove: $$\big((p\rightarrow q) \lor (p \rightarrow r)\big) \rightarrow (q\lor r)\equiv p \lor r \lor q$$ Up until now, we've been using truth-tables to ...
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### Proof ¬q → ¬p from premise p → q using deductive system& Modus ponens

Using the following deductive system D1: ...
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### Every sentence in propositional logic can be written in Conjunctive Normal Form

While reading through Artificial Intelligence: A Modern Approach by Stuart Russell and Peter Norvig, I came upon the following question: Any propositional logic sentence is logically equivalent to ...
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### Functional completeness of $\{\text{or},\text{ xor}, \text{ xnor}\}$

I need to prove the functional completeness of $\{\text{or},\text{ xor},\text{ xnor}\}$ with the help of $\{\text{not},\text{ or},\text{ and}\}$ (which have been already proven to be functional ...
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