# Questions tagged [propositional-calculus]

Appropriate for questions about truth tables, conjunctive and disjunctive normal forms, negation, and implication of unquantified propositions. Also for general questions about the propositional calculus itself, including its semantics and proof theory. Questions about other kinds of logic should use a different tag, such as (logic), (predicate-logic), or (first-order-logic).

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### Is it safe to use $|f(x)|<g(x)\iff -g(x)<f(x)<g(x)$?

Some peoples often use the following. $$|f(x)|<g(x)\iff -g(x)<f(x)<g(x)$$ The weird part occurs when $g(x)<0$, for example: $$10<f(x)<-10\to |f(x)|<-10$$ where $x\in\{\}$ for ...
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### Checking tautology

Given a Boolean formula $\phi$ in CNF form, I'll check whether there exists a clause that can be falsified i.e. check for literals of the form $x \vee \neg x$. If there are not any such literals in a ...
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### why is 'P implies Q' true when P is false and Q is true? [duplicate]

This is the truth table for P implies Q. i get it, When P is True and Q is True then the truth value of this "implication" function is True.('implies' function is performed correctly, if a ...
389 views

### Is there a logical system capturing these subtleties about implication?

I am teaching an introduction to proof course. The truth table for implication is always extremely difficult for students to understand. It feels, to me, that the truth table is an imperfect ...
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### Can B be uniquely determined from A and A->B? Similarly, can A be uniquely determined from B and A->B?

Can $B$ be uniquely determined from $A$ and $A\rightarrow B$? Here's my attempt using the truth table: $A$ $A\rightarrow B$ $B$ T T T T F F F T T/F F F undefined (leads to a contradiction) ...
77 views

### false introduction sequent calculus?

I'm proving the following proposition using sequent calculus. I got stuck at the very top line. My thought is that if the both hypothesis inside the curly bracket are true, then it's false. So I think ...
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### Rules of inference for exclusive disjunction and logical biconditional

Here are the rules of inference in natural calculus of propositions. I'd like to extend this calculus (conservatively) adding exclusive disjunction $\oplus$ and logical biconditional $\sim$ and obtain ...
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### How do I Finish Simplifying this Laws of Propositional Logic Problem?

In my Discrete Mathematics class I have just started learning about the laws of propositional logic. I've got about halfway through a problem but am now stuck for a while now and I don't understand ...
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### When first encountering a set of primitive inference rules, how do we approach the derivation of the very first derivable inference rules?

I'm currently learning Ebbinghaus et. al.'s propositional calculus in their book Mathematical Logic, and I'm trying to derive the very basic rules of inference such as $\land$ introduction, the law of ...
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### Proving Sequent Calculus Statement

I have to prove the sequent $$\vdash (\lnot A \lor \lnot B) \to \lnot (A \land B)$$ using the inference rules for natural deduction listed here (pp. 7-8). I'm super new to natural deduction and ...