# 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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### How to show that a valid rule is not derivable in intuitionistic propositional calculus.

First a word on notation, let the following be an inference rule that takes $\Gamma$ (a set of well-formed formulas) as premises and has $\psi$ (a well-formed formula) as a conclusion. This is ...
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### Expressing “For every positive integer $a$, there exists an integer $b$ with $|b| < a$ such that $|bx| < a$ for every real number $x$” symbolically

I would like to express For every positive integer $a$, there exists an integer $b$ with $|b| < a$ such that $|bx| < a$ for every real number $x$ symbolically. My attempt: For every positive ...
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### How can I know when to negate quantifiers when taking the contrapositive of a statement?

Continued from a question I asked here, since I believed this question deserves its own thread. When taking the contrapositive, I was taught to negate the quantifiers as well. For example, if we have ...
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### How does a negation distribute when a statement is expressed in words?

Consider the following implication: Let $x,y,z$ be integers. If exactly two of the three integers $x,y,z$ are even, then $3x + 5y + 7z$ is odd. The contrapositive of the statement above would be: ...
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### resolvent clause theory

This is about correctness of resolution lemma Let R be a resolvent of two clauses $C_1$ and $C_2$. Then $C_1, C_2 \models R$. Proof By definition $R = (C_1 ā \{L\}) \cup (C_2 ā\{\bar{L}\})$ (for some ...
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### Simplify $¬¬qā§T$ to get $q$. [closed]

I am having issues understanding how to use the laws of propositional logic. Simplify $Ā¬Ā¬qā§T$ to get $q$.
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### Negation of $(p\to\neg q)ā§\neg p$ [closed]

can someone please help me out with the negation of the following: $(p\to\neg q)ā§\neg p$ I keep getting confused at mid-way, so I'd appreciate a step-by-step approach. Thanks!
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### Using Truth Tables to Solve Logic Riddles

I've been traversing the practice exercises in Kenneth Rosen's Discrete Math and Its Applications in preparation for an upcoming class that is heavily proofs-based. Constructing truth tables from ...
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### Simple Logical Simplification - Distributive Law

I can't seem to understand the hopefully simple step of getting from : $(a \land b \land c) \lor (a \land b \land \lnot c)$ to: $(a \land b) \land (c \lor \lnot c)$ This step is in the answers and ...
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### How to prove that propositions are not logically entailed?

Let's consider the next proposition: AāBāØAāØB I used a truth table to show that there exists model1={A=False, B=False} where (AāB) is True but (AāØB) is False. It means AāB does Not entail AāØB. Can you ...
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### Does (A and B) entails (A if and only if B) and what is intuition?

Is the next statement correct: (Aā§B)āØ(AāB) ? Formal definition of entailment is this: Ī±āØĪ² if and only if, in every model in which Ī± is true, Ī² is also true. I used a truth table to show that there is ...
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### A small question concerning the correctness of induction

I think I've read enough about the Peano axioms and I also checked the similar posts here on StackExchange, but all I want to know is why induction can't be validated for all n by writing down the ...
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### Is this a correct translation from english into symbolic logic? [duplicate]

"You can fool some of the people all of the time, and you can fool all of the people some of the time, but you canāt fool all of the people all of the time." (Abraham Lincoln) Let $P$ be &...