# Tagged Questions

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### How to determine whether a set of propositions is consistent?

Definition of consistency is: A set of formulas ⊆ WFF is consistent iff there is no A ∈ WFF such that Σ ⊢ A and Σ ⊢ (¬A). Say you have a set of propositions statements (i.e. $A \lor B \rightarrow C$, ...
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### Convert this solution to inference notation

This is a proof for De Morgan's Law. Could you help me convert this to inference notation so I can understand the proof better? I find it hard reading this, specifically, which line each assumption ...
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### Simple Propositional Logic Explaination?

In this example, the prof states that "Q->R doesn't depend on the assumption Q so he can discharge it, but without assumption Q, he couldn't have concluded with Q->R so the answer still depends on the ...
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### Use of propositional logic connectives in the meta-language

I have a doubt that might seem a bit confusing so i will try to explain it the clearer i can. Suppose we have an expression "A o B" in the meta-language, where 'o' refers to those logical ...
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### Answering questions with truth tables

"With every dinner I have three rules": If I don't drink wine, then I eat soup If I eat soup and drink wine, then I'll have some pudding If I have pudding or don't drink wine, then I'll skip the ...
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### How to show tautologies in FOL using the truth definitions? [duplicate]

Anyone know how to prove these tautologies by way of truth definition? I take it that to solve a), I need to disprove a minimal counter example to the formula/sentence given? If so, how to formally ...
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### Translate n xor expressions to CNF?

I have n xor expressions: a xor b xor c xor d... I want to translate to cnf: The answer of cnf can be found here: http://www.wolframalpha.com/input/?i=a++XOR+b++XOR+c+XOR+d+ I want to write a ...
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### Put $(a \leftrightarrow b) \wedge c$ in DNF

$$(a \leftrightarrow b) \wedge c$$ I'm having problems with this. If I do: $$(a \rightarrow b) \wedge (b \rightarrow a) \wedge c$$ then $$(\neg a \vee b) \wedge (\neg b \vee a) \wedge c$$ But now I'm ...
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### Using the resolution method in logic

Using the resolution method in logic, having these clauses $$\{ \neg M \vee S, \neg S \vee T, \neg W \vee T, W \vee M, \neg T, \neg T \vee S \}$$ is it possible to reach the contradiction directly ...
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### How can I show that three statements are not logically equivalent to another?

I am given three premises and a conclusion. The premises are: \begin{gather} p \lor q \\ p \to \mathord{\sim}q \\ p \to r \end{gather} and the conclusion is $$r$$ I used a truth table and showed ...
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Let $a, b$ be any elements in a Boolean algebra. Prove that the following statements are equivalent: a) $a = ab$ b) $ab^{'}=0$ c) $a^{'}+ b = 1$ [Hint: Show the following chain of implications: ...
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### First-order logic: how-to produce interpretation where a given formula is false?

For example, given Theory T with predicates $$A(x), B(x), C(x,y), D(x,y), x=y$$ axioms $$\exists x.A(x) \land \exists x.B(x) \land \exists xy.C(x,y)\\ \forall x(A(x) \leftrightarrow \neg B(x)),$$ ...
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### Show that $\neg(p \Longleftrightarrow q)$ and $p \Longleftrightarrow \neg q$ are logically equivalent

Given there are 2 logical variables $p$, $q$ . Show that $\neg(p \Longleftrightarrow q)$ and $p \Longleftrightarrow \neg q$ are logically equivalent without using the truth table. And here is my ...
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### Expressing schedule of reinforcement rule using mathematical logic

I am trying to formalize the rules for application of different schedules in a reinforcement learning in special education. Children learn through trials. Each trial is successful if the child ...
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### Are my answers for these propositional calculus questions right?

Formalize the following English sentences as propositional logic formulas: $i)\quad$ "When the front and back doors are closed then the light is off." $ii)\quad$ "Either the lift doors are open or ...
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### Are $p \to (q \to r)$ and $p \to (q \wedge r)$ logically equivalent?

Is $p \to (q \to r)$ logically equivalent to $p \to (q \wedge r)$? I simplified each one, I got $\neg\, p \vee(q \vee r)$ and $\neg\, p ∨(\neg\, q \wedge r)$ respectively. Not sure if my ...
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### How to check the validity of this argument using the rules of inference?

I have this argument : I play basketball and football. If today isn't Saturday, then I play basketball and football. If today is Friday OR today is Saturday, then I don't play football. Therefore, ...
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### Is this a valid proposition?

Consider following two sentences. $x^2 = 1.$ Today is Thursday. The first statement can't be a proposition. because the truth of (1) depends on the value of $x$. For some values of $x$ it is true ...
"When the front and back door are closed then the light is off" p - "front" q - "light is off" r - "back doors are closed" $$(p\land r) \rightarrow q$$ Would this be logically correct?