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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'If…then…' and '…if…' and '…only if…' and 'If… only then…' statements?

Suppose you have two statements A and B and "If A then B". I am trying to think of what this implies and alternative ways of writing this. I think "If A then B" = A$\rightarrow$B = "A is ...
0
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0answers
30 views

Formalize “Statement $A$ is the correct explanation of statement $B$”

If I have two statements. Let say Statement $A$ and Statement $B$. What will be the necessary condition or how to write the following conditions mathematically? Statement $A$ is the correct ...
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1answer
36 views

Why if the antecedent P is false, and the consequence Q true, then the implication P $\Rightarrow$ Q is true? [duplicate]

I know that that's the definition but I wonder why logicians choose that thefinition to be true. It sounds strange to me and I cant make sense of it if someone tell me 'if the sky is red, then I'm ...
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1answer
37 views

Let $\tau$ and $\rho$ be tableaus such that $\tau \leadsto \rho$. Prove that $\tau$ is satisfiable if and only if $\rho$ is satisfiable.

I have this definition: Let $\nu$ be any propositional interpretation. Let $b$ be any branch of a tableau. Say that $\nu$ is faithful to b if and only if for every formula, $A$, on the branch, ...
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1answer
28 views

Can boolean logic compute any sort of mathematical operation?

Computers fundamentally do logical operations on the input and memory they have (as far as I know). Computers are used by mathematicians to do all sorts of mathsy operations (as far as I know). Does ...
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1answer
57 views

What is the difference between Boolean logic and propositional logic?

As far as I can see, they only employ different symbols but they operate in the same way. Am I missing something? I wanted to write "Boolean logic" in the tag box but a message came up saying that if ...
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1answer
39 views

Showing $(¬P\wedge¬Q)\vee(¬P\wedge Q)\equiv¬P\wedge(¬Q\vee Q)$ by distributive law(s)

I want to show that $$(¬P\wedge¬Q)\vee(¬P\wedge Q)\equiv¬P\wedge(¬Q\vee Q)$$ by one of the two Distributivity Laws: $$P\wedge(Q\vee R)\equiv(P\wedge Q)\vee(P\wedge R)$$ $$P\vee(Q\wedge R)\equiv(P\vee ...
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1answer
39 views

Why is that: $P \Rightarrow T$, truth value(P) = ?, but $(P\Rightarrow F) \Rightarrow$ Truth value (P) = F

Why is that: If: P :proposition. T: true statement F: false statement $$P \Rightarrow T $$ In this statement, we can not have for sure the Truth value of P (if P is T or F) , but, in this ...
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2answers
135 views

Completness and Set of Result of One Set ?!?

Dear Everyone on this Wonderful Sites: I'm so glad to participate on this site and ask the first question that mentioned on the Contest some days ago. I ran into a question that wrote this set: ...
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2answers
53 views

a simple but complex old exam question on logic!? [closed]

We are given a set, {$ q \to p, \neg q \to r, \neg r$}. Which of the following is not consequence result of this set? a) $ p \vee q$ b) $ \neg r \vee q$ c) $ (\neg p \wedge \neg q) \to r$ d) $ ...
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2answers
49 views

Do all logic problems have one solution? [closed]

Analyze the logical forms of the following statements: x and y are natural numbers, and exactly one of them is prime. Below are the two answers that I got. The first one is the one the author ...
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0answers
8 views

Is there a way that I can contribute to this site? [migrated]

Hi guys I am studying how to prove it by Velleman. its a great book and I feel that it can really help people out with logic. However, the one thing that this book lacks are the answers to all the ...
2
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1answer
31 views

How does distribution work in logic?

Hi guys A question regarding propositional logic. ¬(¬P∧Q)∨(P∧¬R) = (P∨¬Q)∨(P∧¬R) ...DeMorgan's, Double Negation law = ((P∨¬Q)∨P)∧((P∨¬Q)∨¬R) ...Distribution law = (P∨¬Q)∧((P∨¬Q)∨¬R) ...
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1answer
22 views

Can I use two inferred clauses to get the empty set?

In resolution can I use two inferred clauses to reach the empty set? Consider this set of clauses: $\{ p \lor q,\neg p \lor r, \neg p \lor \neg r, p \lor \neg q\}$ \begin{align*} \quad p \lor ...
2
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2answers
41 views

How do I show that $\forall x \ P(x) \vee \forall x \ Q(x)$ and $\forall x (P(x) \vee Q(x))$ are NOT logically equivalent?

Show that $\forall x \ P(x) \vee \forall x \ Q(x)$ and $\forall x (P(x) \vee Q(x))$ are not logically equivalent. Can someone give a hint?
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1answer
20 views

$P(x)$ consists of the integers -2, -1, 0, 1 and 2. Write out each of these propositions using disjunctions, conjunctions and negations.

Suppose that the domain of the propositional function $P(x)$ consists of the integers -2, -1, 0, 1 and 2. Write out each of these propositions using disjunctions, conjunctions and negations. a) ...
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2answers
81 views

Truth table of proof by contradiction

The following is the truth table for an implication: $(T\Rightarrow T) = T$ $(T\Rightarrow F) = F$ $(F\Rightarrow T) = T$ $(F\Rightarrow F) = T$ Now, in an implication involved in a proof by ...
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2answers
36 views

The propositional logic expression for ∃x∀yP(x,y)

Where u.d. of x is {1,2,3} and y is {a,b} The given answer is ((1,a)Λ(1,b)) V ((2,a)Λ(2,b)) V ((3,a)Λ(3,b)) But I get the expression ((1,a)V(2,a)V(3,a)) Λ ((1,b)V(2,b)V(3,b)) Why is my one wrong ...
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3answers
34 views

Proving logical equivalences

The question is to prove $\neg (p \wedge q) \to (p \vee r)$ equivalent to $p \vee r$ So far, I got $¬[¬(p \wedge q)] \vee (p \vee r)$ - implication $(p \wedge q) \vee (p \vee r)$ - ...
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1answer
20 views

Simpliest Propositional Equivalences proof question

I'm solving some propositional equivalences questions and I just want to make sure that following two logics. If, $p \land q = q \land p$ $p \vee q = q \vee p$ in any case, are correct because ...
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1answer
33 views

Show that $\neg$ and $\wedge$ form a functionally complete collection of logical operators.

Earlier this day I ask about the assignmet: Show that $\neg$ and $\wedge$ form a functionally complete collection of logical operators. I was given the hint that I could use De Morgan law to show ...
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5answers
104 views

Logical equivalent of $p\to(q\to p)$

Is Logical equivalent of $p\to(q\to p)$, $p\to(p\wedge q)$ or $p\to(p\vee q)$? I have a truth table: $$\begin{array}{c|c|c|c} p&q&p\wedge q&p\vee q&q\to p&p\to(q\to ...
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1answer
28 views

Show that $\neg$ and $\wedge$ form a functionally complete collection of logical operators [duplicate]

Show that $\neg$ and $\wedge$ form a functionally complete collection of logical operators Can someone give a hint?
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1answer
45 views

How can I show logically equivalence without a truth table

Show that $(p \rightarrow q) \wedge (p \rightarrow r)$ and $p \rightarrow (q \wedge r)$ are logically equivalent. I tried to do this making a truth table but I think my teacher wants me to solve it ...
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1answer
25 views

Logic question in propositional calculus

How do we prove the following formula for all natural numbers $n$ in propositional calculus $[(q_{1}\vee q_{2}...q_{n})\wedge((q_{1}\Longrightarrow r)\wedge(q_{2}\Longrightarrow ...
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0answers
65 views

Proving $\vdash (p\to q)\lor (q\to r)$ using natural deduction

I'm trying to prove the following: $\vdash (p\to q)\lor(q\to r)$ using only intuitionistically valid rules. I've tried a few different ways, and I think my problem is that I'm not sure what ...
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2answers
55 views

Value V on some formula in Logic

I want to calculate, how many value $v$ on {$p,q,r$} has, such that sentence $(p \to (q\wedge r)) \to r$ gets value $0$? I solve it via truth table, any other methods for solving such questions? or ...
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2answers
23 views

How to express sample space

I have been given No answers though please!
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2answers
65 views

Question about propositional logic

I was just learning the truth table of the propositional logic . I understand the truth table for the conjunction and disjunction because they make sense in the real life. The conjunction A∧B means "A ...
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0answers
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how many symantec equations is there in propositional calculus with n boolean variables?

how many symantec equations is there in propositional calculus with n boolean variables? The answers are: 1) 3^n 2) n 3) 2^(2n) 4) 2^n I think the answer is 2^n. Do you think its correct? ...
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1answer
43 views

Admissible rule in classic logic [closed]

The classical propositional logic admits the concept of admissible rule, and would like some examples of propositional calculus with the 'admissible rule', on wikipedia I don't quite understand...
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0answers
31 views

Monotonic operators in classical logic

Which means monotony for a logical operator, and affinity, in propositional calculus affinity..., here on wiki do not quite understand!!
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0answers
30 views

Can the OR function be linearly separated?

I have two questions regarding linear functions and propositional calculus: 1) How do you decide if, for example, the OR function can be linearly separated? The answer is Yes, however I don't know ...
2
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1answer
94 views

Predicated needed for proof using structural induction

I have a set, $F$, of boolean formulas defined inductively as follows: $X_{i} \in F, \: \forall i \in \mathbb{N} \: \text{(variables)}\\ A \in F \implies \neg A \in F\\ A, B \in F \implies A \land B ...
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2answers
46 views

Proof by cases and contradiction. Is this valid?

Say i have a hypothesis of the following form: $P \lor Q$ and a conclusion $\neg A$. I try a proof by contradiction; so I assume $A$. Now what I am trying to do is break the hypothesis into cases, so: ...
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1answer
37 views

Deduction theorem with undischarged statement

I am reading "Mathematical logic" by Ian chriswell and Hodges and at one point in the text they mention the deductive theorem (page 17) which states; If $\Gamma \cup \left \{ \phi \right \} \vdash ...
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2answers
64 views

How can I indicate a truth table if its Valid or Invalid?

Construct a truth table for Destructive Dilemma using the general symbolic notation for the rule of inference, T for true value, F for false value. Indicate whether valid or invalid. Is this the ...
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2answers
67 views

Are there some techniques for checking whether a statement implies another without truth tables?

Are there some techniques for checking whether a statement implies another without truth tables? For example, I was asked whether $P\Longrightarrow P_{1}$ given the following statements: $$P: [p ...
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1answer
46 views

How to prove a tautology using proof by contradiction?

I am trying to learn proof by contradiction. How would i go about proving that ((A => B) and (C => D)) => ((A => D) or (C => B)) is a tautology, ...
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2answers
91 views

The logical law of closed systems of sentences

Consider the usual logical connectors $\wedge, \vee, \supset, \neg$ (i.e., "and", "or", material implication, negation) and the "stroke" $/$ defined as $p / q := (\neg p) \vee (\neg q)$. In his book ...
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1answer
20 views

Prove conjecture using premises

I have three premises with me defined: $(B \land L) \implies A$ $(A \land D) \implies \lnot H$ $\lnot J \implies (D \land \lnot H)$ I need to prove the following conjecture with the help ...
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5answers
111 views

Why is $P \to Q \equiv \neg P \vee Q$?

By truth table, we know that $P \to Q$ is equivalent to $\neg P \vee Q$. But I'm trying to understand why this work? How can connective "or" be implication. I tried some examples but I still can't ...
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2answers
45 views

Is the following propositional function well defined?

My question is fairly simple: Is $(P \wedge Q)(x)$ equivalent to $P(x)\wedge Q(x)$? Reason I'm asking, is that when I asked my tutor he said the statements weren't equivalent because if $P(x) = "x\ ...
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1answer
99 views

Proving a property about logical entailment

I have an intuitive idea that, given some set of formulas $Γ$, and two formulas $A, B \not\in Γ$, $((Γ\cup{A}) \models B)↔(Γ \models (A→B))$. I can rationalize this as, if the left side of the ...
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1answer
51 views

Is an argument valid simply if its form is valid?

Can I conclude that an argument is valid if its argument form is valid? I realize that a false premise may lead to an incorrect conclusion (which is not what I'm asking). I see a lot of questions ...
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1answer
61 views

Logical implication and valid arguments question

The following is a valid argument: $[[p \lor (q\lor r)]\land \neg q] \rightarrow (p\lor r)$. Determine the rows of the table crucial for assessing the validity of the argument and which rows can be ...
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0answers
40 views

Glivenko's theorem for propositional logic: $\neg\neg A, \neg\neg(A \rightarrow B) \vdash \neg\neg B$. [duplicate]

In proving Glivenko's theorem for propositional logic I have found myself not able to prove the following: $\neg\neg A, \neg\neg(A \rightarrow B) \vdash \neg\neg B$. The only inference rule I have is ...
0
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1answer
55 views

Proving De Morgan's Law with Natural Deduction

Here is my attempt, but I'm really not sure if I've done it right; as I'm just about getting the hang of Natural Deduction technique. Have I done it correctly? If not, where did I make errors and ...
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0answers
46 views

Dual formula in propositional logic

There's something I don't understand in my course on propositional logic. In the case of x being a variable, the definition of its dual is x* = x. Right. However, further in the course, there's a ...
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1answer
102 views

Finding a formula for the number of equivalence classes using $m$ variables and $\rightarrow$

I need to find a formula for the number $n_m$ of equivalence classes of the set of propositional logical formulas only containing the propositional variables $p_0,...,p_m$ and only using the ...