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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Two easy questions of propositional logic [on hold]

if $ M_1\vee M_2$ is unsatisfiable can we say $M_1 \models\neg M_2?$ if $M\models\psi$ then does$\neg\psi\models \neg Μ?$
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32 views

Prove that a boolean function using only $\vee$ and $\wedge$ must attain the value $1$ at least once

Please give me feedback on this Prove that a boolean function constructed only by using $\vee$ and $\wedge$ (without using $\sim$ ) must attain the value $1$ at least once.
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1answer
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Proving $(p\to q)\land(p\to r) \equiv p\to(q\land r)$ using logic laws — short cut or incorrect?

Working through this problem: Using logic laws, show that the following are logically equivalent: $$(p\to q)\land(p\to r)\qquad\text{and}\qquad p\to(q\land r).$$ The way I did the problem is ...
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1answer
31 views

Exam preparation: logic - problems on (maximally) consistent sets

I am preparing for an exam on aspects of Logic related to propositional and first order logic. One of my revision exam questions is . I have attempted this question but I am really struggling with ...
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2answers
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How do I simplify just using 2 logic operations?

I need simplify the following proposition to 2 logic operations using the laws of the algebra of propositions. Write each step on a separate line with the algebra law you used as a justification. ...
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1answer
52 views

Can't get Logical Error

Firstly I 'll mention Absorption laws : $((\sim p) \vee q) \wedge (\sim q)=(\sim p)$ $((\sim p) \vee q) \wedge p=q$ Also, $p \Longrightarrow q = (\sim p) \vee q$ And, $p \Longleftrightarrow q = ...
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1answer
38 views

If there are Predicates before Predicate Calculus, why is it called such?

In my understanding, predicates are synonyms of relations: mappings of an ordered set (a,b) to the set of values "True" and "False" Well, propositional calculus comes before predicate calculus, and ...
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2answers
79 views

Why $ M\models \forall x ( \alpha \to \beta)$ Is False? [on hold]

if M be a model and $\alpha$ and $ \beta$ be two formula the following is False: $ M \models \forall x ( \alpha \to \beta)$ if and only if $ M \models \forall x \alpha$ has conclusion $ M \models ...
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1answer
46 views

Proving these are logically equivalent?

How to prove that these are logically equivalent using laws? a. $p ↔ q ≡ (p ∧ q) ∨ (¬p ∧ ¬q)$ I used the Conditional law and DeMorgan's Law and eventually arrived at $-(p ∨ q) ∧ -(p ∨ q)$ but ...
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0answers
39 views

Proof by contradiction that $P \rightarrow Q$ is true

Let $\tau$ be a closed tableau. Prove that $\tau$ is not satisfiable. So let's say the statement can be expressed by $P \rightarrow Q$. To prove that this statement is true, we look at the assumption ...
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1answer
23 views

Logical Equivalence of Wffs in Sentence, Predicate Logic using Tables, Interpretations Resp.

just curious if there is a formal name for the results that: a) Two wffs in Sentence Logic are equivalent iff their truth tables are equal , as binary functions of {T,F} b) Two wffs A,B in ...
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2answers
70 views

If it rains, John is sick. It didn't rain. $\vdash$ John wasn't sick. Is this valid?

If it rains, John is sick. It didn't rain. $\vdash$ John wasn't sick. I would say that this is false since the weather isn't directly influencing John's health. Am I right or wrong? Should I use ...
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1answer
68 views

What is the _simplest_ way to solve problems of this kind?

Two doors with talking doorknockers - one always tells the truth and one always lies. One door leads to death other to escape. Only one question may be asked to either of the door knockers. What would ...
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3answers
70 views

What is the negation of the statement “Every odd nmber is divisible by 2”.

Intuitively,I think it is "no odd number is divisible by 2" or it could be "every odd number is not divisible by 2". Is this a trivial question or is there more to it? What is the correct answer? BTW ...
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9answers
197 views

Why is the negation of $A \Rightarrow B$ not $A \Rightarrow \lnot B$?

The book I am reading says that the negation of "$A$ implies $B$" is "$A$ does not necessarily imply $B$" and not "$A$ implies not $B$". I understand the distinction between the two cases but why is ...
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0answers
34 views

Non Satisfiability of disjuction

Problem: If S1,S2 are (possibly infinite) sets of propositional formulas where their union: S1VS2 is not satisfiable, prove that there exists an ψ such that S1|=ψ and S2|=¬ψ. Can we say that if ...
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1answer
53 views

Let $\tau$ be a closed tableau. Prove that $\tau$ is not satisfiable.

Let $\tau$ be a closed tableau. Prove that $\tau$ is not satisfiable. Okay can prove this by contradiction. So we say that a tableau $\tau$ is $\textit{satisfiable}$ iff there exists an ...
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1answer
81 views

direct hint to showing a formula is valid?

we know A formula is logically valid (or simply valid) if it is true in every interpretation. These formulas play a role similar to tautologies in propositional logic. which one could direct me to ...
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1answer
50 views

Show $\models(\phi\rightarrow(\psi\rightarrow\theta))\rightarrow((\phi\rightarrow\psi)\rightarrow(\phi\rightarrow\theta))$

Question: Show $\models(\phi\rightarrow(\psi\rightarrow\theta))\rightarrow((\phi\rightarrow\psi)\rightarrow(\phi\rightarrow\theta))$ Answer: (1) Let, ...
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1answer
43 views

how we can prove that argument $P_1,P_2,…,P_n $?

I ran into a one claims on LOGIC. how can add more direction or hint to me? if we have an argument $P_1,P_2,...,P_n $ such that $ n>3$ ($p_i$ is premise) why $P_1,P_2,....,P_n,P_1$ is ...
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37 views

How we can prove that the logical result of a set is effectively enumerable? [duplicate]

How we can prove that the logical result of $\{(p_i \vee $~ $p_{i+1}$$) $$: i \in \mathbb{N} \}$ is effectively enumerable ? Update: as one user requests, I add my method. I use truth table for ...
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1answer
40 views

Proving tautology without using truth tables

I have a statement (P∧Q∧(R∧P⇒~Q))⇒~R that I need to prove tautology without using truth tables. I understand I'll be using inference rules. Here's what I've tried ...
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2answers
35 views

Maximal consistency proof for set of propositional logic with specific restriction?

I ran into struggle when I comes to one sentence on logic. Why the set of all propositional that under any valuation has value 1 is not maximal consistent ? ...
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Interpretation and truth table is enough to showing validity or a better way?

I'm so glad that find this useful site. anyway, I ran into some challenging ways to find a formula is valid. Here is two example in my note that called valid. I ran into such a problem with making ...
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1answer
67 views

RAA elimination and inference a theory ?!

Can somebody explain the why if we eliminate RAA rule in natural deduction system on propositional logic, why ~$(p \wedge $~$p)$ is not inference from the ...
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2answers
38 views

Valid Formula in First Order Logic

I am a little confused about the validity of first order logic formulas. How we can using formal notation to prove the following is VALID? $ \exists x \exists ...
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3answers
80 views

'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 ...
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0answers
32 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
41 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
55 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
31 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
59 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
40 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
163 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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51 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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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 ...
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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
23 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 ...
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2answers
49 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
39 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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87 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
35 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
35 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
111 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
26 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 ...