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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1answer
95 views

Determine whether {⇒, ¬} is functionally complete. [closed]

Show that {⇒, ¬} is functionally complete. And also show that ⇒ is not functionally complete. I'm quite stumped on this one, any help appreciated. Thanks
2
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2answers
89 views

Are the following logically equivalent? $\;p \rightarrow (q \rightarrow r) \text{ and }\ (p \rightarrow q) \rightarrow r$

Determine whether the following pair of statements are logically equivalent or not... $$p \rightarrow (q \rightarrow r) \;\;\text{ and }\;\; (p \rightarrow q) \rightarrow r$$ I am new to logic ...
-1
votes
1answer
17 views

How to identify invalid proposition

In propositional logic, how do i identify if a [compound/non-compound] proposition is valid or not? do the parenthesis matter, even if they start and do not end etc...? for example: ...
1
vote
1answer
63 views

Complicated FOL Formula {∃a,c(a≠c) ∧ ∀a,c[(a≠c)⇒(h(a,c)⟺ ¬h(c,a))] ∧ ∀a,c[h(a,c) ⇒ ∃b(h(a,b)∧h(b,c)∧b≠c)]} ⇒ ¬{∃a∀b[b≠a⇒ h(a,b)]}

In preparing for an exam, I'm working through old exam questions and am now trying to figure out if the following first-order formula is valid and if not, then give a model that does not satisfy the ...
0
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2answers
19 views

simplification of a propositional statement

Write the formula which is equivalent to the formula $$\neg (p\leftrightarrow(q\to(r \lor p)))$$ and contains the connectives AND ( $\land$ ) and NEGATION ( $\neg$ ) only.
1
vote
1answer
31 views

If $\Sigma$ satisfies $\alpha$ and also not-$\alpha$ then $\Sigma$ is not satisfiable?

Why is it true that if $\Sigma$ satisfies $\alpha$ and also not-$\alpha$ then $\Sigma$ is not satisfiable? Is it true at all? it doesn't make any sense to me and I would like to know more about that ...
0
votes
1answer
54 views

How to prove this using natural deduction

⊢ P ∨ ¬P I found this question on the net. I know the solution but i find it complicated. How should i approach to this sort of question? Or can you provide me another solution ?
3
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0answers
61 views

Is the following derivation of Gödel's Paradox generalizable to an intuitionistic one using provability?

I will follow Goedel's original proof of Goedel's paradox located on pg. 264 of Hao Wang's A Logical Journey: From Gödel to Philosophy. With this in sight, and to also avoid confusion, I will also ...
5
votes
1answer
119 views

How to formally prove the negation of a statement “A if and only if B”?

Motivated by this question, I'm trying to establish a logical proof to the fact that the following statement is false: $2x+1$ is prime if and only if $x$ is prime. There are several ways to ...
4
votes
1answer
56 views

Rewriting $X\leftrightarrow Y$ using only $\neg$ and $\lor$

Note: The book I'm using doesn't have any solutions/answers so I will be posting some of the questions I'm unsure about in the hope that someone will check it for me. Question: Re-write ...
2
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1answer
42 views

Statement calculus

Turn the statement 'either $X$ or $Y$' into an iterated composition. I'm not sure if my answer is correct, can someone please check for me? : $$\text{either }X\text{ or }Y \equiv (X\vee Y)\wedge ...
3
votes
2answers
54 views

Establishing the validity of an argument.

I've been trying to determine the validity of a particular argument for some time now and I've had no luck in figuring it out. The argument in question goes as follows: \begin{align} & p \wedge q ...
1
vote
2answers
33 views

proof for propositional logic

I am unable to prove the following proposition logic. $(p \lor \neg r) \land (r \lor \neg p) \leftrightarrow (p \leftrightarrow q) \land (q \leftrightarrow r)$ My solution is given in the image. ...
2
votes
1answer
44 views

Consistency vs Inconsistency in a set of sentences: which is more common

I'm curious whether there is any research in the "probability" that a set of sentences in a first-order logic is consistent. Obviously, there are an infinite number of inconsistent sets and an ...
0
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2answers
32 views

Propositional formula, consisting of $p, q, r$ is true iff only one of them is true

I have some difficulties in building a formula $\phi(p, q, r)$, which is true iff only one of the variables is true. I suppose that it's reasonably to start trying, using the truth table, but ...
4
votes
0answers
49 views

(Co)homology of propositional logic

Sorry if this is a rather vague question, but it seemed like something that might be interesting. Let $P$ be a family of propositions, and let $\mathcal L(P)$ be the set of all compound propositions ...
1
vote
2answers
123 views

Prove $( \lnot C \implies \lnot B) \implies (B \implies C)$ without the Deduction Theorem

The issue is Exercise 1.47 (d) in Elliot Mendelson's "Mathematical Logic". The exercise is to prove $(\lnot C\implies\lnot B)\implies(B\implies C)$ by using the three axioms $(A1,A2,A3)$ without using ...
3
votes
2answers
57 views

Proof, is $\lnot p \land \lnot q \vdash p \iff q$?

I have derived the proof to some extent, mentioned below:- $$\begin{array}{rll} 1. &\lnot p \land \lnot q &\text{Premise} \\ 2. &\lnot p ...
2
votes
2answers
105 views

Is “It is raining or it is not raining.” a tautology?

Is the following proposition a tautology: "It is raining or it is not raining." I is obviously always true, so one thinks that it should be a tautology. However, i thought a tautology has free ...
3
votes
3answers
129 views

If $B$ is a model for the set of positive consequences of $\Gamma$, then there's $A \subseteq B$ such that $A \models \Gamma$

I'm working through Chang & Keisler again and got stuck on the following exercise (1.2.14) about propositional logic. First, consider a set $\mathscr{S}$ of sentence symbols of arbitrary ...
1
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1answer
92 views

Easy question on Logic and Modes Ponens

I got confused with these: using ONLY this three axioms and Modus Ponens:$$1. \ F \implies (G\implies F) \\ 2. \ (F \implies (G\implies H))\implies ((F \implies G)\implies (F \implies H)) \\ 3. \ ...
2
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4answers
197 views

The Order of Mixed Quantifiers

How can we derive the implication: $$ ∃y∀xP(x,y) \implies ∀x∃yP(x,y) $$ In other words, when quantifiers in the same sentence are of the same type (all universal or all existential), the order in ...
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1answer
62 views

Proving a Tautology Formally [closed]

I wish to prove: $(\neg p\leftrightarrow q)\leftrightarrow\neg(p\leftrightarrow q)$
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2answers
27 views

Does $r \implies \neg q$, $q$ give $\neg r$?

In resolution, if we have a premise such as $r \implies \neg q$ and we know that $q$ is true, can we infer $\neg r$? If yes what is the rule called
1
vote
1answer
32 views

Semantic tableau software

Is it possible to find software to perform semantic tableaus (as described in http://en.wikipedia.org/wiki/Method_of_analytic_tableaux) automatically? Right now I am proofing it by hand.
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0answers
49 views

Prove formula's tautology

Prove that a formula that only consists of variables, logical negation and logical equality, and in which all variables and negation appear for an even number of times, must be tautological.
0
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1answer
25 views

Solution of a symbolic logic problem with Separation of Cases inference rule

$$(( S \land \lnot P ) \lor ( Q \land R )) ∴ ( \lnot P \lor Q )$$ I am trying to solve this symbolic logic problem ^^ with the separation of cases inferences rule but I am having trouble.
0
votes
1answer
46 views

Solve this logical inference

I have the logic inference: Hypotheses: $A \implies (B \lor C)$ $A \lor (D \land B)$ Conclusion: $D \implies C$ I have these equivalence formations: Hypotheses: $A \lor (D \land B)$ $\lnot D ...
1
vote
1answer
59 views

Solve this tautology

Hypotheses: not $q$, $p$ or not $s$, $p \rightarrow$ ($d$ and $q$), $e \rightarrow s$ Conclusion: not $e$ I have thus far, but unsure how to proceed. I am looking forward to solve it using ...
1
vote
1answer
25 views

Problem with simplification in discrete math

I am doing my homework in discrete mathematics and I need your help.. I can' t find the way how to simplify this equation. I had to get Minimal Disjunctive Normal Form by just simplifying minimal ...
2
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1answer
23 views

Absorption Law with Negation

Would absorption law work for statements with neagations in them like $( \neg q \land \neg r) \lor r$?
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0answers
37 views

proving $ (A \rightarrow C) \rightarrow ((A\rightarrow B) \wedge (B\rightarrow C))$

I looking for proof of $ (A \rightarrow C) \rightarrow ((A\rightarrow B) \wedge (B\rightarrow C))$ in the foloowing logic (SJ logic in paper of Greg Restall , Subintuitionistic logic) $$⊢A→A$$ ...
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2answers
39 views

To prove $A\rightarrow B, C\rightarrow D \vdash (A\vee C)\rightarrow (B\vee D)$ with natural deduction [closed]

How to prove this statement? $ A\rightarrow B, C\rightarrow D \vdash (A\vee C)\rightarrow (B\vee D)?$ in inference rule? tnx!
6
votes
5answers
209 views

Is a proposition about something which doesn't exist true or false?

Let S = {x | x is not an element of x } The set S doesn't exist. Then, would a proposition such as "The cardinality of S is 1," be true or false? Equivalently, I could have made a proposition, "the ...
2
votes
1answer
79 views

proving $ (A \rightarrow B \vee C )\rightarrow((A\rightarrow B) \vee (A\rightarrow C))$

I'm looking for a way to prow $ (A \rightarrow B \vee C )\rightarrow((A\rightarrow B) \vee (A\rightarrow C))$ from the following axioms and rules $$\vdash A \rightarrow A$$ $$\vdash A \wedge B ...
0
votes
1answer
29 views

Showing logical equivalence of these two formulas

I have the following statement in propositional logic: (¬g v s1 v ¬s2) ^ (¬g v ¬s1 v s2) ^ (¬g v s1 v s2) (1) I want to show equivalence to this statement: ...
0
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0answers
29 views

Determine the truth values

Let P(x) : x^2 ≤ 4. Determine the truth values of the following propositions. Assume the domain for the variable is all positive integers: 1, 2, 3, 4, 5, and so on. ...
2
votes
1answer
30 views

translating phrases into propositional logic

translate the following into propositional logic: students attend the annual meetings where s: students A: attend annual meetings my first intuition is: s -> ...
0
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4answers
45 views

Can $(A \lor B) \land (\lnot A \land \lnot C)$ be more simplified?

Can $(A \lor B) \land (\lnot A \land \lnot C)$ be more simplified/expanded? With a kind of distributive property?
1
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2answers
31 views

Resolution on set of clauses

Given this set of clauses: $\neg \phi = (\neg T \lor \neg Y)\land (S \lor \neg X ) \land (\neg X \lor Z \lor \neg Y) \land(X \lor T) \land (Y \lor U) \land (Y \lor \neg V)\land \neg S \land V$ I ...
0
votes
4answers
53 views

proof for a problem in propositional logic

I cant find a proof for given problem: $$p \to ( q \to p) ≡ \lnot p \to ( p \to q ) $$ Please give proof to prove above statement.
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0answers
84 views

Let Γ = {p∧q,(¬p)∨q,p∨r}. Is it true that Γ ⊢ r?

I"m not sure how to solve this type of question. Here is the problem in more detail, and a similar problem: I know that given this set of formulas I'm supposed to show if its possible to deduce r ...
0
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2answers
81 views

Finding a formal deduction from an empty set of premises

I can't seem to make sense of any of this. I'm given a set of axioms schemes, modus ponens as the inference rule and I'm supposed to find a formal deduction. The question (question 1) is here. It ...
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votes
1answer
80 views

Which if the following three propositions are logically equivalent? [closed]

Which if the following three propositions are logically equivalent? $(p \wedge q) \Rightarrow (p \wedge r)$ $p \wedge (q \Rightarrow (p \wedge r)) $ $(\lnot p) \vee (\neg q) \vee (r \wedge p)$ ...
2
votes
4answers
92 views

Question about logical implication $P\to Q$ [duplicate]

Having come across mathematical logic, a question suddenly came into my mind. We commonly know that the truth value of $P\to Q$ given as: $\begin{matrix} P&Q&P \Rightarrow Q \\ ...
2
votes
3answers
42 views

How to show that if $\neg b = a \land d$ then $a \land \neg b = \neg b$ and $b \land \neg a = \neg a$

I'm new to boolean algebra and am having trouble proving the following simple theorem. Many thanks for any help. If $\neg b = a \land d$ then $a \land \neg b = \neg b$ and $b \land \neg a = \neg a$. ...
2
votes
1answer
81 views

Problem with proving formally tautology using given rules

Using the rules below prove that the following assumeptions leads to the following conclusion by tautology. $A\vee B \vee C, A\to C, B\to C \Rightarrow C$ What I did: $A\vee B \vee ...
1
vote
1answer
36 views

Is the True clause considered the proof of resolution refutation

So, basically I have the sentence $$ (P \Rightarrow (Q \Rightarrow R)) \Rightarrow ((P \Rightarrow Q) \Rightarrow (P \Rightarrow R))$$ and it was asked to prove it by resolution refutation. On the ...
0
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0answers
33 views

Logic negate and simplify

Negate and Simplify: [(pvq)->~r]v~q Can someone show me step by step how to go about this. I am a little confused about negating over an implication.
1
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1answer
36 views

Conversion to CNF - eliminate implications

On the web I found a solution to an exercise on resoulution. Basically, it asks to use resolution refutation to prove $$ (P \Rightarrow (Q \Rightarrow R)) \Rightarrow ((P \Rightarrow Q) \Rightarrow (P ...