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
63 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
votes
1answer
56 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
155 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 ...
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2answers
50 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
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1answer
61 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
45 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
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0answers
74 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
245 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
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2answers
66 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
174 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
151 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
102 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
votes
4answers
353 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 ...
0
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1answer
102 views

Proving a Tautology Formally [closed]

I wish to prove: $(\neg p\leftrightarrow q)\leftrightarrow\neg(p\leftrightarrow q)$
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2answers
32 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
113 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.
1
vote
1answer
74 views

Simplification problem with discrete mathematics

I am trying to achieve this equation: $$x_1x_4 \lor x_1x_2x_3\lor (¬x_1)x_3(¬x_4)$$ I start with: $$(x_1 \lor (¬x_4))(x_3\lor x_4)((¬x_1)\lor x_2\lor x_4)$$ Then I do simplify in the following ...
2
votes
1answer
71 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
175 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.
1
vote
1answer
72 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
46 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
votes
1answer
82 views

Absorption Law with Negation

Would absorption law work for statements with neagations in them like $( \neg q \land \neg r) \lor r$?
0
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0answers
43 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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votes
5answers
343 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
98 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
32 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: ...
2
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1answer
41 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 -> ...
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4answers
58 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?
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2answers
71 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
66 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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2answers
227 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 ...
2
votes
4answers
143 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
45 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
112 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 ...
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vote
1answer
94 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 ...
1
vote
1answer
82 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 ...
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1answer
80 views

Propositional calculus logic question

In my assignment I have the following question: For every proposition $\theta$ let $E(\theta)$ be the set of basic propositions. Prove the following: For every two propositions, $\alpha$ and ...
5
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1answer
121 views

Compactness Theorem / Set made of formulas of infinite size

Could someone give me an example of an infinite countable set, where formulas contained in it are under the form of a conjunction or disjunction of infinite size, for which the compactness theorem ...
2
votes
1answer
98 views

Structural Induction, Propostitonal formulae problem

I am kind of overwhelmed by this question. Can anyone give me some hints about where to start? Propositional formulae PF are inductively defined over the Boolean constants B := {1, 0} (true and ...
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3answers
63 views

Prove the two logic expressions are equal

Prove $\neg(a \lor b)$ is the same as $(\neg a \land \neg b)$ It makes sense when I think about it, but how does one prove it? Also is there a relationship with the above and saying: $(a \implies ...
2
votes
2answers
224 views

Every element of the empty set has three toes true or false? [duplicate]

This is a bonus question that we have and I cannot figure it out. Any help would be great! Is the proposition Every element of the empty set has three toes true or false? Explain your answer
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0answers
31 views

In a formal language, how does one show that $\neg \neg \bot \neq( \phi \wedge \psi) $ [duplicate]

In a formal language, how does one show that $\neg \neg \bot \neq( \phi \wedge \psi) $ Or how do one go about showing that the former is not a proposition. I've just started reading Dalen's Logic and ...
0
votes
3answers
41 views

Question about negating implied propositions

I'm negating this proposition: "If you study you will not fail." I'm using proposition P: "You study" and proposition Q: "You will fail." The original statement can be written as "$P → ¬Q.$" My ...
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4answers
84 views

Basic logic question: Can $\neg p \implies p$ be true?

Can $\neg p \implies p$ be true? How about $p \implies \neg p$? I was told yes, but it doesn't make sense to me. Any help would be appreciated!
0
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1answer
38 views

Given $p \rightarrow q$ and p are true, show $q ∨ r$ is true using rules of inference

I have a question from computing mathematics which I am not really able to prove. Given that $p \rightarrow q$ and $p$ are true, show that $q \lor r$ is true using rules of inference. Any ...
2
votes
2answers
338 views

Proof of a theorem in Hilbert's system

I have been trying to prove that the propositional formula $ \big( \alpha \rightarrow \lnot \beta \big) \rightarrow \big((\alpha \rightarrow \beta) \rightarrow \lnot \alpha \big)$ is a theorem in ...
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4answers
117 views

Need Hints Prove “$((\neg \alpha \to \alpha) \to \alpha) $” Using Axiom 1,2,3 and MP and deduction theorem

$((\neg \alpha \to \alpha) \to \alpha) $ Hi, I am trying to prove this. Can someone gives me some hints to start the question... My friend told me I might need to use deduction theorem here, but I ...
1
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1answer
23 views

Proposotional logic derivation

Show that (φ ∧ ψ) ↔ ¬(φ → ¬ψ) is derivable. I have derived ¬(φ → ¬ψ) from (φ ∧ ψ) by assuming (φ → ¬ψ) and (φ ∧ ψ) and deducing a contradiction. By cancellation of the hypotheses I can then conclude ...
0
votes
1answer
49 views

Conjuctive normal form of $(p\wedge(q\implies r))\implies s$

I am asked to write this in CNF without using truth tables. This is what I worked out so far: $$(p\wedge(q\implies r))\implies s \\ \neg(p\wedge (q\implies r)) \vee s\\ (\neg p \vee \neg(\neg q \vee ...
1
vote
1answer
49 views

Prove that simple conditional statement is tautology

This should be pretty easy, but I don't know how to turn the conditional statement into a tauntology. The statement is: $$ (p \land q) \to p$$ I am able to turn it into: $$ (\lnot p \lor ...