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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3
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1answer
30 views

Simplifying propositional logic

Hi I asked a question a few hours ago which has been solved but I got stuck on another exercise so I thought I'd reach out for some help. I have the premise: $((A \to B) \land (\lnot A \to C))$ ...
1
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1answer
25 views

How can i prove $p\to (q \vee r) \equiv (p \wedge \sim q) \to r$?

please Help me in this question i have tried to solve it like this: $$p \to (q \vee r) \equiv (p \wedge \sim q)\to r$$ $$p \vee \sim (q \vee r) \equiv \sim(p \wedge q)\vee r$$ $$p \vee \sim q \wedge ...
2
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2answers
45 views

Help with natural deduction (Propositional logic)

I'm trying to get to $(\neg A \to C)$ from the following formula: $$(A \wedge B) \vee (\neg A \wedge C)$$ I have attempted the following: $$((A \wedge B) \vee \neg A) \wedge ((A \wedge B) \vee C ...
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3answers
46 views

Verifying logic without drawing truth tables

Want to know is there a way to solve these sort of problems without drawing truth tables? I found that it's kinda time consuming drawing truth table for each question. Help pls. Check the images ...
0
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1answer
26 views

Proposionate problem [on hold]

The Initial Condition of the problem is like this: 1200 : 480 and 800 : 320 I need an equation(or equations) for changing the value but 320 is fixed. for ...
3
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2answers
71 views

Why is the implication “If pigs could fly, I'd be king” a true implication? [duplicate]

Let $P$ = "Pigs can fly" and $Q$ = "I'm king". Apparently, there's a rule stating that $P \implies Q$ is true, if $P$ is false. In this example, $P$ is indeed false, because pigs cannot fly. But how ...
0
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1answer
39 views

What does “resolve away” exactly mean in propositional logic?

I have never had logic classes so I always struggle with the assignments that concern this interesting field. I was reading the slides about resolution theorem proving and there was a step-by-step ...
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0answers
22 views

Translation of English statements to logical expression using nested quantifier and predicates.

I have come across few doubts solving Exercise of Propositional logic and predicates. Here are they, Doubt 1 ...
2
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1answer
30 views

Are the following logical statements equal? Solution verification

We were requested to rewrite the following statement: \begin{equation*} ((\phi \rightarrow(\psi \lor \lnot X)) \land (\phi \rightarrow (\psi \land X))) \end{equation*} using $\exists, \land, \lnot $ ...
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3answers
51 views

How is “p implies q” same as “q unless not p”?

I want to know how is "p implies q" same as "q unless not p"? ie how is "$p\Rightarrow q$" same as "$q$ unless $\neg p$" ?
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3answers
43 views

Prove that the set of sentences $\{A \land (B \lor C), (\lnot C \lor H) \land (H \rightarrow \lnot H), \lnot B\}$ is inconsistent

Prove that the set of sentences $\left\{A \land (B \lor C), (¬C \lor H) \land (H \to \lnot H), \lnot B\right\}$ is inconsistent. I'm confused because it doesn't look like any of the forms I've ...
2
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2answers
37 views

A Natural-Deduction proof of $ \{ \neg N,\neg N \to L,D \leftrightarrow \neg N \} \vdash (L \lor A) \land D $.

I would like to prove $\{ \neg N,\neg N \to L,D \leftrightarrow \neg N \} \vdash (L \lor A) \land D $. My work until now is as follows: $$ \begin{array}{l|ll} 1 & \neg N ...
0
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1answer
40 views

Prove that the following argument is valid

I'm asked to show the following arguments are valid: P1) $[E \lor (L \lor M)] \land (E \leftrightarrow F)$ P2) $L \rightarrow D$ P3) $D \rightarrow \neg L$ C) $E \lor M$ Our work (so far): P2) ...
1
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1answer
41 views

Natural deduction proof: {A → B, B → (C & D), ¬C v ¬D} ⊢ ¬A

$ 1- {A → B, B → (C \land D), ¬C \vee ¬D} ⊢ ¬A$ Our work (so far): $1- A → B$ $2- B → (C \land D)$ $3- ¬¬A$ $4- A$ $5- B$ (from 1,4) $→E$ $6- B$ $7- C \land D$ (from 2,6) $→E$ This is ...
4
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3answers
58 views

Logic: Can you drop parentheses in a conjunction?

In propositional logic, $p \land (q \land r) = (p \land q) \land r$ , where $p, q$ and $r$ are propositions. Does this mean $p \land (q \land r) = p \land q \land r$ ? If so, why?
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2answers
27 views

Negating statements with quantifiers in them

First statement, ∀ odd integers n, ∃ an integer k such that n = 2k + 1 Second statement, ∃ m ∈ ℝ such that ∀ n ∈ ℝ, m · n = n Before the negation, I'd like to ask tips on how to translate this ...
1
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1answer
5 views

Consequence of compactness lemma

Let $\Gamma=\Sigma \cup \left\lbrace p_i,i\geq 1 \right\rbrace$ a countable set of propositional formulas. Assume also that for every boolean evaluation $u$ that maps every member of $\Sigma$ to true ...
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2answers
25 views

What is the name of the Boolean function whose output is always one?

For example: f = a.b.c.d + !a.!b.!c.!d + a.!d + !a.b.!c + !b.d + b.c.d + a.b.!c.d + !a.c.!d = 1 ! is logical NOT, . is logiacal AND and + is logical OR. The ...
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0answers
50 views

Proof verification+proposition

Given 2 function $F(p,v)$ and $\frac{dF}{dv}=g(p,v)$ Differentiate F(p,v) with respect to v give $F_pf+F_v$ Formula 1 $$\frac{dF}{dv}=F_p\left(\frac{dp}{dv}\right)+F_v=g\\ ...
0
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1answer
34 views

Logic : How to determine whether these propositions are contradictory ?

http://postimg.org/image/iips2lwdj/ The question asks to draw a truth table with the values of three propositions (linked), and following this, to "Show that the three propositions are ...
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1answer
34 views

Prove or disprove a sentence using HPC

according to HPC: Let S be a set of sentences and α that is not in S. Prove or disprove : If $S\cup\{\alpha\} \vdash \beta$ and $S\cup\{\neg \alpha\} \vdash \beta$ then $S\vdash \beta$. It ...
0
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4answers
44 views

Use tableau to convert formula to DNF/CNF form

Is there any method that can be used to convert any formula do a DNF/CNF form using only the truth table? For example if I have the following formula p → ¬(q∨r) How can I convert it into DNF? ...
3
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2answers
40 views

Write $(p↔q)$ in DNF

I have the following formula: $(p↔q)$ and I have to write in DNF (disjunctive normal form) This is where I got so far: $(p↔q) = ((p→q)∧(q→p)) = ((¬p∨q)∧(¬q∨p))$ but here I got stuck. How ...
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1answer
35 views

Every positive formula is satisifiable

We say that a propositional logical formula is positive if it does not include the negation connective ¬ anywhere in it (but it may still use ∧, ∨, ↔, →, and propositions). Show that all ...
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2answers
29 views

Proving that a set with a quaternary logical connective is functionally incomplete (i.e. inadequate)

I am stucked at trying to prove that the set $\{N\}$ of one logical connective is inadequate where $N$ is a quaternary connective that is defined as follows: $N(w,x,y,z)=((x\land y)\land(w\lor z))$ ...
2
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1answer
27 views

Question regarding using the natural deduction system

I have the following: Premise: ((V → ¬W) ∧ (X → Y)) Premise: (¬W → Z) Premise: (V ∧ X) |- (Z ∧Y) The part I want to know is how do I go about separating ...
2
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1answer
59 views

Argue that if a sentence has a proof, then it is a tautology

This is a corollary of the soundness theorem, which states that for a set of formulas $\Phi$ (of propositional logic) and a formula $\alpha$ : $$\Phi\vdash\alpha\Longrightarrow\Phi\vDash\alpha$$ What ...
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1answer
33 views

Equivalence of two biconditionals of propositional metalogic

In application to propositional metalogic, I am told that the following two biconditionals are equivalent: (i) Γ is satisfiable iff every finite subset of Γ is satisfiable. (ii) Γ ⊨ α iff some ...
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1answer
76 views

Disjunctive normal form and shannon normal form

Consider the formula (( true | (a <-> b)) & ((c | b) ^ a ^ b)). transform the formula into disjunctive normal form for the variable ordering a ≤ b ≤ c ≤ d. Also transform to Shannon normal form ...
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0answers
22 views

Construction of atomically closed tableu from a closed tableu

Suppose we have a closed tableu with at least one branch $\theta$ that contains $X$ and $\neg X$ where X is non-atomic formula. My strategy could be that of exploring the cases of X being an ...
0
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1answer
56 views

Complete operator base logic [closed]

Show that $F={0,\to}$ is a complete operator basis by giving equivalent formulas for negation,conjunction and disjunction over F.
3
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1answer
52 views

What is the set of propositional formulas?

What is the set of propositional formulas? I am not sure if I understand this
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4answers
65 views

Filling in a missing portion of a truth table

I have the following truth table: $$ \boxed{ \begin{array}{c|c|c|c} a & b & c & x \\ \hline F & F & F & F \\ F & F & T & F \\ F & T & F ...
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1answer
43 views

Prove that the intersection of definable sets is definable

Hello I have a question : $F$ is a family of definable sets. Prove that the intersection of all the sets in the family is definable. ($F$ could be infinite) Definition (Definable): a set $K$ of ...
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1answer
49 views

Find indefinable set that is included in definable set.

Find $K\subseteq \operatorname{Ass} $ and $ K'\subseteq K$ such that $K$ is definable but $K'$ is not. Definition (Definable): a set $K$ of assignments is definable if there is a set of formulas A ...
2
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1answer
81 views

I'm wodering if this statement is provable in logic $ \lnot \alpha \to \lnot \lnot \lnot \alpha ) $

I've encountered this statement in my final exam $$ \lnot \alpha \to \lnot \lnot \lnot \alpha ) $$ there was no open parenthesis and from what I know this is invalid (not a well-formed formula) so ...
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4answers
621 views

Why is a statement “vacuously true” if the hypothesis is false, or not satisfied?

Why isn't a conditional statement said to "not apply" instead of be "vacuously true" if the hypothesis is not satisfied? That would seem more appropriate.
2
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2answers
45 views

Propositional Calculus, Can someone answer the following?

Can somebody please solve the following equations: \begin{align} 1. \quad (A \rightarrow B)\land (A\rightarrow \neg B)=\lnot A \quad \quad \\ \end{align} What I have got for it so far is ...
2
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3answers
64 views

Solving this logical puzzle by resolution doesn't work for me

In this document there is a logical puzzle: If the unicorn is mythical, then it is immortal. If the unicorn is not mythical, then it is a mortal mammal. If the unicorn is either immortal or a ...
3
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4answers
129 views

Proving $ \neg ( \neg \alpha \wedge \neg \neg \alpha )$

I'm training to prove this statement , but first I need to know if this statement can be proved in : 1 - both in classical and Intuitionistic logic ( in this case i need to provide demonstration in ...
2
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4answers
78 views

How to prove this tautology using equivalences?

I am trying to prove that the following is a tautology: $(A \implies (B \implies C)) \implies ((A \implies (C \implies D)) \implies (A \implies (B \implies D)))$ To make progress, I thought I'd ...
2
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1answer
55 views

Proving that a set with a ternary logical connective is functionally incomplete (i.e. inadequate)

I am stucked at trying to prove that the set $\{\lnot ,G\}$ of logical connectives is inadequate where $G$ is a ternary connective that gives $T$ (True) if most of its arguments are $T$. For example: ...
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2answers
110 views

Propositional Logic: Proof involving only the symbols $\{\rightarrow,F \}$

I feel like I literally tried everything. I'm exhausted, and could really use some help. I was instructed to prove some logic proposition using only the symbols $\{\rightarrow,F \}$. Let me first ...
3
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1answer
70 views

Satisfiability proof of formulas with pure literals

Let $\varphi$ be any propositional formula in negation normal form (NNF). A literal $\ell$ is pure in a formula $\varphi$, if the complement of $\ell$, $\ell^c$, does not occur in $\varphi$, where ...
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1answer
46 views

How to evaluate the single turnstile symbol ($\vdash$) in propositional logic?

Wikipedia says, that: $x \vdash y$ means y is provable from x (in some specified formal system). But what do you actually check or calculate, when you have $(a \land \lnot b) \vdash a$? Has $(a ...
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1answer
48 views

Rewrite predicate formulas in propositional calculus

Suppose that the universe of discourse of the atomic formula P(x,y) is the set {0,1,2,3,4,5}. Write each of the following propositions using dis-junctions, conjunctions and only one negation: 1) ∃x ...
3
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1answer
37 views

Prove that ≿ is transitive iff ≻ and ∼ are transitive

Let ≿ be a complete preference relation (as in game theory). How to prove that ≿ is transitive if and only if ≻ and ∼ are both transitive? My reasoning is as follows. ...
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1answer
47 views

I am trying to use proof of sequence correctly to make valid

Here I am trying to use a proof sequence so that the argument is valid (hint: the last A’ has to be inferred). (A → C) ∧ (C → B') ∧ B → A' Here are my steps I tried but not sure if this is correct ...
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0answers
64 views

Trying to justify each step correctly in proof sequence

I am trying Justify each step in the proof sequence below for correctly with [A → (B ∨ C)] ∧ B' ∧ C' → A' So I justified my steps here but I am not sure at 1 to 3 if I did it correctly. A → (B ∨ ...
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1answer
40 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.