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.

learn more… | top users | synonyms (1)

3
votes
1answer
55 views

No truth function that is expressed by a formula that uses only implication and equivalence connectives

I proved the following statement by induction: Let $A$ be a propositional formula which uses only the connectives $→$ and $↔$. Prove (by induction on the complexity of $A$) that if every ...
-1
votes
1answer
50 views

Difference between DNF and CNF

I'm stuck on a particular question, about Propositional Logic. Let $A$ be the propositional formula $((\lnot p \rightarrow q) \leftrightarrow\ (\lnot q \rightarrow \lnot r))$. Find a propositional ...
0
votes
2answers
74 views

Prove propositional logic by resolution.

Prove $$[(p→q) \wedge (qr→s)]\to [pr→s],$$ which is the same as $$[(\lnot p\lor q) \wedge (\lnot (qr) \lor s)]\to [\lnot (pr) \lor s]$$ I believe it can just be done with algebra rules, but I got ...
2
votes
1answer
62 views

Lindenbaum-Theorem only concerning sentential logic provable in ZF?

Is the Lindenbaum-Theorem of sentential logic (= propositional logic) provable in ZF (i. e. without the axiom of choice)? Lindenbaum's theorem of sentential logic states that every set $\Sigma$ of ...
1
vote
1answer
59 views

Proving by induction on the length of a propositional formula?

I'm having a little trouble understanding the following proof question because I'm unsure what defines the 'length' of a propositional formula, I've seen multiple definitions whether it's the number ...
1
vote
1answer
22 views

Writing in disjunctive normal form using logical laws

I'm having trouble converting the below formula to disjunctive normal form using logical laws. I found the DNF using truth tables but I am having issues using just logical laws. Here is the formula: ...
1
vote
1answer
44 views

$(Ǝx)H(x) \dashv \vdash (Ǝy)H(y)$?

I can prove the statement using the natural deduction, but I keep getting confused about this sequent, so it would be very thankful if someone can help me to understand this concept of predicate ...
0
votes
0answers
36 views

Finding a truth function

I wanted to find a truth function $f$ if it exists that make the formula below true: $((p\to \lnot(q \oplus \lnot p)) \to (\lnot r \oplus (q \to p)))$ Where the $\oplus$ operator is defined as: ...
1
vote
3answers
35 views

Showing tautology without a truth table.

Show that the conditional statement is a tautology without using a truth table. $a)$ $(p \wedge q) \rightarrow p$ My suggestion would be getting rid of the implication first, so $(p \wedge q) ...
1
vote
2answers
26 views

Natural deduction proof for : p → ( c ∨ b) , b → s ⊢ ( p ∧ ¬s)→ c

I am trying to prove the following statement but I am getting stuck at the 6th line and I'm unsure how to continue. p → ( c ∨ b) , b → s ⊢ ( p ∧ ¬s)→ c p → ( c ∨ b) (premise) b → s (premise) ...
0
votes
3answers
58 views

Proving $(p \oplus q) \oplus r=p \oplus (q \oplus r)$

I was assigned to prove the associative law on xor. $(p \oplus q) \oplus r=p \oplus (q \oplus r)$ I'm sure $(p\oplus q)=(p∧¬q)∨(¬p∧q)$ But I got stuck on $(p \oplus q) \oplus ...
2
votes
1answer
51 views

Creating a proposition from a truth table using only ~ ⋀ and v

I have to find a simple expression for the third column in the truth table using only the logical connectives I've mention above. There are two questions that are involved here. Problem 1: Truth ...
1
vote
0answers
34 views

Construct theory with a condition

I would need some help here. I'm preparing for finals from mathematical logic and as I am browsing through some exercises, I often found these types: Let's say we have 2 propositions $\phi$ and ...
1
vote
1answer
25 views

Can you give a simple CDCL example?

I am trying to understand how Conflict-Driven Clause Learning works. After reading through the lecture slides, wikipedia article and some additional slides I found online I realized that I still can't ...
1
vote
2answers
26 views

Can I use De Morgan's law in the third step as shown below to solve this problem?

$(p \rightarrow q) \wedge (\neg p \rightarrow q)$ $\equiv(p \rightarrow q) \wedge (\neg p \rightarrow q)$ $\equiv(\neg p \vee q) \wedge (p \vee q)$ $\equiv \neg(\neg (\neg p \vee q) \vee \neg(p ...
3
votes
1answer
45 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
vote
1answer
34 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
votes
2answers
66 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 ...
0
votes
3answers
58 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 ...
3
votes
2answers
97 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
votes
1answer
43 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 ...
1
vote
0answers
55 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
votes
1answer
31 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 $ ...
1
vote
3answers
71 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 $\neg p$" ?
-1
votes
3answers
55 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
votes
2answers
44 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
votes
1answer
45 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
vote
1answer
47 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 ...
5
votes
3answers
65 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?
0
votes
2answers
29 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
vote
1answer
6 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 ...
1
vote
2answers
28 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 ...
0
votes
1answer
39 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 ...
1
vote
1answer
41 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
votes
4answers
69 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
votes
2answers
46 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 ...
1
vote
1answer
39 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 ...
1
vote
2answers
36 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
votes
1answer
32 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
votes
1answer
67 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 ...
1
vote
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 ...
0
votes
1answer
89 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 ...
0
votes
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 ...
-1
votes
1answer
60 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
votes
1answer
57 views

What is the set of propositional formulas?

What is the set of propositional formulas? I am not sure if I understand this
1
vote
4answers
68 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 ...
1
vote
1answer
54 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 ...
1
vote
1answer
52 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
votes
1answer
91 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 ...
10
votes
4answers
667 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.