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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Propositional logic problem: Sales, expenses and happiness of the boss

Either sales will go up and the boss will be happy, or expenses will go up and the boss won’t be happy. Therefore, sales and expenses will not both go up. I know the solution is that the ...
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3answers
67 views

If A implies B and C implies B, do A and C together imply B? [on hold]

If A implies B and C implies B, do A and C together imply B? I need a clarification regarding this question.
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2answers
60 views

Principle of explosion: Other arguments?

I've come across a proof-theoretic argument for explosion on Wikipedia, which is as follows: $A \ \ \wedge\sim A$ $A$ $ \sim A$ $ A \lor B$ $B$ $(A \ \ \wedge \sim A) \implies B$ I've thought of ...
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2answers
28 views

Difference between “necessary” and “necessary but not sufficient”?

This is from Discrete Mathematics and Its Applications: Let $p, q,$ and $r$ be the propositions: $\quad p:$ Grizzly bears have been seen in the area. $\quad q:$ Hiking is safe on the ...
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1answer
23 views

Expressing the converse, contra-positive, and inverse of conditional statements

This problem is from Discrete Mathematics and its Applications Here is my book's definition on converse, contrapositive, and inverse And the common ways to express an implication For this ...
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3answers
312 views

Is there a quicker way to check if this proposition is self contradictory?

I have been trying to refresh my memory with regards to classical logic. As a result, I am currently going over the basics. The following proposition seems to be false in all possible worlds. ...
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2answers
60 views

Can anyone help me with a solution? [on hold]

Write down the assumptions in a form of clauses and give a resolution proof that the proposition $$\Big((p \rightarrow q) \land ( q \rightarrow r) \land p \Big) ...
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2answers
53 views

Can someone verify my assertion from this english sentence? [duplicate]

This is from Discrete Mathematics and its Applications This is the book means when mentions a list of common ways to express conditional statements After going through the list, I immediately ...
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1answer
26 views

Need to prove that a conditional statement is a tautology

The conditional statement is $[(p \rightarrow q) \land (q \rightarrow r)] \rightarrow (p \rightarrow r)$ Here are the steps I took in an attempt to prove the above statement a tautology, but I ...
2
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1answer
37 views

Show that the conditional statement is a tautology without using a truth table

I have been attempting to use identities to get to the answer but I am unable to get anywhere. Here is the equation I am trying to prove tautological without using truth tables: $[(p\rightarrow q) ...
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0answers
18 views

Is it necessary to write out the whole truth table to show system specification is consistent?

This is an example from Discrete Mathematics and its Applications Basically the way I see this problem is "is there a combination of propositions that will make all of these specifications true". ...
2
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1answer
33 views

odd logical structures

How you find contrapositive and converse of these sentences. Only if John chops down the tree, will he be a lumberjack. You can't win if you don't fight. All people that root for the Ducks are from ...
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2answers
40 views

Propositional Logic : Absorption - Why is it so?

Why is the Absorption Law of Propositional Logic so ? p $\lor (p \land q) \equiv$ p Would appreciate an intuitive explanation and not one using a Truth Table
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2answers
34 views

Is my deduction of $t$ being true logically correct?

According to the problem on my homework (yes, this is my homework), number 42 in chapter 2.3 of Discrete Mathematics with Applications by Susanna S. Epp, the following are true: \begin{align} ...
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1answer
31 views

Use logical equivalencies to classify as tautology, contradiction, or contingency.

Classify the following as tautologies, contradictions or contingencies using logical equivalences. Can anyone let me know what I'm missing or doing wrong? I got stuck, here is what I have so far: ...
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5answers
307 views

Which can be logically inferred from the given statements?

All women are entrepreneurs. Some women are doctors. Which of the following conclusions can be logically inferred from the above statements? (A) All women are doctors. (B) All doctors are ...
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0answers
29 views

Expansion Of A algebric term

While doing a coding for software I fell upon in the need to expand the following expression $(A \land B) \land (C \land (D \lor (E \land f)) \land (g \lor h \lor i))$ I tried it and result I got is ...
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4answers
106 views

Necessary but not sufficient in logic

I am working through sample questions and am having a bit of trouble understanding the solution. Write using logical connectives: p : Grizzly bears have been seen in the area. q : Hiking is safe on ...
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1answer
32 views

Are these two statements logically equivalent?

Are the statements $D \Rightarrow H \vee S$ and $(D \Rightarrow H) \vee (D \Rightarrow S)$ logically equivalent?
3
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1answer
32 views

prove that $\Sigma \vdash \phi_1$ and $\Sigma \vdash \phi_2$ leads to $\Sigma \vdash \phi_1 \wedge \phi_2$.

I try to prove that if $\Sigma \vdash \phi_1$ and $\Sigma \vdash \phi_2$ then $\Sigma \vdash \phi_1 \wedge \phi_2$. Notice that, the ONLY rule of inference of the system is modes ponens and the set ...
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2answers
47 views

Discrete Math - Determine if the argument is valid

Can you guys please check my work and syntax. Question: Determine if the argument is valid. p $\rightarrow $ q $\underline{\urcorner{q}}$ $\therefore \urcorner$p Answer: T $\rightarrow $ T ...
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0answers
76 views

Truth Tables in Real Life

Are truth tables something that can be used in real life, or are they merely something that philosophers would have used? And by real life I mean outside of mathematics. I already know that we use ...
2
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1answer
31 views

Partial truth table and proving or disproving tautology

Let $p,q$ be elementry statements and $\alpha,\beta,\gamma$ be statements. (sorry if this is the wrong translation). Prove/disprove: is $p,q\Rightarrow \gamma$ tautology? is ...
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2answers
40 views

Associativity and De Morgan's for more than 2 literals

Do logical operators have meaning when used with more than 2 literals "associatively", e.g.: $(A \land B \land C)$? I.e., are statements such as $(A \land B \land C)$ meaningful, as opposed to $((A ...
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1answer
87 views

Proving each conditional statement is a tautology

I'm having trouble trying to show that each of the conditional statement below is a tautology without using a truth table. I'm assuming you would have to use logical equivalence to figure this out. I ...
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1answer
28 views

Logically determining the validity of a statement

I'm having some trouble determining if the following statement may be considered valid. if the apples are on sale, I will buy the apples. the apples are not on sale. ∴ I will not buy the apples. ...
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2answers
28 views

Logical disjunction truth table

The truth table for a logical disjunction shows that there is only one situation where the result can be false, being when both statements are false. As long as one statement is true, the result is ...
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2answers
62 views

Notation or verbiage for the opposite of 'iff'? [duplicate]

Given the statement $X \implies Y$ and $Y \implies X$, we have the common notation $X \iff Y$. Ok so is there an opposite of this concept? Suppose I have $X$ doesn't imply $Y$, nor does $Y$ imply ...
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1answer
70 views

Need help understanding discrete mathematics logic

I am having a heck of a time understanding Discrete Mathematics. I have tried this myself and put my answer below. If anyone could help me if my answer is incorrect could you please explain to me what ...
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1answer
94 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
85 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 ...
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votes
1answer
16 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: ...
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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 ...
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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.
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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 ...
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1answer
51 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
60 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
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1answer
118 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
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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
votes
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
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2answers
53 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
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
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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 ...
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2answers
30 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
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 ...
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2answers
122 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
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
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2answers
104 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. \ ...