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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How to express the statement “not all rainy days are cold” using predicate logic?

Which is the correct formula for the sentence? “not all rainy days are cold” Can anyone tell me how to read the 4 options correctly?
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2answers
28 views

Proving equivalency using boolean algebra laws of logic

I have a question on my exam papers relating to proving equivalences using the laws of logic, but I'm not sure how to work it out as I don't have the solution paper. Can someone explain to me the ...
0
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2answers
27 views

Principle of propositional congruence

Let $\varphi$ be a propositional formula, defined as a formula containing propositional symbols and connectives only, and let $\psi,\chi$ be propositions. I read the following principle of ...
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2answers
31 views

Propositional Logic Tautology Proof

I have question about a proposition that I need to prove is a tautology: $((p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow (p \rightarrow r)$ I have tried negating the first large bracket, ...
4
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2answers
39 views

Propositional Logic Help

I need to prove that $(\neg p \wedge (p \vee q)) \rightarrow q $ is a tautology using Laws of Logic (not truth tables). This is what I tried: $\equiv (( \neg p \wedge p) \vee (\neg p \wedge q)) ...
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2answers
32 views

Logical equivalence: Which side is better to start to obtain the other?

How to resolve this with steps please: $$p \to (q \lor r) \equiv (p \to q) \lor (p \to r)$$ I just don't get how with less variable we can have more after or with more we can have less?
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2answers
30 views

Find logic expression for given truth table

So I was given this truth table and I need to find a logical expression for the formula to give such a result (where there can be two or three 2-place connective expressions (e.g. $A \lor B$ counts as ...
0
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2answers
46 views

Weird logic question I need help with!

The professor tells Jim: "It is necessary that you get at least a B on the final in order to pass the course". Jim gets a B. What can she conclude? a) He passed b) He can conclude nothing... I ...
2
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1answer
29 views

Prove $ \vdash \alpha \to \alpha $ in minimal logic of Hilbert

$ \vdash \alpha \to \alpha $ I'm trying to find a way solving this statement using minimal logic of Hilbert which have only two axiom's K & S and one only rule the modus pones (MP) : ...
2
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1answer
19 views

Proving a variable true through rules of inference

Question: Use rules of inference to show that if $(p → q) ∧ (q → p),\; t ∨ q,\; t ∨ p,\; (p ∧ q) → t$, then $t$ is true. Work So Far: $$\text{1. }(p \implies q) \land (q \implies p)\text{ | ...
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3answers
56 views

Propositional Logic : Why is ((p $\Rightarrow$ r) $\land$ (q $\Rightarrow$ r)) $\equiv$ ((p $\lor$ q) $\Rightarrow$ r)

I was working my way through some Propositional Logic and had the following doubt : Why is this true : ((p $\Rightarrow$ r) $\land$ (q $\Rightarrow$ r)) $\equiv$ ((p $\lor$ q) $\Rightarrow$ ...
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1answer
26 views

$p\to\neg q, q \vdash \neg p$- natural deduction

I have the following proposition: $$p\to\neg q, q\vdash \neg p$$ Using the following formulas on propositions is easy enough: $$\frac{\psi \qquad \psi\to\varphi}{\varphi}\quad \to_e$$ ...
0
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1answer
47 views

prove $( \lnot \lnot p \Rightarrow p) \Rightarrow (((p \Rightarrow q ) \Rightarrow p ) \Rightarrow p )$ with intuitionistic natural deduction

I'm trying to prove this statement with intuitionistic natural deduction using inference rules like this example : this is the statement I'm trying to solve : $$( \lnot \lnot p \Rightarrow p) ...
2
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1answer
26 views

$p \to (q\vee\neg r), \neg q, r ⊢ \neg p$ - Natural deduction- elimination with $\neg$ operator

I have the following proposition: $$p \to (q\vee\neg r), \neg q, r ⊢ \neg p$$ The only part I have trouble with is the : $$p \to (q\vee\neg r)$$ Clearly the first step is to eliminate $q$ or $\neg ...
2
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3answers
50 views

What is the correct form of De Morgan's Law in logic?

According to wikipedia (link), Morgan's Law is: $$¬ (P \wedge Q) \Rightarrow (¬P) \vee (¬Q)$$ But if you scroll down to 8.2.2 on this page (link), it says that Morgan's Law works as follow: $$¬ (P ...
0
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1answer
36 views

Prove tautology without truth using a truth table. [duplicate]

I am struggling to prove, without using truth tables, that the statement is a tautology. [(p→q)∧(q→r)]→(p→r) My work so far... ¬[(¬p∨q)∧(¬q∨r)]∨(¬p∨r) ...
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1answer
31 views

Proof that the formula $((p\to q)\land (q\to r)\land p)\to r$ is a tautology [duplicate]

Write down the assumptions in a form of clauses and give a resolution proof that the formula is a tautology. $((p\to q)\land (q\to r)\land p)\to r$ I got information that i need to use here ...
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0answers
30 views

Truth table for $(p \implies q) \lor (q \implies r) \lor (r \implies p)$ : What should my next step be?

I am working on a truth table for $(p \implies q) \lor (q \implies r) \lor (r \implies p)$ This is what I have done so far: My next step would be to do the disjunction from the first two ...
4
votes
4answers
85 views

Show that $(p \to q) \lor (q \to p)$ is a tautology

i tried to prove that $(p \to q) \lor (q \to p)$ is tautology i used p and not-q as conditions. (Premises 1 and 5) I managed to get to a solution but I'm not sure if it's right. can you please check ...
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2answers
37 views

How can i get a tautology truth table from using 3 variables?

I am looking to use the variables p, q and r to create a truth table which concludes to a tautology.
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1answer
40 views

Proving an “OR” statement

If one wants to proof $P\vee Q$, is it sufficient to proof $\lnot P \rightarrow Q$? Because it makes intuitively more sense to me that $P\vee Q$ would be logically equivalent with $(\lnot P ...
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3answers
29 views

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 ...
0
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3answers
96 views

If $A\rightarrow B$ and $ C \rightarrow B,$ does $(A \land C )\rightarrow B$ [closed]

If A implies B and C implies B, do A and C together imply B? I need a clarification regarding this question.
2
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2answers
64 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 ...
3
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2answers
32 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
38 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
325 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
73 views

Can anyone help me with a solution? [closed]

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) ...
0
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2answers
60 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 ...
0
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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
50 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) ...
0
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0answers
20 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
votes
1answer
34 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
43 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
3
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2answers
37 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
35 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
313 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
32 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
108 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 ...
1
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1answer
35 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 ...
0
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2answers
51 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
81 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
34 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 ...
2
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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
116 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
29 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
29 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 ...
2
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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
78 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 ...