-1
votes
0answers
44 views

Inference in First Order Logic [on hold]

Suppose we have $ E \bigwedge R \Rightarrow B$ $ E \Rightarrow R \bigvee P\bigvee L $ $ K \Rightarrow B$ $ \neg (L \bigwedge B ) $ $ P \Rightarrow \neg K $ which of them cannot inference from ...
2
votes
0answers
37 views

Proposition into spoken language

Given: $\sim( p \leftrightarrow (q \vee r) )$ $p:$ It's raining $q:$ The sun is shining $r:$ There are clouds in the sky. Translate the proposition into spoken language. ...
3
votes
5answers
141 views

Is $'' \sum_{n = 1}^{\infty} (-1)^n \; \text{is a real number}''$ an invalid statement or a false proposition?

So we're beginning an introductory logic course and my professor is giving examples for valid statements/ propositions - meaningful statements that are either true or false but not both. So he puts ...
1
vote
1answer
33 views

Laws of equivalence needed to prove $\;q \leftrightarrow (¬p ∨ ¬q) ≡ (¬p ∧ q)\;?$

I'm not sure which laws should be applied and how I can tell for myself how to discern which laws I should use - any and all help is appreciated.
-6
votes
1answer
40 views

Logical argument [closed]

Let A, B, and C be propositions. Let ∧ denote logical AND, let ∨ denote logical OR, and let ¬ denote logical NOT. Argue that if (𝐴∨𝐵) ∧(¬𝐵∨𝐶) is true, then (𝐴 ∨𝐶) must be true as well.
3
votes
3answers
83 views

If $(A \vee B) \wedge (¬B \vee C)$ is true, then $(A \vee C)$ must be true … can I argue that?

If $(A \vee B) \wedge (¬B \vee C)$ is true, then $(A \vee C)$ must be true ... can I argue that? I don't see how I can argue that $(A \vee C)$ must be true because can't we have $(T \vee T) ...
3
votes
2answers
49 views

How to prove the following using direct proof

$[(\sim p \vee q) \wedge p ] \Rightarrow q $ What should be done next in order to apply direct proof to the example above? The following process has been already done but seemingly it's incorrect: ...
0
votes
2answers
85 views

Understanding logical form of “Nobody in the calculus class is smarter than everybody in the discrete math class”

I'm self studying How to Prove book and have been working out the following problem in which I have to analyze it to logical form: Nobody in the calculus class ...
0
votes
1answer
20 views

Analyzing Logical Forms involving quantifiers

I have been solving the following problem from How to Prove book: Analyze the logical forms of the following statement: Everyone likes Mary, except Mary ...
1
vote
2answers
51 views

Validity in propositional calculus.

I have read some of the answers on similar questions but I can't really get my head around this. So, here are 2 questions I need to answer. Show using a truth table: That the inference ...
1
vote
1answer
61 views

Thinking logically instead of Venn diagrams

I hit upon the following identity while reading the book How to Prove: $$(A \cup B) \backslash B \subseteq A$$ Now if I solve this logically I can reduce this like this: $$ \begin{gather*} x \in (A ...
0
votes
2answers
37 views

What is the Equivalent formula of $((a\to b) \to ((a \to c) \to (c \to a)))$

Need help to solving a logic. The question is to find an equivalent to the following logic. $((a\to b) \to ((a \to c) \to (c \to a)))$ Thanks in advance for help.
0
votes
3answers
43 views

Find an equivalent to $(p\lor q) \to (p \lor r)$

I need some help regarding solving a logic. The question is to find an equivalent to the following logic. $(p\lor q) \to (p \lor r)$ Thanks in advance for help.
0
votes
2answers
50 views

How to find the equivalent formulas of $\neg ((p\land q) \to (p \land r))$ [closed]

I have following formula: $\neg ((p\land q) \to (p \land r))$ I need to find equivalent formulas of above expression. Thanks in advance for the help.
3
votes
1answer
34 views

Why is the assumption needed in this conditional introduction?

In the first derivation detailed here, why must we include a subderivation with $P$ as an assumption? We can derive $Q$ (4) from $S \land Q$ (2) without the help of $P$ (3); and then since we have ...
6
votes
1answer
63 views

Unique decomposition of wffs when left and right parentheses are indistinguishable

I'm working through Enderton's book A Mathematical Introduction to Logic 2nd Edition for self study. Section 1.3 Exercise 7 Suppose that left and right parentheses are indistinguishable. Thus, ...
2
votes
1answer
51 views

Simplifying ambiguous statements

I have been working on the following question from Velleman's How to prove book: Let S stand for the statement “Steve is happy” and G for “George is happy.” What English sentences are ...
3
votes
2answers
37 views

Forming up Complex logical forms from simple one

This is another problem I have been working from Velleman's How to prove book. Let P stand for the statement “I will buy the pants” and S for the statement “I will buy the shirt.” What English ...
0
votes
3answers
48 views

Logical form of Either and Neither: Alice in room

This is one of the problem I have been working: ...
1
vote
2answers
38 views

Method of verifying answers

I have been reading Velleman's How to prove it book and solving problems of the exercise in it. What concerns me is that I cannot verify if actually my solutions are correct. The book has only ...
-2
votes
2answers
71 views

Creating Truth tables [closed]

What is the truth table for the logical expression? $$ (p \land (p \to q) \land r) \to ((p \lor q) \to r) $$ Frankly, I'm lost.
2
votes
1answer
30 views

Does introduction and elimination rule for an operator determine uniquely its truth table?

My question is regarding the inference of a truth table for an operator given how it behaves according to introduction and elimination. This follows from an exercise I read, and it got me thinking if ...
1
vote
1answer
43 views

Conjuctive Normal Form

In Boolean logic, a formula is in conjunctive normal form or clausal normal form if it is a conjunction of clauses, where a clause is a disjunction of literals; otherwise put, it is an AND of ORs. I ...
1
vote
2answers
38 views

Reducing $ab' + cb + ac$ to $ab' + cb$

Boolean expressions $I = ab' + cb + ac$ and $J = ab' + cb$ have the same truth table. Then why expression $I$ can't be reduced to expression $J$?
1
vote
1answer
36 views

How to prove this logical equivalence using different laws?

Prove that $﹁p → (q→r)$ and $q → (p∨r)$ are logically equivalent using different laws. this is my answer: $﹁p → (q→r) = q → (p∨r)$ $(q→r) = ﹁q∨r$ implication equivalence $﹁p → (q→r) = p∨(﹁q∨r)$ ...
4
votes
1answer
108 views

How to prove Post's Theorem by induction?

The proof of post's theorem is given in my textbook in two pages of explanation using a non-induction method. I was told that ,using induction on length of the proof, one can get a simpler and more ...
0
votes
2answers
74 views

Is it always a tautology?

If any two compound propositions $P$ and $Q$ are equivalent, then is the proposition formed from their biconditional $P \leftrightarrow Q$ always a tautology?
1
vote
1answer
34 views

Correctly understanding truth table problem?

I'm typing up a solution set for an "intro to proof" course. One of the problems asks the student to "construct a truth table for $(P \implies Q) \implies (\neg P)$." I interpreted this as requesting ...
3
votes
0answers
38 views

Proving a graph has a property if all finite subgraphs have that property

Given a graph $G=(V,E)$ and an integer $k\in\mathbb N$, we will say that $G$ is $k$-good if: for every division $V=\bigcup_{i\in I} U_i$ such that $i\not=j \Rightarrow U_i\cap U_j =\emptyset$ and ...
1
vote
2answers
73 views

How to deal with equivalences in proofs?

There is a part I need clarification on regarding the use of equivalence and its symmetry. From what I understand in regards to symmetry is that: $ (p \equiv q) \equiv (q \equiv p) $. Given p and q ...
0
votes
1answer
41 views

How to prove validity of following sequent [closed]

How to prove validity of following: Premises: $p\rightarrow q$, $s\rightarrow t$, Conclusion: $(p \lor s) \rightarrow (q\land t)$
0
votes
4answers
191 views

Logic Confusing Problem

I Read one logic book, can my two conclusion are true? 1- Suppose for each valuation v, we have such n that can we say we have such n that 2- Suppose for each ...
2
votes
1answer
111 views

About $\Sigma=\{p_2\to p_1, p_3\to p_2,\, \dots\,\}$ . . .

Suppose $$\Sigma=\{p_2\to p_1, p_3\to p_2,\, \dots\,\}.$$ Which of the following is true? Explain your answer. For any $n$, $$\Sigma\cup\{p_n, \neg p_{n+1}\}$$ is complete and ...
-1
votes
1answer
60 views

Quick Truth Table in Logic Problem

Suppose We Have: How can quickly detect how many "1" are in the truth table of above formula? (without drawing Truth Table). i think by using some inference. any idea? we know there are 11 "1"s ...
-1
votes
4answers
112 views

Which of $\varphi$ or $\lnot \varphi$ can be expressed by using only the $\rightarrow$ connective? [closed]

if we have: $$\varphi = \lnot(p\land q\to r) $$ (original screenshot) a) we can write $\varphi$ in equivalence just by using $\to$ connective. b) we can write $\lnot\varphi$ in equivalence ...
1
vote
1answer
59 views

Prove A or (A and B) is equivalent to A [duplicate]

Prove $A \lor (A \land B) \Leftrightarrow A$ without using truth table. The proof may involve expanding $B$ into $B \land B$ or possibly $B \lor B$. I am stuck after playing with distributive law ...
0
votes
1answer
45 views

Logic Pure Subset Problem

for example if we define : $$ \$(p,q,r) = (p\to q)\land(\neg p\to r)$$ how we can inference that set $\{\$,\top,\bot\}$ is Full Functional and not any pure subset of this be full functional.
1
vote
3answers
90 views

Can a statement in FOL be equivalent to two separate independent statements?

This may seem like a dumb question, and it certainly seems dumb to me asking it, but I'm running into a contradiction. I'm looking at the problem of finding a statement $\phi$ such that $\psi$ and ...
1
vote
1answer
32 views

From statement to logic

I have a problem with the modelling of the following statement in propositional logic (warning, I translated it from italian): Martha is not a singer, and she doesn't play violin or flute, but not ...
4
votes
3answers
191 views

Simplifying a categorical proof of constructive dilemma

In axiomatic propositional calculus the following axiom schema captures constructive dilemma: $\newcommand{\lif}{\supset} \renewcommand{\land}{\&}$ \begin{equation} (a \lif c) \lif ((b \lif c) ...
1
vote
1answer
45 views

Rules of inference: The Rules of Disjunctive Syllogism and Double Negation

I have a question about the use of Double Negation in relation to this problem I found in my textbook examples. Problem: $\;¬(r \land t) \lor u$ $\;r \land t$ Therefore, $u$. In my textbook it ...
2
votes
1answer
21 views

Proposition Question

I am trying to translate this into propositional symbols but (for me) it's so complicated. Can someone help me figure this out. "If it rains then I will carry a sharp object and I will start laughing ...
0
votes
2answers
67 views

How to express $\lnot (a < b < 0)$ or the contrapositive of this statement?

I can't seem to get the negation, $\lnot (a < b < 0)$, right. I thought I could break it into 3 parts: a < b, a < 0, b < 0, but that leaves me with a > b or a > 0 or b > 0 (greater or ...
2
votes
5answers
87 views

Showing that $\lnot Q \lor (\lnot Q \land R) = \lnot Q$ without a truth table

I've done a truth table after reducing it to this and it seems to be equal to $\neg Q$: $$\lnot Q \lor (\lnot Q \land R) = \lnot Q$$ But when I try to show it without a truth table (with just ...
3
votes
2answers
94 views

Why is removing the negation worse than adding it?

Natural Deduction Rule (¬I): Natural Deduction Rule (RAA): My book [Ian Chiswell & Wilfrid Hodges, Mathematical Logic (2007)]presents these two rules and then adds: The use of (RAA) can ...
2
votes
1answer
61 views

A simpler derivation of ($\phi \lor (\neg \phi)$)

In Chiswell&Hodges [Ian Chiswell & Wilfrid Hodges, Mathematical Logic (2007)] they use this derivation to prove ($\phi \lor (\neg \phi)$): A page earlier they used a simpler derivation that ...
1
vote
1answer
41 views

Is there a proof of this statement about deductions?

Is there a proof of the following statement: you cannot prove with natural deduction theorems that are unprovable in a Hilbert-style proof system? The logic in discussion is either propositional logic ...
1
vote
1answer
93 views

Prove that the disjunctions of all conjucts is a disjunctive normal form

Question: I am attempting the following exercise from An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof by Peter B. Andrews: X1408. Prove that if $\mathbf{A}$ is a wff ...
0
votes
0answers
67 views

alternative Compactness theorem proof

I'm attempting a problem which requires me to prove the compactness theorem for propositional logic ![enter image description here][1]in a slightly different way to normal. I'm struggling to ...
4
votes
1answer
35 views

Discrete 101: Validity of proof: Finding that p→q ∨ ¬r, q→p∧r, therefore p→r is invalid.

I'm sorry to bother with what apparently is a very easy Basic Logic question, but in my class'es notes there's an example that the professor probably explained in class: ...