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 is the first premise true if variable $B$ is true?

Hi this is my second time on here so I am not quite able with the tools yet but I will try and do my best. I am currently studying How to Prove It by Daniel J. Velleman page [19] On page 19 number 2 ...
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4answers
163 views

Iff Interpretation

I understand that (1) "$A$ if and only if $B$" ($A\iff B)$ means that (2) "$A$ implies $B$ and $B$ implies $A$" $(A\implies B)\land (B\implies A)$. The phrase "$A$ if and only if $B$" sounds as ...
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37 views

Looking for in depth material on a formal propositional calculus using only the NAND connective

I am looking for secondary literature on a formal propositional calculus which has the NAND connective as its sole connective. I am coming upon many pages which briefly state that Nicod had shown ...
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3answers
101 views

Understanding iff [duplicate]

I'm having difficulty understanding why it is appropriate to use if and only if, something I thought I had a firm grasp on. From Lara Alcock's book, How to Study as a Mathematics Major: ...
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1answer
37 views

Should this be conditional or biconditional?

You can access Internet from campus only if you are a CS major or you are not a freshman How can the above English sentence be translated into a logical expression? I think this is ...
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41 views

why not include another assumption within natural deduction

question: prove $p → q \vdash ¬q → ¬p$ is valid. The answer is: $1. p → q~~~~\textsf{premise}$ $2. ¬q~~~~~\textsf{assumption}$ $3. ¬p~~~~~\textsf{MT }1,2$ $4. ¬q → ...
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2answers
74 views

Ideas for a history of math paper (with an emphasis on the mathematics), having to do with 19/20th century logic?

So I'm currently taking a history of math course and I need to write a 15 page paper in place of my final. It's a 400 level course (high undergrad) so the paper needs to have emphasis on the ...
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7answers
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Show that (p ∧ q) → (p ∨ q) is a tautology?

I am having a little trouble understanding proofs without truth tables particularly when it comes to → Here is a problem I am confused with: Show that (p ∧ q) → (p ∨ q) is a tautology The first ...
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1answer
25 views

Help understanding $P\land Q$ derivation in implicational propositional calculus?

According to its formulation, the implicational propositional calculus uses implication equipped with a tautologically false proposition $F$ to achieve soundness. Thus, consider the following ...
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2answers
28 views

formulating English sentences to logic

Consider the following sentences "We will play outside tomorrow, if there will be no rain" "We will play outside tomorrow, only if there will be no rain" Let's denote: $A$ = "play outside ...
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1answer
32 views

Help understanding the difference between imperative and logical conditionals?

How would you define a truth set $A$: for all $x \in B$ that satisfies $Q(x)$ where $B$ is another truth set that satisfies $P(x)$? I'm trying to formalize the natural intuition of if-then as distinct ...
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1answer
113 views

Equivalence classes under logical equivalence by 13 valuations

Let L be the set of 5 propositional variables. Under the equivalence relation given by logical equivalence, how many equivalence classes of propositional terms are given the value TRUE by 13 ...
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20 views

Formulas of propositional logic, two sets and refutation.

Let $\alpha$ and $\beta$ be two formulas of propositional logic and set $S_\alpha$ and $S_\beta$ be the sets of clauses representing $\neg \alpha$ and $\neg \beta$, respectively. Show that if ...
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3answers
31 views

Logical Equivalence of two propositions [closed]

Are the following two propositions logically equivalent? $p \rightarrow (\neg q \land r)$ and $\neg p \lor \neg(r \rightarrow q) $ For this one, I'm pretty sure that they are not equivalent ...
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1answer
16 views

Proving the negation of conditional via propositional calculus

https://www.dropbox.com/s/2dpk7fvae668phn/Screenshot%202016-03-03%2018.13.42.png?dl=0 Hi, I'm trying to prove the negation of a conditional. Basically, prove ¬(α → b) is equivalent to α ∧ ¬b. I've ...
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3answers
924 views

Proof of (P → Q) from ¬P?

I'm trying to figure out how to prove P → Q from just ¬P. I can deduce it using informal logic. Since the only way a conditional is False is in the case of T → F, if P is False, P → Q must always be ...
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1answer
31 views

Resaults of conditional statments

Is there a way to represent multiple conditional statements in a truth table and find the results of them: e.g., if person $A$ passes the exam, then person $B$ re-enters the exam. if person $A$ ...
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how to prove the following formula via natural deductions $a ∧ ¬a \vdash b ∧ ¬b$

Hi I am trying to prove the following formula via natural deduction and this is what I have so far. I am not sure however if this is entirely correct. If I could get some verification and be pointed ...
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1answer
36 views

Tautology example [closed]

How to verify if this logical statement is a tautology? AB+BCD+AC=NOT(AB+AC) I have seen all the posibilities but I am not sure that is tautology. And I have to solve it and to do all the ...
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0answers
45 views

Generating all logic propositions

I'm looking for a way of generating all logic propositions (propositional calculus) in an "algorithmic" way. The equivalence is symbolic, so $\neg\neg a \neq a$ and $\neg a \lor b \neq a \implies b$, ...
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1answer
25 views

Prove Tautology

Can you please prove the equation (~pVr)^(pVq)->(qVr) without using truth table. I have tried and ended up half way ~[(~pVr)^(pVq)]V(qVr) ~(~pVr)V~(pVq)V(qVr) (p^~r)V(~p^~q)VqVr (p^~r)VrV(~p^~q)Vq
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1answer
24 views

How to specify the corresponding line of a truth table in a formula?

How to specify the corresponding line of a truth table in a formula: $$p \to (\neg q \lor (q \to p))$$ $p$ evaluates to $F$ and $q$ evaluates to $T$. I want to know the method followed to find ...
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1answer
30 views

How to handle degrees (numerical attributes) in logic? How to model “quantitative changes lead to qualitative changes”?

I am using logics (propositional, predicate, modal) to model one domain, but there are variables that have non-boolean domains, these variables are degrees (it is sufficient that they are degrees, ...
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2answers
54 views

What will the declarative sentence No shoes, no shirt, no service be in propositional logic?

I need to write the following declarative sentence in propositional logic. No shoes, no shirt, no service. My solution is: ~p,~q, ~r , is it correct or do i need to use implication -> instead
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1answer
51 views

All hill-stations have a lake. Ooty has two lakes?

All hill-stations have a lake. Ooty has two lakes. Which of the statement(s) below is/are logically valid and can be inferred from the above sentences? $(i)$ Ooty is not a hill-station. $(ii)$ ...
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1answer
41 views

Propositional Resolution - How to prove?

I was given the following argument which is valid. I must prove it using propositional resolution. ((A $\rightarrow$ B) $\rightarrow$ C) (C $\rightarrow$ (D · E)) $∴ $ (B $\rightarrow$ D). So ...
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2answers
34 views

Solving rules of inference questions from Discrete Maths Rosen and I am confused on a step

So the question is, Show that the premises “It is not sunny this afternoon and it is colder than yesterday,” “We will go swimming only if it is sunny,” “If we do not go swimming, then ...
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0answers
28 views

Consistency of a Knowledge base (resolution)

To prove Q is a logical consequence of a knowledge base KB, it is possible to add not(Q) to KB and perform resolution, and reach an empty clause. But how is it possible to show Q is not a consequence ...
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1answer
92 views

Logical Equivalence as an equivalence relation

I've been given the following problem for homework and I'm struggling with where to begin "Suppose we have a set of 5 propositional variables, denoted L. By considering logical equivalence, what is ...
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3answers
32 views

Show that a set of logical connectives is expresively complete

I've been trying to figure this out for hours now, there doesn't seem to be ample resources online for my skill level to solve such a question: Show that a set of connectives {∧,¬} is expressively ...
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2answers
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substitutional interpretation of quantifiers: examples?

About the differences between propositional logic and (first order) predicate logic, given that if my basis is propositional logic I have to substitute the universal and existential quantifiers with ...
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46 views

Propositional logic vs predicate logic: examples?

About the difference between the propositional logic and the (first order) predicate logic-> can you give me one or more remarkable examples which underly the differences and the similarities ...
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2answers
45 views

If $P \iff Q$ is true and $P \land Q$ is false, do we arrive at a contradiction?

If $P \iff Q$ is true and $P \land Q$ is false, do we arrive at a contradiction? So I know that $P$ is false and $Q$ is false. And I also know that the biconditional $P \iff Q$ is equivalent to ...
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35 views

Proof of Replaceability of Equivalent Formulas by Structural Induction

My class discussed the following theorem for which I wasn't able to make it to class. Its proof is supposed to involve structural induction but I am stuck in the inductive step... Let B |=| C. If A' ...
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1answer
39 views

Prove the following formula is a contradiction: (~(A --> B)) ^ (B V ~ A)

I have worked with this truth table now - is everything correct or, do I miss something?
2
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1answer
34 views

Disjunctive Normal Form for Fuzzy Logic

I'm asked to prove that every propositional assertion in Fuzzy Logic, expressed using the standard propositional connectives $\{\land, \lor,\lnot, \rightarrow, \leftrightarrow\} $ can be expressed in ...
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63 views

A riddle in logic and propositional logic

The professor gave the class a riddle Suppose the following two statements are true: I love A or I love B If I love A, then I love B Does it necessarily follow that I love A? Does it necessarily ...
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1answer
40 views

Knights and Knaves. Translating an “Either or” statement into propositional logic

I'm studying for a midterm for my logic course. I was going through the lecture slides and came across this problem. You are on an island of kights and knaves. Knights always tell the truth and knaves ...
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1answer
30 views

Propositional Logic conversion (English to Language of Logic)

I have a confusion about the below proposition logic: Let p and q be the propositions p :It is below freezing. q :It is snowing. Write these propositions using p and q and logical connectives ...
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1answer
27 views

Conditional Proof and lemmas

I'm trying to solve a problem of propositional logic. The problem is this: $(H\vee P \vee L) \wedge (¬H \Rightarrow ¬P \vee ¬L ) \wedge (¬L\Rightarrow ¬P ) \wedge ¬H \Rightarrow L$ So I'm solving ...
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1answer
30 views

If the TV is not on sale, I will not buy it?

If the TV is on sale, I will buy the TV. The TV is not on sale. $\therefore$ I will not buy the TV. $p$: The TV is on sale. $q$: I will buy the TV. First statement above: $p\implies q$ Second ...
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1answer
48 views

Formally prove the following equations using propositional logic

p ∧ ¬p $\vdash$ q ∧ ¬q p ∧ r ⇒ q ∧ r , p ∨ r ⇒ q ∨ r $\vdash$ p ⇒ q I have literally been trying to figure these out all morning and I'm desperately stuck now. We have to prove them using ...
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1answer
74 views

What type of proof is $P\Rightarrow P$?

I have this problem statement: Let $P$ a proposition, now if we suppose $P$ is $\texttt{true}$, and the proof gives $P$. What demonstrates this? Solution: $\ \ P\Rightarrow P$ $\equiv \langle ...
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28 views

Show that the following argument is valid by Chain of Reasoning

I just recently started proving in propositional logic and everything is going well until I came across this problem. I tried to use different methods and it is going me out of nowhere. I read ...
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1answer
44 views

Negation of a Logical Statement; Proper English Translation

Consider the following two propositions: $p$: We can go to Cancun. $q$: We can go to Iceland. Using symbolic notation, a) Form the conjunction ($\land$). $p \land q$: We ...
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1answer
33 views

Propositional calculus resolution on pigeonhole principal

Here are my premises: A v B, C v D, E v F, ~A v ~C, ~B v ~D, ~A v ~E, ~B v ~F, ~C v ~E, ~D v ~F Is this even possible? I can't get it down to a unit clause because a new clause is added back in ...
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0answers
30 views

Defining the differentiability of a multivariable function (if/then)

I'm trying to understand differentiability for multivariable functions and am thoroughly confused by the introduction (and the direction of implications in a certain definition) I'm given the ...
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1answer
49 views

proof verification for converting sentence into propositional logic

Hi I wanted to know if I have translated this sentence correctly, The sentence is; ...
0
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1answer
31 views

Logically inferred from the given statements?

Consider the following statements relating to the level of poker play of four players $P, Q, R$ and $S$. $I. P$ always beats $Q$ $II. R$ always beats $S$ $III$. $S$ loses to $P$ only sometimes ...
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2answers
54 views

Prove each equivalence by using the rules for semantic equivalence

Having some issue with some logic - the examples I've been provided with arn't very helpful so I can have no idea where to start. The question is to prove; ...