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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Proof for association law?

I am new in logic and I getting a little bit confused with maths. Can I do something like this following the Associative Law? $$(p ∨ ¬r) ∨ (r ∨ ¬p) ≡ (p ∨ ¬p) ∨ (r ∨ ¬r)$$ Thank you in advance for ...
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1answer
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Hilbert style proof of double negation introduction and reductio ab adsurdum

I'm trying to prove: $\phi\to\neg\neg\phi$ $(\neg\phi\to\neg\psi)\to((\neg\phi\to\psi)\to\phi)$ Using these axioms with modus ponens and the deduction theorem: A1: $\phi\to(\psi\to\phi)$ A2: ...
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1answer
34 views

Show that for propositional logic $\vdash_i \neg \varphi \Leftrightarrow \vdash_c \neg \varphi$.

As the title says, where $\vdash_i$ is derivations in Intuitionistic logic and $\vdash_c$ is derivations in Classical logic. I am allowed to use a corollary that states that $\vdash_i \varphi ...
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2answers
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Is $((p\rightarrow q) \land \neg p) \rightarrow \neg q$ a tautology?

Is this proposition a tautology? $((p\rightarrow q) \land \neg p) \rightarrow \neg q$ Knowing that $\alpha \rightarrow \beta$ is equivalent to $\neg \alpha \lor \beta$, I have come up with ...
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0answers
34 views

Hilbert style proof for $ \left( \left( A\rightarrow \left( A\wedge \neg A\right) \right) \rightarrow \left( A\rightarrow A \right) \right) $

How can I proof that the following formula is a tautology by using Hilbert calculus? $ \left( \left( A\rightarrow \left( A\wedge \neg A\right) \right) \rightarrow \left( A\rightarrow A \right) ...
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0answers
33 views

In classical logic ~~p -> p? Intuitionistic?

Is the following rule applicable in classical propositional logic? $\sim (\sim p)\rightarrow p$ In my textbook, it shows that $p \rightarrow\sim(\sim p)$ holds for intuitionistic logic but I was ...
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2answers
25 views

Prove/disprove a propositional statement

I have a homework question that I've been struggling with. I need to prove or disprove that: $(p ∧ (q ∨ r)) \to (r ∨ (q ∨ p)) = p ∨ q$ I've already constructed the first step of the proof which is ...
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1answer
36 views

how to give a truth value for the following formula

I am trying give a structure that makes that makes the formula T and a structure that makes the formula F for the following formula ...
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0answers
32 views

Create the following wffs(axiom rules for domain) for the domain of lists over alphabet A

Recall that in the domain of Lists over Alphabets, the function cons(a,x) where a is an element in an alphabet and x is a list, produces a new list with a at the beginning of L. The predicate ...
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22 views

proof verification for natural deduction

Could someone please let me know if I got the following natural deduction correct for the following formula ...
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1answer
22 views

Prove the following using only axioms of propositional logic and the deduction theorem? [see description] [closed]

$\vdash((\alpha\implies\beta)\implies(¬\beta\implies¬\alpha))$ Give a proof for the above theorem using only the three axioms of propositional logic (below), modus ponens, and the deduction theorem. ...
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0answers
38 views

How many ternary functionally complete connectives are there?

Today I was reading up once more on some of the nice results regarding functional completeness, notably Post's celebrated classification theorem with the 5 classes that need to be avoided. (See this ...
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1answer
7 views

Conjunctive Normal Form (CNF) of a propositional formula

These are my notes for Discrete Math. I'm having trouble understanding how to convert the given formulae at the end into CNF. The example seems to have skipped the steps and jumped straight to the ...
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2answers
50 views

$P ⇒ (Q ∨ S)$ , how can I prove $Q$?

I'm asking this in the context of a logical programming language similar to Prolog. Say I have the rule $P ⇒ (Q ∨ S)$ . How would I go about proving the truth value of $Q$, assuming I know the ...
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2answers
55 views

Can an open statement be a tautology?

A tautology is a statement which is true by dint only of the logical connectives contained therein. My question is about a statement which contains an unquantified variable. For example: P: ($x$ ...
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1answer
24 views

Associativity? Can this be applied here?

As the Associativity law says that (A ∧ B) ∧ C ≡ (A ∧ C) ∧ B, can I do something like this? (A ∧ ¬B) ∨ (B ∧ ¬A) ≡ (A ∧ ¬A) ∨ (B ∧ ¬B) I am new with logic and I still don't get this basic ...
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1answer
26 views

Set theory statements vs. propositional statements

I was wondering if statements that hold in general in set theory, such as De Morgan's Laws, always hold in propositional logic as well. If not, what are some examples of such statements that in the ...
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2answers
36 views

Formal logic proof verification

I am trying to prove the following sequent formally. $$P, (P \land Q)\Rightarrow \sim R \vdash R\Rightarrow \sim Q$$ I have come up with the following formal proof, but I am not completely sure if ...
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1answer
36 views

A faster way of proving that a 'theorem' (logic) is true.

Suppose I want to prove that the following is a theorem. $$\left [ \left ( P \vee Q \right ) \Rightarrow R \right ] \Rightarrow \left [ \left ( P \Rightarrow R \right ) \vee \left ( Q \Rightarrow R ...
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4answers
39 views

Showing that $(A \land B)' \land (C' \land A)' \land (C \land B')' \to A'$ without a truth table

Problem: Prove that $(A \land B)' \land (C' \land A)' \land (C \land B')' \to A'$. What I have done so far: $(A \land B)'$ premise $(C' \land A)'$ premise $(C \land B')'$ premise $A' \lor B'$ ...
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2answers
28 views

Proofs Using Tautologies

Let's say I want to formally prove a statement of the form $$p \implies q$$ So I do a bit of work,some re-arranging and eventually I arrive at a statement of the form $$p \implies p$$ which is a ...
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3answers
101 views

Logic Behind Epsilon-Delta Proofs (Single-Variable Calculus)

Most of what I am asking is based off this (fairly popular) article I've read here : https://bobobobo.wordpress.com/2008/01/20/how-to-do-epsilon-delta-proofs-1st-year-calculus/, but most lecturers, ...
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1answer
108 views

Progressing in Propositional Logic

I am self-studying precalculus-level mathematics in perhaps a more formal way than usual, which means that I am reading about logic, sets, proofs, etc. The text I am looking at contains as an example ...
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0answers
28 views

How to show that $\lnot q \equiv (p \lor q) \rightarrow p$?

How I can show that $\lnot q \equiv (p \lor q) \rightarrow p$ are equivalent using Law of Algebra Propositional ? I applied in this order: $(p \lor q) \implies p$ implication DeMorgan ...
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1answer
19 views

Resolution proof involving more than a literal

I want to show that the following clauses are unsatisfiable together using resolution (i.e. obtain a refutation): 1: $\lnot P_1 \lor \lnot P_2$ 2: $P_2 \lor \lnot P_3$ $P_1 \land P_3$ I perform ...
2
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1answer
15 views

conditional proposition vs biconditional proposition

So I have been working on college and am currently in a math class. The following question came up and I chose "->" as the answer. This was marked wrong and I challenged the answer but was told this ...
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2answers
36 views

finding a formula for a given truth table

How would one proceed in finding a formula from a given truth table without resort to the use of disjunctive normal form and karnaugh maps? For example, given ...
0
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1answer
24 views

Equivalence classes in the logical equivalence on some finite set of propositional formulas

I'm having trouble understanding the following problem: Let $S_n$ be the set of all formulas that can be built up with the atoms $\{A_1,...,A_n\}$. How many equivalence classes does the ...
2
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1answer
34 views

Q: Is the set of all binary connectives having an even number of Truth in their truth table is functionally incomplete?

Is the set $TC$ of all binary connectives having an even number of Truth values assigned to the entries of their truth table (i.e. 0, 2 or 4) is functionally incomplete? It's easy to see that the ...
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2answers
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Does the fact that a modal operator distributive over disjunction imply that a modal operator is distributive over conjunction?

If L is an arbitrary operator on two propositions p and q: Does L(p $\vee$ q) $\Rightarrow$ Lp $\vee$ Lq imply L(p $\land$ q) $\rightarrow$ Lp $\land$ Lq?
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1answer
62 views

proof that $\{\rightarrow, \land \}$ is not a complete set of logical connectives

I need some help to prove that the set $\{\rightarrow, \land \}$ of logical connectives is not a complete set. can someone help me to understand what should I do? thanks!
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1answer
20 views

Counterexamples of existentially quantified statements

I just realized I have a serious problem in properly seeing the logical structure that involves counterexamples. Here there is an example: Proposition F: Assume $P$. Then, there is a function $f ...
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32 views

propositional calculus problem, is this right proof? [duplicate]

I would like to confirm my proof.
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propositional calculus problem, how to prove this right or wrong?

$A$$\rightarrow$$(B$ $\vee$ $C$ ) , $B$ $\rightarrow$ $C$ $\vDash$ $A$ $\rightarrow$ $D$ I think it's wrong but I have no idea how to prove.
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1answer
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$\Sigma$ is maximally satisfiable $\iff$ there exists $M$ such that $\Sigma=\{\alpha \mid M\vDash \alpha\}$

A set of formulas $\Sigma$ is maximally satisfiable $\iff$ there exists $M$ such that $\Sigma=\{\alpha \mid M\vDash \alpha\}$. I have easily proved that if $\Sigma$ is maximally satisfiable than ...
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1answer
35 views

Is this a valid propositional natural deduction proof?

I'm new to logic and I tried to solve an exercise. Since there isn't a given answer, I'd appreciate an indication of whether this is correct ...
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1answer
21 views

logic: derive a formula using laws

Let's say I have the following formula: $$(A\wedge\neg C)\vee(B\wedge C)\vee(A\wedge B).\tag{1}$$ It is easy to show following: $$(A\wedge\neg C)\vee(B\wedge C)\vee(A\wedge B)\Leftrightarrow ...
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1answer
41 views

Truth table and induction

It is true that every truth table can be represented by some wff built using only the connectives $\neg, \implies$ and $\iff$ - let's call it "negation-arrow-wff" for convenience. I want to be able to ...
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2answers
31 views

Decide if $((((P\wedge Q)\wedge R)\wedge S)\wedge T)\Rightarrow(\neg P\vee T)$ is a tautology

How can I show that $((((P\wedge Q)\wedge R)\wedge S)\wedge T)\Rightarrow(\neg P\vee T)$ is a tautology? I tried to apply the implication rule $(p\Rightarrow q)\equiv (\neg p\vee q)$ but it doesn't ...
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0answers
13 views

Show that if X implies Y is valid, then X is unsatisfiable or Y is valid

How can I show that if X and Y are two formulas with no propositional variables in common, and (X ⇒ Y) is valid, then either X is unsatisfiable or Y is valid (or both). I know that (X ⇒ Y) is false ...
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1answer
62 views

Formal deduction proof of predicates

I am trying to proof equality is transitive, that is, $\emptyset \vdash \forall x \forall y \forall z ((x=y) \land (y=z) \to(x=z))$ using formal deduction (17 rules) and also other rules (ex. To ...
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1answer
51 views

HPC: Prove that $\vdash A\to \lnot\lnot A$

Prove that $\vdash A\to \lnot\lnot A$ By Deduction Rule we know that it is sufficient to show that ${A}\vdash \lnot\lnot A$ I am also familiar with the formula: $\lnot A \vdash (A\to B)$. So if ...
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2answers
44 views

Modus Tollens Proof

I came across the following proof in the book Logic, by Paul Tomassi: (P & Q) → ~R : R → (P → ~Q) According to the author, the proof should be a simple application of modus tollens. The following ...
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1answer
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property about truth tables

Is the question "show that any truth table is same as the truth table for some wff built from $\neg,\implies,\iff$ only" the same as asking show that any wff is logically equivalent to some wff built ...
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2answers
60 views

Writing proposition with connectives and laws of logic

Question 1): Pei Ann has been dealt two cards from a standard 52 card deck. She holds one in her left hand and one in her right. Let $p$ be the proposition "The card in Pei Ann's left hand is an ...
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1answer
24 views

Converting formula from CNF to DNF

How do i convert this formula from CNF to DNF? $(¬a \vee b) ∧ (¬b ∨ c) ∧ (¬a ∨ ¬c)$ $(¬a ∨ b) ∧ (¬b ∨ c) ∧ ¬(a ∧ c)$ DeMorgan ?
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1answer
34 views

Constructing the $\wedge$ logical connectives using $¬$ and $\leftrightarrow$

I am trying to show that if $p, q$ are distinct propositional variables, then there is no propositional formula $\phi$ such that the only connectives are $\leftrightarrow$ and $¬$ that is ...
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4answers
37 views

equivalent to $A \to (C \leftrightarrow D)$

I am somewhat confused while reading a paper. Are these two statements equivalent? $ A \wedge B \to (C \leftrightarrow D)$ $ [(A \wedge B \wedge C) \to D] \land [(A \wedge B \wedge D) \to C]$ I ...
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1answer
30 views

Propositional Logic; Im lost!

The proof I have to solve is: $$\lnot Z,~ \bigg((\lnot Z \lor S) \lor T\bigg) \implies L \vdash L \lor T$$ Basically I have tried to work backwards trying to prove the contradiction of $L \lor T$ ...
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1answer
26 views

Propositions ,Logic

John made the following statements: 1.I love Lucy 2.If i love Lucy then i love Vivian. Given that john either told the truth or lied in both cases.Determine whether John really loves Lucy.What that ...