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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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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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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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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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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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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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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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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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 ...
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1answer
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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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37 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 ...
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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 ...
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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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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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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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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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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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$\Sigma$ 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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37 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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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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53 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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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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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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64 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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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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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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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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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
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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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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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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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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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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 ...
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“This statement is false” - Propositional Logic

In a text I am reading, the section on Propositional Logic says that a proposition is a statement that is either true or false, but not both true and false. Also, from this lecture online, the ...
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Is mathematical induction the only known example of a higher-order logic?

Mathematical induction is one well known and widely cited example of a second-order logic. I was wondering whether there are other examples of arguments involving higher-order logic in any branch of ...
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Formal proof of predicates

I need to provide a formal proof of the following argument: Premise: $\exists x[P(X)\land\forall y(Q(y) \to \lnot R(x,y))]$ Premise: $\forall x[P(x) \to \forall y(S(y) \to R(x,y))]$ Conclusion: ...
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“This statement is false.” [duplicate]

In propositional logic, a proposition is a statement that is either true or false, but not both. In a text I am reading and in many others, "this statement is false" is not considered a proposition. ...
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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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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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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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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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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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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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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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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
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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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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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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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115 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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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 ...