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Logic on Math StackExchange
- Can the Completeness Theorem be proven constructively in a natural-deductive system? June 5, 2023In Sets, Logic and Computation the following is said of the Completeness Theorem: When completeness is thought of as “whenever $\models A$ then $\vdash A$,” it may be hard to even come up with an idea: for to show that $\vdash A$ we have to find a derivation, and it does not look like the […]zaq
- What is $r^-1(B)$ if $r$ is a rule and $B$ is an expression in a proof system? June 5, 2023In the first paragraph, what does $r^{-1}(B)$ mean? Does it have something to do with relations? e.g if $A$ is a relation then $A^{-1}$ is the inverse of that relation. Its from a paper about the hilbert system: https://www3.cs.stonybrook.edu/~cse541/chapter8.pdflightyourassonfire
- Does this proof of the compactness theorem assumes countable atomic formulas? June 5, 2023In the book Philosophical and Mathematical Logic, the following proof of the compactness theorem is presented: Theorem 2.3 (Compactness theorem). Let $\Gamma$ be a (possibly infinite) set of formulas such that every finite subset of $\Gamma$ has a model. Then $\Gamma$ has a model Proof. Let $\Gamma$ be a (possibly infinite) set of formulas such […]Michael Novak
- Perfect square that last 2 digits are equal, show that these digits are 0 or 4. [closed] June 5, 2023I am trying to solve this question: Let n > 100 be an integer that is a perfect square such that its last two digits in base 10 are equal. Show that these two digits are 0 or 4. My idea use congruencies but I don't figure how to do it. ThanksssMr Arrows
- In a direct proof, do your chain of deductions have to involve the antecedent in any way in order for this to be considered a "direct proof"? June 5, 2023If I am asked to give a direct proof of the implication "if the sky is blue, 7 is prime", would it be considered a valid direct proof to conclude 7 is prime through a chain of implications which have nothing to do with the sky being blue? (the idea being that any implication whose […]God
- Proof for modus tollens and ponens June 5, 2023are modus ponens and tollens axioms if not how can they be derived or proved? if I am good at math then I am good at logic, I am good at math, therofe I am good at logic this seems evident but is there any proof for them? looking at the truth table for implication […]SirMrpirateroberts
- Prove that the introduction rule of implication can be derived from the rules of conjunction and negation. June 4, 2023The introduction rule of implication states that if, by assuming $A$, we can derive $B$, then we can derive $A\rightarrow B$. $Proof$. Suppose $A$ proves $B$. Now suppose $A\land\neg B$. By conjunction elimination, we have $A$ and $\neg$B, but since $A$ proves $B$, we have $B$ and $\neg B$. Now by negation introduction, we get […]lightyourassonfire
- How to prove that these formulas are uniquely readable. June 4, 2023We have the inductively defined set, M, as follows(this is a language over the alpabet $(e,a,\bar{a},b,\bar{b})$: We let $M$ be the smallest set so that $e\in M$ If $x,y\in M$ then $xy\in M$ If $x \in M$ then $ax\bar{a}\in M$ If $x \in M$ then $bx\bar{b}\in M$ How to I prove that this set is […]user394334
- Prove $\exists$-$E$ can be derived from $\forall$-$E$ and $\forall-I$ June 4, 2023Here are the definitions of the rules: This answer helped me to prove the $\exists$-$I$ rule, but im struggling to prove $\exists$-$E$ How to show that the introduction and elimination rules for $\exists$ can be derived from the rules for $\forall$? So basically, im supposed to show that given $\neg\forall x\neg A(x)$ and a subproof […]lightyourassonfire
- Mendelson, Logic, Lemma 2.16: a possible typo or flawed understanding? June 4, 2023The book is Mendelson's Introduction to Logic, 6th edition. Here is the statement of Lemma 2.16: Let $J$ be a consistent, complete scapegoat theory. Then $J$ has a model $M$ whose domain is the set $D$ of closed terms of $J$. Definition of scapegoat theory in his book is: A theory $K$ is a scapegoat […]noballpointpen
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