# How to prove H → M ￢H → ￢M prove H↔M?

## How do you prove a theorem in logic?

To prove a theorem you must construct a deduction, with no premises, such that its last line contains the theorem (formula). To get the information needed to deduce a theorem (the sentence letters that appear in the theorem) you can use two rules of sentential deduction: EMI and Addition.

## What is existential generalization rule?

In predicate logic, existential generalization (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition.

## Who is the mathematician who used to identify if an argument that has quantifiers is valid or not?

Aristotle investigated a restricted class of inferential patterns, which he called syllogisms, in which two categorical propositions served as premises and a third served as a conclusion.

## How do you represent some in first order logic?

But unfortunately, in propositional logic, we can only represent the facts, which are either true or false.
Basic Elements of First-order logic:

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Constant 1, 2, A, John, Mumbai, cat,….
Variables x, y, z, a, b,….
Predicates Brother, Father, >,….
Function sqrt, LeftLegOf, ….
Connectives ∧, ∨, ¬, ⇒, ⇔

## How do you prove an existential quantifier?

The most natural way to prove an existential statement (∃x)P(x) ( ∃ x ) P ( x ) is to produce a specific a and show that P(a) is true for your choice.

## How do you prove universal generalization?

This rule is something we can use when we want to prove that a certain property holds for every element of the universe. That is when we want to prove x P(x), we take an abrbitrary element x in the universe and prove P(x). Then by this Universal Generalization we can conclude x P(x).

## Which sentence will be unsatisfiable if the CNF sentence is unsatisfiable?

Which sentence will be unsatisfiable if the CNF sentence is unsatisfiable? Explanation: The CNF statement will be unsatisfiable just when the original sentence is unsatisfiable.

## What is a valid formula of first-order logic?

A first-order formula F over signature σ is satisfiable if A |= F for some σ-structure A. If F is not satisfiable it is called unsatisfiable. F is called valid if A |= F for every σ-structure A. Given a set of formulas S we write S |= F to mean that every σ-structure A that satisfies S also satisfies F.

## What is WFF in artificial intelligence?

In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.

## Is P QA WFF?

Rule (2) If p is a wff, so is ~p. Rule (3) If p and q are wffs, (p∧q), (p∨q), (pÉq), and (p⇔q) are wffs. Example: This is a wff: p∧(q∨r).
Valid wffs.

Law wff
Commutative Laws (p∨q) ⇔ (q∨p)
(p.q) ⇔ (q.p)
Associative Laws [(p∨q)∨r] ⇔ [p∨(q∨r)]
[(p.q).r] ⇔ [p.(q.r)]

## How do you convert to clausal form?

Algorithm for Converting a Sentence into Clauses (CNF):

1. Step I: Elimination of if-then operator: …
2. Step II: Reduction of the scope of negation:
3. Replace ¬ sign by choosing any of the following: …
4. Step III: Renaming the variable within the scope of quantifiers: …
5. Step IV: Moving of quantifiers in the front of the expression:

## Who gave resolution based inferencing?

Robinson

Resolution was proposed as a proof procedure by Robinson in 1965 [Robinson, 1965] for propositional and first-order logics. Resolution was claimed to be “machine-oriented” as it was particularly suitable for proofs to be performed by computer having only one rule of inference that may have to be applied many times.

## How do you prove resolution?

In order to apply resolution in a proof:

1. we express our hypotheses and conclusion as a product of sums (conjunctive normal form), such as those that appear in the Resolution Tautology.
2. each maxterm in the CNF of the hypothesis becomes a clause in the proof.

## How does resolution work as a proof procedure?

Resolution is used, if there are various statements are given, and we need to prove a conclusion of those statements. Unification is a key concept in proofs by resolutions. Resolution is a single inference rule which can efficiently operate on the conjunctive normal form or clausal form.