# Tag Info

Accepted

### "Modus moron" rule of inference?

It's true that if $P$ is false then $P\Rightarrow Q$ is true. But the question is not asking if $P\Rightarrow Q$ is true, it's asking you if you can infer $P$ from $P\Rightarrow Q$ and $Q$. Let's be ...
• 152k
Accepted

### Does every proof need an axiom saying it works?

This is a complicated question to answer for multiple reasons. We really have to say something about the foundational crisis to give a full account of the story here. Here is a caricatured brief ...
• 431k

### "Modus moron" rule of inference?

This pattern is a logical fallacy called Affirming the Consequent, though I often call it Modus Bogus. To show it is not a valid inference, here is a simple Refutation by Logical Analogy: If I have ...
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### "Modus moron" rule of inference?

Even if affirming the consequent is not valid, other logical rules still work. Other logical rules still work. Therefore, affirming the consequent is not valid?
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### "Modus moron" rule of inference?

Not stated in the other answers so far is that you misunderstood the meaning of logical validity, which means that, in every situation where the premises hold, the conclusion also holds. Now you may ...
• 59.1k

### Why not ban nested quantifiers over the same variable?

It could be done, as described in the other answers. But it would not make capture-avoiding substitution easier to define, so there's no real benefit. On the other hand, we would lose the sometimes ...

### "Modus moron" rule of inference?

As others have said, the question is asking whether it is necessarily the case that whenever $P \Rightarrow Q$ and $Q$ are true, $P$ must be true. My personal favoured instantiations of $P$ and $Q$ ...
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### Why not ban nested quantifiers over the same variable?

The issue is that that kind of rule requires awareness of the contents of smaller wffs. Think about the other rules - none of them say anything about "if such-and-such a symbol appears inside...". The ...
• 18.1k

### "Modus moron" rule of inference?

if P⇒Q and Q it does not matter whether P or ¬P That's the point. It doesn't matter whether $P$ or $\neg P$: The statement becomes true for $P$ and $\neg P$ either. Therefore, you can not deduce ...
• 10.7k

### "Modus moron" rule of inference?

The other answers have perfectly settled this question, however I thought I would share another funny example. It is not exactly the same logical structure, but very similar. This example originates ...
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Accepted

• 17.2k
Accepted

### What if we don't accept ex falso quodlibet?

See Paraconsistent Logic for a family of logics that reject Ex Falso (aka: Principle of Explosion). See also: Walter Carnielli & Marcelo Esteban Coniglio, Paraconsistent Logic: Consistency, ...
• 94.9k
Accepted

### Proof of double negation elimination using natural deduction

The rules you list comprise a natural deduction system for intuitionistic logic. The sequent $\lnot A\to \bot\vdash A$ is not valid in intuitionistic logic, so it is not provable in your system. To ...
• 78.7k

### natural deduction: introduction of universal quantifier and elimination of existential quantifier explained

Example Let $\Gamma$ the set of first-order Peano axioms: no variables free. 1) $\Gamma \vdash \exists x (x = 0)$ --- easily provable 2) $\Gamma, x=0 \vdash x=0$ --- obvious 3) $\Gamma \vdash x=0$ ...
• 94.9k

### Calculus of Natural Deduction That Works for Empty Structures

The easiest (and in my opinion cleanest) way to do this is to augment the context. In the sequent calculus you presented, you have the left-hand of the sequent being a set $Γ$ of formulae. Instead of ...
• 59.1k
Accepted

### How to prove the validity of $¬∃x P(x) ⊢ ∀x ¬P(x)$ in predicate logic

We need contradiction in the proof (but it is not a poof by contradiction) : 1) $\lnot \exists x \ Px$ --- premise 2) $\quad\quad| \quad Px$ --- assumed [a] 3) $\quad\quad| \quad \exists x \ Px$ ---...
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### Is this proof correct? (natural deduction)

You were right to doubt your proof; it's not quite right. The main mistake is that you are effectively closing two subproofs at once once you go from 2.2.3 to 3, but you can only close one subproof ...
• 101k

### Why not ban nested quantifiers over the same variable?

In your first example $y$ does not occur free in step 2 and in your second example $x$ does not occur free in step 1, so the problem is not with the variable occurrences in the antecedents to the ...
• 49.2k
Accepted

### What is the utility, exactly, of the Deduction Theorem?

The theorem says that to prove an implication it is enough to assume the hypothesis and proceed to prove the conclusion. Proofs of that kind tend to be more natural than proofs that conclude the ...
• 79.6k
Accepted

### Relationship between sequent calculus and Hilbert systems, natural deduction, etc

To define a logic you need to specify a language of formulas, and then you need to provide either 1) a semantics, or 2) a proof system (i.e. a collection of rules of inference). For commonly ...
• 25.2k
It is the principle of explosion, also known as ex falso quodlibert: from contradiction, anything follows. In natural deduction, it says that if $\mathcal{D}$ is a derivation with conclusion $\bot$ ...