# Tag Info

### If a proposition is necessarily true, does it follow that it's a tautology?

In effect, your question is asking whether logical necessity is the only kind of necessity. It is fairly standard to hold that there are many kinds, of which logical necessity is only one. There is ...
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### Is there a fallacy for "A doesn't work very well, therefore B is better than A"?

All formal fallacies are non sequitur, giving them fancy names doesn't make them more "specific", if it works it should make them more "descriptive". So the actual problem is that ...
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### Is there only one viable definition of the logical connectives?

I would say that the plethora of notions of disjunction make the answer to this a resounding no. You can find an overview of them on the Stanford Encyclopedia of Philosophy, but let me give a sketch ...
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### If a proposition is necessarily true, does it follow that it's a tautology?

But what is a necessary truth? "A proposition that is true in all possible worlds." Well, is it possible for there to be something that exists in all possible worlds? Given the variety of ...
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### Is there only one viable definition of the logical connectives?

There is a represenation theorem which says that binary truth functionals are equivalent to Truth Tables. So, I believe as long as we have Truth Tables, our operators being truth functionals is forced....
1 vote

### Is there only one viable definition of the logical connectives?

Outside of its utility within the language it is expressed, there is not an absolute method of endorsement. That is because logical operators are constituents of logical systems. In fact, we can even ...
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### Is there only one viable definition of the logical connectives?

Sounds like a question category theory could answer. I’ll ask some people I know. I think you’re asking about a more essential “motivating criterion” from which the truth functions would follow. One ...
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### What is the most accurate definition of logic?

It seems you are thinking of "classical logic". From Classical Logic/SEP Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-...
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### Formula deduction in K logic

Here's a few hints, rather than a full proof: Assume the negation of your sentence. Derive ((□A → □□A) ∧ A) → □A and also ¬(A → □A) From ¬(A → □A) derive A and ¬□A From ((□A → □□A) ∧ A) → □A and ¬□A ...
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### Why is "the present king of France is bald" considered false?

Let P = The present king of France is bald. If ∃x(Fx) = There exists a King of France. P → ∃x(Fx) ¬∃x(Fx) [We know France has no kings] Ergo, ¬P [Modus Tollens 1, 2] So, does that mean, ¬P = The ...
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### Is this a paradox or a mistake?

@n@ is an expression which Godel number is n. @n@ --> P is a different expression, so its Godel number is different. Therefore, there is no such n for which the equation n = #@n@ --> P# may hold....
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### Existential import in "No A is B"

After much thinking ... I recalled an explanation from a book on logic. If No A are B (universals, this is a universal negative) is granted existential import, we run into the following problem: No A ...
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### Do "No S is P" and "All S are not P" mean the same?

Update: post discussion with Bumble It depends. The propositions E ("No S is P") and A* ("All S are not-P") may or may not be logically equivalent, depending on your interpretation ...
1 vote

### How do you prove transitivity of equality?

Let φ(x) be (A=x & x=C). Then φ(B) is (A=B & B=C), which is the hypothesis. Thus, φ(B) is true. Therefore, B=C & φ(B). Now by substitution φ(C). Therefore, (A=C & C=C). Thus, A=C. ...
• 1,220
Accepted

### How do you prove transitivity of equality?

Given A = B and B = C, use substitution with A := B, B := C, φ(x) := (A = x). This is similar to the way you prove transitivity of Martin-Löf's identity type.
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### Can causality be translated into logical formalism to analyze it via mathematical logic?

Causality is an influence by which one event, process, state, or object (a cause) contributes to the production of another event, process, state, or object (an effect) where the cause is partly ...
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### Can causality be translated into logical formalism to analyze it via mathematical logic?

A implies B - If A is true at time t, then B is true at time t. A causes B - If A had not been true at t₁, then B wouldn't have been true at t₂ = t₁ + Δt (Δt > 0). The human mind only has one ...
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### Is there a paradox in the proof of Godel's incompleteness theorem?

Wittgenstein's criticism can be summed up as "it is not true that Goedel's proof is purely syntactic, in fact it cannot be". Wittgenstein was indeed very critical to the whole Aristotelian, ...
1 vote

### Can causality be translated into logical formalism to analyze it via mathematical logic?

Non liquet but ... both logic and causality are understood/framed as this follows - (non) sequitur and post (hoc ergo propter hoc) - from that.
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1 vote

### Is "we can't do good because bad exists" a logical fallacy?

“I can’t believe” other people do this thing when other things should be done instead. That’s technically a personal opinion. The actual fallacy would come later based on what they decide to do about ...
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### Is there a paradox in the proof of Godel's incompleteness theorem?

In an appendix to Part I of Remarks on The Foundations Of Mathematics, Wittgenstein criticized the following argument: I imagine someone asking my advice; he says: “I have constructed a proposition (...
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### Is there a paradox in the proof of Godel's incompleteness theorem?

Gödel was right. O'Connor 2005 meets every known objection: it is constructive and finite, indeed it runs on commodity hardware in reasonable time; it includes Rosser's trick, it is not relative to ...
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1 vote

### Existential import in "No A is B"

I have doubts about the existential import in categorial propositions of the form "No A is B" under Aristotelian interpretation. In "No A is B", it is assumed that both A and B are ...

### Is there a paradox in the proof of Godel's incompleteness theorem?

You do not understand the incompleteness theorem. It does not require "coding", and it does not depend on "actual infinity", and it does not "hide" any paradox. You ...
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1 vote

### What is the relation between the uniformity of nature and determinism?

I read about Hume's argument that the uniformity of nature is a necessary condition for inductive reasoning to be valid, but we only have inductive reasons for believing in the uniformity in the first ...
Accepted

### Difference between premise and antecedent, and difference between proposition and statement?

"Is P the antecedent and Q the consequent?" Yes. Grammarians also sometimes use the terms protasis and apodosis respectively. It is not correct to say premise and conclusion. A conditional ...
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### Interpretation of the Implication Sign

Implication has a number of different but related senses. In logic we often wish to use the term to indicate a sufficient, but not necessary condition. So, at its simplest, "A implies B" ...
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### Question regarding conjunction and necessity in modal logic

This has been answered in the comments. Here is the solution. All tautologies are theorems. A ∧ B → A is a tautology. ⊢ A ∧ B → A ⊢ □[A ∧ B →A] [3,NEC] ⊢ □[A ∧ B →A] → (□[A ∧ B] →□A) [distribution] ⊢...
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### The application of logic to knowledge seems problematic

"why can we assume that knowledge abides the proof theoretic laws of logic?" There are indeed many forms of knowledge that are not strictly speaking modeled well by grammatical ...
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### The application of logic to knowledge seems problematic

I took an engineering course called Knowledge-Based Systems back in 1990. These are also known as expert systems, but I think expert systems include implicit knowledge stored in artificial or ...
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### The application of logic to knowledge seems problematic

Your question touches on the foundations of logic and epistemology, particularly the distinction between material implication and logical implication, and how knowledge conforms (or not) to the laws ...
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### The application of logic to knowledge seems problematic

The application of logic to knowledge seems problematic Sure it is. Logic comes from logos (shared concept among philosophy and religion); the enclosed concept of reasoning is only a part of it: ...
• 2,515
1 vote
Accepted

### If a wff is satisfable in P, then it is also satisfable in K?

Suppose I(phi) = 1. Then consider the frame with only one world that agrees with I everywhere. ie, V(w, sentence) := I(sentence). It follows easily by definition of the forcing relation that M,w ...
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1 vote

### Looking for a formal proof that x=x isn't a contingency

A follow-up proof outline ∃x(¬x = x) [Assume for reductio ad absurdum] ⊥ Ergo, ¬∃x(¬x = x) [1 to 2, RAA] ∀x(x = x) [3 QN] QED Note: Since The Law of Identity is not contigent, it's necessary. The ...
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1 vote

### Looking for a formal proof that x=x isn't a contingency

I am unable to prove it formally, but I can use the metalanguage English, and a meta-axiom to prove it. Perhaps someone else can improve on this. Metalanguage: English Undefined binary relations: is-...
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### On Relations Versus Relational Properties

In predicate logic you can take an (n+1)-arity predicate and rewrite it using n-arity predicates without loss, but only down as far as n=2. So, for example, you can rewrite a ternary predicate using ...
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Accepted

### Importance of Logical Notation

The matter of mathematical style is not necessarily trivial. That SEP article quotes a certain Granger[68] such that: These different ways of grasping a concept, of integrating it in an operative ...
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### On Relations Versus Relational Properties

A relation between two sets X and Y is a subset R of the Cartesian product R subset X x Y Here R is the set of all ordered pairs (x,y) from XxY such that x relates to y. Not all relations are ...
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### The application of logic to knowledge seems problematic

There is epistemic logic, for a more general example. With regards to the issue of A → B, then with a knowledge operator k, we have to distinguish between (among other things): (kA & (A → B)) → ...
• 15.9k
1 vote

### Looking for a formal proof that x=x isn't a contingency

Let me provide a complementary answer to @Mauro. Classically we attribute contingency to sentences. While we can change our deductive system to allow proofs of formulas with free variables (see Alex ...
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### Looking for a formal proof that x=x isn't a contingency

How to prove that the proposition denoted by x=x isn't a contingency, using Hao Wang's Axiom of Identity. We have to use the formula ¬(z=y) as Φ(z). Thus, Φ(y) is ¬(y=y) and Φ(x) is ¬(x=y). The ...
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### Question regarding disjunction and necessity in modal logic

The question is, is the following a theorem of any modal logic? ⊢ A ∨ B → □A ∨ □B To me it seems to be a consequence of the truth functional definition of OR. If 'A or B' is true then A must be true ...
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### Question regarding disjunction and necessity in modal logic

⊢ (A ∨ B) → (□A ∨ □B) is certainly not generally true, but it is a theorem of a rather weird modal system called Ver. The system Ver consists of the system K together with the axiom ⊢ □P In other ...
• 26.4k
Accepted

### Can natural deduction be incorporated into SQML?

You can't use the necessitation rule like that. It is a rule of proof rather than a rule of inference, so it is better written as: ⊢ P ------ NEC ⊢ □P A rule of inference would look like ...
• 26.4k
Accepted

### How can I derive ~a=b→☐~a=b in SQML?

I’ll provide a sketch of the proof in SQML. 1. a=b→□(a=a)→□(a=b) II 2. □(a=a) RX, NEC 3. a=b→□(a=b) 1,2 Prop. Logic 4. □(a=b→□(a=b)) 3 NEC 5. ◊(a=b)→◊□(a=b) 4, system K ...
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### How can I derive ~a=b→☐~a=b in SQML?

How can I derive ~a=b→☐~a=b in SQML? There's an alternative to RX and II, that allows you to prove this quickly, that I discovered recently. It's a modal logic version of Hao Wang's axiom of ...
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### What is the proper form of universal instantiation?

You say, "C is a general constant iff ∀x [x=C]" but in standard logics, the only thing this says is that the domain of interpretation contains a single value, C. You seem to think you can ...
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### Can you imagine the illogical?

No, you cant imagine anything illogical (as in something self-contradictory) since it is nonsensical, the meaning destroys itself as the terms can only be conjointly posited by excluding each other. ...