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In Appendix of Godel' proof by Nagel, there is a proof being left to the reader. The proof is how to prove the property of being a tautology is hereditary under the Rule of Substitution? I have no idea about this question just according to four formulas and the definition about tautology.

The author want to give a absolute proof of Consistency about the elementary logic of propositions which is formalized.

Formalization steps:

Step 1: build a complete catalogue of signs, such as $\sim$,$\land$,$\lor$,$\subset$,$a$,$b$... Use the signs to combine a sentence(formula). For example, $(p\lor q)\supset(p\lor q)$

Step 2: declare the Formation Rules. The rules tell how to get a sentence. Each sentential variable ($p$/$q$/$t$) counts as a sentence. And if $S$ is a sentence, $\sim S$ is also a sentence. And if $S_1$ and $S_2$ are sentences, $(S_1)\land(S_2)$,$(S_1)\lor(S_2)$,$(S_1)\supset(S_2)$ are all sentences.

Step 3: Two Transformation Rules are declared. Rule of Substitution (for sentential variables), says that from a sentence containing sentential Variables it is always permissible to derive another sentence by uniformly substituting sentences for the variables. The second Transformation Rule is the Rule of Detachment (or Modus Ponens). This rule says that from two sentences having the form $S_1$ and $S_1\supset S_2$ it is always permissible to derive the sentence $S_2$.

Step 4: Choose Axioms.

  • $(p\lor p)\supset p$
  • $p\supset (p\lor q)$
  • $(p\lor q)\supset (q\lor p)$
  • $(p\supset q)\supset ((r\lor p)\supset (r\lor q))$

Define the property of being a tautology

Remember that a sentence of the calculus is either one of the letters used as sentential variables (we will call such sentences elementary) or a compound of these letters, of the signs employed as sentential connectives, and of the parentheses. We agree to place each elementary formula in one of two mutually exclusive and exhaustive classes $K_1$ and $K_2$. Sentences that are not elementary are placed in these classes pursuant to the following conventions:

i) A formula having the form $S_1 \lor S_2$ is placed in class $K_2$ if both $S_1$ and $S_2$ are in $K_2$; otherwise, it is placed in $K_1$.

ii) A formula having the form $S_1 \supset S_2$ is placed in $K_2$, if $S_1$ is in $K_1$ and $S_2$ is in $K_2$; otherwise, it is placed in $K_1$.

iii) A formula having the form $S_1 \land S_2$ is placed in $K_1$, if both $S_1$ and $S_2$ are in $K_1$; otherwise, it is placed in $K_2$.

iv) A formula having the form $\sim S$ is placed in K2, if $S$ is in $K_1$; otherwise, it is placed in $K_1$.

We then define the property of being tautologous: a formula is a tautology if, and only if, it falls in the class $K_1$ no matter in which of the two classes its elementary constituents are placed.

Here gives my answer after some thinking

Suppose there is a tautology, note it $f(q)$, $q$ is a elementary formula. Then I assert $f(q)$ is a tautology whether $q$ belongs to $K_1$ or $K_2$. In this case, tautology is tautology when $q$ is substituted by no matter what formula.

And I think the elementary formula were chosen from $K_1$ and $K_2$ arbitrarily can prove the assertion.

cylia
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  • Do you have a hyperlink to that particular page and some effort on what you tried? – Dominique Jan 03 '24 at 12:00
  • Welcome to Math.SE! Your question needs some more detail, as we don;t just have a copy of Nagel's book lying around ... and even if we did, it is just good practice to include some of these details and make this a self-contained question. So at the very least you should add these 'four formulas' that you are referring to your post. And it wouldn't hurt to briefly state what the book means by tautology, heriditary property, and the Rule of Substitution. Finally, it always helps to get more and better feedback if you tell us about your thinking and efforts with this. – Bram28 Jan 03 '24 at 12:00
  • @Dominique Yes, this is the hyperlink about the question: link. (page 45 and page 109). The whole proof is too long. The author wants to giving a absolute proof of Consistency about the elementary logic of propositions which is formalized. – cylia Jan 03 '24 at 12:39
  • @Bram28 I am sorry for ambiguous description. But the total proof is too long. I will give more detailed description later. – cylia Jan 03 '24 at 12:44
  • @cylia ... so there is a proof given in the book? I thought it was left to the reader and you were wanting to know how to do it? – Bram28 Jan 03 '24 at 13:35
  • @cylia: thanks for the URL, I added it to your question for readability purposes. – Dominique Jan 03 '24 at 13:39
  • @Bram28 Yes, I understand the proof the book gives, only a step stuck me about tautology. – cylia Jan 03 '24 at 13:41
  • Thank you very much. @Dominique – cylia Jan 03 '24 at 13:53
  • Good work on updating your post and making it into a self-contained question! – Bram28 Jan 03 '24 at 14:01
  • The specific question you have is about what in logic is commonly known as the Substitution Theorem: If you have a formula that is a tautology, then if you systematically replace some propositional variable within that formula with some propositional logic formula, then the result is a tautology as well (e.g. Since $P \lor \neg P$ is a tautology, $(B \lor \neg C) \lor \neg (B \lor \neg C)$ is a tautology as well.). So your question is: how do you prove that Theorem? Well, I would do some Googling and looking around! – Bram28 Jan 03 '24 at 14:02
  • @Bram2B Yes, and I want know this proof in this formal system. It means the proof doesn't depend on meaning of signs. – cylia Jan 03 '24 at 14:11
  • I have never seen a completely formal proof of the Substitution Theorem ... that would be a beast ... and it would provide you with no further understanding of the Theorem! I would stick to just trying to understand the mathematical proof ... that is already non-trivial. Indeed, I am sure the main proofs in this book are all mathematical, not 100% formal. That's what a book like this does: it provides mathematical proofs about formal proof systems. But it doesn't formalize those mathematical proofs themselves ... little is gained from that. – Bram28 Jan 03 '24 at 14:16
  • @Bram28 The book says the proof about Consistency is a metamathematical proof. – cylia Jan 03 '24 at 14:44
  • I would call it a meta-logical proof, but sure a meta-mathematical proof is fine to. But what it is certainly not is a formal logic proof :) – Bram28 Jan 03 '24 at 17:01
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    See this post – Mauro ALLEGRANZA Jan 08 '24 at 08:02
  • @MauroALLEGRANZM Thank you very much, but in that post I see a proof using truth table. I want to get a proof for the formal system which has no definitions about a truth table. – cylia Jan 09 '24 at 12:12
  • If I recall the key idea from Kleene's proof in Introduction to Metamathematics, the idea goes that we refer to how formulas can get built up. For instance, say we consider the tautology (P)->((Q)->(P)), and it's substitution instance S1: ((A)->(B))->((Q)->((A)->(B))). S1 gets built up from (A)->(B) and (Q). So, (A)->(B) is true or false, and, of course, (Q) is. But, that means we could temporarily use a meta variable for (A)->(B) say X, and then we have (X) -> ((Q)->(X)). Does that not convince us though that the rule of substitution is hereditary? Uniform substitution or simultaneous – Doug Spoonwood Jan 13 '24 at 06:56
  • @DougSpoonwood I am not sure. The book give a definition about tautology. It doesnt care about logic true or false in that formal system. – cylia Feb 09 '24 at 08:52

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