MHB Proving First Order Logic in Machover's Text

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The discussion focuses on proving specific first-order logic statements as outlined in Maurice Machover's text. Participants express difficulty in understanding the implications of the statements, particularly the conditions under which certain logical equivalences hold. Key points include the interpretation of the entailment relations and the substitution of terms in logical formulas. The need for clarity in the application of set theory principles to these proofs is emphasized. Overall, the conversation highlights the complexities involved in formal logic and the importance of precise definitions and operations.
pooj4
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Trouble working through Set theory, Logic, and their Limitations by Maurice Machover. Particularly these

1. $\sigma \vDash \alpha \rightarrow \forall x\alpha$ where $x$ does not occur in a free $\alpha$

2. $\sigma \vDash s_1 = t_1 \rightarrow ... \rightarrow s_n = t_n \rightarrow fs_1...s_n=ft_1...t_n$

3. $\sigma \vDash \forall x \alpha \rightarrow \alpha(x/t)$ (appealing to the fact that generally $\alpha(x/t)^\sigma = {\alpha}^{\sigma(x/t^\sigma )})$
 
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What exactly has to be done?
 

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