Isn't that the unproblematic case? The formula is a tautology and hence is true under any valuation. Particularly, $[x \to x]_v = 1$ and then $f\big( \neg(x \to x)\big) = 1$ and no issues there. What am I missing?

@lafinur I was referring more to the original problem. Suppose, for instance, that $\Gamma = \{x\to x\}$ where $x$ is a sentence symbol. Then $\Gamma' = \{\neg(x\to x)\}$. Is this consistent?

@lafinur Regarding your argument, it looks like you're misunderstanding what a valuation does. It doesn't assign a true/false value to every formula, but rather a true/false value to every sentence symbol (assuming we are talking about sentential logic here). Then the truth value of every formula is determined by that assignment. The $f$ you've defined is not a valuation at all.

For instance, consider $\Gamma = \{x\land y, \neg(x\lor y)\}$. By your reasoning, one might say "define $f$ such that $f(x\land y) = 1$ and $f(\neg(x\lor y)) = 1$, then $f(\gamma) = 1$ for all $\gamma\in\Gamma$ and therefore $\Gamma$ is consistent". But this $\Gamma$ certainly is not consistent. Really you should be assigning values for $f(x)$ and $f(y)$, and the truth value of each formula involving $x,y$ is determined by these truth assignments. You'll find that no assignment of $x,y$ makes both formulas of $\Gamma$ true.

Your last two comments seems to hit the nail in the head in what comes to my confusion. If valuations refer to propositional atoms (or sentence symbols) and not formulas - it makes sense that this is the case now that I think about it -, any $\Gamma$ consisting of a tautological formula is consistent with $\Gamma'$ necessarily inconsistent.

Thank you very much!

@lafinur Sure thing, glad it helped! :-)

@lafinur You wouldn't be working through a model theory book, would you? Maybe Chang and Kiesler?

Not quite. I'm studying Logic and Structure, by Van Dalen. I'm a computer science student

Would you recommend those?

Neato. Just for fun?

"Chang and Kiesler" is one book, "Model Theory 3rd ed". I really like that one myself, it's really dense though

No, although it'd sound better to say yes. I have an exam on a couple of weeks on logics. I am quite fond of the subject though.

But I also work at a neuroscience laboratory so most of my "for fun" studies are on neuroscience-related stuff

Most of my mathematics/logics is, although very much enjoyed, what my university courses dictate

Understood hehe