## Franklin's Notes

### Notation for Sentential Logic

We restrict ourselves to the logical connectives $\neg$ and $\land$ in sentential logic because it makes inductive proofs on sentences much easier. However, writing out every proposition using only parentheses and these two connectives would be way too much trouble, so we use the following abbreviations:

• $(\varphi \lor \psi)$ stands for $(\neg((\neg\varphi)\land(\neg\psi)))$ (i.e. "$\varphi$ or $\psi$")

• $(\varphi\rightarrow\psi)$ stands for $((\neg\varphi)\lor\psi)$ (i.e."$\varphi$ implies $\psi$")

• $(\varphi \leftrightarrow\psi)$ stands for $((\varphi\rightarrow\psi)\land(\psi\rightarrow\varphi))$ (i.e. "$\varphi$ is equivalent to $\psi$")

Chang and Kiesler note that we could have actually used "neither... nor..." as our only logical connective, since all of the others can be built up from it, but it is a much more "unnatural" logical connective, so they consider it more convenient to use the more intuitive connectives "not" and "and" at the price of making inductive proofs slightly more time-consuming.

Additionally, we use lowercase Greek letters $\varphi,\psi,\theta...$ to refer to sentences in a language, and uppercase Greek letters $\Sigma,\Gamma,\Delta...$ to refer to sets of sentences (also called theories) in a language. Note that these are not symbols of the language itself, but "placeholders" or "variables" for symbols/sentences/sets of sentences.