We discussed the connection between logic and language, and the role of predicates in particular. We practiced formalizing three of the axioms of set theory (the axioms of extensionality, empty set, and pairing).

1. How finite sequences are defined in set theory.
2. The fact that if $A$ is an alphabet of size $\leq \kappa$, $\kappa$ infinite, then there are $\leq \kappa$ finite sequences that are constructible from the alphabet $A$. This implies that there are only countably many formulas in a given countable alphabet.
3. The types of symbols needed to express the Goldbach Conjecture: variables, logical symbols, equality, nonlogical symbols (constant symbols, operation symbols, predicate symbols), and punctuation symbols.
4. The types of symbols needed to express the continuity of a function: variables, logical symbols, equality, nonlogical symbols (constant symbols, operation symbols, predicate symbols), and punctuation symbols.
5. The construction of formulas in a 1-sorted language, $\mathcal{L}$. We completed the description of $\mathcal{L}$-terms.

1. Terms.
2. Atomic formulas.
3. Formulas and sentences.
4. Formula trees.

We discussed:

1. The language of Sentential Logic (= propositional logic = 0th-order logic). (Introduced by Stoic philosopher Chryssipus ~250 BCE.)
2. Sentential formulas. Wffs.
3. Formula trees.
4. Truth assignments. Tautologies and contradictions.

1. Unique readability of propositional formulas.
2. Completeness of sets of connectives.

1. Completeness of sets of connectives.
   The first major topic was Post's Classification. Then we discussed a corollary to Post's Classification: a set $\Omega$ of logical connectives is complete iff the following five statements are true:
   • $\Omega$ contains an operation that fails to preserve the relation $\{0\}$.
     (I.e., some $f(x_1,\ldots,x_n)\in\Omega$ satisfies $f(0,\ldots,0)\neq 0$.)
   • $\Omega$ contains an operation that fails to preserve the relation $\{1\}$.
     (I.e., some $f(x_1,\ldots,x_n)\in\Omega$ satisfies $f(1,\ldots,1)\neq 1$.)
   • $\Omega$ contains an operation that fails to preserve the relation $\leq$.
     (I.e., for some $f(x_1,\ldots,x_n)\in\Omega$ and some $a_i\leq b_i$, $i=1,\ldots,n$, $f(a_1,\ldots,a_n)\not\leq f(b_1,\ldots,b_n)$.)
   • $\Omega$ contains an operation that fails to commute with negation. (I.e., some $f(x_1,\ldots,x_n)\in\Omega$ fails to satisfy $\neg f(x_1,\ldots,x_n)=f(\neg x_1,\ldots,\neg x_n)$.)
   • $\Omega$ contains an operation that fails to commute with ternary addition modulo $2$. (I.e., there is some $f(x_1,\ldots,x_n)\in\Omega$ and tuples $\mathbf{a}, \mathbf{b},\mathbf{c}$ such that $f(\mathbf{a}\oplus \mathbf{b}\oplus\mathbf{c})\neq f(\mathbf{a})\oplus f(\mathbf{b})\oplus f(\mathbf{c}))$. )
2. Disjunctive Normal Form.. (We briefly mentioned Conjunctive Normal Form, too.)

1. Soundness.
2. Completeness.
3. Decidability.

Quiz 2 solutions.

Quiz 3 solutions.

Next, we began a discussion of Models and Truth. We discussed the possibility of multisorted models, but I proposed that we avoid multisortedness by introducing extra predicates. Then we introduced valuations and explained how they can be used to decide truth in a model. We reviewed this handout from August 26 (skipping the exercises), and the first two pages of this handout.

Quiz 4 solutions.

For Item (i), we discussed how Galois Connections ecode binary relations following these notes. Terminology associated to this point of view include the concept of an elementary class and the concept of a theory. We also defined the concept of elementary equivalence.

For Item (ii) we discussed how to define new constants, operations, and predicates in a structure. In one example, we explained how to define $\{0\}$, $\{1\}$, $\{2\}$, $[0,\infty)$, $\leq$, and $[1,2]\cup [3,4]$ in $\langle \mathbb{R}; \cdot,+\rangle$.

Midterm Review Sheet.

Quiz 5 solutions.

The midterm will be held on Friday, October 9, in class, from 12:20-1:10pm.

No quiz next Monday. No HW assignment due next Wednesday.

No quiz today. No HW assignment due Wednesday.

1. homomorphism
2. strong homomorphism
3. isomorphism
4. automorphism
5. embedding
6. substructure

We discussed a proof calculus for first-order logic. (Page 112.) The only deductive rule is modus ponens. The sets of axioms is the closure under $\forall$ of the set of formulas that are:

1. First-order tautologies,
2. Universal specification,
3. Quantifier distribution,
4. Quantifier introduction,
5. $(x=x)$, and
6. $(x=y)\to (\alpha=\alpha')$ if $\alpha'$ is obtained from $\alpha$ by replacing zero or more $x$'s with $y$'s.

Quiz 6 solutions.