Chapter 2

2.1 Mathematical Statement

A mathematical statement (also called proposition) is a statement that is true or false in an absolute sense.

Implication and Double-Implication

$S\Rightarrow T$ signifies that S implies T. This has nothing to do with causality, it only tells us something about the truthyness of both statements.

$S\Rightarrow T$ can only be false if $T$ is false while $S$ is true. As long as $S$ is false, $T$ does not need to be false as well for the whole statement to be true.

$S\iff T$ means that S is only true if T is true and likewise. This is the same as $S\iff T\text{and}T\iff S$.

Propositional Logic

2.5 Formulas

A correctly formed expression involving the propositional symbols $A,B,C...$ and logical operators is called a formula (of propositional logic).

Every such formula can be seen as a function with the domain $f:\{0,1{\}}^n\to \{0,1\}$. We can also express every function of that type as a logical formula as there is a function table for it.

Types for the different expressions

• $\equiv$ formula → statement (used to link different PL formulas, not part of PL itself)
• $\leftrightarrow$ formula → formula
• $\iff$ statement → statement and equivalently for
• $\models$ formula → statement
• $\to$ formula → formula
• $\Rightarrow$ statement → statement

2.1 Rules of Propositional Logic

1. $A\wedge A\equiv A$ and $A\vee A\equiv A$ (idempotence);
2. $A\wedge B\equiv B\wedge A$ and $A\vee B\equiv B\vee A$ (commutativity of $\wedge$ and $\vee$);
3. $(A\wedge B)\wedge C\equiv A\wedge (B\wedge C)$ and $(A\vee B)\vee C\equiv A\vee (B\vee C)$ (associativity);
4. $A\wedge (A\vee B)\equiv A$ and $A\vee (A\wedge B)\equiv A$ (absorption);
5. $A\wedge (B\vee C)\equiv (A\wedge B)\vee (A\wedge C)$ (first distributive law);
6. $A\vee (B\wedge C)\equiv (A\vee B)\wedge (A\vee C)$ (second distributive law);
7. $\neg \neg A\equiv A$ (double negation);
8. $\neg (A\wedge B)\equiv \neg A\vee \neg B$ and $\neg (A\vee B)\equiv \neg A\wedge \neg B$ (de Morgan’s rules).
9. $A\leftrightarrow B\equiv (A\to B)\wedge (B\to A)$ (double implication);

2.6 Logical Equivalence

Two formulas $F$ and $G$ (in propositional logic) are called equivalent, denoted as $F\equiv G$, if they correspond to the same function, i.e., if the truth values are equal for all truth assignments to the propositional symbols appearing in $F$ or $G$. Examples:

• $A\wedge B\equiv B\wedge A$ (commutativity)
• $\neg (\neg A)\equiv A$ (double negation)
• $A\to B\equiv \neg A\vee B$ (implication as disjunction)

2.7 A Logical Consequence

A formula $G$ is a logical consequence of a formula $F$ denoted $F\models G$ if for all truth assignments to the propositional symbols appearing in F or G, the truth value of $G$ is $1$ if the truth value of $F$ is 1.

2.8 Tautology

A formula $F$ (in PL) is called a tautology or valid if it is true for all truth assignments of the involved propositional symbols. One often writes $\models F$. The symbol $\top$ is used to denote a tautology.

2.9 Satisfiable

A formula $F$ (in PL) is called satisfiable if it is true for at least one truth assignment of the involved propositional symbols. It is called unsatisfiable otherwise (denoted $\perp$).

2.2 Tautology and Unsatisfiability

A formula F is a tautology if and only if $\neg F$ is unsatisfiable.

2.3 Tautology

For any formulas $F$ and $G$, $F\to G$ is a tautology if and only if $F\models G$.

First Introduction to Predicate Logic

2.10 $k$-ary Predicates

A $k$-ary predicate $P$ on $U$ is a function $U^k\to \{0,1\}$.

This predicate assigns each element of $U^k$ to a truth value.

We can use functions on $U$ and constants (fixed elements from $U$).

2.11 Quantifiers $\exists$ and $\forall$

For a universe $U$ and a predicate $P(x)$ we define the following logical statements:

• $\forall xP(x)$ stands for $P(x)$ is true for all $x$ in $U$.

• $\exists xP(x)$ stands for $P(x)$ is true for some $x$ in $U$, i.e. there exists an $x$ in $U$ for which $P(x)$ is true.

  Note that quantifiers bind stronger than logical operators!

The quantifiers can also be nested, where the order of the quantifiers matters.

Free parts

A formula of predicate logic has some free parts that are usually left open for interpretation. These could be predicates or the universe.

Some statements in PL are true for all interpretations, some depend on the specific interpretation…

• $G=Q(x)$ is satisfiable, since the interpretation Q is always true satisfies it. This is true, even though there is no quantifier or universe or predicate defined.

• We can even say that $H=Q(x)\models G=\exists xQ(x)$ since if there is an interpretation that makes $H$ satisfiable, there is at least one $x$ in that universe that satisfies $Q(x)$.

The other way around however is not true however! That is because if we were to provide an interpretation for that formula we would need to provide:

• A universe
• A function for the predicate
• A specific value for $x$!

If there is no quantifier, then $x$ has to be replaced by a constant from the universe!

Satisfiability, Tautology and Unsatisfiability

Tautology

A formula is a tautology (or valid) if it is true for all interpretations (all choices of universe and predicates).

Satisfiable

A formula is satisfiable if there is at least one interpretation under which it is true.

Unsatisfiable

A formula is unsatisfiable if it’s never true under any interpretation.

These definitions are distinct from those of Aussagenlogik, where they are about truth-assignments to symbols.

We are not allowed to use $\top$ or $\perp$ in formulas, to replace statement that are true or false under our interpretation. For example, in $U=\mathbb{N}$, $x\ge 0\wedge x=5⟹\top \wedge x=5⟹x=5$ but this is wrong as $x\ge 0$ is only equivalent to $\top$ in this specific universe. We instead can just write the implication directly.

Rules of manipulation

Equivalence in predicate logic

Two formulas are equivalent ($\equiv$) if the evaluate to the same truth table for any interpretation of the symbols in the formula.

Distributivity of $\forall$

$\forall xP(x)\wedge \forall xQ(x)\equiv \forall x(P(x)\wedge Q(x))$

Distributivity of $\exists$

$\exists x(P(x)\wedge Q(x))\models \exists x(P(x)\wedge \exists xQ(x))$

Negation of $\exists$ and $\forall$

• $\neg \forall xP(x)\equiv \exists x\neg P(x)$
• $\neg \exists xP(x)\equiv \forall x\neg P(x)$

Formulas vs. Mathematical Statements

If for a formula $F$ the interpretation is fixed then this can be a mathematical statement. It is then also meaningful to say that the formula is true or false (we can then use them in proofs).

2.4 Tautology and Logical Consequence

For any two formulas $F$ and $G$, if $F\models G$, then $F\text{ is valid}\Rightarrow G\text{ is valid}$.

Proof Patterns

Composing implications works by transitivity.

2.12 Transitivity of Implication

The proof step of composing implications is as follows: If $S\Rightarrow T$ and $T\Rightarrow U$ are both true, then $S\Rightarrow U$ is also true.

2.5 Transitivity of Implication

$(A\to B)\wedge (B\to C)\models A\to C$.

Direct Proof of an Implication

2.13 Direct proof

A direct proof of an implication $S\Rightarrow T$ works by assuming $S$ and then proving $T$ under this assumption.

Indirect Proof of an Implication

2.14 Indirect Proof (Proving the Contraposition)

An indirect proof of an implication $S\Rightarrow T$ proceeds by assuming that $T$ is false and proving that $S$ is false, under this assumption. This works because $\neg B\to \neg A\models A\to B$

Modus Ponens

2.15 Modus Ponens

A proof of a statement $S$ by use of modus ponens proceeds in three steps:

1. Find a suitable mathematical statement $R$
2. Prove $R$
3. Prove $R\Rightarrow S$

2.7 Modus Ponens

$A\wedge (A\to B)\models B$.

Case Distinction

2.16 Case Distinction

A proof of a statement $S$ by case distinction proceeds in three steps:

1. Find a finite list $R_1,\dots ,R_k$ of mathematical statements (the cases)
2. Prove that at least one of $R_i$ is true (at least one case occurs)
3. Prove that $R_i\Rightarrow S$ for $i=1,\dots ,k$.

One proves for a complete list of cases that the statement S holds in all the cases.

2.8 Case Distinction

For every $k$ we have $(A_1\vee \cdots \vee A_k)\wedge (A_1\to B)\wedge \cdots \wedge (A_k\to B)\models B$

If a single one of the $A_i$ is true then $A_i\to B$ also true thus $B$ is also true.

Proofs by Contradiction

2.17 Proof by contradiction

A proof by contradiction of a statement $S$ proceeds in three steps:

1. Find a suitable statement $T$.
2. Prove that $T$ is false.
3. Assume $S$ is false and prove that $T$ is true (a contradiction).

2.9 Contradiction

$(\neg A\to B)\wedge \neg B\models A$

Existence Proofs

2.18 Existence Proof

Consider a universe $X$ of parameters and consider for each $x$ in $X$ a statement $S_x$. An existence proof is a proof of the statement that $S_x$ is true for at least one $x\in X$. An existence proof is constructive if it exhibits an $a$ for which $S_a$ is true, otherwise it’s non constructive.

Existence Proof via the Pigeonhole principle

2.10 Pigeonhole Principle

If a set of $n$ objects is partitioned into $k<n$ sets, then at least one of those sets contains at least $\lceil \frac{n}{k}\rceil$ objects.

Example Among 100 people, there are at least nine who were born in the same month. Proof: Set $n=100$ and $k=12$ (number of months). Then $\lceil 100/12\rceil =9$. By the pigeonhole principle, at least one month must contain at least 9 birthdays.

Proofs by Counterexample

Counterexample

Consider a universe $X$ or parameters and consider for each $x$ a statement $S_x$. A proof by counterexample is a proof of the statement that $S_x$ is not true for all $x\in X$ by exhibiting an $a$ called counterexample such that $S_a$ is false.

Proof by Induction

We prove that a statement $P(x)$ is true for $P(0)$ the first element of the set $x$ is in.

We then prove the induction step which shows that $P(n)\Rightarrow P(n+1)$.

2.11 Induction

For the universe $\mathbb{N}$ and an arbitrary unary predicate $P$ we have $P(0)\wedge \forall n(P(n)\to P(n+1))\Rightarrow \forall nP(n)$