Introduction: What IS Mathematical Logic?

Mathematical Logic IFall 2019

Robert Rynasiewicz

September 4, 2019

Arguments

Def. An argument is an ordered pair 〈Σ, τ〉 s.t.(i) τ is a sentence (called the conclusion), and(ii) Σ is a set of sentences (called the premises).

N.B. Σ may be infinite or empty.

Notation

Σ ∴ τ means: 〈Σ, τ〉 is an argument.

If Σ = {σ1, . . . , σn}, then

σ1, . . . , σn ∴ τ

means the same.

Arguments (cont.)

Can also write:

σ1

...

σn

————∴ τ

especially when considering arguments in natural languages.

Sample Argument

If Alice studies, then Alice gets good grades.

If Alice does not study, then Alice enjoys college.

If Alice does not get good grades, then Alice does not enjoy college.

—————

∴ Alice enjoys college.

I But is this a “good” argument?

Validity / Logical Consequence

Def. Σ ∴ τ is valid iff every possible situation in which each member ofΣ (each premise) is true, τ (the conclusion) is also true.

I.e., every model of Σ is a model of τ .

Notation. Σ |= τ indicates that the argument Σ ∴ τ is valid, in whichcase τ is said to be a logical consequence of Σ.

Def. Σ ∴ τ is sound iff Σ ∴ τ is valid and each member of Σ is true.

Philosophy and Science: want sound arguments.

Logic: cares only about validity.

Exercise

Is the following argument valid?

If Alice studies, then Alice gets good grades.

If Alice does not study, then Alice enjoys college.

If Alice does not get good grades, then Alice does not enjoycollege.

—————

∴ Alice enjoys college.

Hint: Can you find a counterexample, i.e., a situation in which all thepremises are true but the conclusion false?

Try: Alice studies, gets good grades, but fails to enjoy college.

Symbolically . . .

(S → G )

(¬S → E )

(¬G → ¬E )

————

∴ E

Counterexample: S , G , but ¬E

Some Terminology

Def. ¬ϕ is called the negation of ϕ.

Def. A sentence of the form (ϕ→ ψ) is called a (material) conditional.

I ϕ is said to be the antecedent of the conditional.

I ψ is said to be the consequent of the conditional.

Def. The converse of (ϕ→ ψ) is the conditional (ψ → ϕ).

Def. The contrapositive of (ϕ→ ψ) is the conditional (¬ψ → ¬ϕ).

N.B. The converse of the converse of (ϕ→ ψ) is (ϕ→ ψ).

But the contrapositive of the contrpositive of (ϕ→ ψ) is (¬¬ϕ→ ¬¬ψ).

Relevant Truth Tables

Negation:

ϕ ¬ϕT FF T

Material Conditional:

ϕ ψ (ϕ→ ψ)

T T TT F FF T TF F T

Symbolically, again

(S → G )

(¬S → E )

(¬G → ¬E )

————

∴ E

Counterexample: S , G , but ¬E

Another Exercise

Is the following argument valid?

(S → G )

(¬S → E )

(¬G → ¬E )

————

∴ G

Counterexample: ?????

First rule of logical practice:

I To establish that an argument is invalid , produce a counterexample.

I To establish that an argument is valid , prove the conclusion fromthe premises.

Informal Proof of G

Proof (by reductio ad absurdum):

Two strategies:

I. To show a sentence of the form ¬ϕ, suppose ϕ and derive acontradiction. (Both classically and intuitionistically permissible.)

II. To show a sentence of the form ϕ, suppose ¬ϕ and derive acontradiction. (Classically, but not intuitionistically permissible.)

Modus Ponens

(MP) From (ϕ→ ψ) and ϕ infer ψ.

Schematically:

(ϕ→ ψ)

ϕ

————

ψ

Modus Tollens

(MT) From (ϕ→ ψ) and ¬ψ infer ¬ϕ.

Schematically:

(ϕ→ ψ)

¬ψ

————

¬ϕ

Informal Proof of G (cont.)

1. Assume (S → G ). [Premise]

2. Assume (¬S → E ). [Premise]

3. Assume (¬G → ¬E ). [Premise]

4. Suppose that G is false, i.e., suppose ¬G . [reductio supposition]

5. ¬E [3,4 MP]

6. ¬S [1,4 MT]

7. ¬¬S [2,5 MT]

8. G [6,7 contradiction]

Formal Proof of G in Fitch-Style Natural Deduction

1 (S → G )

2 (¬S → E )

3 (¬G → ¬E )

4 ¬G

5 ¬E MP, 3, 4

6 ¬S MT, 1, 4

7 ¬¬S MT, 2, 5

8 G Reductio, 6, 7

Informal Proof of G w/o MT

1. Assume (S → G ). [Premise]

2. Assume (¬S → E ). [Premise]

3. Assume (¬G → ¬E ). [Premise]

4. Suppose that G is false, i.e., suppose ¬G . [reductio supposition]

5. ¬E [3,4 MP]

6. Suppose S [interior reductio supposition]

7. G [1,6 MP]

8. ¬S [4,7 contradiction]

9. E [2,8 MP]

10. G [5,9 contradiction]

Formal Proof of G w/o MT

1 (S → G )

2 (¬S → E )

3 (¬G → ¬E )

4 ¬G

5 ¬E MP, 3, 4

6 S

7 G MP, 1, 6

8 ¬G Repetition, 4

9 ¬S Reductio, 7, 8

10 E MP, 2, 9

11 G Reductio, 5, 10

The Character of Proof

Notation. Σ `S τ indicates that τ is provable from Σ in the system ofderivation S .

Query. What counts as a system of derivation?

Modern concept due to Gottlob Frege (1848-1925).

Figure: Frege circa 1879

The Character of Proof (cont.)

Figure: Proof from Frege’s Begriffsschrift (1879)

Frege’s goal in the Begriffsschrift (1879): eliminate all anschaulicheelements from mathematical proof.

A proof Π is:

I a finite syntactic objects s.t.

I there is a decision procedure as to whether Π is a proof of τ from Σ.

Deductive Soundness and Completeness

Σ `S τ : SYNTACTIC matter

Σ |= τ : SEMANTIC matter

A necessary condition for a system of proof S :

Deductive Soundness of S . If Σ `S τ , then Σ |= τ .

A desideratum for a system of proof S :

Deductive Completeness of S . If Σ |= τ , then Σ `S τ .

Why MATHEMATICAL Logic?

I In order for decision procedures be applicable to purported proofs inS , the language must be precisely formalizable, i.e., we need to useformal languages.

I In order to have a soundness theorem or even the hope of acompleteness theorem, the relation Σ |= τ must be no less precisethan the relation Σ `S τ . This requires formal semantics.

I Upshot: sentential logic, elementary (i.e., 1st-order) logic, andhigher-order logic are precise mathematical objects.

I Thus, logic becomes a branch of mathematics.

I But not just that. Other branches of mathematics (graph theory,group theory, geometry, arithmetic, set theory) are formalizable in(elementary) logic.

I This allows us to prove meta-theorems about mathematicaltheories.

I Examples?

Theories

Def. Fix a given language L. Let Sent(L) be the set of sentences of thatformal language, and let Σ ⊆ Sent(L). Then

Cn(Σ) =df {τ ∈ Sent(L) | Σ |= τ}.

Def. A theory T in a given language L is a set of sentences of L closedunder logical consequence, i.e., T is a theory iff T = Cn(T ).

Two Methods of Specifying Theories

Axiomatically. Let Σ be some decidable set of sentences of L.Take

T = Cn(Σ).

Model-Theoretically. Let Str(L) be the class of (semantic) structuresfor L, and let K ⊆ Str(L). Take

T = Th(K) =df {σ ∈ Sent(L) | σ is true in each structure A ∈ K}.

Examples

I Let Kfin = {A ∈ Str(L) : |A| is finite}.I Let N = (N, 0,S ,+,×) and K = {N}. Then Th(K) is the set of all

truths of arithmetic.

Properties of Theories

Properties a theory T may or may not have:

I T is complete, i.e., for every sentence σ of L, either σ ∈ T or¬σ ∈ T .

I T is decidable, i.e., there exists a decision procedure fordetermining for any given sentence σ of L whether σ ∈ T .

I T is axiomatizable, i.e., there exists a decidable set A of sentencess.t. T = Cn(A).

I T is finitely axiomatizable (f.a.), i.e., there exists a sentence α s.t.T = Cn(α).

I T is (absolutely) consistent, i.e., T 6= Sent(L).

I T is consistent relative to T ′, i.e., T is interpretable in T ′ (or,equivalently if T ′ has a model, then so does T ).

Some Results to Take Home

Godel’s 1st Incompleteness Theorem. Th(N) is neither decidable noraxiomatizable, i.e., any axiomatizable set of arithmetic truths isincomplete (including Peano arithmetic).

Godel’s 2nd Incompleteness Theorem. No axiomatizable set ofarithmetic truths can prove its own consistency unless it is inconsistent(in particular, Peano arithemetic).

What Is Peano Arithmetic?

PA is the logical closure of the following axioms.

(S1) ∀x : 0 6= Sx

(S2) ∀x∀y(Sx = Sy → x = y)

(A1) ∀x : x + 0 = x

(A2) ∀x∀y : x + Sy = S(x + y)

(M1) ∀x : x · 0 = 0

(M2) ∀x∀y : x · Sy = x · y + x

(Induction Scheme) For any formula ϕ(x) (with perhaps other freevariables) the universal closure of the following is an axiom:

(ϕ(0) ∧ ∀x(ϕ(x)→ ϕ(Sx)))→ ∀xϕ(x).

Billion Dollar Question: Is PA is consistent?

Strong Induction

The induction scheme of Peano arithmetic is usually called weakinduction on N.

There is a second form of induction on N that is often used, viz., stronginduction, or course-of-values induction:

∀n((∀m < n)ϕ(m)→ ϕ(n))→ ∀nϕ(n).

In “words”:

If the following holds for any n ∈ N,

if (∀m < n)ϕ(m), then ϕ(n),

then∀n ϕ(n).

“Equivalence” of Strong and Weak Induction on N

Lemma. Strong induction is a theorem of Peano arithmetic. Moreover, ifthe (weak) induction scheme is replaced by strong induction, theresulting theory is still Peano arithmetic.

Proof. HOMEWORK.

What We Assume

You can’t define and prove results about logical systems w/o using logicor any mathematics.

We assume what any mathematician standardly assumes, viz., elementary(1st-order) logic and Zermelo set theory (Z).

Zermelo Set Theory

The only non-logical primitive is the set membership relation ∈.

All other set-theoretic notions defined ultimately in terms of ∈. E.g.,

x ⊆ y =df ∀z(z ∈ x → z ∈ y).

(Extensionality) Sets having the same elements are identical.

(Separation Schema) For any set A and any property P( ) involvingultimately only ∈, there exists the set {x ∈ A | P(x)}.

Zermelo Set Theory (cont.)

N.B. A property alone defines a class. Some classes are “too big” tocount as sets. E.g., Take the class C = {x | x /∈ x}.

If C is set, we are led to a contradiction: C ∈ C iff C /∈ C . So, C is saidto be a proper class.

How does that help? Although sets are allowed to be elements of othersets (and of classes), classes are not allowed to be elements of classes(and certainly not sets). And the variable in expression such as x in{x | x /∈ x} ranges over only sets.

Proposition. There is no set of all sets.

Proof. Suppose there is, call it U. Then D = {x ∈ U | x /∈ x} is a setand we arrive at the contradiction D ∈ D iff D /∈ D.

Further Axioms of Z

(Unordered Pairs) For all x and y , {x , y} is a set. (An ordered pair(x , y) is defined to be {{x}, {x , y}}.)

(Union) For any set A,⋃A is a set, where⋃

A =df {x | ∃y ∈ A s.t. x ∈ y}.

Def. S(x) = x ∪ {x}.

Def. 0 = ∅, 1 = S(0), 2 = S(1), etc.In general, n + 1 = {0, . . . , n}.

(Infinity) There exists a set A s.t. ∅ ∈ A and for all x , if x ∈ A, thenS(x) ∈ A.

(Power Set). For any set A, P(A) =df {B | B ⊆ A} is a set.

(Foundation/Regularity) Every non-empty set x has a member y s.t.x ∩ y = ∅.

Axioms beyond Z

(Replacement Schema) Complicated to state and not needed forordinary math.

(Axiom of Choice) Comes in a variety of forms. Here are two.

I Any set can be well-ordered (where a well-ordering is a strict linearordering s.t. every non-empty subset has a least element.

I Let A be any set of non-empty sets. Then there is a “choice”function f : A→

⋃A s.t. f (X ) ∈ X for each X ∈ A.

Z as a Foundation for All of Mathematics

We’ve seen how to construct N in Z. Given N, there are standard ways toconstruct

I Z from N,

I Q from Z,

I R from Q, and

I C from R.

Furthermore,

I any group (or other algebraic structure) is isomorphic to some set,and

I any topological space is homeomorphic to some set.

Details for the last two claims

A group is an ordered pair (G , ◦), where ◦ : G × G → G s.t.

1. ◦ is associative, i.e., for all x , y , z ∈ G ,

x ◦ (y ◦ z) = (x ◦ y) ◦ z ,

2. There exists a unique e ∈ G s.t. for any x ∈ G ,

x ◦ e = x = e ◦ x ,

and

3. for each x ∈ G there exists a y ∈ G s.t.

x ◦ y = e = y ◦ x .

Group details (cont.)

Def. Let (G , ◦) and (G ′, ◦′) be groups and φ : G → G ′. Then ϕ is agroup homomorphism iff φ(x ◦ y) = φ(x) ◦′ φ(y), for all x , y ∈ G .

Def. φ is furthermore a group isomorphism iff φ is both 1-1 and onto.

Explanation. Group (G , ◦) is isomorphic to some set S iff

1. there exist sets X and f s.t. S = (X , f ) is a group, and

2. there exists a mapping φ : G → X s.t. φ is a group isomorphismfrom (G , ◦) to (X , f ).

Topology details

Def. Let T ⊆ P(S). Then (S , T ) is a topological space iff

1. ∅,S ∈ T ,

2. T is closed under arbitrary unions, and

3. T is closed under finite intersections.

Terminology. The members of T are called open sets.

Def. Let (S , T ) and (S ′, T ′) be topological spaces and φ : S → S ′.Then φ is continuous iff φ−1[U ′] ∈ T for each U ′ ∈ T ′.

Def. φ : S → S ′ is a homeomorphism iff φ is bi-continuous, i.e., φ is1-1, onto, and continuous and so is φ−1.

Explanation. Topological space (S , T ) is homeomorphic to some set Aiff

1. there exist (hereditary) sets B and C ⊆ P(B) s.t. A = (B, C) is atopological space, and

2. there exists a mapping φ : S → B s.t. φ is a homeomorphism from(S , T ) to (B, C).

Final Somber Remark

I Peano Arithmetic is interpretable in Z.

I Thus, Godel’s 2nd incompleteness theorem applies to Z as well.

I If nobody knows whether PA is consistent, then nobody knows if Zis consistent.