 # Classical Mathematical Logic: The Semantic Foundations of Logic

Richard L. Epstein
with contributions by Lesław W. Szczerba
Pages: 544
https://www.jstor.org/stable/j.ctt7rg5q

1. Front Matter
(pp. i-vi)
(pp. vii-xvi)
3. Preface
(pp. xvii-xviii)
4. Acknowledgments
(pp. xix-xx)
5. Introduction
(pp. xxi-xxii)

Classical mathematical logic is an outgrowth of several trends in the 19th century.

In the early part of the 19th century there was a renewed interest in formal logic. Since at least the publication ofLogic or the Art of Thinkingby Antoine Arnauld and Pierre Nicole in 1662, formal logic had meant merely the study of the Aristotelian syllogisms. Richard Whately,1827, and others began a more serious study of that subject, though still far below the level of clarity and subtlety of the medieval logicians (compare Chapter XXII).

Shortly thereafter, George Boole, Augustus De Morgan and others began...

6. I Classical Propositional Logic
(pp. 1-26)

In general we understand well enough what it means for a simple sentence such as ‘Ralph is a dog’ to be true or to be false. For such sentences we can regard truth as a primitive notion, one we understand how to use in most applications, while falsity we can understand as the opposite of truth, the not-true. Our goal is to formalize truth and falsity for more complex and controversial sentences.Which declarative sentences are true or false, that is, have atruth-value? It is sufficient for our purposes in logic to ask whether we can agree that a particular...

7. II Abstracting and Axiomatizing Classical Propositional Logic
(pp. 27-52)

Propositions, as I’ve presented them, are written or uttered. There can only be a finite number of them. But though finite, we can create more propositions at any time: the supply is unlimited. So rather than restricting our formal or semi-formal language to, say, all wffs using fewer than 47 symbols made up from p0 , p1 , . . . , p 13 or the realizations of those in a particular model, it is more general and reflects better our assumption on form and meaningfulness (p. 16) to include all well-formed formulas in our logical analyses. This will also...

8. III The Language of Predicate Logic
(pp. 53-68)

Propositional logic allows us to justify our intuition that certain inferences are valid due solely to the form of the propositions. For example:

Ralph is a dog and George is a duck.

Therefore,Ralph is a dog.

This can be justified as valid by arguing that the first proposition has the form of two propositions joined by ‘and’, which we can analyze via our formal models of ∧ .

But now consider:

(1) All dogs bark.

Ralph is a dog.

Therefore,Ralph barks.

This is clearly valid: It’s not possible for the premises to be true and the conclusion false....

9. IV The Semantics of Classical Predicate Logic
(pp. 69-98)

Our goal now is to give meanings to thelogicalorsyncategorematicvocabulary, which consists of the connectives ┐,→, ∧, ∨, the quantifiers, ∀, ∃, and the variables, x₀, x₁, . . . . We need to settle on interpretations that will not vary from proposition to proposition, nor from realization to realization. Predicates and names— thenonlogicalorcategorematicparts of speech—are what give content to propositions, and we need to explain how our interpretations of these connect to the meanings of the logical vocabulary in determining the truth or falsity of propositions.

We saw that names play...

10. V Substitutions and Equivalences
(pp. 99-112)

In the last chapter we gave a formal definition of truth in a model. Here we’ll see that the assumptions that motivated that definition are respected in our models. At the same time, I’ll draw out some consequences for our further work.

I’ll leave the proof of the following two theorems to you, using Theorem IV.2.

If x is not free in A, then ∀x A and ∃x A are both semantically equivalent to A .

∀x ∀y A is semantically equivalent to ∀y ∀x A .

∃x ∃y A is semantically equivalent to ∃y ∃x A .

Alternating quantifiers,...

11. VI Equality
(pp. 113-120)

Essential to our analyses of the truth of a wff is our ability to distinguish one object from another as the reference of a name or a variable. Though this is a semantic notion, we can formalize it within the language of predicate logic without creating self-referential paradoxes.

The phrases ‘— is identical to —’, ‘— is the same as —’, and ‘— equals —’ are roughly synonymous in our ordinary speech. Sometimes ‘is’ can also be understood as ‘is the same as’, as in ‘7 · 2 is 14’ or ‘Marilyn Monroe was Norma Jean Baker’. We can abstract...

12. VII Examples of Formalization
(pp. 121-138)

In this text we’re interested in formalizations of reasoning from mathematics. But the general methods of formalizing can be introduced more clearly for ordinary reasoning, which will also alert us to some limitations of predicate logic. A full discussion of formalizing ordinary reasoning with criteria for what constitutes a good formalization can be found in myPredicate Logic.

1. All dogs bark.

$\forall X$(x is a dog → x barks)

Predicales: ‘— is a dog’, ‘— barks‘

names: none

$\forall {x_1}({P_0}({x_1}) \to {P_1}({x_1}))$

AnalysisThe grammatical subject of this example is ‘dogs’. That is not a name of an individual nor a pronoun...

13. VIII Functions
(pp. 139-152)

In attempting to place the infinitesimal calculus on an intellectually solid foundation in the 19th century, the notion of limit was refined along with a new theory of collections viewed as sets. In this view there was no longer room for the conception of functions as processes. The functionsin x, for example, was no longer a process, represented as a continuous curve, but a relationship between real numbers, where 0 is related to 0; π/2 is related to 1; π is related to 0; and so on. A function was reduced to or reconceived of as a set of...

14. IX The Abstraction of Models
(pp. 153-166)

In this chapter we’ll further abstract the semantics of predicate logic in order to be able to use mathematics to study predicate logic. To do that, we’ll have to consider more carefully what it means to classify a collection as a thing.

Consider again the model discussed in Chapter IV. E:

$L(\neg , \to , \wedge , \vee ,\forall ,\exists ;{P_0},{P_1},...,{c_0},{c_1},...)$

realizations of name and predicate symbols; universe

$L(\neg , \to , \wedge , \vee ,\forall ,\exists ;$is a dog, is a cat, eats grass, is a wombat, is the father of; Ralph, Dusty, Howie, Juney)

universe: all animals, living or toy

assignment of references; valuations of atomic wffs; evaluation of$\neg , \to , \wedge , \vee ,\forall ,\exists$

Recall the classical abstraction of...

15. X Axiomatizing Classical Predicate Logic
(pp. 167-182)

In this chapter we’ll see how to axiomatize the valid wffs and the consequence relation of classical mathematical logic. We defined the general notion of syntactic consequence in Chapter II.C.2 and derived some properties of that. Here we will give a specific axiom system.

Again, we start with an idea of how to prove completeness and then add axioms that allow that proof. But rather than add as an axiom every wff that’s needed, I’ll establish syntactically that certain wffs are theorems (Section A.2). In some cases these will be only sketches of how we can give a derivation, relying...

16. XI The Number of Objects in the Universe of a Model
(pp. 183-190)

Thesizeof a model is the number of elements in its universe. In particular, a model is finite if its universe is finite, infinite if its universe is infinite. In this chapter we’ll see whether we can syntactically characterize how large a model of a theory must be. We’ll look at models of L(P₀, P₁, . . . , f₀, f₁, . . . , c₀, c₁, . . . ), which I’ll call ‘L’, and models of L with ‘=’, which I’ll call ‘L(=)’.

In Example 17 of Chapter VII (p. 131), we defined in the language L(=)...

17. XII Formalizing Group Theory
(pp. 191-206)

We now turn to testing how suitable classical mathematical predicate logic is for formalizing mathematics. In this and succeeding chapters we’ll look at formalizations of particular mathematical theories. In doing so we’ll also develop new methods of analyzing reasoning.

What exactly is it we wish to formalize? What is a group?

Consider a square. If we rotate the square any multiple of 90º in either direction it lands exactly on the place where it was before. If we flip the square over its horizontal or vertical or diagonal axis, it ends up where it was before. Any one of these...

18. XIII Linear Orderings
(pp. 207-224)

In this chapter we’ll look at formalizations of the theory of linear orderings. This will lead us to consider when we should identify two models as the same.

What is an ordering? Consider some examples.

1.The natural numbers

The usual ordering of the natural numbers corresponds to the way we learn to count. We can define this ordering in terms of addition:n<mDefthere is somek≠ 0 such thatm+k=n. There is a least element in this ordering, 0, but no greatest element.

2.The integers,Z

3.The set...

19. XIV Second-Order Classical Predicate Logic
(pp. 225-262)

Consider the semi-formal proposition:

(1)$\forall {x_1}$(x1is a dog v ┐ (x1is a dog))

In any model this is true or false.

Consider now a formal version of (1):

(2)$\forall {x_1}({P_1}({x_1}) \vee {P_1}({X_1}))$We do not say that this is a proposition. But given a model of the formal language we do speak of (2) as true or false, understanding it as true or false for that interpretation of the predicate symbol ‘P1’.

But we say more: (2) is true in every model. That is,

For every interpretation of ‘P1’, ‘$\forall {x_1}({P_1}({x_1}) \vee \neg {P_1}({x_1}))$’ is true.

The self-reference exclusion principle (Chapter IV.D) a...

20. XV The Natural Numbers
(pp. 263-290)

Our goal in developing predicate logic has been to have a logic suitable for formalizing mathematics. We’ve seen some examples of formalization in the previous chapters, but a crucial test of our logic is how well we can formalize the most fundamental part of mathematics we all reckon we understand: arithmetic.

We’ll start with a simple theory of counting and progress to theories that encompass more of the arithmetic of the natural numbers. In trying to answer questions about decidability and completeness, we’ll have to consider what functions we can compute within the theory, which will lead us to consider...

21. XVI The Integers and Rationals
(pp. 291-302)

We’ve taken the natural numbers as given. Supposedly we know what they are: 0, 1, 2, 3, . . . . We know how to add and multiply natural numbers.

It doesn’t seem a major leap to suppose we also know how to subtract natural numbers, so we take the integers as unproblematic: . . . – 3, – 2, – 1, 0, 1, 2, 3, . . . . Nor does it seem unreasonable to assume we can divide natural numbers. So we take the rational numbers as given, though we cannot so easily list those.

Mathematicians long ago made it clear...

22. XVII The Real Numbers
(pp. 303-330)

In ancient times mathematics was about whole numbers and their ratios.

But about the time of Pythagoras someone discovered that lengths of certain line segments could not be expressed as ratios of whole numbers. For example, consider the diagonal of a square whose side has unit length.

The length d of the diagonal cannot be a ratio of whole numbers relative to the unit length. That’s because by Pythagoras’ theorem, d² = 1² + 1² = 2 . So if d =m/n, wheremandnhave no common factor, then 2 =m²/n² . So 2n² =...

23. XVIII One-Dimensional Geometry
(pp. 331-362)

Formalizing the geometry of flat surfaces has a long tradition. Euclid’sElements(ca. 300 B.C.) was already a summary of many generations’ work (seeHeath, 1956). Euclid, and others until quite recently, saw their work not as a formalization but as a discovery of truths and an attempt to reduce those many truths to a few from which all others could be deduced. But what were those truths about?

Points and lines are the basic “things” in geometry. Euclid defined them as:

Apointis that which has no parts.

Acurveis a length without width.

Alineis...

24. XIX Two-Dimensional Euclidean Geometry
(pp. 363-402)

In this chapter we’ll continue our geometric analysis of the real numbers by formalizing the geometry of flat surfaces. Our goal is to give a theory that is equivalent to the theory of real numbers presented in Chapter XVII.

Our axiomatization of two-dimensional geometry will use the same primitives as for one-dimension: points and the relations of betweenness and congruence. Lines and other geometric figures and relations, which others often take as primitive, will be definable. Roughly, since two points determine a line, we can define a line as all those points lying in the betweenness relation with respect to...

25. XX Translations within Classical Predicate Logic
(pp. 403-412)

We’ve seen many examples of translations in the text. Here we’ll make some general definitions and investigate some of the properties of translations. Then we’ll return to the translations we’ve already seen in the text as examples.¹

Since we’re generally interested in translating named theories, such as Q , I’ll use boldface capitals to stand for theories here.

We require the maps to be constructive because if we allow nonconstructive maps, then we can map any theory T into any theory R with a 1-1 mapping that takes the wffs of T to the wffs of R, and the wffs...

26. XXI Classical Predicate Logic with Non-referring Names
(pp. 413-436)

When we gave semantics for classical predicate logic, we took as a simplification that every name must refer. In this chapter we’ll see how we can lift that restriction. This will allow us to model reasoning about partial functions in mathematics.

Lifting the restriction that names refer is often considered part of a program calledfree logic: ridding logic of existential assumptions that are built into its semantics. Along with no longer requiring names to refer, it’s sometimes suggested, we should also lift the assumption that the universe of a model must contain at least one object. It is not...

(pp. 437-460)

The liar paradox in its simplest form is the assertion:

This sentence is false.

On the face of it, if it is true then it is false, and if it is false then it is true. In giving semantics for classical predicate logic, we chose to avoid dealing with the liar paradox by adopting the self-reference exclusion principle (Chapter IV.D): We exclude from consideration in our logic sentences that contain words or predicates that refer to the syntax or semantics of the language for which we wish to give a formal analysis of truth.

At the time I presented this...

28. XXIII On Mathematical Logic and Mathematics Concluding Remarks
(pp. 461-464)

One of the goals of this book has been to try to clarify the relation of classical mathematical logic to mathematics. By ‘mathematics’ I do not mean a body of knowledge, but the practice of mathematics, particularly reasoning in mathematics.

Mathematical logic is a formal model for reasoning in mathematics. Though it may be motivated by reflecting on mathematical practice, it is not considered wrong when we find that some or many mathematicians do not reason in accord with it. No surveys of mathematicians, no inspection of papers of mathematicians could establish the incorrectness of mathematical logic. So, it seems,...

29. Appendix: The Completeness of Classical Predicate Logic Proved by Gödel’s Method
(pp. 465-474)
30. Summary of Formal Systems
(pp. 475-486)
31. Bibliography
(pp. 487-494)
32. Index of Notation
(pp. 495-498)
33. Index
(pp. 499-522)