# Logic

## Logic Summary

History of logic - Syllogism WP - Aristotle (384–322 BC ancient Greek philosopher and scientist WP) - George Boole (November 1815 – 8.12.1864 English mathematician, educator, philosopher and logician WP) - Gottlob Frege (8.11.1848–26.7.1925 German philosopher, logician, mathematician) - Principia Mathematica 23rd in top 100 English-language nonfiction books of 20th century - Modern logic is fundamentally a calculus whose rules of operation are determined only by the shape and not by the meaning of the symbols it employs, as in mathematics - WP
Timeline of mathematical logic WP

Mengenlehre - von Georg Cantor 1874-1897 begründet, Richard Dedekind 1872 und 1888, Giuseppe Peoano 1889 Klassenlogik-Kalkül, Peano-Axiome, ∈ vom griechischen ε von epsilon des Wortes ἐστί, Gottlob Frege Kalkül 1893, Bertrand Russell entdeckt darin Widerspruch 1902 Russelsche Antinomie, aufgrunduneingeschränkter Mengenbildung, daher Frühformen der Mengenlehre als naive Mengenlehre bezeichnet - 20. Jhdt setzen sich Cantors Ideen durch, gleichzeitig Axiomatisierung der M., konnte Widersprüche überwinden - 1903/1908 Russells Typentheorie, Mengen haben höheren Typ als ihre Elemente, macht problematische M.-Bildungen unmöglich, Ausweg aus Widersprüchen, heigt in Principia Mathematica von 1910-1913 Leistungsfähigkeit der Typentheorie, letzlich aber unzulänglich, setzt sich wegen Kompliziertheit nicht durch - Ernst Zermelo 1907 awiomatische M., widerspruchsfreie Begründung der M. von Cantor und Dedekind durch Ersetzungsaxiom 1930, Einfügung in Zermelo-Fraenkel-System, kurz ZF-System, keine Widersprüche bisher ableitbar - Gödels Unvollständigkeitssatz 1931 - Pseudonym Nicolas Bourbaki Gruppe wollte Mathematik auf Mengenlehre-Basis neu darstellen, ab 1939 Umsetzung - 1960er Jahre klar dass ZF-M. als Grundlage zur Mathematik geeignet, kurz sogar in Grundschulen - auch weitere M. - WP
Introduction to Formal Logic by Mark Thorsby - 38 videos YT

## Introuction to Logic

### and to the Methodology of the Deductive Sciences

Introduction to Logic and to the Methodology of the Deductive Sciences – 4th edition 6.1.1994 by Alfred Tarski (Author), Jan Tarski (Editor) - Alfred Tarski 14.1.1901–26.10.1983 Polish logician, mathematician and philosopher WP - Banach–Tarski paradox WP - The Banach–Tarski Paradox YT - Michael Stevens (educator) 23.1.1986 American educator, comedian, speaker, entertainer, editor, and Internet personality, best known for creating and hosting the popular education YouTube channel, Vsauce - WP - Showing My Desk to Adam Savage YT - vsauce.com - TW 762K

Contents
From Author's Prefaces to Previous Editions - From the original edition (1936) - ix
First Part. Elements of Logic. Deductive Method
I - On the Use of Variables - 3
II - On the Sentential Calculus - 17
III - On the Theory of Identity - 49
IV - On the Theory of Classes - 63
V - On the Theory of Relations - 81
VI - On the Deductive Method - 109
Second Part. Applications of Logic and Methodology in Constructing MathematicalTheories
VII - Construction of a Mathematical Theory: Laws of Order for Numbers - 145
VIII - Construction of a Mathematical Theory: Laws of Addition and Subtraction - 159
IX - Methodological Considerations on the Constructed Theory - 181
X - Extension of the Constructed Theory: Foundations of Arithmetic of Real Numbers - 201
Index - 213-229 end of book

From Author's Prefaces to Previous Editions - From the original edition (1936)
ix - math seems to laymen a dead science - entirely erroneous view, intense manifold development, expanding in all possible directions, hight width depth, hight: on soil of old theories new problems and results, width: other branches, phenomena, new theories, depth: foundations become more firmly established, mehtods perfect, principles stabilized - my intention is giving an idea of growth in depth - acquaint reader with most important concepts of independent discipline of mathematical logic, existing barely a century, permeating whole of mathematics - tried to present basic principles enployed in construction of theories, forming another discipline: methodology of mathematics, and how to use them in practice
not easy carry plan within small book without presupposing reader specialized mathematical knowledge or training in argumentation of an abstract character - care avoiding scientific inexactitudes, use language deviating as lttle as possible from language of everyday life, had to give up special logical symbolism - idea of systematic treatment had to be abandoned - shorten abundance of questions - in contemorary science all views are presented - methods of solution chosen simple insofar as possible, not personal choice - not in illusion having succeeded in overcoming all of this
From the first American edition (1941)
present book partially modified and extended edition of 1936 - present to layman clear idea of modern logic - originally to stabilize foundations of mathematics, in present phase wider aims: to relate to whole of human knowledge - one goal is to perfect and sharpen deductive method ... - ... present edition discusses all former skipped basic themes, logical symbolism added - passages and sections marked with * asterisks at beginning and end contain more difficult material ... R
Editor's Preface
xv - changes described ... - *** shows change of topic within section - ...
xix - A Short Biographical Sketch of Alfred Tarski - R

Introduction to Logic - First Part - Elements of Logic - Deductive Method
I - On the Use of Variables - 1 Constants and variables
p3 - every scientific theory is system of sentences accepted as true, called laws or asserted statements, accompanied by arguments or proofs, and such statements are called theorems (Ch VI) - among terms and symbols occuring in theorems we distinguish between constants and variables - in arithmetic called "number" as 0, 1, or sum + etc., meaning remains unchanged - in highschool often word "algebra" instead of arithmetic used, in higher math reserved for algebraic equations - in this book number means real number without imaginary or complex ones - for variables choose lower-case letters like a, b, c, ..., x, y, z -
p4 - opposed to constants they don't possess meaning by themselves - question: does 0 have such and such a property? answer affirmative or negative possible, may be true or false, but meaningful - question: is zero an integer? can't be answered meaningful - we can't attribute independent meaning to variables, be careful, in some books not correct
2 - Expressions containing variables — sentential and designatory functions
variables no meaning by themselves - phrases like x is an integer are not sentences, although they are gramatically, don't express assertions, can't be confirmed or refuted -
p5 - if replace x with 1 result is true sentence, by 1/2 false - these expressions called sentential function, but mathematicians are reluctant to do so, using term "function" in different sense, instead the word condition is employed - sentential functions, also called sentences if no danger of misunderstanding, or formulas, are composed entirely of mathematical symbols, not everyday words: x + y = 5 ...

4 - Universal and existential quantifiers; free and bound variables
p9 - phrases like for any x, y,... and there are x, y,... such that are called quantifiers, former a universal, latter an existential quantifier - ...
words like "every, all, a certain, some" exhibit a very close connection with quantifiers - connection becomes obvious observing expressions like all men are mortal or some men are wise have about the same meaning as sentences which are formulated with the help of quantifiers, according to the following schemes: for any x, if x is a man, then x is mortal and there is an x, such that x is both a man and wise

2 • x

13 The symbolism of sentential calculus; compound sentential functions and truth tables
p34 - ... - replace the sentential connectives:
not; and; or; if..., then...; if, and only if; by the symbols:
$\sim$; $\wedge$; $\vee$; $\to$; $\leftrightarrow$

Vocabulary
abundance - Fülle, Reichtum, Überfluss, Menge ... - x
acquaint - bekanntmanchen, unterrichten, vertraut machen ... - ix
ambiguity - Doppeldeutigkeit, Ambiguität, Zweideutigkeit, Vieldeutigkeit, Doppelsinn, Mehrdeutigkeit ... - ambiguous - x
appropriate - angemessen, geeignet, entsprechend ... - p5
arbitrary - willkürlich, beliebig ... - x
asserted - behauptet, erklärt ... - assertion - Behauptung, Erklärung ... - p3+4
comprehend - begreifen, erfassen, umfassen, verstehen, beinhalten, einschließen, enthalten, nachvollziehen ... - x
engrained - tief verwurzelt, tief eingeprägt, tiefsitzend, tief eingewurzelt - xxii
erroneous - falsch, irrig, fälschlich, fehlerhaft, irrtümlich - ix
feasible - machbar, durchführbar, praktikabel, möglich, ausführbar, brauchbar, realisierbar ... - p12
imply - bedeuten, implizieren, beinhalten, einbeziehen, einschließen, voraussetzen ... - p12
layman - Laie, Nichtfachmann ... - ix
manifold - mannigfach, mannigfaltig, mehrfach, vielfältig, facettenreich - ix
permeate - durchdringen, durchsetzen, eindringen, einziehen -
petrified - versteinert ... - ix
questionnaire - Fragebogen, Formular ... - p5
remark - Bemerkung, Anmerkung, Beachtung, Hinweis - p12
rigid - rigid(e), starr, steif, unbiegsam, unelastisch ... - ix

## Logic

Logic 1999/2002 Edition by Paul Tomassi - most accessible and user-friendly introduction to formal logic currently available to students

Contents
Chapter One: How to Think Logically 1
Chapter Two: How to Prove that You Can Argue Logically #1 31
Chapter Three: How to Prove that You Can Argue Logically #2 73
Chapter Four: Formal Logic and Formal Semantics #1 121
Chapter Five: An Introduction to First Order Predicate Logic 189
Chapter Six: How to Argue Logically in QL 265
Chapter Seven: Formal Logic and Formal Semantics #2 333
Glossary 375 - Bibliography 399 - Index 403

## Logic

Mathematical logic WP - Logic WP
Handbook of Philosophical Logic 2nd Edition by Dov M. Gabbay (Editor), Franz Guenthner (Editor) - Vol. 1-18
Reading materials for mathematical logic mathoverflow.net
Teach Yourself Logic: A Study Guide (and other Book Notes) - logicmatters.net
Introduction to Logic YT

{< a, b > | a ∈ A, b ∈ B}.

## A short introduction to formal logic

v0.3.2, 20.7.2012 by Dan Hicks - born and grew up in Northern California - dhicks.github.io - TW 430 - Dan Hicks: Two Views of Scientific Practice YT - about LaTeX, PDF, and other apps
Contents
1 The idea of logic - 1
2 Some common argument patterns - 12
3 Working with arguments in the wild - 28
Note for instructors 37
Fig. 1: Categorical syllogisms 38
Fig. 2: Other argument patterns 39 end of book

1 The idea of logic
p1 - argument as reason
1.1 Arguments and argument patterns - p2 (1) All humans are mortal. (2) Socrates is a human. (3) ∴ Socrates is mortal. (1,2) argument is premise-comclusion form - (1) (2) premises, (3) conclusion - premises are reasons given to support conclusion - if believe 1 and 2 then reason to believe 3 - premises and conclusion are steps of argument - 2 basic rules for writing arguments in premise-conclusion form: 1st: Every step is a declarative sentence (true or false) - not declarative: a. Is a human. not grammatically correct, no subject - b. Make tacos! imperative, command - c. Is it raining? question d. I like corn. tricky, can be read as declarative or optative (expression of wish) - 2nd: Premises come first, conclusions come last - first at least one premise, then conclusion (exceptions not covered here) -
p3 - (3) statred with symbol read 'hence' or 'therefore', signals a conclusion - (1, 2) after (3) indicate logical support, step (3) is supported by step (1) and (2)
compare earlier argument with this one: (4) All cats are mammals. (5) Whiskers is a cat. (6) ∴ Whiskers is a mammal. arguments look similar - replace nouns and names with algebra letters: (7) All As are Bs. (8) x is a A. (9) ∴ x is a B. we say to similar arguments of this kind: they have the same logical form or pattern - other examples: (10) If Obama is President then Biden is VP. (11) Obama is President. (12) ∴ Biden is VP. (10′) If it’s raining then the sidewalks will be slippery. (11′) It’s raining. (12′) ∴ The sidewalks will be slippery. (10′′) If p then q. (11′′) p (12′′) ∴ q (The is a ‘prime’, so for example (10′) is ‘ten-prime’ and (11′′) is ‘eleven double-prime’.) - only knowing about the argument's form or pattern whithout knowing the content, we can see that premises are related to the conclusion in the same way as they are in other arguments with same patterns - move from natural languages arguments to their patterns is like moving from drawing shapes to geometry - study of argument-patterns goes by two names: formal logic — because it studies logical form — and mathematical logic — because it’s studied mathematically - can be good at logic without knowing anything about math, therefore author wrote pamphlet so that anyone can use it -

Vocabulary
generic - generisch, allgemein, gattungsmäßig, gewöhnlich, spezifisch - p3
mammal - Säugetier, Säuger
pamphlet - Pamphlet, Broschüre, Heftchen, Flugschrift, Flugblatt, Streitschrift, Prospekt, Aufklärungsschrift - p2

## A Mathematical Introduction to Logic

A Mathematical Introduction to Logic 3rd Edition by Herbert Enderton WP - Probability for Life Science - complete, Lecture 1-28 YT
John Baez - Books advice: After basic schooling, the customary track through math starts with a bit of: 1 Finite mathematics (combinatorics) - 2 Calculus - 3 Multivariable calculus - 4 Linear algebra - 5 Ordinary differential equations - 6 Partial differential equations - 7 Complex analysis - 8 Real analysis - 9 Topology - 10 Set theory and logic - and 11 Abstract algebra --- not necessarily in exactly this order

Contents
Preface - IX
Introduction - XI
Ch 0 - Useful Facts abaut Sets - 1
Ch 1 - Sentential Logic - 11
Ch 2 - First-Order Logic - 67
Ch 3 - Undecidability - 187
Ch 4 - Second-Order Logic - 282
Suggestions for Further Reading - 307
List of Symbols - 309
Index - 311-317 End of Book

Preface
pIX - ...

Ch 0 - Useful Facts about Sets
p1 - ... read only if necessary for next chapters ...

Ch 1 - Sentencial Logic
p11 - will construct formal (not natural or informal) language with precise formation rules, into which we can translate English sentences - ...
Section 1.1 - The Language of Sentential Logic
p13 -

Vocabulary
incorporated - eingearbeitet, eingebaut, eingebunden, eingetragen, verbunden, inkorporiert, vereinigt ... - p11
potassium - Kalium - p11

## Handbook of Philosophical Logic

### Symbols

(\ ) space after backslash equivalent to space after normal text --- space character or blank \text{ } --- text hello \text{hello}
$\neg$ (\neg) - $\wedge$ (\wedge) - $\vee$ (\vee) - $\to$ (\to or \rightarrow) - $\leftrightarrow$ (\leftrightarrow) - $\bot$ (\bot) - six artificial symbols used in propositional logic, called truth-functors - p4
$\phi$ (\phi) and $\psi$ (\psi) used for sentences - p4
iff - if and only if - p4

Testing Aerea
$\phi\ \phi$

Preface to 2nd edition
Ch 1 - Elementary Predicate Logic - Introduction
p1 - Elementary first-order predicate logic is child of many parents, at least 3 - simplest, most powerful and applicable branch of modern logic -
1st group traditional logicians - central aim was to schematise valid arguments: string of sentences called premises, followed by 'therefore', followed by single sentence called conclusion - argument valid if premises entail conclusion, means if premises true then conclusion - typical scheme: 1. a is more X than b. b is more X than c. Therefore a is more X than c. substituting for example planets, or mixed: 2. a) sun > earth. earth > moon. therefore sun > moon. b) sun brighter than fire. fire brighter than earth. therefore sun brighter than earth. arguments like (2) called instances of schema (1) - traditional logicians collect schemes like (1 - Aristoteles gives 24 schemes (19 mentioned by himself), these arguments are called formal logic - today formal vs informal, mathematical correct vs incorrect - watershed between classical and modern logic 1847, when George Boole (2.11.1815 Lincoln, Lincolnshire, England - 8.12.1864 Ballintemple, Cork, Ireland) published calculus with infinitely valid argument schemas of arbitrarily high complexity - today Boole's calculus known as propositional logic - other researchers Augustus De Morgan 1806-1871 and C.S. Peirce 1839-1914 - examples of people i enemies to people j at time k, and other people overdrawing their bank accounts
p2 - 2nd group called Proof Theorists - among them Gottlob Frege (1848-1925), Hilbert, Russell, Herbrand, Gentzen - aim to systematise mathematical reasoning, all assumtions made explicit, all steps rigorous - ... - devised notation and proof theory of first-order logic, earliest calculus Ferge's "Begriffschrift" 1879, also first work discussing quantifiers -
3rd group called Model Theorists - aim study mathematical structures ... - Schröder, Löwenheim, Skolem, Langford, Gödel, Tarski, notion of first-order property already 1895 by Schröder - ...
other groups ...

I: Propositional Logic
1 - Truth Functors
p4 - $\neg$ (\neg) - $\wedge$ (\wedge) - $\vee$ (\vee) - $\to$ (\to or \rightarrow) - $\leftrightarrow$ (\leftrightarrow) - $\bot$ (\bot) - six artificial symbols used in propositional logic, called truth-functors - all symbols have agreed meanings, can be used in English or can have artificial language built around them
will first explain $\vee$, remainder will then be easy
$\phi\text{ phi }\psi\text{ psi}$

Vocabulary
arbitrarily - beliebig, willkürlich, eigenmächtig --- arbitrary - willkürlich, beliebig, frei wählbar, arbiträr, tyrannisch, unbegründet - p1
devise - erfinden, ersinnen, konstruieren, erdenken, entwerfen, formulieren, konzipieren ... - p2
entail - verursachen, nach sich ziehen, bedingen ... - p1
utterance - Äußerung --- utter - äußern, aussprechen, von sich geben, in den Mund nehmen - p4
watershed - Wendepunkt, Wasserscheide, Eizugsgebiet ... - p1

## LaTeX

LaTeX Symbols artofproblemsolving.com - LaTeX Commands

The default math delimiters are $...$ and $$...$$ for displayed mathematics, and $$$...$$$ for in-line mathematics. Note in particular that the $...$ in-line delimiters are not used by default
\lt and \gt instead of < and > inside of TeX

TeX Demo 1:
When $a \ne 0$, there are two solutions to $ax^2 + bx + c = 0$ and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$

Demo 2:
$$(2\pi h)^{-d}\iint_{\{H(x,\xi) <\tau\}} dx d\xi$$

Demo 3 (display mode multi-size):
$$\sum \int \oint \prod \coprod \bigcap \bigcup \bigsqcup \bigvee \bigwedge \bigodot \bigotimes \bigoplus \biguplus$$

Demo 4 (inline mode):

$\sum \int \oint \prod \coprod \bigcap \bigcup \bigsqcup \bigvee \bigwedge \bigodot \bigotimes \bigoplus \biguplus$

Symbols - Mac keyboard German
√ = alt v --- ∫ = alt b --- ~ = alt n --- º = alt j --- ∆ = alt K --- – = alt - --- ∞ = alt , --- ≈ = alt x --- ª = alt h --- ƒ = alt f --- ∂ = alt d --- ± = alt + --- π = alt p --- ø = alt o --- Ω = alt z --- ∑ = alt w --- « = alt q --- ¿ = alt ? --- ≠ = alt = --- ¡ = alt 1 ---

### Greek Alphabet

Number Letter Name Spell ancient / modern

1 Α α alpha, άλφα /a/ /aː/
2 Β β beta, βήτα [b] / [v]
3 Γ γ gamma, γάμμα [ɡ] [ŋ] / [ɣ] ~ [ʝ]
4 Δ δ delta, δέλτα [d] / [ð]
5 Ε ε epsilon, έψιλον [e]

6 Ζ ζ zeta, ζήτα [zd] / [z]
7 Η η eta, ήτα [ɛː] / [i]
8 Θ θ theta, θήτα [tʰ] / [θ]
9 Ι ι iota, ιώτα [i] [iː] / [ʝ] [ɲ]
10 Κ κ kappa, κάππα [k] / ~ [c]

11 Λ λ lambda, λάμδα [l]
12 Μ μ mu, μυ [m]
13 Ν ν nu, νυ [n]
14 Ξ ξ xi, ξι [ks]
15 Ο ο omicron, όμικρον [o]

16 Π π pi, πι [p]
17 Ρ ρ rho, ρώ [r]
18 Σ σ/ς sigma, σίγμα [s] / ~ [z]
19 Τ τ tau, ταυ [t]
20 Υ υ upsilon, ύψιλον [y] [yː] / [i]

21 Φ φ phi, φι [pʰ] / [f]
22 Χ χ chi, χι [kʰ] / [x] ~ [ç]
23 Ψ ψ psi, ψι [ps]
24 Ω ω omega, ωμέγα [ɔː] / [o]