Elementary Logic W.V.O. Quine
Revised edition
Preface , 1980
‘Mathematical Logic’
‘Elementary Logic’, 1940
‘Methods of Logic’
Godel
Quantification theory admits of a complete proof procedure
Boston, March 1980
Preface to Revised Edition
Minimum essentials
‘Methods of Logic’
Skolem and Herbrand
Testing procedure
Proof technique
Modernization of terminology
Cantor
Hilbert and Bernays
Harvard, Massachusetts, August 1965
Preface to the 1941 edition
‘Modern Logic’
Quantification theory: ‘some’, ‘every’, ‘no’, and pronouns
Statement composition more akin to algebra than geometric axioms
‘analyzing’
…get along with two symbols: ‘and’ & ‘not’
Cooley, ‘Outline of Formal Logic’
Equivalence, validity, implication
Wohlstetter, A.
Frames = schemata
Hempel
Cambridge, January, 1941
Elementary Logic
Introduction
The logic which is commonly described vaguely, as the science of necessary inference.
Certain basic locutions to begin with:
‘if’, ‘then’, ‘and’, ‘or’, ‘not’, ‘unless’, ‘some’, ‘all’, ‘every’, ‘any’, ‘it’, etc…may be called logical
Logical structure
A statement is logically true
1) every microbe is an animal or a vegetable
2) every Genevan is a Calvinist or a Catholic
Same logical structure
A statement is logically true if it is true by virtue of its logical structure
‘Every microbe is either an animal or not an animal.’
Two statements are logically equivalent if they agree in point of truth or falsehood by virtue solely of their logical structure i.e. if no uniform revision of the extra logical ingredients of the statements is capable of making one of the statements true and one false.
The statement:
‘If something is neither animal nor vegetable it is not a microbe.’
For example, is logically equivalent to 1) one statement logically ‘implies’ another if from the truth of one we can infer the truth of the other solely of the logical structure of the two statements
The statement:
Every Genevan is a Calvinist
Thus logically implies (2)
(3) some animals are microbes
Errors can occur even at the level of simplicity
i.e. change ‘microbes’ to ‘azaleas’
And observing that (1) remains true while (3) becomes false
The reader …may be assured nevertheless that there are more and more complicated cases, without end, when logical truth and equivalence and implication are hidden from all men save those who have special techniques at their disposal. Logic is concerned with developing such techniques.
Aristotle
‘formal logic’
The past century brought radical revisions of concepts and extensions of method
A vigorous new science of logic
Not repudiated
More efficient
Logic, in its modern form, may conveniently be treated as falling into three parts.
In the theory of ‘truth functions’, first, we study just those logical structures which emerge in the construction of compound statements from simple statements by means of the particles ‘and’, ‘or’, ‘not’, ‘unless’, ‘if….then’, etc.
In the theory of ‘quantification’, next, we study more complex structure, wherein the aforementioned particles are mingled with the generalizing particles such as ‘all’, ‘any’, ‘some’, ‘none’.
In the theory of ‘membership’, finally, we turn to certain special structures involved in discourse about universals, or abstract objects.
This trichotomy afforded the basic plan of Quine’s ‘Mathematical Logic’
Theory of ‘membership’ sometimes placed outside logic and regarded as the basic ‘extra logical branch of mathematics’
A. Tarski
Takes this point of view
A question merely of how far we choose to extend the catalogue of ‘logical locutions’
According to the wider version, logic comes to include mathematics
According to the narrower version, a boundary survives between logic and mathematics
A task of analyzing ordinary statements, making implicit ingredients explicit, and reducing the whole to systematically manipulate form. It is this interpretive task as opposed to the calculative
The intersection of these two dichotomies divides the book into four chapters.
Statement Composition
Truth Values
Statements are sentences , but not all sentences are statements
Declarative statement – ‘I am ill’ is intrinsically neither true nor false
‘he is ill’ – sometimes true, sometimes false
‘Jones is ill’ – not clear whether Jones refers to Henry Jones or Lee St. Jones, Tulsa Jones or John J. Jones of Wenham, Mass.
‘it is drafty here’, may be simultaneously true for one speaker and false for a neighboring one.
‘Tibet is remote’ is true in Boston and false in Darjeeling.
‘spinach is good’, if uttered in the sense of ‘I like spinach’, rather than ‘spinach is vitaminous, is true for a few speakers, and false for the rest.
The words ‘I’, ‘he’, ‘Jones’, ‘here’, ‘remote’ and ‘good’ have the effect in these examples of allowing the truth value of a sentence to vary with the speaker or scene or context
Words which have this effect must be supplanted by unambiguous words of phrases before we can accept a declarative sentence as a statement
To have a ‘truth value’
Revision
Time
Nazis, e.g.
Tenseless
A statement is a sentence which is uniformly true or uniformly false independently of context, speaker, and time and place of utterance
Connectives – ‘and’, ‘or’, ‘if’, ‘then’, neither’, ‘nor’, etc. to combine single statements to form compound statements & their compounds
Components
‘and’
…is true just in case both of the component statements are true
‘and’
‘neither’
‘nor’
‘or’
Increasingly difficult
Necessary to develop a systematic technique
Modern Logic at its most elementary level is concerned with the development of such techniques
Conjunction
‘.’ = ‘and’
Conjunction
Compound
Conjunction of its components
Shakespeare
Denial
~ = ‘not’
~ tilde
‘not’
Denial of statements
Periphrasiso
‘it is not the case that’
Parentheses
Whereas conjunction combines statements two or more at a time, denial applies to statements one at a time
Denied jointly
Verify
‘neither’ ‘nor’
‘Or’
The connective ‘or’ or ‘either’ in its most usual sense , yields a statement which is false just in case the corresponding ‘neither’….’nor’ statement is true
Construed
‘either’, ‘or’
‘neither’, ‘nor’
Just in case
….inclusive sense, according to which the compound is true whenever one or more of the components are true, but ‘or’ is sometimes used rather in a so-called ‘exclusive’ sense, according to which the compound is true only in case exactly one of the components is true.
‘or both’ or ‘but not both’
‘both’
‘but not both’
The inclusive sense of ‘or’ is perhaps the commoner of the two
‘or’ is always to be understood in the inclusive sense if there is no explicit indication to the contrary
Discourse not only for joining statements but also for joining nouns, verbs, adverbs, etc….
‘But’, ‘Although’, ‘Unless’
Choice ‘and’, ‘but’ ‘although’ is indifferent to the truth-value of the resulting statements
‘unless we hear from you to the contrary
‘unless’ seems to answer to ‘or’
Agreeing in general to understand ‘unless’ in the inclusive sense
Epochs
‘If’
If…then
Conjunction and denial
A possible causal or psychological connection
The ‘motivation’ of the statement
Income tax evasion
The prefix ‘only if’ is the reverse of ‘if’
In toto
General and Subjunctive Conditionals
Quantification subjunctive
Indicative
Truth – functional
The truth value of the compound depends only on the truth values of its components
‘Because’, ‘hence’, ‘that’
Truth of a because compound requires not only truth of the components but also some sort of ‘causal connection’ between the matter which the two compounds describe.
Believes, doubts, says, denies, regrets, ‘is surprised’, etc…
The rigor and precision of a science is indeed measured by the extent to which its formulations are free from statement compound of non-truth-functional kind.
The philosophical problem of ‘cause’
Reduction to Conjunction and Denial
Conjunction and denial
Those two basic devices
If p then q
Paraphrase
~(p .~ q)
(2) ~(J or S. ~ neither A nor M and D unless C and T)
(3) ~(~(~J.~S) .~neither A nor M and D unless C and T)
Grouping
The faultiness of ordinary language
Systematic and unambiguous
Two versions
1) ~A . ~(M . D)
2) the tripartite conjunction ‘~A.~M.D’
Meet and declare
m . d
Making a ‘conjecture’ as to most likely intention of supposed speaker
Construe
Decide between two versions
Idiomatically
Verbal Cues to Grouping
Intended grouping sometimes have to be guessed or sometimes inferred from unsystematic cues
‘that’
‘either’/ ‘or’
Both
‘it is the case that’
Two ocurrences of ‘that’ are surely co-ordinate
‘that’ clause is the other compound of the ‘and’ compound…
Inserting ‘both’
The ‘but’ compound
Conjunction
‘it is not the case that’
Furthermore:
‘also’, ‘else’
and/or compounds
Suggest by way of unambiguous idiomatic renderings
Paraphrasing Inward
The task of translating an elaborate compound into terms of conjunction and denial consists ….in discerning the intended groupings
Verbal cues or guesswork
First paraphrase the main connective of the whole compound – then paraphrase the main connective of a continuous segment of verbal text which is marked off by logical signs; and continue thus as long as verbal expression of statement composition remain
Be marked off by logical signs if it abuts on logical signs at both sides
If…then
The form ‘if p then q’
Translating ‘if p then q into conjunction and denial
We pick a continuous verbal segment which is bounded in by logical signs
Truth Functional Transformations
Substitution in Truth-Functional Schemata
Statement connectives
Truth functional onesssss
The truth- functional structure
Truth – functional equivalence and implication
(p) and (q)
p, q, r, s
Statement letters
Truth-functional schemata
Introduction
Substitution in a given schema S
A) whatever is introduced at one occurrence of a letter is introduced also at all other ocurrences of that letter throughout S
B) the final results in a statement or truth-functional schema
All occurrences throughout
Joint substitution
Instances
Instance of that schema
Corresponding instances
Substitution
Equivalent Schemata
Truth – functionally equivalent
Saying the same thing in different language
Replacement
Principle of replacement
Parentheses
Truth value
Either denial or conjunction of its predecessor
Transformations
Forward transformation
Backward transformation
Proofs of Equivalence
Tacit transformation
P v q p or q
P É q if p then q
From verbal or end of verbal
Readiness for formal transformation or computation
Denials of conjunction resolve always to alternations
Duality
Truth-tables
T for truth ^ for false
deMorgan’s laws
Normal Schemata
Distributive law
a(y1 + y2 + …+yn) = xy1 = xy2 = …+xyn
literals
condensed notation
Validity
A valid schema is one whose truth table marks every row T
Valid if and only if
Test of truth-functional truth
Inconsistency and Truth-Functional Falsity
A schema whose instances are all false – is called inconsistent
Implication between Schemata
Equivalence is mutual implication
Truth-Functional Implication
p. 66
Quantification
‘Something’
(1) London is big and noisy
E = ‘there is’
Quantifiers
E(y) or (v)
E(x) = there is something such that
E(x) E(y) – quantifiers
Quantification
Variables and Open Sentences
‘u’, ‘v’, ‘w’, ‘x’, ‘y’, ‘z’, ‘u’, ‘v’,’u’’,’v’’, ‘u’’’, etc.
Variables
Variants of ‘some’
‘a five legged calf exists’
‘Some’ Restricted
There is something such that or = (Ex)
We must continue to some degree to guess intentions
‘No’
(2) – ‘nothing bores George’
~(Ex) x bores George
~(Ex) – there is nothing such that
‘it is not the case that’
‘Every’
~(E)~ no matter what x may be
~(Ex)` Affirms x is true of every entity
False of none
Variants of ‘Every’
‘every object’ ‘every entity’
‘all’ or ‘all the’ instead of ‘every’
General conditionals
Persons
(Ex) there is an entity such that
Time and Place
‘all squares of odd numbers are odd’
‘no squares of odd numbers are even’
Quantification in Context
93
Quantificational Inference
F,G predicate letters
‘Fx’, ‘Fy’, ‘Gx’, ‘Gy’, ‘Fxy’, ‘Gxx’, ‘Fxyz’, etc.
Atomic open schemata
Quantificational schemata
Truth-functional schemata
Predicates
Predicate schemata
Restraints on Introducing
101
Substitution Extended
104
Validity Extended
‘existential’
‘existential closure’
Equivalence Extended
109
Inconsistency Proofs
Universal instantiation
Existential instantiation
Logical Arguments
Quantificational schemata
Identity and Singular Terms
(1) Tom married Sadie
(2) (Ex) x married Sadie
F2
(x) ~ Fx
~Fz
Membership
Set theory
X e y
X ‘is a member’ of y
Russell’s paradox:
(x)[x e z = ~ (x e x)]
tomreading 
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