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Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Classical Propositional Logic
Daniel Bonevac
January 16, 2013
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Logic
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Logic is the science of inference.
Given a body of information, we can sensibly ask,
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Is that information consistent?
What follows from it?
What additional information would be consistent with it?
What information would imply it?
We can also ask whether individual sentences are truths of
logic, contradictions, or contingent statements that give real
information about the world.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Implication
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A logic is a theory of inference.
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The most fundamental concept of logic is implication or
entailment, symbolized by |=.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Notation
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We will use capital Italic letters (generally from the beginning
of the alphabet) to stand for sentences or formulas or a
formalized language.
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I’ll generally use capital letters from the end of the alphabet for
sets of sentences or formulas.
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So, we can think of a logic as a theory specifying when
X |= A .
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Generally we’ll restrict ourselves to finite sets unless
otherwise indicated.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Syntax and Semantics
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Logics generally characterize implication in two ways.
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They formalize a language that reveals relevant aspects of
structure, allowing the identification of valid patterns of
reasoning, and
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specify a semantics, or theory of meaning, for that language.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Semantics
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The semantics usually, but not always, takes the form of a
theory of truth.
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It characterizes implication, specifying when X |= A .
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But it does not always characterize it in a very useful form.
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We want to be able to test arguments for validity; we want to
be able to construct arguments that are valid.
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We want, in other words, to be able to identify valid inferences.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Logical Systems
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Logics thus also specify valid patterns of inference or provide
tests of valid inference that depend on form.
They might be axiomatic, specifying axioms and rules of
inference that allow us to move from axioms to theorems.
They might be natural deduction systems, which forgo axioms
and consist entirely of rules of inference.
They might be tableaux systems, that test for validity using
trees.
Or they might take other forms.
If the inference from X to A passes the test (or is possible
given the axioms and rules, etc.), we can write X ` A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Implication
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Implication itself is a semantic notion; `, provability, is a
syntactic or structural notion.
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The goal of logic is typically to find a syntactic method that
matches the semantics.
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This is not always possible, as we shall see.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Vocabulary
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The language of propositional logic is remarkably simple.
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Its vocabulary consists of propositional parameters (also
called sentence letters) p0 , p1 , ...., connectives ¬, ∧, ∨, ⊃, ≡,
and parentheses ( and ).
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Syntax
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The syntax: The set of formulas is the smallest generated by
the inductive clauses:
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Propositional parameters are formulas;
If A and B are formulas, so are ¬A , (A ∧ B ), (A ∨ B ), (A ⊃ B ),
and (A ≡ B ).
The conditional ⊃ is often written →; the biconditional ≡, as ↔.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Semantics
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A model, interpretation, or assignment v is a function from
propositional parameters to truth values—in classical logic, {0,
1}, where 0 represents falsehood and 1 represents truth.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Truth Clauses
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We can extend v to a function from formulas to truth values by
way of an inductive truth definition:
I v (¬A ) = 1 ⇔ v (A ) = 0 (and = 0 otherwise; hereafter I omit
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this)
v (A ∧ B ) = 1 ⇔ v (A ) = v (B ) = 1
v (A ∨ B ) = 1 ⇔ v (A ) = 1 or v (B ) = 1
v (A ⊃ B ) = 1 ⇔ v (A ) = 0 or v (B ) = 1
v (A ≡ B ) = 1 ⇔ v (A ) = v (B )
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Implication
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A set X of formulas implies a formula A (X |= A ) iff every
assignment making every member of X true also makes A
true.
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A is logically true or valid iff ∅ |= A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Adequacy
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How well does classical propositional logic represent items in
natural language? Negation is meant to mirror ‘not’;
conjunction, ‘and’; disjunction, ‘or’; and the conditional, ‘if’. Do
they succeed?
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Bivalence
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The principle of bivalence states that each sentence is either
true or false.
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Classical logic also assumes that no sentence is both true
and false.
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Many-valued and paraconsistent logics reject those
assumptions.
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For now, assume that every sentence is either true or false,
and not both.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Adequacy
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Negation, conjunction, and disjunction are reasonably
accurate representations of the meanings of natural language
connectives.
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Some people think that there is a tensed aspect of
conjunction; compare ‘John fell down and got up’ to ‘John got
up and fell down’, or ‘Joan got married and got pregnant’ to
‘Joan got pregnant and got married’.
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But this effect occurs with or without ‘and’; it seems to be a
function of narrative discourse, not of conjunction.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Adequacy
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Some also think that disjunction is ambiguous between
inclusive (p or q or both) and exclusive (p or q but not both)
senses.
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But that’s a hard case to make.
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Many putative examples of exclusive disjunction involve
permission, which is notoriously tricky to analyze.
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Others rely on background information. (‘The car was either a
Dodge or a Plymouth.’ Given the background information that
no car is both, inclusive and exclusive senses are equivalent.)
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
The Conditional
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The chief problem with classical propositional logic, as a
theory of natural language meaning, lies with the conditional.
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The material conditional is not a very accurate representation
of the meaning of ‘if’ in English.
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Much of this course, in fact, will be devoted to finding better
representations.
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The search for a logic of the conditional adequate to natural
language conditionals is a, and perhaps the, main motivation
for developing nonclassical logics.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
History
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Tableau or tree-based systems for evaluating arguments stem
from Gerhard Gentzen’s 1936 dissertation (Investigations into
Logical Deduction).
There are many differences, however, between Gentzen’s
method and the increasingly popular tableaux technique.
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First, he thought of the method as syntactic.
Second, and more importantly, he made explicit use of the
deduction symbol; the basic unit in a Gentzen sequent
calculus is a sequent (or, in Belnap’s Abelard-inspired
terminology, consecution) of the form A1 , ..., An ` B.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
History
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In 1958, E. W. Beth and K. J. J. Hintikka independently
realized that a simplified version of Gentzen’s method
reflected semantic truth clauses very closely and yielded an
intuitively and computationally appealing way to evaluate
arguments for validity.
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Tableaux make metatheoretic proofs easy, because the rules
for each connective so closely match its semantics.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Strategy
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The general strategy underlying tableaux rests on the fact that
X |= A iff X , ¬A (which abbreviates X ∪ {¬A }) is contradictory.
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If X |= A , every model of X is a model of A .
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But then no models of X are models of ¬A , so, X , ¬A is
unsatisfiable, that is, contradictory.
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Tableaux in effect formalize W. V. O. Quine’s Main Method
(Methods of Logic, 1950): to find out whether X implies A ,
assume X and ¬A , and try to deduce a contradiction.
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To assess whether X |= A , we construct a tableau with X , ¬A
as its initial list.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Tableau Rules
Define the rules as in Priest:
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Negation
¬¬A
A
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Conjunction
A ∧B
A
B
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Negated Conjunction
¬(A ∧ B )
¬A
¬B
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Disjunction
A ∨B
A
B
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Negated Disjunction
¬(A ∨ B )
¬A
¬B
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Conditional
A ⊃B
¬A
B
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Negated Conditional
¬(A ⊃ B )
A
¬B
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Biconditional
A ≡B
A
B
¬A
¬B
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Negated Biconditional
¬(A ≡ B )
A
B
¬B
¬A
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Closure
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A tableau branch is closed iff some formula and its negation
both appear on it.
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A tableau is closed iff all its branches are.
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A tableau is completed iff every rule that can be applied on it
has been applied.
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Throughout, assume that X is finite.
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Say that X ` A iff there is a completed, closed tableau with
X , ¬A as initial list.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Soundness and Completeness
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We can now ask whether our rules are sound, that is, whether
X ` A ⇒ X |= A , and also whether we have enough rules,
that is, whether X |= A ⇒ X ` A .
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These are relatively new questions. Throughout the history of
logic, soundness was an intuitive notion, and asked
rule-by-rule; the assumption seems to have been that a logical
system is sound if and only if all its rules are sound.
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As we will see, this is usually, but not always, true; Quine’s
system for predicate logic is sound, but not rule-by-rule.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Truth Preservation
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The intuitive idea behind the soundness theorem is
truth-preservation.
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A valid inference is truth-preserving: If the premises are true,
the conclusion is true.
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If our logical system consists in a set of rules, we might show
that each rule is truth-preserving, and then use induction to
show that any proof using those rules yields valid conclusions.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Fidelity
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A tableau system requires a variation on this strategy. We
need to show that closed tableaux establish validity; if
something can be proved from a set of premises by way of
trees, it follows from them.
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We define the fidelity of an assignment to a branch, and then
show that the rules are fidelity-preserving.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Fidelity
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Say that an assignment v is faithful to b iff v (A ) = 1 for every
formula A on b.
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An assignment is faithful to a branch if it makes every formula
on the branch true.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Soundness Theorem
The Soundness Theorem: X ` A ⇒ X |= A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Soundness: Proof
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Proof: We’ll prove the contrapositive: If X |=
/ A , then X 0 A .
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Assume that X |=
/ A.
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Then there is an assignment v such that v (B ) = 1 for all
B ∈ X but v (A ) = 0, so v (¬A ) = 1.
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We show that X 0 A .
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Consider a completed tableau with the initial list X , ¬A .
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Notice that v is faithful to that initial list. It makes everything in
X true and it also makes ¬A true.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Soundness Lemma
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Soundness Lemma: Rules preserve fidelity.
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If v is faithful to b, and a tableau rule is applied to b, v is
faithful to at least one resulting branch.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Proof of Lemma
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Proof: Assume that v is faithful to b, to which a tableau rule is
applied. There are as many cases as rules. Here are two
representative ones.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Conjunction Case
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Case 1: The rule for A ∧ B is applied.
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Then A and B both appear on the branch.
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Since v (A ∧ B ) = 1, by the truth definition v (A ) = v (B ) = 1.
I
So, v is faithful to the resulting branch.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Negated Conjunction Case
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Case 2: The rule for ¬(A ∧ B ) is applied.
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Then two branches result, one with ¬A , and one with ¬B.
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By the truth definition, since v (¬(A ∧ B )) = 1, v (¬A ) = 1 or
v (¬B ) = 1.
I
Thus, v is faithful to at least one of the resulting branches.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Proof by Induction
With lemma in hand, we can now proceed to show that v is faithful
to an open branch on the completed tableau by induction on the
number of rules applied to complete the tableau.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Base Case
Base: No rules need to be applied. Then the initial list is itself a
completed tableau, and v is faithful to it.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Inductive Hypothesis
Inductive step: Assume that v is faithful to an open branch up to
the nth rule application. By the lemma, v is faithful to at least one
resulting branch.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Inductive Step
I
So, by induction, v is faithful to a branch on the completed
tableau.
I
If it were closed, no assignment could be faithful to it, since no
assignment makes both a formula and its negation true.
I
So, the branch is open, and thus X 0 A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Completeness
I
A logical system is (strongly) complete if and only if every
valid inference can be proved valid within the system. That is,
a system is complete iff, whenever X |= A , X ` A .
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(It is weakly complete if and only if every logical truth is
provable within it; if, that is, |= A only if ` A .)
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Surprisingly, no one seems to have asked the question of
completeness until David Hilbert and Wilhelm Ackermann
raised it in their Principles of Theoretical Logic (1928) and
Kurt Gödel, in his 1930 dissertation (The Completeness of the
Axioms of the Functional Calculus of Logic), answered it.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Strategy
I
The central strategy of the completeness theorem is to show
that valid inferences can always be proved valid by showing
the contrapositive.
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Assume that an inference cannot be shown to be valid. In a
tableau system, this means there is at least one open branch
of any tableau constructed to evaluate the inference.
I
Derive, from this open branch, a set of formulas—here, the
set of literals, that is, atomic formulas and their
negations—that appear on the branch.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Strategy
I
Construct, from this set of formulas, a semantic
interpretation—here, the interpretation assigning p truth if it
appears on the branch and falsehood if ¬p appears on the
branch.
I
Show that the interpretation is a countermodel of the original
inference, thus concluding that it was not valid.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Strategy
Putting this in a diagram:
Open branch → Set of literals → Induced assignment →
Countermodel
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Open Branches
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We can use open branches to build models and
countermodels.
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An open branch of a tableau not only tells us that there is an
assignment making the initial list true, but also specifies it.
I
Say that an assignment v is induced by an open branch b iff,
for every propositional parameter p on b, v (p ) = 1, and for
every ¬p on b, v (p ) = 0.
I
Induced assignments thus respect the values implicitly
assigned to propositional parameters on the branch.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Induced Assignments
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Induced assignments, that is, make all propositional
parameters and their negations on a branch true; faithful
assignments make every formula on it true.
All faithful assignments are induced, by definition, but whether
all induced assignments are faithful depends on the tableau
rules.
Indeed, showing that all induced assignments are faithful is
the key lemma in the completeness theorem.
It accomplishes steps (3) and (4), showing that the
assignment induced by the branch is a countermodel of the
original inference.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Completeness Theorem
The Completeness Theorem: X |= A ⇒ X ` A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Proof of Completeness
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Proof: We prove the contrapositive.
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Suppose that X 0 A .
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Then there is no completed closed tableau with X , ¬A as
initial list.
I
So, every completed tableau with X , ¬A as initial list has at
least one open branch b.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Induce an Assignment
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Any completed open branch b, moreover, induces an
assignment.
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For any propositional parameter p, let v (p ) = 1 if p is on b
and v (p ) = 0 if ¬p is on b.
I
Assign any other parameters any value you like.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Completeness Lemma
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Completeness Lemma: All induced assignments are faithful.
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If v is induced by b, v is faithful to b.
That is, if b, a completed open branch of a tableau, induces v,
then:
I if A is on b, v (A ) = 1;
I if ¬A is on b, v (A ) = 0.
I
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Proof by Induction
I
Let b, a completed open branch, induce v.
I
We show that for every A on b, v (A ) = 1, and for every ¬A
on b, v (A ) = 0, by induction on the complexity of A , that is,
the number of connectives in A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Base Case
I
Base: Say A is a propositional parameter on b.
I
Then v (A ) = 1, since b induces v.
I
Say ¬A on b is ¬p for some propositional parameter p.
I
Then v (A ) = 0, since b induces v.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Inductive Step
I
Inductive step: Assume that v (D ) = 1 for every D on b with
fewer than n connectives and that v (D ) = 0 for all ¬D on b
with fewer than n connectives .
I
Assume that A has n connectives.
I
There are nine cases to consider:
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Negation
I
Case 1: A = ¬B is on b.
I
Since the result holds for B, v (B ) = 0, so, by the truth
definition, v (A ) = 1.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Conjunction
I
Case 2: A = (B ∧ C ) is on b.
I
By the tableau rules, if A , that is, B ∧ C is on b, then so are B
and C.
I
But they contain fewer connectives than A , so, by inductive
hypothesis, v (B ) = v (C ) = 1.
I
By the truth definition, therefore, v (A ) = 1.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Negated Conjunction
I
Case 3: A = ¬(B ∧ C ) is on b.
I
By the tableau rules, if A , that is, ¬(B ∧ C ) is on b, then either
¬B or ¬C is on b.
I
Both B and C have fewer connectives than A , so, by the
inductive hypothesis, v (B ) = 0 or v (C ) = 0.
I
By the truth definition, then, v (B ∧ C ) = 0, and
v ¬(B ∧ C ) = 1.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Disjunction, Conditional, Biconditional
I
The other six cases are similar.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Completing the Completeness Proof
I
We have shown that,
I
if X 0 A , every tableaux with the initial list X , ¬A has an open
branch;
I
that such a branch induces an assignment v; and
I
that every induced assignment is faithful to the branch.
I
Since open branch b has the initial list X , ¬A ,
I
v, being faithful to it, makes every member of X true, but A
false. So, X does not imply A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Natural Deduction Systems
I
Leon Henkin developed his method in 1948 (“The
Completeness of the First-Order Functional Calculus”) to
prove completeness without immersion in the apparatus of
recursion theory.
I
It will be convenient to limit our language to negation and
conjunction.
I
We will not begin with a specific logical system in mind, but
construct what we need for a complete system from the proof.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Proofs
I
A proof system consists of a (possibly empty) set of axioms
and a nonempty set of rules of inference.
I
A proof is a finite sequence of formulas each of which is an
assumption, an axiom, or follows from previous lines by a rule
of inference.
I
There is a proof of A from X (X ` A ) iff there is a proof
including as assumptions formulas in X and terminating in A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Strategy
I
The general idea of the completeness proof is similar to
tableau completeness proofs.
I
The completeness theorem says that X |= A ⇒ X ` A .
I
So, assume that X 0 A , and construct a model of X , ¬A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Strategy
I
Let’s assume, then, that X 0 A .
I
It follows that X , ¬A are consistent; there is no B such that
X , ¬A |= B , ¬B.
I
Our strategy is to construct a model of X , ¬A by first
extending X , ¬A to a maximally consistent set.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Lindenbaum’s Lemma
Lindenbaum’s Lemma: Every consistent set of formulas extends to
a maximal consistent set.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Proof of Lindenbaum’s Lemma
I
Proof. Let Y be a consistent set of formulas.
I
Our language is countable, so we can enumerate the formulas
of the language: A1 , ..., An , ....
I
We construct a series of sets:
Y0 = Y
Yn+1 = Yn ∪ {An }, iff that is consistent; if not,
Yn+1 = Yn ∪ {¬An }
S
Yω = Yn
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Maximality
I
Obviously, Yω ⊇ Y . Yω is also maximal: for every formula A ,
A ∈ Yω or ¬A ∈ Yω .
I
The argument: A appears at some stage in our enumeration
of the formulas of the language.
I
Say that A is formula n.
I
Either A is consistent with Yn or not.
I
If so, A ∈ Yn+1 ⊆ Yω .
I
If not, ¬A ∈ Yn+1 ⊆ Yω .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Consistency
I
I
I
I
I
I
I
Moreover, Yω is consistent.
The argument: Suppose not. Then Yω ` B , ¬B for some B.
Since proofs are finite, and Y is consistent, there must be
some first finite stage n + 1 at which Yn+1 ` B , ¬B.
Yn+1 = Yn ∪ {An }, iff that is consistent; since it is inconsistent,
Yn+1 = Yn ∪ {¬An }.
So, Yn ∪ {¬An } ` B , ¬B.
But then Yn ` An , provided that our deduction system has
rules of indirect proof and double negation.
Since Yn ∪ An ` B , ¬B, however, Yn ` ¬A , so Yn is
inconsistent, contradicting the assumption that Yn+1 is the first
finite stage at which an inconsistency appears.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Closure
I
Also, Yω is closed under proof; Yω ` A ⇒ A ∈ Yω .
I
Let Yω ` A but A < Yω .
I
By maximality, ¬A ∈ Yω , so Yω ` A , ¬A ; Yω is inconsistent.
I
But we have already demonstrated its consistency.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Countermodel
I
Since, by the Lemma, every consistent set extends to a
maximal consistent set, X , ¬A does so.
I
Call that set Xω .
I
We construct a model v from this set as follows:
v (p ) = 1 ⇔ p ∈ Xω .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Countermodel
Claim: v |= A ⇔ A ∈ Xω .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Base Case
I
Proof: By induction on the complexity of A .
I
For propositional parameters, it is trivial.
I
Assume that the claim holds for all formulas less complex than
A.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Negation
I
Case 1: A = ¬B.
I
v |= A ⇔ v |= ¬B ⇔ v 2 B.
I
By inductive hypothesis, that is true iff B < Xω .
I
Since that set is maximal, that is true iff ¬B ∈ Xω .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Conjunction
I
Case 2: A = (B ∧ C ). v |= B ∧ C ⇔ v |= B and v |= C.
I
By inductive hypothesis, that is true iff B , C ∈ Xω .
I
So, that is true iff B ∧ C ∈ Xω , provided that our deduction
system has rules corresponding to conjunction introduction
and exploitation.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Completeness
I
We have now shown that we can construct a model making all
of X true but A false, provided that we have rules of indirect
proof, double negation, conjunction introduction, and
conjunction exploitation. So, X 2 A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Compactness
I
As a quick corollary to the completeness theorem, we obtain:
I
Compactness: X |= A ⇔ there is a finite X0 ⊆ X such that
X0 |= A .
I
In an alternate form, a set is satisfiable iff every finite subset
of it is satisfiable.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Completeness
I
There is a sense in which completeness is a limiting result.
I
Syntactic methods, such as tableaux and proof systems, have
limited complexity.
I
Semantic characterizations of entailment can easily have
greater complexity.
I
A completeness result thus shows that the syntactic system is
strong enough—a positive result—but also, correspondingly,
that the semantic system is weak enough to be captured by a
set of rules.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Effective Procedures
I
Say that a method for answering a question—determining
whether an object has a property, or whether an object
belongs to a set—is effective iff it is mechanical and infallible.
I
It is mechanical in the sense that it is entirely rule-governed; it
is algorithmic. No creativity is required.
I
It is infallible in the sense that it always gives the correct ’yes’
or ’no’ answer after a finite time.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Decidability
I
A property (or set) is decidable iff there is an effective method
(called a decision procedure) for determining whether an
object has that property (or belongs to that set).
I
Examples: validity in propositional logic; validity in first-order
logic with only monadic predicates; quantifier-free arithmetic.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Decidability
I
I
I
I
I
Tableaux constitute a decision procedure for classical
propositional logic.
A tree always has finitely many branches, and each branch is
finite.
The tableau terminates after finitely many steps.
And then it either closes or yields a model of the formulas on
the trunk. If the tree is testing the validity of a formula or
argument form, that model is a countermodel of the formula or
argument form.
We have thus shown that classical propositional logic is
decidable.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Semi-decidability
I
Sometimes, a method is only half infallible—it always gives a
correct positive answer after a finite time, say, but may run on
infinitely if the answer is negative.
I
Such a method is positively semi-decidable.
I
A method that always gives a correct negative answer after a
finite time, but may run on infinitely if the answer is positive, is
negatively semi-decidable.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Enumerability
I
A property is (effectively) enumerable iff there is an effective
method of generating or listing all and only its instances.
I
An enumerable set is the range (or the domain) of a decidable
relation.
I
Any finite set is enumerable.
I
But there are also infinite enumerable sets.
I
Any decidable set, for example, is enumerable.
I
Examples of enumerable properties that are not decidable:
validity in full first-order logic. (Church proved first-order logic
to be undecidable in 1936.)
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Enumerability and Semi-decidability
I
An enumerable property is positively semi-decidable: take the
method to be waiting for the object to emerge on the list.
I
If it is an instance, it will be on the list at some finite stage.
I
Conversely, any positively semi-decidable property is
enumerable: test objects one by one, and add to the list those
for which the method yields a ’yes’ answer.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Craig’s Little Theorem
I
Craig’s Little Theorem: If P and ¬P are both enumerable, then
both are decidable.
I
Start listing instances of P and of ¬P.
I
Any given object must appear on one list or the other
(assuming bivalence) after a finite time.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Axiomatizability
I
A theory is a set of sentences closed under logical
consequence.
I
Let Cn(X ) = {A : X |= A }.
I
For any theory T , T = Cn(T ).
I
T is finitely axiomatizable iff there is a finite X such that
T = Cn(X ), and axiomatizable iff there is a decidable X such
that T = Cn(X ).
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Axiomatizability
I
A first-order theory is axiomatizable iff it is enumerable.
I
We want sets of axioms to be decidable; we want to be able to
determine effectively whether or not something is an axiom.
I
Similarly, we want proofs to be decidable; we want an effective
method for determining whether something is a proof.
I
If a theory is axiomatizable, then the set of its theorems is
enumerable, for it is the range of a decidable relation (that of
proof) on a decidable set (of axioms).
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Partial Enumerability
I
Say that a property is partially enumerable iff it has an infinite
enumerable subset.
I
Gödel’s incompleteness theorem (1931) states that truth in
arithmetic is only partially enumerable; arithmetic is not
axiomatizable.
I
But it is partially axiomatizable; that is, there are decidable
sets of axioms that are true in arithmetic and whose
consequences are all true in arithmetic.
I
But there are truths of arithmetic that are not consequences of
them.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Partial Enumerability
I
The general idea of Gödel’s proof is to assign symbols,
formulas, sequences of formulas, and thus proofs code
numbers in a systematic way, and then show that there are
arithmetical formulas that represent a sequence being a proof
of a formula, a formula’s bring provable, etc.
I
It is possible to show that there is an arithmetical predicate Pr
such that Pr [A ] if and only if A is provable in Peano arithmetic
(where [A ] is the code number of A ).
I
That is, PA ` Pr [A ] if and only if PA ` A .
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Partial Enumerability
I
But then it is possible to construct a sentence G such that
PA ` G ≡ ¬Pr [G ]—that says, in other words, that it is not
provable.
I
If G, “I am not provable,” is true, it is not provable, and there is
a true sentence of arithmetic not provable from the axioms.
I
If it is false, then it is provable, and the axioms and rules are
unsound, allowing us to prove something false.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Immunity
I
A property is immune iff it is infinite but has no infinite
enumerable subsets.
I
Such a property is not even partially enumerable.
I
Any axiom or axiom set would have exceptions; it would have
consequences that are not instances of the property.
I
Example: the set of ethical truths, from an intuitionist point of
view; the set of truths about God, in Aquinas.
Daniel Bonevac
Classical Propositional Logic
Logic
The Language of Classical Propositional Logic
Natural Language
Tableau Systems
Soundness
Completeness
A Henkin-style Completeness Proof for Natural Deduction
Computability
Cardinality
I
Consider the set of natural numbers. It is countable, but its
power set—the set of all subsets of the set of natural
numbers—is uncountable.
I
Countably many of those subsets are decidable; countably
many others are enumerable but not decidable.
I
Uncountably many are partially enumerable. And uncountably
many are immune.
Daniel Bonevac
Classical Propositional Logic