Philosophical Devices
Week 11 – Soundness & Completeness
---------------------------------------------------------------------------------------------------------------- Syntactic Consequence: A formula is a syntactic consequence (within some formal
system) of a set of formulae iff there is a formal proof (in the system) of the former
from the latter
Semantic Consequence: A formula is a semantic consequence (within some formal
system) of a set of statements iff there is no model in which all members of the set
are true and the formula is false

Fundamental Question:
What is the relation between syntactic and semantic consequence (for a given logic)?
Soundness: Can we construct proofs in the logic only in cases where the truth of the
premises guarantees the truth of the conclusion? I.e., is every case of ⊢ a case of ⊨?

o
Proving Soundness: go through every rule of inference, appeal to the semantic
definitions of the relevant connectives, and show that none of them take us
from truths to falsehoods
Inference Rules for Propositional Logic (PL)
–Intro: From P, Q; PQ
-Elim: From PQ; P or to Q
-Intro: From P or Q; PQ
-Elim: From PQ, PR, QR; R
-Intro: From PQ, PNot-Q; P
-Elim: From Not-Not-P; P
-Intro: If assuming P lets you move via -Elim: From P, PQ; Q
any of the other inference rules to Q; PQ
Semantic Definitions for logical connectives in PL

P
Q
PQ
T
T
T
T
F
F
F
T
T
F
F
T
P
T
T
F
F
Q
T
F
T
F
P Q
T
F
F
F
P
P
T
F
F
T
P
Q
PQ
T
T
T
F
T
T
F
T
T
F
F
F
Completeness Question: Can we construct a proof in every case where the truth of the
premises guarantees the truth of the conclusion? I.e., is every case of ⊨ a case of ⊢?
o
Proving Completeness: This is much, much harder. A standard move is to prove
the contrapositive (i.e. suppose that ⊬and show this entails ⊭)
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Soundness & Completeness
Note that following this procedure would tell us that Propositional Logic is sound &
complete. For other logics, we would need to evaluate their respective Inference Rules &
the semantic definitions of their logical connectives; i.e. for

Predicate Logic (QL): concerned with arguments whose validity depends on the
meanings of the truth-functional connectives, the quantifiers, and (sometimes)
identity
we need to test the semantic definitions & inference rules associated with ‘’, ‘’, and ‘=’.

Second-Order Logic (2L): concerned with arguments whose validity depends on the
above plus additional variables that range over ‘properties’
We’d need to test the semantic definitions & inference rules associated with the 2nd
Order ‘’ and ‘’ – more on this in a second
----------------------------------------------------------------------------------------------------------------So what?
Testing for soundness & Completeness amounts to testing a logic to see if it really does
what it says on the tin – whether it serves as a guide to constructing valid arguments,
arguments that preserve truth.
-
Checking for soundness: does the logic permit a proof only when the truth of the
premises genuinely guarantee the truth of the conclusion?
-
Checking for completeness: does the logic always permit a proof when the truth
of the premises genuinely guarantees the truth of the conclusion?
Two questions:
(1) What would it tells us about a logic if we discovered it wasn’t sound?
(2) What would it tell us about a logic if we found out it wasn’t complete?
----------------------------------------------------------------------------------------------------------------Methodological Problem?
The IE-rules in the table above were carefully designed to capture the way ‘good
logicians’ – i.e. people who employ valid arguments – argue, and we’re evaluating them
when we look for proofs of soundness and completeness. Yet how can we use proofs
when figuring out what proofs are acceptable is the issue at hand? In other words,
How can we use argumentative skills to evaluate what argumentative skills are
acceptable?
-
Papineau’s Reply: Laying out the inference rules is tantamount to ‘describing the
internal structure of our argumentative practice’. Meanwhile, the semantic
perspective is that of an outsider evaluation – sentences are taken to not just be
positions in a symbol game, but genuine, interpreted expressions. And this
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Soundness & Completeness
second perspective lets us figure out whether our moves are ‘well suited to
moving us from truths to truths.’
The fact that the rules preserve truth vindicates accepting them – i.e. explains why it is
good to buy into these rules (if you want to be in the business of making valid
arguments in the first place)
----------------------------------------------------------------------------------------------------------------Undecidability
 Decidability: A logical system is decidable iff there is an effective method for
determining whether arbitrary formulas are theorems of the logical system;
otherwise it is undecidable
o Neat connection: the definition of a decidable logical system can be given
either in terms of effective methods or in terms of computable functions, which
are generally considered equivalent per Church's thesis. The upshot is that, if
a logic is decidable, then it is possible to program a computer to check
whether any arbitrary sentence is a theorem in the system!
-
Recursive: a class is recursive when it can be defined by two properties – (i) a
simple base case (or cases); and (ii) a set of rules which reduce all other cases
toward the base case
-
Recursively enumerable: a class is recursively enumerable iff there is an
algorithm that enumerates (i.e. lists) the members
PL is decidable: there is at least one straightforward, mechanical procedure for
determining whether any given sentence is a logical truth of the language (In fact, there
are at least two such procedures: one is to construct a truth table; another is the
tableaux (or ‘tree’) method).
QL, on the other hand, is undecidable: there is no mechanical procedure for determining
whether any given sentence is a logical truth of QL. There is a mechanical procedure for
listing all the QL logical truths, but an infinite list doesn’t settle the initial question - if
the sentence is a theorem, you will eventually find it, but if it isn’t, no matter how far
down the list you go, you’ll never be in a position to determine whether the sentence is
either simply further down or not on the list at all
Two questions:
(1) Papineau says that things would be different if, to go along with the infinite list of
all theorems, we also had an infinite list of all the non-logical truths of QL. What
difference would it make?
(2) Why couldn’t we systematically generate a list of all the non-logical truths of QL?
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Philosophical Devices
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Soundness & Completeness
Second Order Logic is incomplete
(with apologies to Dr. Krämer)
Lemma One: If F is a subset of G and if G is recursively enumerable, then F is recursively
enumerable:
(1) Suppose that F is a subset of G and that G is recursively enumerable.
(2) G is the range of a total or partial recursively computable function ƒ on the
natural numbers.
(3) F, being a subset of G, is within the range of ƒ.
(4) So F is also recursively enumerable.
Lemma Two: The set of theorems of 2L is recursively enumerable:
(1) Whether or not a sequence of sentences is a derivation in 2L is effectively
computable – there is a step-by-step procedure that will determine in a finite
number of steps whether a given sequence of sentences is a derivation in 2L or
not.
(2) Thus, whether a derivation is a derivation of a logical theorem is computable.
(3) Hence, the set of theorems of 2L is recursively enumerable.
Lemma Three: The set of valid sentences of 2L is not recursively enumerable:
(1) Suppose that the set of valid sentences of 2L is recursively enumerable.
(2) Let PA2 be the conjunction of the seven axioms of 2nd order Peano arithmetic.
(3) 2nd order Peano arithmetic is complete.
(4) Thus, for every 1st order sentence Φ of arithmetic, (PA2  Φ) is a valid sentence.
(5) The set of such conditionals is recursively enumerable. (Follows from (1) and
Lemma One.)
(6) So the set of sentences of true arithmetic is recursively enumerable.
(7) But every sentence in the language of 1st order arithmetic or its negation is an
element of true arithmetic.
(8) Hence, true arithmetic is decidable – to determine whether or not a given
sentence Φ is an arithmetic truth just read down the list of true arithmetic
sentences until one reaches either Φ or Φ.
(9) But true arithmetic is NOT decidable, per Gödel’s incompleteness theorem!
∴ Theorem – 2L is incomplete:
(1) The set T of theorems of 2L is recursively enumerable
(2) The set V of valid sentences of 2L is not recursively enumerable
(3) Thus, by Lemma One, V is not a subset of T
(4) So there must be a valid sentence of 2L that is not a theorem of 2L
(5) By the definition of completeness, 2L is therefore incomplete