Intuitionistic Logic Lecture 4
Intuitionistic Logic 4
Owen Griffiths
oeg21@cam.ac.uk
9/3/16
Last week
We considered Fitch’s paradox of knowability. This has been used by
intuitionists to argue that if we have some sympathy with the verificationist
thought that all truths are knowable, then we should be intuitionists, or accept
the ridiculous conclusion that all truths are known. The intuitionist is still,
however, committed to the claim that every sentence of the form ‘A ∧ ¬KA’ is a
contradiction. Another solution to the paradox is to restrict knowability, but we
saw that the two major solutions here face problems.
1 Logical inferentialism
This week, we’ll consider a distinct argument for intuitionistic logic: from
inferentialism and considerations of harmony.
What do the logical constants mean? A plausible thought is that the meaning of a
logical constant is given by its inference rules: its I-rule(s) and its E-rule(s). The
meaning of →, for example, is given by:
1
i
A
m
A→B
j
B
n
A
A→B
→I i–j
B
→E m, n
How does this work? If we follow Wittgenstein’s thought that meaning is use,
and we believe that the rules capture the use of the logical constant in question,
then we’ll think that they capture its meaning. This view is logical inferentialism.
Not just any old I- and E-rules will succeed in defining a logical constant,
however. Arthur Prior, in ‘The Runabout Inference Ticket’ (1960) made this point
very succinctly with the example of tonk:
m
m
A
A tonk B
tonkI m
A tonk B
B
tonkE m
Adding tonk to a deductive system has disastrous consequences. It trivialises
every system into which it is added:
1
P
2
P tonk Q
tonkI 1
3
Q
tonkE 2
P and Q were totally arbitrary, so tonk allows us to derive anything from
anything, hence its being a runabout inference ticket.
The logical inferentialist evidently needs some constraints in place to rule out
tonk and, to avoid the charge of being ad hoc, the constraints had better be
motivated. Let’s start with the motivation.
Florian Steinberger, in ‘What harmony could and could not be’ (2011, p. 619)
expresses a principle that could be useful here:
Principle of innocence It should not be possible, solely by engaging in
deductive logical reasoning, to discover hitherto unknown (atomic) truths
we would have been incapable of discovering independently of logic.
The thought is that logic alone should not create novel grounds for asserting
hitherto unknown atomic sentences. And, in the case of tonk, it looks like that’s
exactly what’s going on.
2
What would be nice would be a precise test that can be applied to a pair of
inference rules to check whether they are innocent. We now generally call this
testing for harmony. Roughly, pairs of inference rules are harmonious just if you
don’t get out of the E-rule any more than you put in with the I-rule. Let’s look at
Dummett’s suggestion in The Logical Basis of Metaphysics (p. 247–8):
it should not be possible, by first applying one of the introduction
rules for c and then immediately drawing a consequence from the
conclusion of that introduction rule by means of an elimination rule
of which it is the major premiss, to derive from the premisses of the
introduction rule a consequence that we could not otherwise have
drawn.
Let’s call this the levelling of local peaks test for harmony. Consider conjunction:
..
.
..
.
Γ
Γ
Π1
..
.
..
.
m
..
.
A
m
A
n
B
n+1
A∧B
∧I m, n
n+2
A
∧E n + 1
Π1
Π2
How about tonk?
..
.
..
.
Γ
m
A
m+1
A tonk B
tonkI m
m+2
B
tonkE m+1
Π
Clearly, the local peak w.r.t tonk on line m + 1 cannot be levelled. We have no
guarantee of a proof of B before line m + 1. So tonk was clearly crucial in bringing
B about, and (by innocence) logic shouldn’t do that.
3
2 Intuitionism
Now an obvious question presents itself: what other rules are harmonious? I
leave it as an exercise to see that the standard rules for ∨, →, ↔, ∃ and ∀ are
harmonious. Negation is the interesting case. Consider a local peak with respect
to negation:
..
.
..
.
Γ
Π1
m
..
.
¬A
n
⊥
Π2
n+1
¬¬A
¬I m–n
n+2
A
DNE n + 1
This local peak cannot be levelled. But this is classical negation. Intuitionists
usually accept the classical rule of negation introduction but, as you know, reject
DNE. In its place, they use ⊥I as the elimination rule for negation.
..
.
..
.
Γ
m
..
.
A
Π1
Π2
k
..
.
A
l
⊥
..
.
..
.
Γ
m
..
.
A
n
⊥
Π1
Π3
Π3
n
¬A
¬I k–l
n+1
⊥
⊥I m, n
So intuitionistic negation is harmonious. We therefore have a distinct argument
for intuitionistic logic: if you accept the principle of innocence, then you should
be an intuitionist.
4
3 Replies
3.1 Multiple conclusions
You are familiar with logics that allow arguments to have multiple premises but
only a single conclusion. But there are multiple conclusion logics which break with
this standard practice. There are many multiple-conclusion systems available,
but they generally define logical consequence as follows:
Multiple-conclusion consequence P1 , ..., Pn |= C1 , ..., Cn iff every model that
makes all of P1 , ..., Pn true makes at least one of C1 , ..., Cn true.
How can this help the classicist to reply to the intuitionist here? The negation
rules for classical multiple-conclusion logic are as follows:
i
A
m
Θ, A
j
∆, ⊥
n
∆, ¬A
∆, ¬A
¬I i–j
Θ, ∆
¬E m, n
These rules are harmonious, as shown in e.g. Stephen Read’s ‘Harmony,
autonomy and classical logic’ (2010).
Here’s Ian Hacking, from ‘What is Logic?’ on this situation:
Gentzen noticed that it is convenient to make statements of the form
Γ ⇒ Θ, where Θ may have several members. On the intended
reading, this will be valid only if some member of Θ is assigned the
value true whenever each member of Γ is assigned the value true. It is
well known that in Gentzen’s calculi, with his rules, intuitionist logic
[rather than classical logic] results from insisting that Θ have at most
one member. I shall not discuss this seemingly magical fact here.
So, if we formulate classical logic in a multiple-conclusion system then,
seemingly by magic, we get harmonious classical negation.
Peter Milne discusses this in his ‘Harmony, purity, simplicity and a “seemingly
magical fact” ’ (2002):
the difference between classical logic and intuitionistic logic is that
the former, but not the latter, sanctions the introduction of ¬ and → to
5
a position subordinate to an occurrence of ∨. This is the explanation
of Hacking’s seemingly magical fact. (p. 515)
The thought is that, since we are reading the conclusions disjunctively, we aren’t
really introducing negation with the introduction rule. If we were doing that, it
would be introduced as the main logical constant. Other criticisms focus on
whether multiple-conclusion arguments are present in natural language (see
Steinberger’s ‘Why conclusions should remain single’).
3.2 Bilateralism
The other best-known response is inspired by Timothy Smiley’s 1996 paper
‘Rejection’. Imagine a speech community for whom every sentence is structured
into a propositional content and a force-indicator. We will consider two force
indicators: one for assertion and one for rejection.
Ian Rumfitt makes this more precise in his 2000 paper ‘ “Yes” and “no” ’. Instead
of ‘Today is Wednesday’, our imagined speech community would say ‘Today is
Saturday? Yes!’; instead of ‘I don’t like ice cream’, they would say ‘I like ice
cream? No!’. The ‘Yes!’ and ‘No!’ here are force markers, which we will
represent with ‘+’ and ‘-’. Therefore ‘+P’ represents the assertion of ‘P’ and ‘-P’
the rejection of ‘P’.
Now let’s imagine that our imagined speech community want to do some logic.
They might give the following clauses to cover conjunction:
m
+P
n
+Q
m
∧I m, nn
+(P ∧ Q)
+(P ∧ Q)
m
∧E m
+P
+(P ∧ Q)
∧E m
+Q
Clearly these rules for ∧ are harmonious. How about negation?
m
m
−P
+¬P
¬I m
+¬P
−P
¬E m
Now let’s write A∗ to represent the result of changing A’s force-marker
(whatever it is) to the other. Then the following looks reasonable:
Reversal Principle If Γ, A ⊢ B, then Γ, B∗ ⊢ A∗ .
6
The ¬E rule tells us that from +¬¬P we can infer −¬P. Now, by Reversal, we can
get from −¬P to +P. So DNE is provable in this system. So the system is classical
logic. And yet, as we saw, the rules are harmonious. Indeed, they are also quite
elegant, since we have no need for a primitive ⊥.
3.3 Identity
Finally, if we take harmony seriously, do we respect all of the expressions we
usually want as logical constants? Plausibly not. Consider identity. The usual
rules are:
a=a
m
Fa
n
a=b
Fb
=I
=E m, n
It’s hard to know how we could begin to apply tests for harmony in this case:
the I-rule only concerns a subclass of cases governed by the E-rule. Stephen
Read, ‘Identity and Harmony’ (2004) suggests modifying the I-rule:
i
Fa
j
Fb
m
a=b
=I′
In this rule, F must not appear amongst any undischarged assumptions other
than Fa. This appears to be harmonious with =E. But, as I prove in my
‘Harmonious rules for identity’ (2014), Read’s rule is precisely as strong as the old
=I rule.
To see why, consider the conditions under which we’ll get an introduction of
a = b for mixed a and b. We must either have identity amongst our premises, or a
contradiction. But in those cases, we could have introduced identity even with
the old rules. And, if harmony is a measure of inferential powers, then =I and
=I′ are either both harmonious with respect to =E, or neither are.
If neither are, then we’ve made no progress. If both are, then harmony must
have some other component, e.g. a presentational component. But it’s not clear
that this is motivated by the principle of innocence.
7
Owen Griffiths
oeg21@cam.ac.uk
9/3/16
Last week
We considered Fitch’s paradox of knowability. This has been used by
intuitionists to argue that if we have some sympathy with the verificationist
thought that all truths are knowable, then we should be intuitionists, or accept
the ridiculous conclusion that all truths are known. The intuitionist is still,
however, committed to the claim that every sentence of the form ‘A ∧ ¬KA’ is a
contradiction. Another solution to the paradox is to restrict knowability, but we
saw that the two major solutions here face problems.
1 Logical inferentialism
This week, we’ll consider a distinct argument for intuitionistic logic: from
inferentialism and considerations of harmony.
What do the logical constants mean? A plausible thought is that the meaning of a
logical constant is given by its inference rules: its I-rule(s) and its E-rule(s). The
meaning of →, for example, is given by:
1
i
A
m
A→B
j
B
n
A
A→B
→I i–j
B
→E m, n
How does this work? If we follow Wittgenstein’s thought that meaning is use,
and we believe that the rules capture the use of the logical constant in question,
then we’ll think that they capture its meaning. This view is logical inferentialism.
Not just any old I- and E-rules will succeed in defining a logical constant,
however. Arthur Prior, in ‘The Runabout Inference Ticket’ (1960) made this point
very succinctly with the example of tonk:
m
m
A
A tonk B
tonkI m
A tonk B
B
tonkE m
Adding tonk to a deductive system has disastrous consequences. It trivialises
every system into which it is added:
1
P
2
P tonk Q
tonkI 1
3
Q
tonkE 2
P and Q were totally arbitrary, so tonk allows us to derive anything from
anything, hence its being a runabout inference ticket.
The logical inferentialist evidently needs some constraints in place to rule out
tonk and, to avoid the charge of being ad hoc, the constraints had better be
motivated. Let’s start with the motivation.
Florian Steinberger, in ‘What harmony could and could not be’ (2011, p. 619)
expresses a principle that could be useful here:
Principle of innocence It should not be possible, solely by engaging in
deductive logical reasoning, to discover hitherto unknown (atomic) truths
we would have been incapable of discovering independently of logic.
The thought is that logic alone should not create novel grounds for asserting
hitherto unknown atomic sentences. And, in the case of tonk, it looks like that’s
exactly what’s going on.
2
What would be nice would be a precise test that can be applied to a pair of
inference rules to check whether they are innocent. We now generally call this
testing for harmony. Roughly, pairs of inference rules are harmonious just if you
don’t get out of the E-rule any more than you put in with the I-rule. Let’s look at
Dummett’s suggestion in The Logical Basis of Metaphysics (p. 247–8):
it should not be possible, by first applying one of the introduction
rules for c and then immediately drawing a consequence from the
conclusion of that introduction rule by means of an elimination rule
of which it is the major premiss, to derive from the premisses of the
introduction rule a consequence that we could not otherwise have
drawn.
Let’s call this the levelling of local peaks test for harmony. Consider conjunction:
..
.
..
.
Γ
Γ
Π1
..
.
..
.
m
..
.
A
m
A
n
B
n+1
A∧B
∧I m, n
n+2
A
∧E n + 1
Π1
Π2
How about tonk?
..
.
..
.
Γ
m
A
m+1
A tonk B
tonkI m
m+2
B
tonkE m+1
Π
Clearly, the local peak w.r.t tonk on line m + 1 cannot be levelled. We have no
guarantee of a proof of B before line m + 1. So tonk was clearly crucial in bringing
B about, and (by innocence) logic shouldn’t do that.
3
2 Intuitionism
Now an obvious question presents itself: what other rules are harmonious? I
leave it as an exercise to see that the standard rules for ∨, →, ↔, ∃ and ∀ are
harmonious. Negation is the interesting case. Consider a local peak with respect
to negation:
..
.
..
.
Γ
Π1
m
..
.
¬A
n
⊥
Π2
n+1
¬¬A
¬I m–n
n+2
A
DNE n + 1
This local peak cannot be levelled. But this is classical negation. Intuitionists
usually accept the classical rule of negation introduction but, as you know, reject
DNE. In its place, they use ⊥I as the elimination rule for negation.
..
.
..
.
Γ
m
..
.
A
Π1
Π2
k
..
.
A
l
⊥
..
.
..
.
Γ
m
..
.
A
n
⊥
Π1
Π3
Π3
n
¬A
¬I k–l
n+1
⊥
⊥I m, n
So intuitionistic negation is harmonious. We therefore have a distinct argument
for intuitionistic logic: if you accept the principle of innocence, then you should
be an intuitionist.
4
3 Replies
3.1 Multiple conclusions
You are familiar with logics that allow arguments to have multiple premises but
only a single conclusion. But there are multiple conclusion logics which break with
this standard practice. There are many multiple-conclusion systems available,
but they generally define logical consequence as follows:
Multiple-conclusion consequence P1 , ..., Pn |= C1 , ..., Cn iff every model that
makes all of P1 , ..., Pn true makes at least one of C1 , ..., Cn true.
How can this help the classicist to reply to the intuitionist here? The negation
rules for classical multiple-conclusion logic are as follows:
i
A
m
Θ, A
j
∆, ⊥
n
∆, ¬A
∆, ¬A
¬I i–j
Θ, ∆
¬E m, n
These rules are harmonious, as shown in e.g. Stephen Read’s ‘Harmony,
autonomy and classical logic’ (2010).
Here’s Ian Hacking, from ‘What is Logic?’ on this situation:
Gentzen noticed that it is convenient to make statements of the form
Γ ⇒ Θ, where Θ may have several members. On the intended
reading, this will be valid only if some member of Θ is assigned the
value true whenever each member of Γ is assigned the value true. It is
well known that in Gentzen’s calculi, with his rules, intuitionist logic
[rather than classical logic] results from insisting that Θ have at most
one member. I shall not discuss this seemingly magical fact here.
So, if we formulate classical logic in a multiple-conclusion system then,
seemingly by magic, we get harmonious classical negation.
Peter Milne discusses this in his ‘Harmony, purity, simplicity and a “seemingly
magical fact” ’ (2002):
the difference between classical logic and intuitionistic logic is that
the former, but not the latter, sanctions the introduction of ¬ and → to
5
a position subordinate to an occurrence of ∨. This is the explanation
of Hacking’s seemingly magical fact. (p. 515)
The thought is that, since we are reading the conclusions disjunctively, we aren’t
really introducing negation with the introduction rule. If we were doing that, it
would be introduced as the main logical constant. Other criticisms focus on
whether multiple-conclusion arguments are present in natural language (see
Steinberger’s ‘Why conclusions should remain single’).
3.2 Bilateralism
The other best-known response is inspired by Timothy Smiley’s 1996 paper
‘Rejection’. Imagine a speech community for whom every sentence is structured
into a propositional content and a force-indicator. We will consider two force
indicators: one for assertion and one for rejection.
Ian Rumfitt makes this more precise in his 2000 paper ‘ “Yes” and “no” ’. Instead
of ‘Today is Wednesday’, our imagined speech community would say ‘Today is
Saturday? Yes!’; instead of ‘I don’t like ice cream’, they would say ‘I like ice
cream? No!’. The ‘Yes!’ and ‘No!’ here are force markers, which we will
represent with ‘+’ and ‘-’. Therefore ‘+P’ represents the assertion of ‘P’ and ‘-P’
the rejection of ‘P’.
Now let’s imagine that our imagined speech community want to do some logic.
They might give the following clauses to cover conjunction:
m
+P
n
+Q
m
∧I m, nn
+(P ∧ Q)
+(P ∧ Q)
m
∧E m
+P
+(P ∧ Q)
∧E m
+Q
Clearly these rules for ∧ are harmonious. How about negation?
m
m
−P
+¬P
¬I m
+¬P
−P
¬E m
Now let’s write A∗ to represent the result of changing A’s force-marker
(whatever it is) to the other. Then the following looks reasonable:
Reversal Principle If Γ, A ⊢ B, then Γ, B∗ ⊢ A∗ .
6
The ¬E rule tells us that from +¬¬P we can infer −¬P. Now, by Reversal, we can
get from −¬P to +P. So DNE is provable in this system. So the system is classical
logic. And yet, as we saw, the rules are harmonious. Indeed, they are also quite
elegant, since we have no need for a primitive ⊥.
3.3 Identity
Finally, if we take harmony seriously, do we respect all of the expressions we
usually want as logical constants? Plausibly not. Consider identity. The usual
rules are:
a=a
m
Fa
n
a=b
Fb
=I
=E m, n
It’s hard to know how we could begin to apply tests for harmony in this case:
the I-rule only concerns a subclass of cases governed by the E-rule. Stephen
Read, ‘Identity and Harmony’ (2004) suggests modifying the I-rule:
i
Fa
j
Fb
m
a=b
=I′
In this rule, F must not appear amongst any undischarged assumptions other
than Fa. This appears to be harmonious with =E. But, as I prove in my
‘Harmonious rules for identity’ (2014), Read’s rule is precisely as strong as the old
=I rule.
To see why, consider the conditions under which we’ll get an introduction of
a = b for mixed a and b. We must either have identity amongst our premises, or a
contradiction. But in those cases, we could have introduced identity even with
the old rules. And, if harmony is a measure of inferential powers, then =I and
=I′ are either both harmonious with respect to =E, or neither are.
If neither are, then we’ve made no progress. If both are, then harmony must
have some other component, e.g. a presentational component. But it’s not clear
that this is motivated by the principle of innocence.
7