Medieval Period (400 A.D.-1600 A.D.

):
        The 11th -15th centuries were the main time of activity regarding logic in this period.  The
influence of Aristotle dominates the medieval logicians, who wrote commentaries on him and on
others who had commented on him such as Boethius and Porporhy.   Among the important
logicians were Peter Abelard (1079-1142) with 4 works on logic, William of Sherwood (ca.
1200-1271) who developed mnemonic verses as an aid in learning the syllogisms (Barbara, aka
AAA, being the best known), William of Ockham (died 1349) (best known for Ockham’s Razor
which suggests the importance of simplicity), and J. Buridan (mainly known for Buridan’s Ass,
involving decisions in cases of equal preference, but which does not apparently actually occur in
his writings).  The development and emergence of universites during this period is important for
the study of logic.  Textbooks and manuals on the subject began to appear.  One of the more
important textbooks was the “Port Royal Logic” of Arnauld and Nicole, appearing in 1662, in
which logic is the “art of managing one’s reason right in the knowledge of things, both for the
instruction of oneself and of others”.
 
Early Modern Period (1600-1850 A.D.):
          Leibnitz (1646-1716) is considered a great logician and his work exhibits a respect for
traditional “Aristotelean” logic but also an interest in general theories of arrangements, plans for
an “ideal” language, and general science of method.   The German philosopher Kant (1724-1804)
made the distinction between types of statements a key to understanding his philosophy; he
distinguished between analytic statements whose truth can be determined on the basis of the
meanings of the words in the statements, and synthetic statements, which require a direct appeal
to experience.   Bolzano (1781-1848) continued to examine the analytic-synthetic distinction in
his chief work Wissenschaftslehre.
         
 Modern and Contemporary Period (1850-present)
        The 19th and 20th centuries have involved great activity and discovery in logic, including
the “rediscovery” of the Stoic type of logic or logic of propositions.  De Morgan  (1806-1871)
discovered the theorems that  bear his name and that are now routinely part of the logic of
propositions.  George Boole (1815-1864), considered the founder of symbolic logic, used
symbols to depict arguments; he wrote the “Analysis of the Laws of Thought” and
“Mathematical Analysis of Logic”, in which he argues that math is the basis of logic; and his use
of numbers to    
express the truth values of compound statements (conjunctions, disjunctions, etc.)  directly
influenced the development of computers.
        The British logician John Venn (1834-1923) developed circular diagrams used as a tool to
test the validity of syllogisms.  J. S. Mill (1806-1873), another British philosopher, was
particularly interested in inductive arguments and gave an account of methods for checking such
arguments, known in fact as “Mill’s Methods”.  His countryman Charles Dodgson (1832-1898)
wrote “Symbolic Logic” and “The Game of Logic”, but is better known under his pen name
Lewis Carroll (“Alice in Wonderland”).  In the United States C. S. Pierce (1839-1914), an
initiator of American pragmatism,  was the earliest influential logician.  Pierce stated that “few
persons care to study logic, because everybody conceives himself to be proficient enough in the
art of reasoning already.  But I observe that this satisfaction is limited to one’s own ratiocination,
and does not extend to that of other men.”
        G. Frege (1848-1925) claimed that logic is the basis of math, and specifically aimed to
reduce or derive arithmetic from logic; also, he developed the predicate calculus (quantification
theory) which brought the categorical (Aristotelean) and propositional (Stoic) traditions
together.  His greatest work is perhaps the Begriffsschrift. 
        Bertrand Russell (1872-1970)  and Alfred North Whitehead (1861-1947) continued the
development of the predicate calculus in their Principia Mathematica (1910-1913).  Russell’s
criticism of some of Frege’s ideas led to the development of a paradox involving the “set of all
sets”, known in its more popular version as “the Barber paradox”: in a certain city a barber
shaves the heads of all those people and of only those people who do not shave themselves, . . .
but then who shaves the barber?   Reflections on this paradox led Russell to develop his “theory
of types”. 
         Other contributions in this century have been from Wittgenstein (1819-1951), one of the
developers of “truth tables”, K. Godel (1916-     ), known for his “ incompleteness theorem”,
and Lofti Zadeh (1916-     ), who is associated with the development and formalizing of “fuzzy”
logic in 1965.  Rudolph Carnap (1891-1970), who defends a thesis of extensionality in his
Logical Syntax of Language (1934) attempted to give precise definition to the distinction
between analytic and synthetic statements and was associated with the philosophy of  logical
empiricism and it famous verifiability principle, according to which a synthetic statement is
meaningful only if it is verifiable.
Another of Carnap’s works is The Logical Structure of the World (1928).
       The development of non-Euclidean geometry, many- valued logics, proof theory and
systems theory, and of course computers and information technology have had far-reaching
impact and significance for logic and critical thinking.
 
Difference of Traditional and Modern Symbolic Logic

Traditional logic, modern logic and natural

language
Wilfrid Hodges
Herons Brook, Sticklepath,
Okehampton, Devon EX20 2PY, England
wilfrid.hodges@btinternet.com
DRAFT June 2009. The paper is for a Festschrift and this draft has
removed a number of personal references.
1 The questions
. . . Wikipedia [38] defines: traditional logic is ‘a loose name for the way
of doing logic that began with Aristotle, and that was dominant until the
advent of modern predicate logic in the late nineteenth century’. It is of
great interest to place the transitions between traditional and modern logic.
In this paper I will say where I think the main differences lie. In my last
section I will comment on . . . the relationship between some traditional
argument forms and natural language argument.
The strength of traditional logic is sometimes measured in terms of the
valid inference patterns that it recognises. Among other patterns:
(1) “Some P R all Q” implies “All Q are R-ed by some P”.
and the pattern behind some inferences that De Morgan studied:
(2) “All horses are animals. So, all horse tails are animal tails.”.
This is not a new measure; it was widely used in the mid 20th century
to gauge the strength of various early logicians, among them Aristotle,
Boethius and Ockham. But I have my doubts about it. We first need to
see how traditional logic used its inference patterns. Until we know that,
we don’t know that we are comparing them with the right things in modern
logic.
1
From early times Euclid’s Elements was considered one of the pinnacles
of human reasoning. (For modernmathematicians this is odd: Archimedes
was clearly light-years ahead of Euclid. But that wasn’t the traditional perception.)
Aristotelian logicians believed that syllogisms accounted for all
forms of logical reasoning, and hence by implication for all the logical steps
in the Elements. Some platonists dissented: geometry also needs spatial intuitions.
But itwas never clear—perhaps even in the case of Kant—whether
the claim was that spatial intuition gives new logical methods rather than
geometrical axioms. So apparently neither aristotelians nor platonists saw
any knock-down refutation of the claim that all the reasoning steps in Euclid
can be justified in terms of syllogisms. If, as historians of logic, we
want to maintain that traditional logic falls far short of justifying Euclid’s
inferences, then we need to explain why for two thousand years nobody
called attention to the emperor’s clothes.
For example one sees the view expressed that traditional logic is obviously
unable to handle geometry because traditional logic is monadic and
geometry involves relations. But bear in mind who we are talking about:
the list includes Proclus, Ibn S¯ın¯a, T¯us¯ı, Leibniz, Wallis, John Bernoulli, Euler,
Gergonne. Most of these people were leading specialists in geometry,
logic or both, and all of them were competent in both fields. Any view like
this one, that by implication makes them out to be a bunch of intellectual
time-servers, should never have been allowed off the starting block.
There is a lot to discuss below. In cutting down the length I may have
said too little about some questions. I hope to develop several of the points
below in greater detail elsewhere, if it turns out that they are both new and
correct.
2 Aristotle’s use of logic
In the mid fourth century BC, Aristotle introduced the idea that one can
check arguments by showing that they conform to one of a set of valid
argument patterns. The main patterns that he used are the categorical syllogisms,
for example
(3)
If A belongs to no B but to some C, it is necessary that B does
not belong to some C.
(Cf. Prior Analytics i.5, 27a32, [2] p. 44.) Today we find it more convenient
to phrase this pattern as follows:
(4) No B is an A. Some C is an A. Therefore some C is not a B.
2
Aristotle himself also considered modal syllogisms, and later traditional
logicians added propositional syllogisms to the stock of valid argument
patterns.
Howexactly did one check an argumentwith the help of these patterns?
Aristotle said very little about this in his surviving texts. But he, or perhaps
his editor Andronicus, collected some information about it in the later sections
of Book One of the Prior Analytics. There we learn that, faced with a
natural language argument like
(5)
God doesn’t have times that need to be set aside for action. God
does have right moments for action. Therefore some right moment
for action is not a time that needs to be set aside for action.
Aristotle would find the premises and the conclusion, and then write out
the syllogistic terms together with letters to stand for them:
(6)
A: thing that God has. B: time needing to be set aside for action.
C: right moment for action.
(This is from Prior Analytics i.35f. Aristotle’s text needs careful dissection,
and for a smooth exposition I’ve permuted some of his material.) Aristotle
refers to the correlation of terms and letters as ‘setting out the terms’
(´ekthesis tˆon horˆon, not to be confused with the method of argument known
as ´ekthesis). Then he would indicate which syllogistic mood the argument
was supposed to conform to, unless this was obvious. (For the syllogism
(6) the mood is (5) above.)
So far, his practice seems almost identical with what we do today in
elementary logic classes. For example Gamut [13] (p. 67) sets out terms as
follows:
(7)
B2: lies between
b: Breda, t: Tilburg, e: Eindhoven
Frommodern experiencewe know what comes next, though Aristotle’s
text doesn’t discuss it. The student has to check that the relevant valid
sequent becomes a correct paraphrase of the original argument if we read
the nonlogical constant symbols in the sequent asmeaning what the setting
out of terms says theymean. Oneway to check this is to rewrite the sequent
with the symbols replaced by their natural language equivalents; the result
is a natural language argument, maybe with some logical symbols like ‘∧’
for ‘and’. For future reference I call this rewriting the formal paraphrase of the
argument. For example the argument (5), with (4) as the relevant sequent,
3
has the formal paraphrase
(8)
No (time needing to be set aside for action) is a (thing that God
has). Some (right moment for action) is a (thing that God has).
Therefore some (right moment for action) is not a (time needing
to be set aside for action).
Then the student uses her linguistic intuitions to check that the original
argument and the formal paraphrase mean the same thing. This step of
confirming the paraphrase is essential to complete the checking of the original
argument, though today it is usually done silently and withoutwriting
out the formal paraphrase explicitly.
Besides the syllogisms, Aristotle also had a more sporadic collection of
argument principles which he called topoi, ‘topics’ in English (cf. his book
Topics, [2] pp. 167–277). He and his successors never integrated the topics
with the syllogisms. In fact the list of topics varied fromlogician to logician,
and there were persistent questions about whether topics are for validating
arguments or for finding them, and whether they describe the forms or
the subject-matter of arguments; Green-Pedersen [14] documents all this.
But some topics certainly were used as patterns for valid arguments. One
famous example, which we can trace back at least to Boethius [3] Book 3,
1198A4, takes the following form in Peter of Spain in the 13th century:
(9)
If one of the correlated things is posited, the other is posited.
(Peter of Spain, Tractatus [31] V para. 28.)
Peter illustrates it with the inference
(10) There is a father; therefore there is a son.
The inference (3) is a close analogue of a topical inference in Aristotle (‘If
knowledge is a belief, then also the object of knowledge is an object of belief’,
Topics ii.8 114a18f, [2] p. 190).
For completeness one should also mention Aristotle’s conversion rules.
These are one-premise inferences such as ‘Every horse is an animal; therefore
some animal is a horse’.
3 Hilbert’s observation
In lectures in G¨ottingen in 1917–1921, David Hilbert assembled for the first
time the syntax and semantics of first-order predicate logic (of course using
4
earlier work of Peirce and others). In 1928 these lectures were published,
after light editing, as a textbook with Ackermann [17]. Hilbert introduced
monadic first-order logic by showing how it handles syllogisms ([17] ii.3).
To introduce relational logic he showed how it handles the topical argument
(10) above (or more strictly its converse, [17] iii.1, iii.3).
It might seem strange that Hilbert advertises first-order logic by showing
it can handle arguments that Boethius could already manage in the 6th
century. But see exactly what Hilbert says about this:
(11)
“If there is a son, then there is a father,” is certainly a logically
self-evident assertion, and we may demand of any satisfactory
logical calculus that it make obvious this self-evidence, in the
sense that the asserted connection will be seen, by means of the
symbolic representation, to be a consequence of simple logical
principles. ([17] iii.1)
Here Hilbert points to two differences between his modern logic and traditional
logic.
First, his logic finds ‘simple logical principles’ that are adequate to support
both syllogisms and relational topics. The traditional logicians never
analysed their principles down to this level of simplicity. Hilbert is clearly
right about this, and the point is quite fundamental.
Second, Hilbert’s ‘logical calculus’ allows one to express complex ideas
in a form where we can read off their implications by applying the simple
logical principles. This is true but less telling. Having a formal calculus
makes possible a number of logical tricks that were completely beyond the
traditionals, for example nonstandard analysis and computerised proofchecking.
But at the level of basic logic, the formal language is a convenience
and a protection against error, and not much more than this. Weierstrass
could reason logically and securely with ideas of great complexity,
using only mathematical German and no logical calculus at all.
Michael Friedman [12] p. 474 notes that for reasoning about the continuum
we need sentences of the form ∀ . . . ∀∃, and he speaks of ‘the absence
of such logical forms’ in Kant’s logic. This is too fast. Certainly Kant had no
formal calculus containing propositions of this form; but why should they
have been missing from his German? Maybe he couldn’t apply his logic to
sentences of this complexity, but this is hardly because he lacked a formal
language. . . .
5
4 Local formalising
Consider this, from Ibn S¯ın¯a’s autobiography:
(12)
As for the Elements of Euclid, I read the first five or six propositions
. . . , and thereafter undertook on my own to solve the entire
remainder of the book. . . . I read Logic and all the parts of philosophy
once again. . . . I compiled a set of files for myself, and
for each argument that I examined, I recorded the syllogistic premisses
it contained, the way in which they were composed, and
the conclusions which they might yield, and I would also take
into account the conditions of its premisses [i.e. their modalities]
until I had Ascertained that particular problem. . . . So I continued
until all the Philosophical Sciences became deeply rooted
in me and I understood them as much as is humanly possible.
. . . Having mastered Logic, Physics and Mathematics, I had now
reached Theology. . . . (Gutas [15] pp. 26–28.)
I draw two implications from this passage. The first is that Ibn S¯ın¯a believed
he could reduce Euclid’s Elements to syllogisms. He claims this
explicitly only for Aristotle’s Organon, but the Organon already contains
enough mathematics to occupy about a hundred pages of Thomas Heath’s
book on mathematics in Aristotle [16]. He would hardly have added that
he had ‘mastered Mathematics’ if he thought that the Elements contained
arguments that resisted this treatment.
The second is that Ibn S¯ın¯a verified (‘Ascertained’ in Gutas’ translation)
the arguments in ‘Logic and all the parts of philosophy’ by breaking them
down into pieces that he could treat as syllogisms, and then recording each
syllogism separately on a file card. This is very much in tune with the
way he deploys syllogisms when he is analysing philosophical theses; see
Gutas’ analysis in [15] pp. 177–181. In short, Ibn S¯ın¯a verifies (and hence
formalises) each syllogism separately. In a typical logic course today we treat
arguments quite differently: we choose a symbolism that fits the whole argument,
andwe deduce the conclusion fromthe premises by formal derivations
that can be very much more complex than a single syllogism.
Let us refer to Ibn S¯ın¯a’s procedure, where each syllogism is checked
separately, as local formalisation; the modern approach is global formalisation.
To avoid misunderstandings, let me clarify at once that the contrast local/
global is not the same as the contrast small-scale/large-scale. For example
Aristotle in Prior Analytics i.25 discusses arguments that can be analysed
into a tree of syllogisms. The conclusion of the argument is the con-
6
clusion of the bottom syllogism, which (to adapt a term from Aristotle) we
can call the ‘leading’ (k´urios) syllogism. The leading syllogism could well
take control of the large-scale form of the argument. Thus
(13)
We will prove that right moments for action don’t have to be
times that need to be set aside for action For this we first show
that God doesn’t have times that need to be set aside for action
(and here follows a string of subsidiary syllogisms). Next we show
that God does have right moments for action (and here another
string of subsidiary syllogisms). We conclude that right moments
for action etc. etc.
But the leading syllogism is just one syllogism, like each of the subsidiary
syllogisms. Local formalising in this case tells us that the choices of domains
and terms for formalising the subsidiary syllogisms don’t have to
agree with the domain and terms chosen for formalising the leading syllogism.
(The previous sentence is my point, not Aristotle’s.)
Other texts confirm that local formalising is typical of the aristotelian
tradition. Here is Proclus in the 5th century, describing the procedures of
the commentator Iamblichus in the early 4th century:
(14)
That method of dividing the dialogue seems tome to be themost
perfect of all, which the philosopher Iamblichus also fixed on.
This bases itself on the actual subject-matter and gathers into
three sections the whole substance of the work, and relates to
this the division according to syllogisms and that which takes
account of the forms of discourse. . . . How then do we say that
the dialogue is divided into its immediate and most basic parts?
. . . [T]here comes first therefore one section which removes ignorance
from the Reason and does away by means of a copious variety
of syllogisms with all those obstacles to knowledge which
exist as the result of generation.
(Proclus In Alcibiadem Fr. 2, in [8] p. 73.) The syllogisms correspond to the
‘immediate and most basic parts’ of the argument being studied.
Leibniz saw formalising as a kind of paraphrase. (For example, raw
natural language arguments are not ‘transmutatae in aliam formam, scholarum
more’, [22] p. 36.) The paraphrases that particularly interested him
were thosewhere the salient parts of an inference are brought into the nominative
case. We will see where this viewpoint comes from in §6 below. The
effect is that in general an analysed piece of reasoning consists of interleaved
steps: ‘grammatical’ steps of paraphrasing, and ‘syllogistic’ steps of
7
logical reasoning. Thus:
(15)
It should also be realized that there are valid non-syllogistic inferences
which cannot be rigorously demonstrated in any syllogismunless
the terms are changed a little, and this altering of the
terms is the non-syllogistic inference. There are several of these,
including arguments from the direct to the oblique — e.g. ‘If Jesus
Christ is God, then themother of Jesus Christ is themother of
God’. And again, the argument-formwhich some good logicians
have called relation-conversion, as illustrated by the inference:
‘If David is the father of Solomon, then certainly Solomon is the
son of David’. . . . ([23] 479f.; his italics.)
And in the context of his plan for a universal language:
(16)
So we can do without adverbs, most conjunctions and all interjections;
and even cases and times and persons. This is grammatical
analysis . . . After this purely grammatical analysis comes
Logical analysis. ([22] p. 353.)
The paraphrases that Leibniz describes are all at the level of single sentences.
They bring the premises of an inference to a form where the traditional
rules of logic apply. So again we have local formalising.
Now if syllogisms are patterns for the ‘immediate and most basic parts’
of an argument, do they correspond to modern valid sequents, or to modern
inference rules?
The distinction between valid sequents and inference rules is twentieth
century; we can reasonably project it back to Frege, but hardly further. So
there is some anachronism in thinking of syllogisms as corresponding to
either of the two. But once the question is raised, it must surely be clear
that they correspondmore closely to inference rules than to valid sequents.
They are more complex than we like our rules of inference to be; but this is
precisely Hilbert’s observation that traditional logic failed to analyse down
to simple logical principles.
In fact it could be argued that the profusion of more complicated inference
patterns in the later centuries of the aristotelian tradition is less a sign
of logical progress, and more a testimony to the failure to reach primitive
rules that would have allowed these patterns to be generated. But I come
back to this question at the end of the paper.
Inference rules normally contain just one active variable. First-order
translations of categorical syllogisms normally have a single active variable
8
too, which is why Hilbert uses monadic logic to express them. But modern
inference rules usually carry along a set of individual parameters, either
universally quantified or as formal parameters. For example Shoenfield
[34] p. 21 has an inference rule
(17) If x is not free in B, infer ∃xA→B from A→ B.
where A and B are formulas with any number of free variables. In exactly
the same way, the terms in a natural language syllogistic inference can have
any number of formal parameters in them.
In practice the best aristotelian logicians could handle parameters inside
terms, even if they had no good theoretical account of them. A note
recently published by Mugnai [28] p. 153 shows Leibniz confidently using
a formal parameter ‘Titius’ that is clearly part of the interpretation of
a syllogistic term. (Who on earth is Titius? Exactly so—formal parameters
don’t name individuals.) Ibn S¯ın¯a constantly refers to implicit parameters
in terms, and he has a modality d¯a’iman which is adequate for universally
quantifying them.
In short, the contrast ‘traditional logic monadic, modern logic polyadic’
is an illusion. If Kant wasn’t able to put relational arguments into syllogistic
form, this is only one of many pieces of evidence of the sad decline of
traditional logic in its later centuries.
5 Comparison of local and global formalising
A few things are harder for the local formaliser than they are for modern
logicians. The local formaliser has to break each complex argument down
to simple self-contained pieces. This prevents her from making an assumption
and then discharging it seventeen steps later—a device that Euclid certainly
uses in the Elements, for example when he argues by reductio ad
absurdum. This is why traditional logicians sometimes explained reductio
ad absurdum by taking the false assumption not as a premise but as the
antecedent of a conditional. (One of the clearest accounts along these lines,
before Frege, is by Ibn S¯ın¯a in his Qiy¯as viii.3, cf. [19].)
On the other hand the ability to shift to a new formalisation at each
step of an argument certainlymakes possible some things thatwewouldn’t
dream of doing in a standard first-order formalisation. The clearest cases
are where the reformalisation actually changes the universe of discourse.
Let me give three examples.
9
5.1 Changing the domain to pairs of elements
This changewas used to handle binary relations. Alexander of Aphrodisias
has a clutch of examples in his commentary on Prior Analytics i in the 2nd
century AD. For example:
(18)
“This individual (for example, A) has the same
parents as that one (for example, B); but also B
has the same parents as C; therefore A has the
same parents as C.”
What is left out for there to be a syllogism is the universal
premiss which says ‘All things which have the same parents
as someone are siblings’, to which one adds the divided
premiss made one and saying that A and C have the same
parents as B. In this way it follows that A and C are siblings.
([26] p. 28f.)
Thus Alexander moves to a universe of pairs, so that he can formulate the
syllogistic sentence ‘Every pair (x, y) of people such that both x and y are
siblings of the same person, is a pair such that they are siblings of each
other’.
A more clear-headed example is in Ibn S¯ın¯a:
(19)
When you say ‘If a line falls on two lines in such a way that the
two angles which are on one side etc., then the two lines are parallel’,
this can be paraphrased as a predicative proposition thus:
‘Every pair of lines, on which a line falls in such-and-such a way,
is a parallel pair’. (Ibn S¯ın¯a Qiy¯as [21] 256.12–15.)
Probably Proclus has a similar device inmind in his syllogistically-arranged
commentary on Aristotle’s Physics, [32] I.1:
(20)
Any pair AB of things that touch each other have a common
boundary.
So this device seems to have been standard equipment before the Scholastic
period. Perhaps it’s only my ignorance that prevents me giving Scholastic
examples too.
Ian Mueller [27] p. 42 has described arguments along the lines of (18)
as ‘wrong’. The reason is that in order to bring the premise ‘Every pair . . . ’
10
into play, one needs a further inference of the form
(21)
x and y are both siblings of z; therefore both x and y are siblings
of the same person.
But this inference ‘depends on the relations among the three’ people x, y, z
‘and not on properties of themtaken as pairs’. (He takes a slightly different
example but his point is the same.) Of course Mueller is right that Alexander’s
inference needs to be combined with (21). But under local formalising
there is no problem about doing this. The inference (21) is a special
case of ‘z is a B; therefore something is a B’, with the single active variable
z; the variables x and y are formal parameters. Challenged to justify this
inference by syllogisms, a traditional logician can answer at once with a
syllogism in mood Darapti:
(22) z is a B. z is a thing. Therefore some thing is a B.
This syllogistic argument is embarrassing for modern logicans because it
throws a harsh light on the traditional failure to analyse down to simple
principles; but its validity is impeccable. I take Mueller’s discussion not as
a demonstration that Alexander’s argument is wrong, but as further evidence
for the use of local formalising.
Later in the same article Mueller [27] p. 62f criticises similar arguments
in Alexander and Galen as ‘ridiculous’. I think this is unfair. Formal devices
often do seem silly when they are detached from the formal machinery
where they belong.
Note once again that the dichotomy ‘traditional monadic, modern relational’
is an illusion.
5.2 Changing the universe to situations
This device often appears under the name of ‘reduction of hypotheticals
to categoricals’. Briefly, we translate p → q into the form ‘Every situation
in which p is true is a situation in which q is true’. The reduction was
taken for granted by Ibn S¯ın¯a (at least as a heuristic, for example Qiy¯as [21]
415.10f) andWallis ([37] Thesis Secunda), and advertised by Boole [5] p. 49
(where he switches his Universe to ‘all possible cases and conjunctures of
circumstances’). Particularly interesting is Boole’s comment
(23)
it is necessary that each sign should possess, within the limits of
the same discourse or process of reasoning, a fixed interpretation
([5] p. 26).
11
takenwith the fact that Boole nevermixes syllogistic reasoningwith propositional
reasoning. Apparently he regarded a switch from a quantifier rule
to a propositional rule as a change of ‘discourse or process of reasoning’.
5.3 Incorporating a metadomain
For example Buridan’s version of reductio ad absurdum ([6] 5.10.6) avoids
actually making the false assumption. Instead it deduces the conclusion C
from the metalevel fact that ‘we can prove that the contradictory [of C] is
impossible’.
Global formalising appeared quite suddenly, towards the end of the
19th century, in the work of Frege and Peano. Its success was so great that
most modern logicians assume it was always the aim of logic. I leave aside
Peano for reasons of space, and concentrate on Frege.
Frege’s earlywork in logicwas in aid of showing that the truths of arithmetic
are analytic, i.e. definable and provable in general logic. He found
traditional logic seriously inadequate for showing this. He complained
about the ambiguities of natural language. But he reserved his chief ire
for local formalising. Sometimes he attacked the syllogistic/non-syllogistic
alternation that Leibniz had described:
(24)
[Our doubt about the analytic character of arithmetic] can only
be canceled by means of a gapless chain of deductions, so that
no step could appear in it that is not in accordance with one of
a few inference principles that are recognized as purely logical.
([10] §90).
Sometimes he condemned the changes of ‘viewpoint’, in words that seem
to be aimed at shifts like those discussed earlier in this section:
(25)
We cannot give toomany warnings against the danger of confusing
points of view and switching from one question to another,
a danger to which we are particularly exposed because we are
accustomed to thinking in some language or other and because
grammar . . . is a mixture of the logical and the psychological.
(‘Logic’, [11] p. 6.)
One should add that global formalising, which Frege embraced from his
earliest published work in logic, doesn’t remove all appeals to linguistic
intuition. Instead it concentrates these appeals at the places where axioms
are chosen or definitions are introduced—one has to use intuition to check
12
that the formal definition matches the informal notion that it was intended
to formalise. I believe Frege himself took this fact on board only in late
work, unpublished in his lifetime (‘Logic in mathematics’, [11] p. 208ff).
In the light of all this, does it seem likely that Ibn S¯ın¯a really had the
logical resources to validate the arguments used by Euclid? I think there
are too many rough edges in Ibn S¯ın¯a’s procedures to make this a sensible
question. A more profitable approach would be to take a logical calculus
that certainly is sufficient for Euclid’s arguments, say a suitable Hilbertstyle
calculus with axioms for real-closed fields, and prove a theorem to
the effect that any sequent provable in this calculus is also provable using
such-and-such kinds of syllogism and such-and-such kinds of reformalising
between steps. After the theorem is proved, one can compare the devices
that it invokes with the ones that Ibn S¯ın¯a actually had. At least until
this has been done, I wouldn’t criticise Ibn S¯ın¯a for supposing that he had
in principle all the logical tools needed for the job.
It may be relevant that on Reviel Netz’s analysis ([29] p. 197) the majority
of assertions in two sample sections of Euclid’s Elements are about
equivalences. It was precisely to handle equivalences that Alexander of
Aphrodisias and Ibn S¯ın¯a passed to a domain of pairs, as we saw earlier in
this section.
6 Top-level processing
Setting out of terms (section §2 above) rapidly disappeared from the scene.
After Aristotle, we have little evidence of its use until Boole reintroduced it
in 1854 ([4] pp. 33, 57, 59 for example).
In passing I note that there may be a historically interesting difference
between Aristotle’s version of setting out and Boole’s. Netz ([29] 2.3.1 ‘The
semiotics of letters’) argues, on the basis of the phrases that they use, that
for early Greek geometers a letter didn’t stand for an item in a diagram, it
stood next to the item and thereby served as a reference point for the item.
ThenNetz notes thatAristotle uses exactly those same phrases in his setting
out. This seems to me evidence that Aristotle did actually draw diagrams
like (8), writing the letters next to phrases in order to link them with the
phrases. So the diagrams were essentially macros for constructing the formal
paraphrases, and this might help to explain why they disappeared so
soon.
Boole’s notion was different. In the opening paragraph of his [5] (p. 3)
13
he had announced his allegiance to George Peacock’s theory of interpretations
of the symbols of Symbolical Algebra. In that context, setting out of
terms is not just a matter of using letters as labels; it involves interpreting
the letters as meaning what the associated phrases mean. Boole’s version
of setting out is much closer to model theory than Aristotle’s was.
But to return to ourmain theme: instead of setting out terms, traditional
logicians after Aristotle normally went straight to the formal paraphrase.
This works, provided that the formal paraphrase is in a formwhere one can
read off the terms. Logicians writing in Greek, Latin or German normally
ensured this by using paraphrases of the form
(26) Some/every A is/isn’t a B.
where the subject term A and the predicate term B are picked out by being
in the nominative case. Arabic doesn’t allow this strategy: Arabic translations
of (26) are liable to throw either of A and B into the accusative or
the genitive. So Ibn S¯ın¯a adopts the alternative convention that the formal
paraphrase is a topic-comment sentence (mubtada’-k
¯
abar, a kind that occurs
frequently in Arabic), and he reads off the topic as subject and the comment
as predicate (e.g. [20] 31.17–32.2). The implications of this choice for logic
are curious, but not our concern here. The important point for us is that in
both East and West one formalised by adopting a paraphrase that brought
the sentences to a certain grammatical form where both subject and predicate
were at the top level of the syntactic analysis.
In both East and West one meets a tendency to think that somehow the
formal paraphrase is the ‘real’ form in which the syllogistic inference takes
place, or that the mind operates directly with the formal paraphrase. I use
the name top-level processing for this tendency. In Ibn S¯ın¯a it becomes part
of a full-blown theory about how the mind processes the raw data of inference;
Ibn S¯ın¯a believes that the internal parts of the subject and predicate
are in some sense unavailable to the mind while it syllogises. In Leibniz it
appears as a conviction that the way to handle intractable inferences is to
get the relevant parts into the nominative case.
We are dealing here with something not quite fully articulated. It’s not
just that, as Van Benthem rightly says, syllogisms were
(27)
a method for one-step analysis of statements of any kind into one
layer of quantification ([36] p. 23).
The damaging extra ingredient in top-level processing was the conviction
that for some deeper reason, inferences had to be at the top syntactic level.
14
The problem this created for logic was that some inferences do require
dipping down below the top syntactic level. For example if we say ‘a is
greater than b’, the term ‘b’ will not be in the nominative and it will be
an internal part of the k
¯
abar. So it will supposedly be invisible to logical
processing unless we paraphrase, either bringing ‘b’ itself to top level, or
switching to pairs and taking ‘the pair (a, b)’ as subject. Both these fixes are
possible and were used. But they are a strong hindrance to free-wheeling
arguments with relations, and they interrupt the logical process with an
appeal to linguistic intuition.
Frege was the first to see this clearly. In Begriffsschrift [9] §§3, 9 in 1879,
he discarded the notion of subject and took all arguments of a relation to
be at the same level. He also introduced rules like that for equality ([9]
§20). He wrote it as an axiom, but for ease of comparison I rephrase it as an
inference rule:
(28) (c = d), f(c) ⊢ f(d).
where c can be arbitrarily deep down in the syntax of f(c). (Or rather, he
claimed that in ‘the content of a possible judgement’ we can choose where
we want to split off the replaceable argument; see for example [11] p. 16ff,
and Danielle Macbeth [25] p. 39ff for a discussion. It’s as if hewanted to say
that we can only reason at the top level, but the top level is wherever we
want to put it. The psychological pull of top-level processing is so strong
that Frege has to re-introduce it in metaphor at the same time as he is dismantling
its literal application.)
In his chapter ‘Properties of elective functions’ ([4] p. 60ff) Boole made
substitutions at arbitrary depth in terms. But he was just copying the standard
practice of mathematical analysts, apparently without any realisation
of what this move implies for logic.
7 Monotonicity and natural logic
...
There were some attempts to break away from top-level processing already
in the early fourteenth century. The ineptitude of these attempts is a
neat illustration of the weaknesses of traditional logic in its later centuries.
The Scholastics had noticed that all categorical syllogisms are instances
of a common pattern. Most of these syllogisms contain a premise of the
form ‘Every A is a B’. Then the passage from the other premise _ to the
conclusion is got by replacing A by B in _, or vice versa. If A is replaced
15
by B we say we are using upward monotonicity at A; in the other direction
we are using downward monotonicity at B. Downward monotonicity applies
when B is ‘distributed’ in _; upward monotonicity applies when A
is ‘undistributed’ in _. For each syllogistic sentence it was specified which
terms are distributed; a subject term is distributed if it is universally quantified
and undistributed if it is existentially quantified, while a predicate
term is distributed if the sentence is negative and undistributed otherwise.
Variants of this procedure work for the remaining categorical syllogisms.
The general procedure was sometimes described as dici de omni et nullo.
In the early 14th century some logicians tried to apply monotonicity to
terms that were buried deeper in the syntactic structure of a sentence, and
couldn’t be brought to top level by paraphrasing. These attempts continued
till the mid 19th century; for example De Morgan tried to justify (2) by
a monotonicity argument. But they were largely failures, though at least
three approaches were tried.
First,Walter Burley ([18] p. 95f) broke down the sentence involved until
he reached a piece where the term in question was at top level, so that
he could apply monotonicity straightforwardly. Then he reassembled the
sentence step by step. This is sound procedure, but it involves handling
formal parameters when one removes a quantifier to reach the required
piece of the sentence. Burley was easily confused by formal parameters
([18] p. 92f).
Unlike Burley, some authors really did want a monotonicity rule that
applied directly to terms below the top level. So they had to decide what
was the appropriate generalised definition of ‘distributed’. Aswe saw, ‘distributed’
is defined one way for the subject term and another way for the
predicate term. (In fact a correct statement of the definition for the predicate
works also for the subject; but this wasn’t realised until the 20th century.
Briefly, a term in a sentence is distributed if it occurs only negatively,
and undistributed if it occurs only positively. . . . ) Some logicians tried to
generalise the definition given for subject terms, while others tried to generalise
that given for predicate terms.
ThusDe Morgan in 1847 ([7] p. 114f) explained that a termallows downward
monotonicity if it is ‘used universally’ and upwards if it is ‘used particularly’
(i.e. existentially). Similar ideas can be found earlier in Buridan.
To make this seemto work in the cases where he wanted to apply it, he had
to invent quantifiers where there clearly weren’t any. He was trapped by
his persistent habit of preferring metaphors and analogies to precise statements.
In themid 14th century Richard Billingham ([33] p. 51) declared that we
16
can use upward monotonicity at positively occurring terms. This is correct,
and it used to impress me greatly before I realised that he was putting into
a neat form observations about syllogisms that were made already in the
Abbreviatio Montana from the late 12th century. We don’t know that he had
any intention of applying the rule to anything except predicates in syllogisms.
(See the reference and discussion at [18] p. 87.)
Leibniz did better: he allowed upward monotonicity at a term ‘in every
. . . affirmative propositionwhere the termoccurs as a predicate’ ([24] p. 38).
His examples suggest that he meant a term which occurs as predicate—
possibly of a subclause—in a sentence with no negations. The subclause
could be inside the scope of several quantifiers, universal or existential.
This definitely breaks the top-level restriction, but it is still very limited.
Also Leibniz gave no hint of how to prove its correctness;with complicated
patterns of quantifiers one can hardly rely on intuition to confirm it.
The natural 20th century response to all this is to combine the approaches
of Burley, Billingham and Leibniz. In a suitably regimented formof natural
language, one defineswhich occurrences of terms are positive in a sentence
and which are negative, by induction on the syntax. Then one introduces a
monotonicity rule along the lines:
(29)
Suppose A has only positive occurrences in _(A) and B has only
negative occurrences in (B). Then:
1. From ‘Every A is a B’ and _(A) we can infer _(B);
2. From ‘Every A is a B’ and (B) we can infer (A).
This works, and it wraps up the logic beautifully. Van Benthem [36] gives
some references and some examples.
8 Natural?
But now I want to raise a final question which is only incidentally about
traditional logic.
It’s true that the monotonicity rules of ‘natural logic’ are based on ideas
from traditional logic, and that traditional logic was done in natural language.
But aside from this link via traditional logic, I’m not convinced that
there is any direct connection between natural language and these rules.
Why should we think that this 20th century updating of monotonicity is
‘natural’ in any sense in which natural languages are ‘natural’? . . . It seems
to me that identifying occurrences at any significant depth in a sentence as
positive or negative just isn’t one of our inbuilt skills. I can quote at least
17
one traditional logician in my support: already in the 12th century John of
Salisbury suggested that if you want to use multiple nesting of negations,
you may need to carry around a calculating device to keep track of the
nesting ([18] p. 86).
It’s an empirical questionwhat kinds of reasoning people performsmoothly
and reliably, and what kinds they find ‘natural’. There already is some empirical
evidence out there. For example Oakhill et al. [30] p. 126f gave their
subjects the premises of some syllogisms and asked them to say what conclusions
follow, and found after the experiment that ‘there were too few
cases in which the correct conclusion was produced to warrant an analysis’;
how does this square with a blanket claim that syllogistic reasoning is
‘natural’? (I thank Jane Oakhill for calling this result tomy attention.) How
does this square with a blanket claim that monotonicity reasoning is ‘natural’?
In Oakhill’s case the substitutions are not even deep in the sentences.
There is experimental work to be done here. Let me make two observations
about possible experiments.
The first is that it might clear the air to compare the monotonicity rule
(29) with Frege’s rule (28). They are both rules that apply to occurrences
arbitrarily deep in a sentence. My prediction is that in terms of any reasonable
notion of naturalness (reliability, speed, confidence . . . ) Frege’s rule
will come out more natural than the monotonicity rule. I did make some
preliminary soundings among some experts in psycholinguistics and the
psychology of reasoning, to see whether any relevant work has been done
yet. Nobody could point me to any.
The second is that it really doesn’t make too much sense to ask what
kinds of reasoning are natural without discussing the context where the
reasoning is called for, and the intentions and expectations of the reasoner.
(See Stenning and Van Lambalgen [35] passim.) Our reasoning powers are
highly adaptable. I wouldn’t expect that nature follows Hilbert’s ideal, giving
us a small number of simple reasoning principles that allow us to generate
all the inferences that we need to make. Much more likely is that
nature equips us with a large number of relatively complex reasoning patterns,
corresponding to the lavishness of natural language and the variety
of situations in which we use it. I would expect that if we really do succeed
in identifying those reasoning patterns that are ‘natural’, we will find that
they form a hugely richer collection than the rather stunted list that came
down to us from Aristotle. . . .
18
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[3] Anicius Manlius Severinus Boethius, De Topicis Differentiis, translated
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[7] Augustus De Morgan, Formal Logic: The Calculus of Inference, Necessary
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19
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20
[30] J. V. Oakhill, P. N. Johnson-Laird and Alan Garnham, ‘Believability
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21