=Paper=
{{Paper
|id=Vol-1811/paper8
|storemode=property
|title=Reasoning with Generics and Induction
|pdfUrl=https://ceur-ws.org/Vol-1811/paper8.pdf
|volume=Vol-1811
|authors=Liying Zhang
|dblpUrl=https://dblp.org/rec/conf/clar/Zhang16
}}
==Reasoning with Generics and Induction==
https://ceur-ws.org/Vol-1811/paper8.pdf
Reasoning with generics and induction∗
Liying Zhang
The Institute of Modern Logic, Central University of
Finance and Economics Beijing, 100081
clearliying@126.com
Abstract
Based on the results of reasoning with generics, this paper attempts to
clarify the relationship between generic reasoning and inductive reason-
ing. First, to capture reasoning with generics, logics for obtaining inter-
mediate conclusions are provided; then, the priority order of the subsets
of the premise set are introduced to eliminate the contradictions or in-
compatibility generated by intermediate conclusions. Through these
filters, the final conclusions are obtained. There are different priority
rules on subsets of a premise set corresponding to three kinds of generic
reasoning: reasoning with factual sentences (from generic sentences),
reasoning by deduction and reasoning by induction. If these three kinds
of reasoning are combined, together they can characterise inductive
reasoning. After reviewing how scholars’ have previously interpreted
inductive reasoning, this paper concludes that there is another way
to interpret inductive reasoning with generic reasoning, parallel to the
probability method.
1 Reasoning with generics
Generic sentences, for instance, ‘birds fly’ or ‘ducks lay eggs’, express rules or laws. Unlike universal sentences,
generic sentences tolerate exceptions. Even when there is no positive example in the real world (for example,
‘unicorns have one horn’), we sometimes accept a generic sentence, which makes generic sentences intensional.
Because reasoning with generics tolerates exceptions, it is non-monotonic.
1.1 Interpretation of generic sentences
This paper restricts its scope to i-generics1 and employs the semantics from Mao and Zhou (2003) for generics:
The canonical form of a generic sentence with subject-predicate (SP) structure is ‘(normal S) (normally P)’. The
duty of ‘normal’ is to choose a set of (normal) objects (the term’s extension) for every possible world based on
both a subject sense and predicate sense. ‘Normally’ is used to choose the normal situation.
The sense of the term ‘S’ can be expressed as λxS by λ-expression. From a semantic aspect, a sense is a
function expressed as s, and ‘normal’ is a binary function N (s1 , s2 ) called the normal function. s1 , s2 respectively
∗ This work was supported by “the Fundamental Research Funds for the Central Universities”.
c by the paper’s authors. Copying permitted for private and academic purposes.
Copyright ⃝
In: T. Ågotnes, B. Liao, Y.N. Wang (eds.): Proceedings of the first Chinese Conference on Logic and Argumentation (CLAR 2016),
Hangzhou, China, 2-3 April 2016, published at http://ceur-ws.org
1 Krifka et al. (1995) distinguish between d-generic and i-generic sentences. ‘Dinosaurs are extinct’ is an example of d-generic
sentence, since a type of animal can be extinct but individual members of that type can only be dead. ‘Birds fly’ is a famous example
of an i-generic sentence.
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represent the subject sense and the predicate sense. This means that the normal subject’s sense is ascertained
through the subject sense and the predicate sense. There are two basic restrictive conditions on s. First, for any
sense s1 , s2 , N (s1 , s2 ) ⊆ s1 ; i.e., the normal subject sense (chosen by N ) is included in the subject sense. Second,
for any sense s1 , s2 , N (s1 , s2 ) = N (s1 , s∼ 2 ); i.e., the normal subject sense has a relationship with the predicate
sense regardless of whether the predicate is affirmative or negative. Due to these two restrictions, the drowning
problem2 can be solved, and a cyclical definition caused by choosing a normal subject based on a predicate can
be avoided.
‘Normally’ is represented by ‘>’, which comes from conditional logic used by Pelletier and Asher (1997),
improved by Mao and Zhou (2003). Based on possible worlds semantics, the intension of a generic sentence can
be expressed, and a sentence like ‘unicorns have one horn’ can be interpreted.
Based on the points above, ‘(normal S) (normally P)’ can be interpreted further: for any object x, if x
is normal S regarding P or not P, then, normally, x is P. According to this analysis, a generic sentence SP’s
formalization is ∀x(N (λxSx, λxP x)x > P x), and G(Sx; P x).3 For short, G is called a generic quantifier.
There is also an important interpretation of generics by Cohen (1999) using probability techniques; however,
that method through probability techniques is doomed to be inadequate for interpreting intensional generics.
To ease readers and make the structure of this paper clearer, in section 1.1, 1.2, 1.3, I only give the main ideas
and main conclusions. The technique details are left in section 1.4.
1.2 Reasoning with generics
When reasoning with generics, there are two questions: what can be obtained from generic sentences and how
are generic sentences obtained? These two parts are mixed since from generic sentences and factual sentences
we can obtain both generic sentences and factual sentences. For historical and technical reasons, in this paper
reasoning with generics is divided into two different parts (not a partition): conclusions are factual propositions
or conclusions are generic sentences. The former reasons that generic sentences are common sense, as they
reach conclusions about concrete facts under concrete circumstances. For instance, from ‘birds fly’ and ‘Tweety
is a bird ’, the conclusion is ‘Tweety flies’. The latter reasons from some propositions and reaches general
conclusions represented by generic sentences. This kind of reasoning involves both deduction (in which premises
may contain generic sentences) and induction (in which premises may not contain generic sentences). At first
glance, induction seems more important and interesting; however, in daily life, reasoning by deduction with
generics is more common. For instance, from ‘sparrows are birds’ and ‘birds have feathers’ we get ‘sparrows have
feathers’, one of the thousands of inferences leading to a generic sentence by deduction. Because there is an
enormous amount of knowledge and belief in our brains represented by generic sentences, reasoning with pure
induction in our daily lives approaches an ideal.
Based on the semantics provided in section 1.1., there are different logics corresponding to the different parts of
generic reasoning. Different from classical logic where the logics can be used to capture the deductive reasoning,
in the paper, the logics captured partial reasoning with generics; the logics within the priority ordering on the
premise set will capture the whole reasoning process. This means that logics are used to capture the inferences
contained in each reasoning step. When these inferences’ conclusions are combined, some partial conclusions
may lead to disorder and even conflict. Then, the priority ordering of premise sets work, drawing back some
‘conclusions’ and leaving the final conclusions.
1.2.1 Reasoning with generics when conclusions are factual propositions
A famous example of reasoning with generics in which the conclusions are factual propositions is ‘birds fly’ and
‘Tweety is a bird ’, therefore ‘Tweety flies’. In Mao (2003), logics M and G are constructed to address inferences
that reach factual conclusions from general premises. In M, the default implication MP inference is considered:
(α ∧ (α > β)) > β. Beginning with this formula, G, as the quantifier extension has an important new axiom
GU: Gx(α; β) > ∀x(α > β). Its intuition is: normally, if ‘(normal S) (normally P)’, then ‘S normally P’. With
GU, Mao (2003) transforms the generic quantifier into a universal quantifier. In the substitution example, the
universal quantifier can be eliminated, and from the default MP rule implication, inferred from the generic
proposition and other specific conditions, we reach the factual conclusion accordingly.
2 Both ‘peacocks lay eggs’ and ‘peacocks have colorful feathers’ are true generic sentences, but only female peacocks lay eggs and
only male peacocks have colorful feathers.
3 Sometimes it is written as Gx(α; β).
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1.2.2 Obtaining generic sentences mainly by deduction
Based on the similar interpretation of generics, there is a logic GD for obtaining generic sentences by deduction.
Here, the goal is for the inferences to lead to generic sentences, so we don’t need to consider the transformation
from generic quantifier to universal quantifier. In the formal system, we don’t add the axiom GU; instead, the
relationship between generic sentences is considered by studying the normal function N . We will expand and
enrich N , which has been discussed above, and obtain some inference rules about generic sentences to obtain,
for example, the inference for GAG.
There is an important axiom in GD :
NAM ∀y(α → γ)(y/x) → ∀y(N (λxα, λxβ)y → N (λxγ, λxβ)y)
The corresponding semantic condition (subject-monotonic) is:
For any s1 , s2 , s3 ∈ P(D)W , any w ∈ W , if s1 (w) ⊆ s2 (w), then N (s1 , s3 )(w) ⊆ N (s2 , s3 )(w).
The intuition is that, if in the same possible world w the object sets corresponding to s1 are included in the
object sets corresponding to s2 , then for the same predicate sense s3 , in w, the object sets according to normal
s1 are included in the object sets corresponding to s2 . For instance, in the actual world, there is: ‘sparrows are
birds’, ‘birds fly’, ‘sparrows fly’. Corresponding to ‘fly’ in the actual world, ‘normal sparrow(s)’ are also normal
bird(s)’.
1.2.3 Obtaining generic sentences mainly by induction
If there is no generic sentence in the premise set then to obtain a generic sentence the only option is simple
enumeration. Does that sound too simple? That is it. If there is something more complex, then the premise
set must contain higher-order knowledge or belief, i.e., generic sentences, then the inferences come down to the
scope of GD .
1.3 Priority orders on premise set
There are logics for different parts of generic reasoning. However, this is not enough, for example, based on the
system GD . From {sparrows are birds, birds fly} we get sparrows fly; also, from {penguins are birds, birds fly},
we get penguins fly. But what we can get from {penguins are birds, birds fly, penguins don’t fly}? Introducing
priority orders on a premise set is a feasible and reasonable way to deal with the potential disorder.
The aim is to create an order for the subsets of the premise set. The order among subsets can be defined by
the order among the formulas in the premise set. So, we first define the strict partial order among the formulas
in the premise set. Then, we present normal rules for priority order and special rules for different kinds of
reasoning. As an example, details of the orders on GD are in section 1.4. Next, I will present some special orders
for different parts of reasoning.
1.3.1 Special priority orders for reasoning obtaining factual propositions
Zhou and Mao (2004) have placed priority order on the premise set and shown that their definition can pass
through benchmark examples such as Nixon Diamond, Penguin Principle, etc.
(1) Factual priority: if α > β, ¬β ∈ Γ, then ¬β ≻G α > β,
(2) Sub-category default priority: if α > β ∈ Γ, and α > γ, β > δ ∈ Γ, then α > γ ≻G β > δ.
1.3.2 Special priority orders for reasoning obtaining generics by deduction
Moving from ‘birds fly’ and ‘sparrows are birds’ to ‘sparrows fly’ (GAG type) is a representative instance of
reasoning obtaining generics by deduction. There are two special rules for this kind of reasoning: sub-category
generics priority and generics priority. Our definition can plainly pass through the GAG.
(1) For any formula α, β, γ, δ, if ∀x(α → β) ∈ Γ and Gx(α; γ), Gx(β; δ) ∈ Γ, then Gx(α; γ) ≻G Gx(β; δ)
(sub-category generics priority);
(2) For any formula α, β, if ∀x(α → β) ∈ Γ, and Gx(α; γ) ∈ Γ, then Gx(α; γ) ≻G ∀x(α → β). (generics
priority)
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1.3.3 Special priority orders for reasoning obtaining generics by induction
Now we reach the most difficult one: obtaining generic sentences by induction. As I say in 1.2.3., simple
enumeration is the way to obtain generics by induction. But why, even there are many examples of white swans,
do we not accept ‘swans are white’ to be a true generic sentence by simple enumeration? One explanation
is that the premise contains a generic sentence (or the propositions to reach it) such as ‘poultry feathers are
multiple colours’. This can be obtained by simple enumeration from ‘ducks’ feathers are multiple colours’,
‘parrots’ feathers are multiple colours’, ‘domestic gooses’ feathers are multiple colours’. But why we accept this
conclusion rather than ‘swans are white’ ? They all come from simple enumeration. The answer is again the
ordering of the premise sets. Here, for inferences obtaining generic sentences by induction, the priority order is
category generics priority.4
The special priority order for obtaining generic sentences by induction is:
For any formula α, β, γ, δ, if ∀x(α → β) ∈ Γ and Gx(α; γ), Gx(β; δ) ∈ Γ, then Gx(β; δ) ≻G Gx(α; γ) (category
generics priority).
This has provided an overview of the work on reasoning with generics. Below, we begin to discuss induction
and the relationship between it and reasoning with generics.
1.4 Formalization
1.4.1 Language and semantics
Formal language LG contains a denumerable set of individual variables (Var), a denumerable set of individual
constants and a denumerable set of predicate variables (n > 0), a sentential constant ⊥, a truth-functional
operator →, universal quantifier ∀, parenthesis ), (. These are the symbols of first-order language. Based on
them, we add new symbols: >, λ, N .
We use x, y, z, etc., for any individual variable, c, d, e, etc., for any individual constant, P, Q etc., for any
predicate, t, etc., for any term, α, β, γ, ϕ, etc., for any formula.
α ::= ⊥|P t|α → β|∀x α|α > β|(λxα)t|N (λxα, λxβ)t|
We introduce symbols in LG such as ⊤, ¬, ∧, ∨, ↔, ∃, following the usual definitions.
Gx(α; β) =df ∀x(N (λxα, λxβ)x > β)
Definition 1. W is a non-empty set. Function !: P(W ) × P(W ) → P(W ) is a set selection function on W ,
if for any X, Y, Z, X ′ , Y ′ ⊆ W, ! satisfies:
(1) If X ⊆ X ′ , then !(X, Y ) ⊆ !(X ′ , Y ).
(2) If !({w}, Y ) ⊆ Z for every w ∈ X, then !(X, Y ) ⊆ Z.
(3) If !(X, Y ) ⊆ Z, then !(W, X ∩ Y ) ⊆ Z.
(4) !(X, Y ) ⊆ Y .
(5) If !(X, Y ) ⊆ Z and !(X, Y ′ ) ⊆ Z, then !(X, Y ∪ Y ′ ) ⊆ Z.
(6) If !(X, Y ) ⊆ Y ′ and !(X, Y ′ ) ⊆ Z, then !(X, Y ) ⊆ Z.
Definition 2. W and D are non-empty sets. S = P(D)W , For any s1 , s2 ∈ S,
(1) s1 ⊆ s2 iff. s1 (w) ⊆ s2 (w) for all w ∈ W .
(2) s1 = s2 iff. s1 (w) ⊆ s2 (w) and s2 (w) ⊆ s1 (w) for all w ∈ W .
(3) s1 = s∼ ∼ ∼
2 iff. s1 (w) = (s2 (w)) for all w ∈ W . (A is the supplementary set of A.)
(4) For each w ∈ W, (s1 ∪ s2 )(w) = s1 (w) ∪ s2 (w), (s1 ∩ s2 )(w) = s1 (w) ∩ s2 (w).
(5) Vacant sense s⊤ , s⊤ (w)=D for all w ∈ W .
4 In fact, the generics priority in 1.3.2., above, also works through this process. There are more details in Zhang (2009).
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(6) Full sense s⊥ , s⊥ (w) = ∅ for all w ∈ W .
Definition 3. W and D are non-empty sets, S = P(D)W . N : S × S → S is a normal object selection function
.iff. For all s1 , s2 ∈ S, N satisfies:
(1) N (s1 , s2 ) ⊆ s1 , and
(2) N (s1 , s2 ) = N (s1 , s∼
2 ).
Definition 4. W and D are non-empty sets, S = P(D)W . [, ] : D × S → P(W ) is a function that satisfies:
for all d ∈ D, s ∈ S, [d, s] = {w ∈ W : d ∈ s(w)}.
Definition 5. Quadruple F = is a frame; if W and D are non-empty sets, then N is a normal
object selection function on S and ! is a set selection function on W .
Definition 6. F = is a frame η is interpretation on F , .iff.
(1) For each individual constant c ∈ C, η(c) ∈ D.
(2) For each predicate symbol P, η(P, w) ∈ D.
Definition 7. An ordered pair S = is a structure if F is a frame and η is a interpretation function on
F.
Definition 8. An ordered pair M = is a model; if S is a structure σ is a mapping function from Var
to D. We also call σ an assignment.
We use FM , SM to denote the frame and structure of M , using WM , DM , ηM , σM , etc., to spell out in detail
components of the model M . Given term t, the interpretation of t in M is denoted by tM (the usual definition).
Definition 9. For each formula ϕ, the symbol ∥ϕ∥M is used to stand for the set of worlds in M in which ϕ is
true, satisfying:
(1) ∥⊥∥M = ∅
(2) ∥P t∥M = {w ∈ W : ∈ ηM (P, w)}
(3) ∥α → β∥M = (W − ∥α∥M ) ∪ ∥β∥M
(4) ∥α > β∥M = ∪{X ⊆ W : !(X, ∥α∥M ) ⊆ ∥β∥M }
(5) ∥∀xα∥M = {w ∈ W : for each d ∈ DM , w ∈ ∥α∥M (d/x) }y
(6) ∥N (λxα, λxβ)t∥M = {w ∈ W : tM ∈ N ((λxα)M , (λxβ)M )(w)}((λxα)M ∈ S, is a mapping satisfying, for
each w ∈ W, (λxα)M (w) = {d ∈ DM : w ∈ ∥α∥M (d/x) }).
Proposition 1. For any variable x, any formulae α, β, any model M :
(1) (λx¬α)M = ((λxα)M )∼
(2) (λx(α ∨ β))M = (λxα)M ∪ (λxβ)M
(3) (λx(α ∧ β))M = (λxα)M ∩ (λxβ)M
(4) (λx(α → β))M = ((λxα)M )∼ ∪ (λxβ)M
Proposition 2. For any M, ∥α(y/x)∥M (d/y) = [dM , (λxα)M ]
Proposition 3. For any model M , any w ∈ W, w ∈ ∥∀y(β → γ)(y/x)∥M .iff. (λxβ)M (w) ⊆ (λxγ)M (w).
Corollary 1. For any model M , any w ∈ W, w ∈ ∥∀y(β ↔ γ)(y/x)∥M .iff. (λxβ)M (w) = (λxγ)M (w).
Definition 10. Let M be a model and α be a formula, X ⊆ WM , X ̸= ∅. α is true at the set X (written as
M |=X α) iff. X ⊆ ∥α∥M . When X = {w}, we also say that α is true at w. When X = WM , we say that α is
valid at M ; we use M |= α to denote it.
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Definition 11. Given a formula α, we say that α is valid (written as |= α) iff. M |= α for all M .
Theorem 1. The formulas listed below are valid.
(1) (∀x(α > β) → (α > ∀xβ)) (x is not a free variable in α)
(2) ∀y(N (λxα, λxβ)y → α(y/x))
(3) ∀y(N (λxα, λxβ)y → N (λxα, λx¬β)y)
(4) ∀x(α → β) → ∀x(α > β) "
Theorem 2. (1) If M |= β ↔ γ for all model M , then, M |= ∀y(N (λxα, λxβ)y ↔ ∀y(N (λxα, λxγ)y) for all
model M .
(2) If M |= β ↔ γ for all model M , then, M |= ∀y(N (λxβ, λxα)y ↔ N (λxγ, λxα)y) for all model M . "
Definition 12. A frame is a subject-monotonic frame iff. It satisfies: for any s1 , s2 , s3 ∈ S, if
s1 (w) ⊆ s2 (w), then N (s1 , s3 )(w) ⊆ N (s2 , s3 )(w). A model is a subject-monotonic model
iff. is a subject-monotonic frame.
Theorem 3. For any subject monotonic model M, M |= ∀y(α → γ)(y/x) → ∀y(N (λxα, λxβ)y → N (λxγ,
λxβ)y)).
1.4.2 Logic GD for partial inference
Axiom schemata:
T all tautologies
∀− ∀xα → α(x/t)
∀→ ∀x(α → β) → (∀xα → ∀xβ)
>BF ∀x(α > β) → (α > ∀xβ) (x is not a free variable in α)
CK (α > (β → γ)) → ((α > β) → (α > γ))
>MP (α ∧ (α > β)) > β
TRAN (α > β) → ((β > γ) → (α > γ))
AD (α > γ) ∧ (β > γ) → (α ∨ β > γ)
IC ∀x(α → β) → ∀x(α > β)
N ∀y(N (λxα, λxβ)y → α(y/x))
N¬ ∀y(N (λxα, λxβ)y → N (λxα, λx¬β)y)
NAM ∀y(α → γ)(y/x) → ∀y(N (λxα, λxβ)y → N (λxγ, λxβ)y))
Rules of inference:
MP; ∀+ ;
RCEA From β ↔ γ, infer (β > α) ↔ (γ > α);
RN From β, infer α > β;
RM From α > β, infer (α ∧ γ) > β;
RNEA From β ↔ γ, infer ∀y(N (λxα, λxβ)y ↔ N (λxα, λxγ)y);
RNEC From β ↔ γ, infer ∀y(N (λxβ, λxα)y ↔ N (λxγ, λxα)y).
Some theorems and derived rules:
ID α>α
AM ∀x(α → β) → ∀x((β > γ) → (α > γ))
CI ∀x(α > β) ∧ ∀x(β → γ) → ∀x(α > γ)
CM (α > β) → ((α ∧ γ) > β)
CR (α > β ∧ γ) → ((α > β) ∧ (α > γ))
CC ((α > β) ∧ (α > γ)) → (α > β ∧ γ)
ThMN 1 (α > ⊥) → (α > β)
RN From β, infer α > β;
RCEA From β ↔ γ, infer (β > α) ↔ (γ > α);
RCEC From β ↔ γ, infer (α > β) ↔ (α > γ);
RIC From α → (β > γ), infer (α ∧ β > γ);
RCK From (β1 ∧ . . . ∧βn ) → β, infer (α > β1 ) ∧ . . . ∧ (α > βn ) → (α > β); (n ≥ 1);
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RCI From α > β, β → γ, infer α > γ;
REQ β is a sub-formula of α, from β ↔ γ and α infer α[γ/β].
Some theorems and derived rules about generic sentences:
T hG D 1 Gx(α; α)
T hG D 2 Gx(α ∧ β; α)
T hG D 3 ∀y(N (λxα, λxβ)y ↔ (N (λxα, λx¬β)y)
T hG D 4 ∀x(α > β) → Gx(α; β)
T hG D 5 ∀x(α → β) → Gx(α; β)
T hG D 6 ∀x(α → β) → (Gx(β; γ) → Gx(α; γ))
T hG D 7 Gx(α; β) → Gx(α ∧ γ; β)
T hG D 8 Gx(α ∨ β; γ) → Gx(α; γ)
RGN From β, infer Gx(α; β).
RGIC From α → Gx(β; γ), infer Gx(α ∧ β; γ).
By theorems 1, 2, and 3, the system GD is sound. The completeness of GD can be proved by a variant of the
canonical model method, the details of which can be found in Zhang (2005).
1.4.3 Logic GD for partial inference
1.4.3.1. The general priority order
Definition 13. Γ is any formula set. ≻ is a (general)priority order, iff., ≻ is a 2-ary relation on Γ that satisfies
(strictly partial order): for any formula α, β, γ ∈ Γ,
(1) α ̸ ≻α (not α ≻ α);
(2) if α ≻ β and β ≻ γ, then α ≻ γ.
Reasonability
Definition 14. Let Φ is any formula set. Φ is reasonable, iff., the following two conditions do not occur:
(1) There is a formula α, α ∈ Φ and ¬α ∈ Φ;
(2) There is a formula Gx(α; β), Gx(α; β) ∈ Φ and Gx(α; ¬β) ∈ Φ.
Definition 15. Let ∆ = {α1 , α2 , . . . , αn } is any finite formula set.
!
α1 ∧ α2 ∧ . . . ∧ αn , n ≥ 1;
∧∆ =
⊥, if not.
Definition 16. Let ∆ is any finite formula set, S(L) is a formal system of logic L.
(1) Cn(∆) is L-consequence set of ∆, if Cn(∆) = {α : ⊢ S(L) ∧ ∆ → α};
(2) CN (∆) is reasonable L-consequence set of ∆, if
!
Cn(∆), if Cn(∆) is reasonable,
CN (∆) =
∅, if not.
(3) For any α ∈ CN (∆), ∆ is L-premise set of α. If for any ∆’s proper subset ∆′ , α ∈/ CN (∆′ ), then ∆ is α’s
minimal L-premise set. If there is no confusion, we will abbreviate L-premise set as premise set.
Definition 17. < Γ, ≻> | ∼S(L) Gx(α; β), iff.,
(1) There is ∆ ⊆ Γ, ∆ is the minimal premise set of Gx(α; β), and
(2) For any Λ ⊆ Γ, if Λ is the minimal premise set of ¬Gx(α; β), then ∆ > Λ, and
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(3) For any Λ ⊆ Γ, if Λ is the minimal premise set of Gx(α; ¬β), then ∆ > Λ.
Definition 18. Γ is any formula set. CN (Γ) is the (generic) consequence set under priority order ≻, if CN (Γ) =
{Gx(α; β) :< Γ, ≻> | ∼S(L) Gx(α; β)}.
Proposition 4. Γ is any formula set. ≻ is the priority order on Γ. CN (Γ) is reasonable.
Let Γ is any formula set, for any finite formula set ∆ ⊆ Γ. CN (∆) is a partial conclusion set reasoning from
the premise set Γ. CN (Γ) is the final-conclusion set reasoning (generic sentences) from the premise set.
1.4.3.2. Special priority order for getting generic sentences by deduction
Definition 19. Let Γ is any formula set, ≻G is the G-priority order on Γ, iff., ≻G is the general priority order
on Γ, which satisfies:
(1) For any formula α, β, γ, δ, if ∀x(α → β) ∈ Γ, and Gx(α; γ), Gx(β; δ) ∈ Γ, then Gx(α; γ) ≻G Gx(β; δ)
(sub-category generics priority);
(2) For any formula α, β, if ∀x(α → β) ∈ Γ, and Gx(α; γ) ∈ Γ, then Gx(α; γ) ≻G ∀x(α → β) (generics
priority).
From ≻G , we can obtain the strict priority order on P(Γ) (denoted by >G ). From definitions 18 and definition
19, there is:
< Γ, ≻G > | ∼G Gx(α; β)
This is the reasoning from Γ to Gx (α; β), which obtains generic sentences mainly through deduction, based
on the logic G and the G-priority order on the premise set. If let G is GD , then we can begin the GAG-type
reasoning.
2 Quick review of the history of inductive reasoning
2.1 Induction
What is inductive reasoning? There is a definition on Wikipedia5 : ‘The premises of an inductive logic argument
indicate some degree of support (inductive probability) for the conclusion but do not entail it; that is, they
suggest truth but do not ensure it’. Wikipedia also says: ‘though many dictionaries define inductive reasoning
as reasoning that derives general principles from specific observations, this usage is out-dated’.
Now is the time to add some new ideas to this definition.
What is inductive reasoning? From a premise, we reach a conclusion only with some degree of support, which
means that conclusions can be changed. This is non-monotonic reasoning.
Why do researchers use the qualifier ‘degree’ to define inductive reasoning? Let us review the history. There
are two periods in the history of inductive reasoning: the classical period and the modern period. In the classical
period, there are two representative researchers: Francis Bacon (1561–1626)and John Stuart Mill (1806–1873)
tried to find methods to reach certain conclusions by induction. But in the modern period, started by John
Maynard Keynes (1883–1946) who identified the first inductive logic, researchers no longer think that conclusions
obtained by induction are certain. Based on the development of classic probability theory, researchers employ
probability to express that uncertainty. Today, this idea and methodology are still in use. Due to the popular
use of the qualifier ‘degree’, the concept of probability is employed to define uncertainty. However, probability
is only one way to express uncertainty.
2.2 Core issues of inductive reasoning
Now, let us discuss the core issues of inductive reasoning. First, we express them in common usage.
1. How is the initial probability obtained?6
2. The logic system usually (after the initial probabilities) contains the axioms or rules for how to calculate
probability.
3. Reasoning categories: inductive reasoning is used to reach conclusions about individuals; inductive reasoning
is used to reach conclusions about classes.
5 See in https://en.wikipedia.org/wiki/Inductive reasoning.
6 Some researchers, like Keynes, didn’t provide their method for obtaining the initial probability.
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Table 1 A brief review of the study on inductive reasoning
Periods Representative researchers Problem concerned Technique used to deal with induction
Classical: Bacon, F. (1561–1626), How to get certain - Bacon’s three tables - Mill’s four
Mill, J. S. (1806–1873) conclusions by in- canons
duction.
Modern: Keynes, J. M. (1883–1946) How to get - Proposition logic + probability - First
uncertain conclu- calculation of probability. - No method
sions by induction. for obtaining initial probability.
Reichenbach, H. (1891– - Predicate logic + probability - Fre-
1953) quency probability, obtain the initial
probability by weight, weight posit
Carnap, R. (1891–1970) Modal logic + probability
Cohen, L. J.
Burks, A. W.
...
3 Comparing reasoning with generics and induction
We can now compare the main problems concerning inductive reasoning and reasoning with generics. If uncertain
conclusions are represented as generic sentences, the problem above can be translated as follows:
1. How to obtain the generic sentence by induction.
2. Logic M, G, GD , . . .
3. Reasoning categories: the conclusion is a factual proposition; obtaining generic sentences is done mainly by
deduction.
If we employ Reichenbach (1971)’s theory, things may become clearer. Reichenbach distinguished induction in
primitive knowledge (primitive induction) and induction in advanced knowledge (advanced induction). Advanced
induction is inductive reasoning with ‘weights’.
In our theory, there is simple enumeration and inductive reasoning with weights: obtaining generic sentences
mainly by induction and obtaining generic sentences mainly by deduction. The conclusion is a factual proposition.
4 Reasoning with generics versus probability
Sometimes we have to make a decision quickly, such as a weather forecast or a decision about an emergency, fire,
or earthquake. This kind of reasoning is not within the scope of this study, and in my opinion, probability is the
most helpful and practical way for that type type of inductive reasoning. However, what we are concerned with
in this paper is reasoning about knowledge and beliefs, namely generic sentences. Table 2 compares these two
ways of reasoning within the scope of this study.
Table 2 Comparing reasoning with generics and induction
Generics Probability
Expressiveness intensive extensive
Intuition nature
Scope default reasoning, characterises human statistics, optimal strategy
thinking process
√
Application ?
5 Conclusion
Now, the conclusion is reached naturally. With generic reasoning, there is another way to interpret inductive
reasoning: a method parallel to the probability method.
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