MATHEMATICAL
LOGIC
The main purpose of these logic resources is to
provide a variety of materials for relating the fundamental concepts of
contemporary mathematical logic to concepts that are more manifest. These
resources are written for anyone who wants to think about logic from a manifest
perspective. They are designed as a resource for self-directed personal study
by a student who has access to a mentor. You do not need any previous knowledge
of logic in order to use these units. If have already studied logic, or if you
are currently taking a logic course, these units can provide a more intuitive
grasp of the remote concepts you have encountered and an intuitive basis for
those you are about to encounter. For teachers of logic, these resources can
provide materials or ideas that they can easily adapt and use in their own
teaching.
The initial perspective papers below indicate
what is meant by a manifest approach to mathematical logic. Click to see part of the first paper.
Brief Initial Perspective on Math Logic This paper (or the one below) is a prerequisite to
all the other units.
Initial Perspective on Math Logic This paper is an expanded version of the brief
introduction paper. It is designed for anyone interested in having a broader
initial perspective
The Core Units The core units are named below and most are available
only as Microsoft Word files. Select any highlighted one to download. All of
these units will always be in a developmental stage, and discussion about any
of them would be highly appreciated.
Attribute Logic 0 Primary
Concepts For Attribute Structures
Attribute Logic 1 Informal
Propositional Reasoning For Attribute Structures
Attribute Logic 2 Formal
Propositional Theories For Attribute Structures
Relational Logic 0 Primary
Concepts For Relational Structures
Relational Logic 1 Informal
Propositional Reasoning For Relational Structures
Relational logic 2 Formal Propositional Theories For Relational
Structures (not yet available)
Operational Logic 0 Primary
Concepts For Mathematical Structures
Operational Logic 1 Informal
Propositional Reasoning For Operational Structures
Operational Logic 2 Formal Propositional Theories For Operational
Structures (not yet available)
Quantificational Logic 0 Primary Quantification Concepts
Quantificational Logic 1 Informal Quantification Reasoning For Various
Structures
Quantificational Logic 2 Formal Quantified Theories For Various Structures
(not yet available)
Using Logic Units This paper
gives a perspective on the approach used in these units and some advice on
using them.
There are also a number of reference files that
aren’t directly related to any particular unit.
A Manifest Approach to Mathematical Logic
This approach to logic makes use of some
physical tokens called attribute items. The items most often used are named and
pictured below.
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lbc:
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lbd:
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lrc:
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lrd: |
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sbc:
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sbd:
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src:
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srd: |
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lbc |
lbd |
lrc |
lrd |
sbc |
sbd |
src |
srd |
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l |
u |
l |
u |
l |
u |
l |
u |
Analytic Reasoning Our analysis of Situation 1 below makes explicit
information that was only implicit in the clues. Such analysis is called
analytic reasoning, and is one of the main tools of contemporary mathematics.
Mathematical logic is the study of the concepts and conceptual networks that
are used to study such reasoning. Reasoning that is not analytic is also
extensively used in the study of mathematics, but such reasoning is not
analyzed in mathematical logic, and will not be a topic in these units. Aside
from helping you in the study of mathematics, mathematical logic can help you
understand the relation of analytic reasoning to other types of reasoning and
other types of human endeavor. In fact one of the main achievements of
mathematical logic was to prove analytically that analytic reasoning was not
sufficient to derive the theory for any significant infinite mathematical
structure.
Situation 1 We have 4 clues about an item from BAS which is not
currently observable.
(1) If it is a
diamond then it is not blue.
(2) If it is a
circle then it is large.
(3) It is blue.
(4) It is either a
circle or a diamond.
Manifest Analysis Our dictionary gives the literal meaning of the word
‘manifest’ as "hit by hand’. It then goes on to say that to be manifest is
to be readily perceived by the senses. Thus we refer to the analysis below as
manifest. To determine the desired item merely remove those that do not satisfy
the clues. The remaining one is the item we want. I recommend starting with
(3), since it is easy to use. Note that given these tokens, (4) is redundant.
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lbc |
lbd |
lrc |
lrd |
sbc |
sbd |
src |
srd |
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Clue 3 eliminates the red items. |
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Clue 1 eliminates blue diamonds. |
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Clue 2 eliminates small circles. |
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Of course we could present this same solution
more compactly using names for instead of using tokens for these items.
Clue 3 eliminates
all non-blue items. Keep: lbc lbd sbc sbd
Clue 1 eliminates blue diamonds. Keep: lbc sbc
Clue 2 eliminates small circles. Keep: lbc
Deductive Analysis By (3) and (1), it cannot be a diamond. Thus by (4)
it must be a circle, and so by (2) it is also large. Combining this
information, the item is a large blue circle.
Manifest and Remote Reasoning Ordinary reasoning is often manifest, being guided by
situations providing fairly immediate feedback. Such reasoning is directed to
immediate purposes. Highly manifest reasoning includes reasoning which is
likely to occur when planting a garden, repairing a broken vase, building a
bookcase, playing chess, driving a car to some new destination. We also engage
in more remote reasoning, that is in the kind of reasoning in which our
attention is directed away from our present environment. Ordinary examples
include analyzing the strategy used in an earlier game of chess, predicting the
weather for the coming weekend, defending a proposed tax reform, or thinking
about causes of the Civil War. Even more remote, would be reasoning about
general strategies for chess, weather prediction theories, the general causes
of war. The deductive analysis of Situation 1 was more remote than the manifest
analysis, since we deduced the attributes of the item without tokens and without
even imagining a set from which it was taken.
One advantage of the deductive analysis used
for Situation 1 is that we could infer that the item was a large blue diamond
without looking at a representatives. We did not even need to know which other
items were available. This contrasts sharply with the manifest reasoning which
depends on knowing which items are candidates and is done in contexts which
implicitly supply us with at least part of the information we use. The main
advantage of manifest reasoning is that it provides direct feedback about what
we are doing, however for a situation involving many items, manifest reasoning
is tedious.
What seems remote to one person may seem fairly
close at hand to another. Those familiar with the concepts involved in
deductive reasoning find such reasoning remote only in the sense that it
involves very little feedback from their surroundings. Anyone less familiar
with these concepts might find deductive reasoning more remote. This may not
make the reasoning about a particular situation hard; however for most people
remoteness is one factor which increases difficulty. Perhaps our remote
reasoning abilities are an evolutionary extension of our manifest reasoning
abilities; but in any case, most people obtain a better grasp of remote
reasoning when they can relate it to manifest reasoning. Mathematical logic
studies the kind of reasoning which is about as remote from ordinary feedback
as we can imagine. In order to relate this reasoning to less remote reasoning,
this paper focuses on reasoning about information which can be analyzed by both
manifest and remote methods.
Formal Language Ordinary language uses words to focus on the content
of what we are saying. Mathematical language augment this by using concise
symbols that help us focus on the form our language rather than on its content.
A language composed exclusively of such symbols is called a formal language.
Formal language allows compact representations that focus on the aspects of the
information that are relevant to our reasoning. However symbols can be tedious
to read directly, so it is often useful to use ordinary language in reading
them. For example in ordinary arithmetic we usually read ‘1/2’ as ‘one half’
rather than ‘1 slash 2’. The logical symbols for formal language are Þ for ‘then’, Ø for ‘not’, & for ‘and’. Ú for ‘or’
Below is the remote analysis for Situation 1
using both formal language and ordinary language.
In addition to the logical symbols, we use ‘D’ for ‘it is a diamond’, ‘B’ for
‘it is blue’.
Thus ‘D Þ ØB’ is the clue ‘If the item is a diamond then it is
not blue’.
(1) D Þ ØB clue If it is a diamond then
it is not blue.
(2) C Þ L clue If it is a circle then it is large.
(3) B clue It
is blue.
(4) CÚ D clue It is either a circle or a diamond.
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(5) ØD (1)
(3) It is not a diamond.
(6) C (4) (5) It is a circle.
(7) L (2) (6) It is large.
(8) L&B&C (7) (3) (6)
It is a large blue circle.
Form vs Content The content of a statement is the information it is
intended to propose. The form is its structure as a sentence. The content of ‘CÚD’ is the information about the possible shapes for the item. Its form
is an alternation, with alternatives ‘C’ and ‘D’. Symbols can help us focus on
form and avoid being distracted by content. To infer (6) from (4) and (5), we
focus an alternation and a negation of the second alternative, ignoring the
content of ‘C’ and ‘D’. While symbols can be useful in making form manifest,
they can make content seem remote. So in this paper we give all deductions in
formal language to stress form and in ordinary language to stress content.
However part of becoming skillful at mathematical logic is learning when to
focus on form versus content.
Situation 2 Buying a Car Luke’s Parents need to buy a car from a local dealer.
Blue Dodges are not available. If they buy a Chevy then Luke will be angry.
They agree to buy a blue car. The only other dealer in town is a Chevy dealer.
There only option is to let Luke be angry and buy a blue Chevy. The premises
implicit in the situation are given below, followed by a deduction showing how
this option follow from these premises.
(1) D Þ ØB If
they buy a Dodge then they do not buy a blue car.
(2) C Þ L If they
buy a Chevy then Luke will be angry.
(3) B They will buy a
blue car.
(4) CÚD They
will buy a Chevy or a Dodge.
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(5) ØD (1) (3) They do not buy a Dodge
(6) C (4) (5) They do buy a Chevy
(7) L (2) (6) Luke will be angry
(8) L&B&C (7) (3) (6) Luke
will be angry and they will buy a blue Chevy
Perspective Focusing on form is an extremely important aspect of
mathematical logic. Situation 2 involves information of the same form as that
given in Situation 1. Thus the deductive solutions for these two situations is
identical in form. It is the interpretation of the symbols, rather than the
form of the reasoning, that gives a different content to the conclusion. Using
formal language stresses the fact that the deduction depends on form rather
than content.
For More Download One Of The Initial Perspective Papers