E. M. Segal:
Primer on Logic
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V. Predicate logics (predicate calculus)
Logics which represent individuals, and predicates and
relations. Usually they also contain variables and quantifiers.
Argument:
Second concept called argument (the first being a set of premises leading
to a conclusion). This refers to a component of a proposition or sentence
in a predicate logic. It represents an individual thing. It is something
like a noun or a pronoun. Arguments tend to be represented by lower case
letters in formal logics.
Predicate: something
like a verb or an adjective. It often represents a property, relation or
action. When a predicate is combined with the correct number of arguments
it is a simple sentence or proposition. Often a predicate is represented
symbolically by a capital letter. A given predicate must have a predetermined
number of arguments associated with it. eg. (1) Red, tall, smart are one
argument predicates. "Tb" might mean "The boy is tall" or "The tall boy"
(2) larger than, hit, kissed, are two argument predicates Kbg might mean
"The boy kissed the girl." In multiple argument predicates, the order of
the arguments determines their role in the predicate relation. Lab might
mean "A is larger than B." It could not also mean "B is larger than A."
(3) there are a few three and four place predicates as well, e.g. give,
put. "John gave the book to Mary."
Quantifiers and variables:
One might say something like "Some people are tall."
This might be represented as (($ x)(Px&Tx).
($ x), the existential
operator, means that something represented
by the variable x exists. By itself it is an incomplete expression. Px
and Tx, which are said to be in the scope of the quantifier, say
'x is P' (x is a person), and 'x is T' (x is tall). Neither Px nor Tx are
complete propositions because without being quantified they have no truth
value. You cannot decide whether x is a person until you know what x
refers to. However "There exists an x such that x is a person" is true
if at least one person exists. "There exists an x such that x is a person
and x is tall" is true if there is at least one tall person. Thus the quantified
expression is a proposition.
The sentence "All men are mortal" is represented using
the universal operator ("
x) or often (x). Logicians identified this proposition
as a conditional: (x)(Man x É
Mortalx ). This is usually expressed as
"For all x, if x is a man then x is mortal."
The two propositional representations of the sentences
"Some people are tall" and "All men are mortal" may not look like the English
sentences, but if you think about it you may see that they would be true
in exactly the same circumstances, thus the English sentences and the formal
notation represent the same propositions.
For your information, there are other logics,
some called modal logics, have usually
been about necessity and possibility; but may be about time or space or
beliefs, etc. These logics often are not strictly truth functional. That
is the truth of a whole proposition may not be simple function of the truth
of its parts. Think about this: If a proposition is possible it may either
be true or false John is tall. If it is necessary, it must be true
Either Spot is a dog or Spot is not a dog. If it is not possible,
it must be false, The cat is both alive and not alive. And if it
is not necessary, it may be true or false; Sosa will hit 70 homeruns
in 1998. Formal Logic requires a formal representational system, a
set of axioms, and rules of derivation. It does not require strict truth
functionality.
VI. Logical derivation.
A widespread AI, Cognitive Psychology, and Cognitive
Science principle is that there are procedures which can be implemented
on computers which represent the way that people think. These involve applying
logic type rules to a formal representation base. The psychological study
of reasoning in this case is the discovery of the form of the representation
and the rules of inference that people use when they reason.
a. Incoming information is
transformed into a symbolic representation of the implied proposition.
There are certain formal ways in which propositions are interrelated and
conclusions are derived. These are implemented by the application of logic-like
rules. In information processing systems they are the application
of efficient procedures which represent algorithms or heuristics.
b. Logic (and other) problems are
solved by the use of formally valid derivations on symbolic representations;
apply logical rules directly to 'sentences' which represent propositions.
Rules of inference include such as modus ponens, modus tollens, and hypothetical
syllogism.
c. Errors occur when a) there is an error
in the representation, b) the derivation is not completed, or c) the
problem solver applies the wrong rule.
A problem of application is: it is not always
clear how to get from premises to representations, nor for some problems
what the derivation rules are. The basic ideas for applying rules ties
this process into some of those in computation. Three relevant concepts
are:
Effective procedure--a
procedure which transforms a form (e.g. proposition) into another one in
a well specified way. The concept of 'effective procedure' is one of the
more important concepts in the symbolic sciences, and one which is needed
at least on an informal basis in order to work within any cognitive science.
"An effective procedure is a finite, unambiguous description of
a finite set of operations. The operations must be effective in the sense
that there is a strictly mechanical procedure for completing them"
Algorithm--a sequence
of effective procedures that is guaranteed to solve a problem. or produce
a valid outcome.
Heuristic--an effective
procedure that is likely to solve a problem relatively quickly.
There are times when the goal of identifying an effective
procedure is not reached but the term heuristic is still applied. Heuristics
are often called 'rules of thumb.'
VII. Wason's Selection task
1. If a card has a vowel on one side, it has an even
number on the other <E, K, 4, 7>*
Try to set this up according to formal rules of symbolic
logic.
Other versions
2. Every time I go to Manchester, I go by car <Manchester,
Leeds, Car, Train>
3. If the envelope is sealed it needs a 5 c stamp. <
5c, 3c, sealed, unsealed>
4. If someone is drinking beer, s/he must be over 21
<beer, coke, 25, 17>
5. Every time I eat haddock, I drink gin <haddock,
cod, gin, scotch>
6. If a student is to be assigned to Grover High, then
that student must live in Grover City.
<Grover High, Hanover High & Grover City, Hanover
City> Grover has better schools and higher taxes.
7. Validity judgment: If it rained, then the streets
are wet,
8. Is this correct? All the squeaky mice are in the house
vs. All the squeaky mice must stay in the house.
Issues, Logic problems with the same form are solved quite
differently. A number of alternative explanations have been proposed,
Causal relations and mechanisms,
expectancies,
Permission view. Deontic reasoning moral obligation,
what one ought to do.
*In order, the cards represent <P, ~P, Q, ~Q>
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