Kinds of Natural Deduction: Propositional Calculus, Deduction Rules (Essay Sample)

Natural Deductions
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In proof and logic theory, ND is a type of natural deduction where logical reasoning is articulated through a set of inference rules that are almost similar to natural reasoning. This is in contrast to axiomatic systems where axioms are used to show set of logical laws that govern deductive reasoning. Different kinds of natural deductions exist. The whole concept of natural deduction stemmed from dissatisfaction with systems that were common of philosophers and great thinkers like Hilbert, Russell and Frege. These mathematicians applied these concepts in their Principia Mathematic principles. A moral way of making natural deductions to solving common problems in life, not necessarily mathematical problems was imminent around this time. After a series of seminars in Poland, greatest scientists and thinkers of those days attempted, and maybe succeeded in defining new truth logics that could surpass natural reasoning. In this type of reasoning, an individual must provide evidence for his assertions. A statement must be backed with enough truth so as to be considered true. But perhaps the current natural deduction is the brain works of Gentzen, a great German mathematician in those days. He was influenced with the desire to create consistency in results for number theory. In this regard he introduced an alternative system to cut elimination theory-the sequent calculus. He thus proved both intuitionistic and classical logic. His 1965 Natural Deduction; proof-theoretical became a reference for natural deduction.[Prawitz, Dag. 1965. Natural deduction; a proof-theoretical study. (Stockholm: Almqvist & Wiksell)] [Anderson, John Mueller, and Henry W. Johnstone. 1962. Natural deduction, the logical basis of axiom systems. (Belmont, Calif: Wadsworth Pub. Co).]
Propositional calculus
This type of calculus makes use of statement like and, not, or, if, then, if and only if to represent inferences for different situations.[Nidditch, P H. 1962. Propositional calculus, (New York: Free Press of Glencoe).]
For example: Chelsea has won the Europa league and Liverpool has won the Europa league
Deductions: Chelsea has won the Europa league.
Liverpool has won the Europa league, too
These statements are valid. It is true that both teams have won the Europa league. These are valid inferences, and one fact affects the other. It does not depend on which team one is talking about. These are declarative statements. Variables can be made from such deductions.For example
(P&Q)/Q These are achieved through manipulations of the original constants, may be through multiplications and divisions of the original constants
P*QPC can represent the above references. It is important to define the following:
 Logical constants = these are expressions with constant meanings; they are not variable.
Truth functions = these are constants that express functions from truth values or truth tables.
In PC the language used comprises variables that commonly use P, Q, R & S and logical constants. The standard constants which are considered logical are 
¬ = not
& = and
v = or, +
→ = if…, then…
↔ = if, and only if (iff)
Sentences of propositional calculus include symbols that are combinations(lawful)of variables and constants. Examples to illustrate this are
  ¬P
(‘Chris isn’t tall’)
P → (Q & ¬R)
(‘If Chris is short, then James is tall and Agnes isn’t blonde’)
(P ↔ Q) → ((P → Q) & (Q → P))
(‘If Henry is tall iff Margaret has brown hair, then if Henry is tall, then Margaret is blonde, and if
Margaret is blonde, thenHenry is tall’)
Etc.
 Excursus on scope
Brackets are used in propositional calculus so as to avoid ambiguities of scope.
 Here are further examples of PCs
 Representation meaning
P → P & Q if P, then P and (conditional)
(P → P) & Q stronger
P → (P & Q) weaker
The use of brackets creates different meanings in each situation. If brackets were not used, there would be no clear distinctions. Brackets re used to create distinct meanings and without them the meanings are lost.
 Strong = the expression has strength enough to be considered false.
Weak = the expression is weak enough to be true
 Main constant = constants with wider possible scopes.
Scope =this is the smallest formula that is part of an argument.
Syntax of propositional calculus involves creations that do not really have to obey common grammar rules. In PC codes are used to modify valid inferences upon which constants depend. Syntax is therefore a language that is used to communicate formulas
If:
P, Q, R,… are formulae of PC.
R is a PC formula, then ¬X, the following are options that can be used
Q & Q,
Q v Q,
Q→Q,
Q↔Q are formulae of PC.
Truth Tables
These tables are used to simplify certain conditions that must be met if the inferences are to be valid. It just makes use of the terms that have already been discussed. It also uses “Truth or False” statements to simplify assumptions. The following syntax will help in understanding.Each function is defined by basic functions that have already been inferred. For purposes of understanding, we use A and B as variables. The truth tables for A and B can be represented as shown below. F stands for false, while T stands for true.
¬

&

v

→

↔

¬ Q

Q & R

Q v R

Q → R

Q ↔ R

F T

T TT

T TT

T TT

T TT

T F

T F F

T T F

T F F

T F F

-

F F T

F T T

F T T

F F T

-

F FF

F FF

F T F

F T F

(i) ‘¬Q’ is true if and only if ‘Q’ is false.
(ii) ‘Q & R’ is true if and only if ‘Q’ is true and ‘R’ is true.
(iii) ‘Q v R’ is true if and only if‘Q’ is true or if ‘R’ is true.
(iv) ‘Q → R’ is true if and Q is not affected by R. Q could be true, while R is false. Or R could be true while Q is false.
(v) ‘Q ↔ R’ 
Each truth table details all possible ways in which the given formula could be assigned the value T or F. That is, a truth table exhausts the values a truth function can have, given all possible arguments. In general, for a formula of n variables, its table will contain 2n rows, each specifying a way the ‘world’ could be such that the formula receives a value. So, with ‘¬P’ we have 21 = 2; with two variable formulae, we have 22 = 4. For the formula, ‘P → (Q → R)’, we have 23 = 8. And so on.
Inference rules
Assumption Rule
One can insert any letter at the start of a formula and consistently assume it throughout his deductions for example
A=B 
(ii) Double Negation AB can be double negated so that it becomes A’B’. two negations cancel each other. A could mean presence of rain. When negated the meaning also changes: it now means the opposite i.e. presence of rain. B could mean an event occurring, e.g. going to school. When negated B’ means the opposite i.e. not going to school
Or condition
If one says that a condition C is achieved in the presence of A or B, then, C can be represented as C=A +B. the (+) is used to denote (or)
 Introduction rules
‘True’ judgments are made in instances when one is expected to prove that Q and R are propositions to denote ‘a Q true’ and ‘a R true.’ This can be represented as below:
 
In these rules the propositions are the objects. The rule is an abbreviation of the longer version of the same rule’
 
 Here the first condition is satisfied through ^F rule of formation use previous forms of the conditions for A and B. under null conditions, truth can be derived with or without such conditions. They can be represented as
The truth for the above proposition can be established in different ways as shown, provided the meaning is not lost.
Elimination rules are used to de-construct compound compositions. For example in A^B true, one concludes that both A and B...