METHODS OF LOGICAL DEDUCTION
truth-table for a group of sentences
A truth-table for a group of sentences is a table with columns and rows that lists the resultant truth-values of the given sentences for each of the possible combinations of truth-values to the simple sentences out of which the given sentences are constructed. The vertical columns are headed by the given sentences, and each of the horizontal rows begins with one of the possible combinations of truth-values to the simple components. Auxiliary columns are introduced to contain the intermediate resultant values of the subparts leading up to the resultant values of the given sentences. This kind of tabular display permits one to investigate the logical characteristics of the given sentences by an examination of their possible truth-value behavior.
implication
Some sentences (one or more) together imply another sentence if the corresponding inference from those sentences to that other sentence is a valid argument.
equivalence
Two sentences are equivalent if for every possible interpretation of all the basic components of the two sentences, the resultant value of the one sentence is always the same as the resultant value of the other sentence. For example, the sentence "Not all dogs sing" is equivalent to the sentence "some dogs do not sing" (not all D are S; some D are not S). [Truth-functionally equivalent sentence-pairs form a smaller subgroup within the set of equivalent sentences-pairs.]
deduction
A (direct) deduction for an argument is a (vertical) sequence of sentences (1) beginning with the premisses of the argument, and (2) ending with the conclusion of the argument, and (3) all the sentences in the sequence that come after the premisses (the intermediate conclusions and the final conclusion) are derived from previous sentences in the sequence by the rules of deduction for logic. [When one speak of a deduction for an argument one always means a direct deduction as here described, unless the deduction is explicitly referred to as an indirect deduction or a conditional deduction.]
rules of deduction for logic
The rules of deduction for logic are a set of selected rules, already known to be valid, by means of which one is permitted to derive certain sentences from already given sentences if those sentences satify the conditions specified in the rules. There is an important division of these rules into rules of simple inference and rules of equivalence.
rules of simple inference
These rules permit one to infer a certain conclusion from some given premisses. These rules specify the pattern that the premisses must have and the pattern that the conclusion must have; e.g., the rule
Modus Ponens, p
q , p
q
rules of equivalence
These rules permit one to infer a certain conclusion from a given premiss. These rules specify that some part of the premiss may be changed into an alternative form while the remainder stays fixed; e.g., the rule Double Negation, ~(~p) = p .
proper annotations of deductions
Deductions are to be properly annotated. Such annotations consist of three vertically aligned and coordinated columns: (1) a column that numbers all the lines of the deduction, (2) a column of the sentences that make up the deduction, and (3) a column of the reasons for each step. These reasons list (a) the line-numbers of the previously listed lines involved in the step, and (b) a short label that identifies the rule being used. For example, "12. not M and not S 2, 10, Conj."
RULES OF INFERENCE (or Rules of Substitution)
modus ponens [rule of inference]
From a conditional sentence and from the sentence that is its antecedent part, one may infer the sentence that is its consequent part.
From "If Tom went then Sue went"
and from "Tom went"
infer "Sue went."
MP. p
q , p
q
modus tollens [rule of inference]
From a conditional sentence and from the sentence that is the opposite of its consequent part, one may infer the sentence that is the opposite of its antecedent part.
From "If Tom went then Sue went"
and from "not Sue went"
infer "Not Tom went."
MT. p
q , ~q
~p
hypothetical syllogism (chain argument) [rule of inference]
From a conditional sentence and from another conditional sentence that is the
continuation of the first, one may infer the conditional sentence whose condition is the condition of the first one, and whose result is the result of the second one.
From "If Tom went then Sue went"
and from "If Sue went then Mary went"
infer "If Tom went then Mary went."
Chain [
HS]. p
q , q
r
p
r
dilemma [rule of inference]
From a disjunctive sentence with two choices, and from two conditional sentences that assert the respective results of those choices, one may infer the disjunctive sentence whose two choices are those results.
From "Either Tom or Sue went"
and from "If Tom went then Liz went"
and from "If Sue went then Matt went"
infer "Either Liz or Matt went."
Dilemma. p V q , p
r , q
s
r V s
simplification [rule of inference]
From a conjunctive sentence that asserts two parts, one may infer a sentence that is one of those two parts.
From "Both Tom and Sue went"
infer "Tom went."
Simp. p & q
p
conjunction [rule of inference]
From two sentences, one may infer a sentence that is the conjunction of those two sentences.
From "Tom went"
and from "Sue went"
infer "Both Tom and Sue went."
Conj. p , q
p & q
disjunctive syllogism [rule of inference]
From a disjunctive sentence with two choices, and from a sentence that is the opposite of one of the choices, one may infer the sentence that is the remaining choice.
From "Tom or Sue went"
and from "Not Tom went"
infer "Sue went."
Disj Syll [
DS]. p V q , ~p
q
disjunctive addition [rule of inference]
From a given sentence, one may infer the
weaker disjunctive sentence in which the given sentence is one choice and some other sentence is the other choice.
From "Tom went"
infer "Tom or Sue went."
Disj Add [
Add]. p
p V q
conjunctive syllogism [rule of inference]
From a sentence that is the denial of a conjunction of two assertions, and from a sentence that is one of those assertions, one may infer the opposite of the other one.
From "Not both Tom and Sue went"
and from "Tom went"
infer "Not Sue went."
Conj Syll [
CS]. ~(p & q) , p
~q
reductio ad absurdum [rule of inference]
If a hypothesis (a condition) leads both to one result and also to its denial, then one may infer that the hypothesis is false and that its opposite is true.
From "If Tom went then Sue went"
and from "if Tom went then not Sue went"
infer "It can't be that Tom went."
RAA. p
q , p
~q
~p
Rules of Equivalence
double negation [equivalence rule]
The denial of a denial of a sentence is equivalent to just that sentence. These equivalent items are interchangeable. "It is not the case Tom didn't go" equals "Tom did go". DN. ~(~p) = p
De Morgan's Laws [equivalence rule]
The opposite of the conjunction of two sentences is equivalent to the disjunction of the opposities of the two sentences. "Not both Tom and Sue went" equals "Either not Tom went or not Sue went". Likewise, the opposite of the disjunction of two sentences is equivalent to the conjunction of the opposites of the two sentences. "Not either Tom or Sue went" equals "both not Tom went and not Sue went". Such equivalent items are interchangeable. DeMorg ~(p & q) = ~p V ~q ; also, ~(p V q) = ~p & ~q
contraposition, propositional [equivalence rule]
A conditional sentence is equivalent to the conditional sentence that results by switching the two sides
and taking the opposite of each. These equivalent items are interchangeable. "If Tom went then Sue went"
equals "if not Sue went then not Tom went".
Contrap. p
q = ~q
~p
conditional exchange [equivalence rule]
A
conditional sentence is equivalent to a
disjunctive sentence whose two choices are, one, the opposite of that antecedent part and, two, that same consequent part. These equivalent sentences are interchangeable. "If Tom went then Sue went"
equals "either not Tom went or [in case he did, then] Sue went".
Cond. p
q = ~p V q
biconditional exchange [equivalence rule]
A biconditional sentence is equivalent to the conjunction of the
two conditional sentences formed from its two sides. These sentences are interchangeable. "Tom went if and only if Sue went"
equals "If Tom went then Sue went, and if Sue went then Tom went".
Bicond. p
q = (p
q) & (q
p)
double ifs (Exportation) [equivalence rule]
A conditional sentence whose one antecedent is a
double condition is equivalent to the
single, complex conditional sentence in which the two conditions are stated as separate antecedent parts. "If both Tom and Sue went then Mary went"
equals "If Tom went, then, further, if Sue went, then Mary went"
equals "If Tom went, and if Sue went, then Mary went". Such equivalent complex sentences are interchangeable.
Double Ifs [
Exp]. (p & q)
r = p
(q
r)
double thens [equivalence rule]
A conditional sentence whose one consequent is a
double result is equivalent to the conjunction of the
two conditional sentences that state the two results separately. "If Tom went then both Sue and Mary went"
equals "If Tom went then Sue went; and also, if Tom went then Mary went". Such equivalent items are interchangeable.
Double Thens. p
(q & r) = (p
q) & (p
r)
duplication [equivalence rule]
A conjunction consisting of a sentence conjoined with itself is equivalent to that sentence by itself. "Tom went, and moreover, Tom went" equals "Tom went". Similarly, a disjunction consisting of a sentence disjoined with itself is equivalent to that sentence by itself. "Tom went, or else Tom went" equals "Tom went". Such equivalent items are interchangeable. Dupl. p & p = p ; also, p V p = p
commutation [equivalence rule]
A conjunction is equivalent to that conjunction when its conjuncts are switched. "Tom and Sue went" equals "Sue and Tom went". Similarly, a disjunction is equivalent to that disjunction when its disjuncts are switched. "Tom or Sue went" equals "Sue or Tom went". Such items are interchangeable.
Comm. p & q = q & p ; also, p V q = q V p
association [equivalence rule]
A conjunction with multiple conjuncts that are grouped in one way is equivalent to that conjunction when its conjuncts are grouped in some other way. "Tom went, and Sue and Mary went" equals "Tom and Sue went, and Mary went". Similarly, a disjunction with multiple disjuncts that are grouped in one way is equivalent to that disjunction when its disjuncts are grouped in some other way. "Tom went, or Sue or Mary went" equals "Tom or Sue went, or Mary went". Such items are interchangeable.
Assoc. p & (q & r) = (p & q) & r ; also, p V (q V r) = (p V q) V r
distribution [equivalence rule]
A conjunctive sentence, one of whose components is a disjuction, is equivalent to the disjunctive sentence that results when the one conjunct is conjunctively distributed over the two disjuncts. "Tom went, and either Sue or Mary went" equals "Tom and Sue went, or else Tom and Mary went". Likewise, a disjunctive sentence, one of whose components is a conjuction, is equivalent to the conjunctive sentence that results when the one disjunct is disjunctively distributed over the two conjuncts. "Tom went, or both Sue and Mary went" equals "Tom or Sue went, and also, Tom or Mary went". Such items are interchangeable. Distr. p & (q V r) = (p & q) V (p & r) ; also, p V (q & r) = (p V q) & (p V r)