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Table of Entries
Addition
Affirming the Consequent
Alternation
Alternatives
And
Antecedent
Argument
Argument Form
Association
Assumption
Biconditional
Commutation
Component
Compound Statement
Conclusion
Conditional Proof
Conditional Statement
Conjunct
Conjunction
Connective
Consequent
Consistent
Constructive Dilemma
Contingent
Contradiction
Contradictory
Contrapositive
Contrary
De Morgan's Laws
Deduction
Deductive Argument
Deductive System
Denying the Antecedent
Destructive Dilemma
Discharge
Disjunct
Disjunction
Disjunctive Syllogism
Distribution
Distributive Laws
Double Negation
Entails
Exclusive Or
Exportation
Fallacy
False
Follows From
Formal
Formal Proof of Validity
Grounds
Hook
Horseshoe
Hypothetical Syllogism
If ... then ...
Implication
Implies
Inclusive Or
Inconsistent
Indirect Proof
Induction
Inductive Argument
Infer
Inference
Inferring
Invalid
Logical Consequence
Logically Equivalent
Material Implication
Material Equivalence
Materially Equivalent
Modus Ponens
Modus Tollens
Natural Deduction
Natural Deductive System
Necessary Condition
Negation
... only if ...
Or
Paradoxes of Material Implication
Parentheses
Partial Common Meaning
Premise
Proof by Contradiction
Proposition
Propositional Calculus
Propositional Connective
Propositional Operator
Reasoning
Reductio Ad Absurdum
Relation
Rule of Inference
Rule of Replacement
Scope
Sentence
Simple Statement
Simplification
Sound Argument
Soundness
Specific Form
Statement
Statement Form
Statement Letter
Statement Variable
Strengthened Conditional
Sufficient Condition
Proof
Substitution Instance
Symbolic Logic
Tautologous
Tautology
Transposition
True
Truth-Functional Component
Truth-Functional Compound
Truth Table
Truth Value
Under Construction
Valid Argument
Validity
Wedge
Statement
A linguistic expression that asserts something to be the case,
and that therefore has a truth-value (can be true or false).
"Roses are red or violets are blue" is a statement (in the
English language). "Je voudrais une chambre" is a statement
(in the French language). "Roses" is an English language
expression but it is not a statement, because it does not
assert anything to be the case and so does not have a truth
value.
Argument
An argument consists of a non-empty set of statements called
premises, a possibly empty sequence of statements called
intermediate steps, and a single statement called the
conclusion. Furthermore, each intermediate step is claimed to
follow logically from one or more premises and/or preceding
intermediate steps, and the conclusion is claimed to follow
logically from one or more premises and/or intermediate steps.
Here is an argument with two premises and no intermediate
steps:
All men are mortal. Socrates
Therefore, Socrates is mortal.
is
a
man.
The word "therefore" indicates that an argument is being
presented. "All men are mortal" and "Socrates is a man" are
not claimed to follow from any other statements, so they must
both be premises.
"Socrates is mortal" is being claimed to follow from other
statements.
an argument is being presented.
serves to claim that the last statement follows from one or
more preceding statements, so is the conclusion of an
argument. Neither of the first two statements is claimed to
follow from any other, so they must both be premises.
Here is an argument with three premises and one intermediate
step:
1.
2.
3.
4.
5.
All men are mortal
Socrates is a man
Everything mortal will cease to exist
So, Socrates is mortal (by 1 and 2)
Therefore, Socrates will cease to exist (by 3 and 4)
Again, the word "therefore" indicates that the sequence is
argument with statement 5 as its conclusion. Statement 4
claimed to follow from statements 1 and 2, so is
intermediate step. None of statements 1, 2 and 3 is claimed
follow from any other, so they must be all be premises.
an
is
an
to
Here is a sequence of statements that is not an argument:
1. All men are mortal
2. Socrates is a man
3. Socrates is mortal
It is not an argument, because no statement in the sequence is
claimed to be a logical consequence of preceding ones. Lines 1
and 2 are premises, line 3 is the conclusion.
An argument will not always be written as a sequence in the
way described. For example, the second argument above may be
written like this:
Socrates is mortal, because Socrates is a man
and all men are mortal. But everything mortal
will cease to exist, so Socrates will cease to
exist.
Nevertheless, it is always possible to write an argument as a
sequence in the way described (although it might take some
work).
An argument is still an argument even if it has a false
premise, and even if one of the intermediate steps or the
conclusion is not a logical consequence in the way that it is
claimed to be. For example:
1. All men are mortal
2. Socrates is not a man
3. Therefore, Socrates is not mortal
This is an argument, even though the second premise is false,
and even though the conclusion is not a logical consequence of
the premises. Evaluating an argument for the truth of its
premises and conclusion and for its logical consequence is the
topic of soundness and validity.
Statement Letter
A letter that stands for a particular statement, usually used
for purposes of abbreviation. For example, we can define
statement letters R and V by letting R stand for "Roses are
red" and V stand for "Violets are blue". Then we can
abbreviate the argument
1. If roses are red then violets are blue.
2. Roses are red.
3. Violets are blue.
1. If R then V.
2. R.
3. V.
Or if we define a statement letter I by letting it stand for
"If roses are red then violets are blue", then we can
abbreviate the argument to this:
1. I.
2. R.
3. V.
It is usual to use UPPER CASE letters for statement letters.
Note that a statement letter is different to a statement
variable. They ought not to be confused.
Statement Variable
A placeholder in a statement form or argument form into which
can be substituted a particular statement or statement letter.
The statement form "p or q" contains two statement variables p and q. We can substitute the statement "Roses are red" into
p and the statement "Violets are blue" into q to get the
statement "Roses are red or violets are blue". Or we can
substitute the statement letter R into p and the statement
letter V into q to get the statement "R or V".
The argument form modus ponens
1. If p then q.
2. p.
3. q.
contains two statement variables - p and q. We can substitute
the statement "Roses are red" into p and the statement
"Violets are blue" into q to get the argument:
1. If roses are red then violets are blue.
2. Roses are red.
3. Violets are blue.
Or we can substitute the statement letter R into p and the
statement letter V into q to get the argument:
1. If R then V.
2. R.
3. V.
It is usual to use lower case letters for statement variables.
Note that a statement variable is different to a statement
letter.
A
statement
letter
stands
for
one
particular
statement. A statement variable, however, can be substituted
into by many different statement letters (or statements).
Substitution Instance
A statement is a substitution instance of a statement form if
it can be obtained by substituting statements into the
statement variables of the form. The statement "Roses are red
or violets are blue" is a substitution instance of the
statement form "p or q", because it can be obtained by
substituting "Roses are red" for p and "Violets are blue" for
q. It is not a substitution instance of the statement form "p
and q", because there are no statements that can be
substituted for p and q to get "Roses are red or violets are
blue". Nor is it a substitution instance of the statement form
"p or p", because the same statement or statement letter must
be substituted into every occurrence of the same statement
variable.
An argument is a substitution instance of an argument form if
it can be obtained by substituting statements into the
statement variables of the form. The argument
1. If roses are red then violets are blue.
2. Roses are red.
3. Violets are blue.
is a substitution instance of the argument form
1. If p then q.
2. p.
3. q.
because it can be obtained by substituting "Roses are red" for
p and "Violets are blue" for q. It is not a substitution
instance of the argument form
1. p or q.
2. Not p.
3. q.
because it cannot be obtained from this form by substituting
statements for p and q.
Validity
If it is impossible for an argument to have true premises and
a false conclusion then it is valid, otherwise it is invalid.
The argument:
All men are mortal
Socrates is a man
Therefore, Socrates is mortal
is valid, because it is impossible for the premises ('All men
are mortal', and 'Socrates is a man') to be true and the
conclusion ('Socrates is mortal') to be false.
The argument:
All cats are mortal
Socrates is not a cat
Therefore, Socrates is not mortal
is invalid, because it is possible for the
are mortal', and 'Socrates is not a cat)
conclusion ('Socrates is not mortal') to
the premises are true and the conclusion is
premises ('All cats
to be true and the
be false (in fact,
false).
Note that the validity of an argument is determined by the
possible truth-values of its premises and conclusion, not by
their actual truth values. An argument might have true
premises and a true conclusion and yet be invalid:
All men are mortal
Socrates is a man
Therefore, Socrates is Greek
An argument might have false premises and a false conclusion
and yet be valid:
No men have red hair
Socrates has red hair
Therefore, Socrates is not a man
For this reason, the validity of an argument is no measure of
the truth of its conclusion. Some valid arguments have a true
conclusion, some have a false one. Some invalid arguments have
a true conclusion, some have a false one. The only relation
between the validity of an argument and the actual truth
values of its premises and conclusion is this: if an argument
is valid and has true premises, then its conclusion must be
true. Or: if an argument has true premises and a false
conclusion then it must be invalid.