Symbolic Logic Flashcards
Simple Statement
A statement that does not contain any other statement as a component.
Compound Statement
A statement that contains two or more statements as components.
Component
A part of a compound statement that is itself a statement, and is of such a nature that, if replaced in the larger statement by any other statement, the result will be meaningful.
Conjunction
A truth-functional connective meaning “and,” symbolized by the dot, •.
A statement of the form p•q is true if and only if p is true and q is true.
Conjunct
Each one of the component statements connected in a conjunctive statement.
Truth-functional component
Any component of a compound statement whose replacement there by any other statement having the same truth value would leave the truth value of the compound statement unchanged.
Truth-functional compound statement
A compound statement whose truth value is determined wholly by the truth value of its components.
Truth-functional connective
Any logical connective (e.g. conjunction, disjunction, material implication and material equivalence) between the components of a truth-functionally compound statement.
Truth table
An array on which all possible truth values of compound statements are displayed, through the display of all possible combinations of the truth values of their simple components.
A truth table may be used to define truth-functional connectives; it may also be used to test the validity of many deductive arguments.
Negation
Denial
Symbolised by the tilde or curl.
~p simply means “it is not the case that p”, and may be read as “not-p”.
Disjunction
A truth-functional connective meaning “or”; components so connected are call disjuncts.
There are two types of disjunction: inclusive and exclusive.
Inclusive disjunction
A truth-functional connective between two components called disjuncts.
A compound statement asserting inclusive disjunction is true when at least one of the disjuncts (that is, one or both) is true.
Normally called simply “disjunction”, it is also called “weak disjunction” and is symbolized by the wedge, V.
Exclusive disjunction
A logical relation meaning “or” that may connect two component statements.
A compound statement asserting exclusive disjunction says that at least one of the disjuncts is true and that at least one of the disjuncts is false.
It is contracted with an “inclusive” (or “weak”) disjunction, which says that at least one of the disjuncts is true and that they may both be true.
Punctuation
The parentheses, brackets and braces used in mathematics and logic to eliminate ambiguity.
Conditional Statement
A hypothetical statement; a compound proposition or statement of the form “if p then q”.
Antecedent
In a conditional statement (“if … then …”), the component that immediately follows the “if”.
Sometimes called the implicans or the protasis.
Consequent
In a conditional statement (“if … then …”), the component that immediately follows the “then”.
Sometimes called the implicate or the apodosis.
Implication
The relation that holds between the antecedent and the consequent of a true conditional or hypothetical statement.
Horseshoe
The symbol for material implication, ⊃.
Material implication
A truth-functional relation (symbolized by the horeshoe, ⊃) that may connect two statements.
The statement “p materially implies q” is true when either p is false, or q is true.
Refutation by logical analogy
A method that shows the invalidity of an argument by presenting another argument that has the same form, but whose premises are known to be true and whose conclusion is known to be false.
Variable
or Statement Variable
A place-holder.
A letter for which a statement may be substituted.
By convention, any of the lowercase letters beginning with p, q, etc. are used.
Argument form
An array of symbols exhibiting logical structure.
It contains no statements but it contains statement variables. These variables are arranged in such a way that when statements are consistently substituted for the statement variables, the result is an argument.
Substitution instance
Any argument that results from the substitution of statements for statement variables of a given argument form.
Specific form
When referring to a given argument, the argument form from which the argument results when a different simple statement is substituted consistently for each different statement variable in that form.
Define the term
Invalid
as applied to argument forms.
Not valid.
Characterising a deductive argument that fails to provide conclusive grounds for the truth of its conclusion.
Every deductive argument is either valid or invalid.
Define the term
Valid
as applied to argument forms.
A deductive argument is said to be valid when its premises, if they were all true, would provide conclusive grounds for the truth of its conclusion.
Validity is a formal characteristic. It applies only to arguments, as distinguished from truth, which applies to prepositions.
4 Common Valid Forms
- Disjunctive Syllogism
- Modus Ponens
- Modus Tollens
- Hypothetical Syllogism
Disjunctive Syllogism
A valid argument form in which one premise is a disjunction, another premise is the denial of one of the two disjuncts, and the conclusion is the truth of the other disjunct.
Symbolized as:
p v q,
~p,
therefore q.
Modus Ponens
An elementary valid argument form according to which, if the truth of a hypothetical premise is assumed, and the truth of the anticedent of that premise is also assumed, we may conclude that the consequent of the premise is true.
Symbolized as:
p ⊃ q,
p,
therefore q.
Modus Tollens
An elementary valid argument form according to which, if the truth of a hypothetical premise is assumed, and the falsity of the consequent of that premise is also assumed, we may conclude that the antecedent of the premise is false.
Symbolized as:
p ⊃ q,
~q,
therefore ~p.
Hypothetical Syllogism
A syllogism that contains a hypothetical proposition as a premise. If the syllogism contains hypothetical propositions exclusively, it is called a “pure” hypothetical syllogism; if the syllogism contains one conditional and one categorical premise, it is called a “mixed” hypothetical syllogism.
p ⊃ q,
q ⊃ r,
therefore p ⊃ r
Statement form
A sequence of symbols containing no statements, but containing statement variables connected in such a way that when statements are consistently substituted for the statement variables, the result is a statement.
Disjunctive statement form
A statement form symbolized as p v q
Its substitution instances are disjunctive statements.
Specific form
When referring to a given statement, the statement form from which the statement results when a different simple statement is substituted consistently for each different statement variable in that form.
Tautology
A statement form all of whose substitution instances must be true.
Contradition
A statement form all of whose substitution instances are false.
Contingent
Being neither tautologous nor self-contradictory.
A contingent statement may be true or false; a contingent statement form has some true and some false substitution instances.
Material equivalence
A truth-functional relation (sybmolized by the three-bar sign ≡) that may connect two statements.
Two statements are materially equivalent when they are both true, or when they are both false - that is, when they have the same truth value. Materially equivalent statements always materially imply one another.
List the 4 truth-functional connectives
- And, • (dot)
- Or, v (wedge)
- If-then, ⊃ (horseshoe)
- If and only if, ≡ (tribar)
Proposition type for the truth-functional connective:
And
Conjunction
Proposition type for the truth-functional connective:
Or
Disjunction
Proposition type for the truth-functional connective:
If-then
Conditional
Proposition type for the truth-functional connective:
If and only if
Biconditional
Names of components of propositions of type:
And
Conjuncts
Names of components of propositions of type:
Or
Disjuncts
Names of components of propositions of type:
If, then
Antecedent, Consequent
Names of components of propositions of type:
If and only if
Components
Logical Equivalence
When referring to truth-functional compound propositions, the relationship that holds between two propositions when the statement of their material equivalence is a tautology.
A very strong relation; statements that are logically equivalent must have the same meaning and may therefore replace one another wherever they occur.
Double negation
An expression of the logical equivalence of any symbol and the negation of the negation of that symbol.
p ≡ ~~ p
De Morgan’s theorems
Two expressions of logical equivalence.
The first states that the negation of a disjunction is logically equivalent to the conjunction of the negations of its disjuncts:
~(p v q) ≡ (~p • ~q)
The second states that the negation of a conjunction is logically equivalent to the disjunction of the negations of its conjuncts:
~(p • q) ≡ (~p v ~q)
3 Laws of Though
- Principle of Identity
- Principle of noncontradiction
- Principle of excluded middle
Principle of Identity
This principle asserts that if any statement is true, then it is true.
Every statement of the form p ⊃ p must be true.
Every such statement is a tautology.
Principle of noncontradiction
No statement can be both true and false.
Every statement of the form p • ~p must be false, that every such statement is self-contradictory.
The principle of excluded middle
This principle asserts that every statement is either true or false.
Every statement of the form p V ~p must be true.
Every such statement is a tautology.
