Constructive dilemma
[Copi and Cohen] is a
valid rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
of
propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
. It is the
inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
that, if ''P'' implies ''Q'' and ''R'' implies ''S'' and either ''P'' or ''R'' is true, then either ''Q or S'' has to be true. In sum, if two
conditionals
Conditional (if then) may refer to:
*Causal conditional, if X then Y, where X is a cause of Y
*Conditional probability, the probability of an event A given that another event B has occurred
*Conditional proof, in logic: a proof that asserts a co ...
are true and at least one of their antecedents is, then at least one of their consequents must be too. ''Constructive dilemma'' is the
disjunctive Disjunctive can refer to:
* Disjunctive population, in population ecology, a group of plants or animals disconnected from the rest of its range
* Disjunctive pronoun
* Disjunctive set
* Disjunctive sequence
* Logical disjunction
In logic, ...
version of
modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
, whereas,
destructive dilemma
Destructive dilemmaMoore and Parker is the name of a valid rule of inference of propositional logic. It is the inference that, if ''P'' implies ''Q'' and ''R'' implies ''S'' and either ''Q'' is false or ''S'' is false, then either ''P'' or '' ...
is the disjunctive version of ''
modus tollens
In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. ''Modus tollens' ...
''. The constructive dilemma rule can be stated:
:
where the rule is that whenever instances of "
", "
", and "
" appear on lines of a proof, "
" can be placed on a subsequent line.
Formal notation
The ''constructive dilemma'' rule may be written in
sequent
In mathematical logic, a sequent is a very general kind of conditional assertion.
: A_1,\,\dots,A_m \,\vdash\, B_1,\,\dots,B_n.
A sequent may have any number ''m'' of condition formulas ''Ai'' (called " antecedents") and any number ''n'' of asse ...
notation:
:
where
is a
metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
al symbol meaning that
is a
syntactic consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more statements. A Validity (lo ...
of
,
, and
in some
logical system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
;
and expressed as a
truth-functional
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one ...
tautology or
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
of propositional logic:
:
where
,
,
and
are propositions expressed in some
formal system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
.
Natural language example
:If I win a million dollars, I will donate it to an orphanage.
:If my friend wins a million dollars, he will donate it to a wildlife fund.
:Either I win a million dollars or my friend wins a million dollars.
:Therefore, either an orphanage will get a million dollars, or a wildlife fund will get a million dollars.
The dilemma derives its name because of the transfer of disjunctive operator.
References
{{DEFAULTSORT:Constructive Dilemma
Rules of inference
Dilemmas
Theorems in propositional logic