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In
classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...
, disjunctive syllogism (historically known as ''modus tollendo ponens'' (MTP), Latin for "mode that affirms by denying") is a valid
argument form In logic, logical form of a Statement (logic), statement is a precisely-specified Semantics, semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly Syntactic ambiguity, ambiguous sta ...
which is a syllogism having a disjunctive statement for one of its
premise A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...
s. An example in English: # The breach is a safety violation, or it is not subject to fines. # The breach is not a safety violation. # Therefore, it is not subject to fines.


Propositional logic

In propositional logic, disjunctive syllogism (also known as disjunction elimination and or elimination, or abbreviated ∨E), 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 ...
. If we are told that at least one of two statements is true; and also told that it is not the former that is true; we can
infer 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 it has to be the latter that is true. If ''P'' is true or ''Q'' is true and ''P'' is false, then ''Q'' is true. The reason this is called "disjunctive syllogism" is that, first, it is a syllogism, a three-step
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
, and second, it contains a logical disjunction, which simply means an "or" statement. "P or Q" is a disjunction; P and Q are called the statement's ''disjuncts''. The rule makes it possible to eliminate a
disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
from a logical proof. It is the rule that: :\frac where the rule is that whenever instances of "P \lor Q", and "\neg P" appear on lines of a proof, "Q" can be placed on a subsequent line. Disjunctive syllogism is closely related and similar to
hypothetical syllogism In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises. An example in English: :If I do not wake up, then I cannot go to work. :If I cannot go to work, then ...
, in that it is also a type of syllogism, and also the name of a rule of inference. It is also related to the law of noncontradiction, one of the three traditional laws of thought.


Formal notation

For a
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 for ...
that validates it, the ''disjunctive syllogism'' 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 ass ...
notation: : P \lor Q, \lnot P \vdash Q where \vdash is a metalogical symbol meaning that Q is a syntactic consequence of P \lor Q, and \lnot P. It may be expressed as a truth-functional tautology or theorem in the object language of propositional logic: : ((P \lor Q) \land \neg P) \to Q where P, and Q are propositions expressed in some formal system.


Natural language examples

Here is an example: # I will choose soup or I will choose salad. # I will not choose soup. # Therefore, I will choose salad. Here is another example: # It is red or it is blue. # It is not blue. # Therefore, it is red.


Inclusive and exclusive disjunction

Please observe that the disjunctive syllogism works whether 'or' is considered 'exclusive' or 'inclusive' disjunction. See below for the definitions of these terms. There are two kinds of logical disjunction: * '' inclusive'' means "and/or"—at least one of them is true, or maybe both. * '' exclusive'' ("xor") means exactly one must be true, but they cannot both be. The widely used English language concept of ''or'' is often ambiguous between these two meanings, but the difference is pivotal in evaluating disjunctive arguments. This argument: # P or Q. # Not P. # Therefore, Q. is valid and indifferent between both meanings. However, only in the ''exclusive'' meaning is the following form valid: # Either (only) P or (only) Q. # P. # Therefore, not Q. With the ''inclusive'' meaning you could draw no conclusion from the first two premises of that argument. See
affirming a disjunct The formal fallacy of affirming a disjunct also known as the fallacy of the alternative disjunct or a false exclusionary disjunct occurs when a deductive argument takes the following logical form: :A or B :A :Therefore, not B Or in logical op ...
.


Related argument forms

Unlike '' modus ponens'' and ''
modus ponendo tollens ''Modus ponendo tollens'' (MPT; Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic. It is closely related to ''modus ponens'' and '' modus tollendo ponens''. Overview MPT is usually described as having ...
'', with which it should not be confused, disjunctive syllogism is often not made an explicit rule or axiom of
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 for ...
s, as the above arguments can be proven with a combination of
reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
and
disjunction elimination In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof. It ...
. Other forms of syllogism include: *
hypothetical syllogism In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises. An example in English: :If I do not wake up, then I cannot go to work. :If I cannot go to work, then ...
* categorical syllogism Disjunctive syllogism holds in classical propositional logic and intuitionistic logic, but not in some paraconsistent logics.Chris Mortensen
Inconsistent Mathematics
''Stanford encyclopedia of philosophy'', First published Tue Jul 2, 1996; substantive revision Thu Jul 31, 2008


See also

* Stoic logic


References

{{reflist Rules of inference Theorems in propositional logic Classical logic Paraconsistent logic