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A disjunctive syllogism, also known as modus tollendo ponens (literally: mode which, by denying, affirms) is a valid, simple argument form: This article discusses validity in logic, for the term in the social sciences see validity (psychometric). ...
In logic, the argument form or test form of an argument results from replacing the different words, or sentences, that make up the argument with letters, along the lines of algebra; the letters represent logical variables. ...
- P or Q
- Not P
- Therefore, Q
In logical operator notation: In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...
 - ¬
  where  represents the logical assertion. The logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. ...
Roughly, we are told that it has to be one or the other that is true; then we are told that it is not the one that is true; so we infer that it has to be the other that is true. The reason this is called "disjunctive syllogism" is that, first, it is a syllogism--a three-step argument--and second, it contains a disjunction, which means simply an "or" statement. "Either P or Q" is a disjunction; P and Q are called the statement's disjuncts. A syllogism (Greek: συλλογισμός — conclusion, inference), more correctly a categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises). ...
Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ...
Here is an example:
- Either I will choose soup or I will choose salad.
- I will not choose soup.
- Therefore, I will choose salad.
Here is another example: - Either the Browns win or the Bengals win.
- The Browns do not win.
- Therefore, the Bengals win.
Inclusive versus exclusive disjunction
It should be noted with importance that there are two kinds of logical disjunction: - inclusive means "and/or" where at least one term must be true or they can both be true.
- exclusive ("xor") means one must be true and the other must be false. Both terms cannot be true and both cannot be false.
The popular English language concept of or is often ambiguous between these two meanings, but the difference is pivotal in evaluating disjunctive arguments. OR logic gate. ...
Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ...
This argument:
- Either 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 P or Q (exclusive).
- 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 logical fallacy of affirming a disjunct occurs in a disjunctive syllogism when an argument takes the form: Either A or B (this is the disjunct) A (Affirming the middle term) Therefore, not B The fallacy lies in concluding that B must be false because A is true; in fact...
Related argument forms Unlike modus ponendo ponens and modus tollendo tollens, with which it should not be confused, modus tollendo ponens is often not made an explicit rule or axiom of logical systems, as the above arguments can be proven with a (slightly devious) combination of reductio ad absurdum and disjunction elimination. Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: where represents the logical assertion. ...
Modus tollens (Latin: mode that denies) is the formal name for indirect proof or proof by contrapositive, often abbreviated to MT It is a common, simple argument form: If P, then Q. Q is false. ...
Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις άτοπον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then conclude the original assumption must...
In propositional calculus disjunction elimination is the inference that, if A or B is true, and A entails C, and B entails C, then we may justifiably infer C. The reasoning is simple: since at least one of the statements A and B is true, and since either of them...
Modus tollendo ponens should also not be confused with modus ponendo tollens. A disjunctive syllogism, also known as modus tollendo ponens (literally: mode which, by denying, affirms) is a valid, simple argument form: P or Q Not P Therefore, Q In logical operator notation: ¬ where represents the logical assertion. ...
Other forms of syllogism:
In logic, a hypothetical syllogism has two uses. ...
Wikipedia does not yet have an article with this exact name. ...
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