|
To meet Wikipedia's quality standards, this article or section may require cleanup.
Please discuss this issue on the talk page, or replace this tag with a more specific message. Editing help is available.
This article has been tagged since June 2006. In traditional Aristotelian logic, Deductive reasoning is reasoning in which the conclusion is necessitated by, or reached from, previously known facts. The premises: if the premises are true, the conclusion must be true. This is distinguished from abductive and inductive reasoning, where the premises may predict a high probability of the conclusion, but do not ensure that the conclusion is true. Aristotelian logic, also known as syllogistic logic, is the particular type of logic created by Aristotle, primarily in his works Prior Analytics and De Interpretatione. ...
Abduction, or abductive reasoning, is the process of reasoning to the best explanations. ...
// Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument support the conclusion but do not ensure it. ...
Deductive reasoning may also be defined as inference in which the conclusion is of no greater generality than the premises or inference in which the conclusion is just as certain as the premises. Inference is the act or process of deriving a conclusion based solely on what one already knows. ...
How it works Somebody could say: "Since it is raining, the street must be wet.". However, there is a hidden argument in this statement: "If it's raining then the street gets wet". Using the premise "If it's raining then the street gets wet" you could argue that "Since it's raining the street is wet" but not "the street is wet so it must be raining". Or you could say: "The street is not wet, so it's not raining" (excluding, for the sake of example, the possibility of a tunnel), but not "It is not raining so the street is not wet".
This is because the wet street is an unavoidable product created by the rain but the wet street does not have to be caused by rain. So the basic statement "if something then something else" could logicly be followed "something is so something else must be" and "something else is not so something can not be". These are the first two basic valid reasoning types.
A few examples:
Valid:
- Since Socrates is a man,
- and since all men are mortal,
- therefore Socrates is mortal.
- Since the picture is above the desk,
- and since the desk is above the floor,
- therefore the picture is above the floor.
- Since a cardinal is a bird,
- and since all birds have wings,
- therefore a cardinal has wings.
Invalid: - A truly left wing politician does not tolerate animal cruelty.
- G. Houseman thinks hitting a dog is wrong.
- G. Houseman is a truly left wing politician.
- Every criminal opposes the government.
- Everyone in the opposition party opposes the government.
- Therefore everyone in the opposition party is a criminal.
This is invalid because the premises fail to establish commonality between membership in the opposition party and being a criminal. This is the famous fallacy of the undistributed middle. The fallacy of the undistributed middle is a logical fallacy that is committed when the middle term in a categorical syllogism isnt distributed. ...
| Basic arguments of the propositional calculus | | Name | Sequent | Description | | Modus Ponens | [(p → q) ∧ p] ⊢ q | if p then q; p; therefore q | | Modus Tollens | [(p → q) ∧ ¬q] ⊢ ¬p | if p then q; not q; therefore not p | | Hypothetical syllogism | [(p → q) ∧ (q → r)] ⊢ (p → r) | if p then q; if q then r; therefore, if p then r | | Disjunctive syllogism | [(p ∨ q) ∧ ¬p] ⊢ q | Either p or q; not p; therefore, q | | Constructive dilemma | [(p → q) ∧ (r → s) ∧ (p ∨ r)] ⊢ (q ∨ s) | If p then q; and if r then s; but either p or r; therefore either q or s | | Destructive dilemma | [(p → q) ∧ (r → s) ∧ (¬q ∨ ¬s)] ⊢ (¬p ∨ ¬r) | If p then q; and if r then s; but either not q or not s; therefore rather not p or not r | | Simplification | (p ∧ q) ⊢ p,q | p and q are true; therefore p is true | | Conjunction | p, q ⊢ (p ∧ q) | p and q are true separately; therefore they are true conjointly | | Addition | p ⊢ (p ∨ q) | p is true; therefore the disjunction (p or q) is true | | Composition | [(p → q) ∧ (p → r)] ⊢ [p → (q ∧ r)] | If p then q; and if p then r; therefore if p is true then q and r are true | | De Morgan's theorem (1) | ¬ (p ∧ q) ⊢ (¬p ∨ ¬q) | The negation of (p and q) is equiv. to (not p or not q) | | De Morgan's Theorem (2) | ¬ (p ∨ q) ⊢ (¬p ∧ ¬q) | The negation of (p or q) is equiv. to (not p and not q) | | Commutation (1) | (p ∨ q) ⊢ (q ∨ p) | (p or q) is equiv. to (q or p) | | Commutation (2) | (p ∧ q) ⊢ (q ∧ p) | (p and q) is equiv. to (q and p) | | Association (1) | [p ∨ (q ∨ r)] ⊢ [(p ∨ q) ∨ r] | p or (q or r) is equiv. to (p or q) or r | | Association (2) | [p ∧ (q ∧ r)] ⊢ [(p ∧ q) ∧ r] | p and (q and r) is equiv. to (p and q) and r | | Distribution (1) | [p ∧ (q ∨ r)] ⊢ [(p ∧ q) ∨ (p ∧ r)] | p and (q or r) is equiv. to (p and q) or (p and r) | | Distribution (2) | [p ∨ (q ∧ r)] ⊢ [(p ∨ q) ∧ (p ∨ r)] | p or (q and r) is equiv. to (p or q) and (p or r) | | Double negation | p ⊢ ¬¬p | p is equivalent to the negation of not p | | Transposition | (p → q) ⊢ (¬q → ¬p) | If p then q is equiv. to if not q then not p | | Material implication | (p → q) ⊢ (¬p ∨ q) | If p then q is equiv. to either not p or q | | Material equivalence (1) | (p ↔ q) ⊢ [(p → q) ∧ (q → p)] | (p is equiv. to q) means, (if p is true then q is true) and (if q is true then p is true) | | Material equivalence (2) | (p ↔ q) ⊢ [(p ∧ q) ∨ (¬q ∧ ¬p)] | (p is equiv. to q) means, either (p and q are true) or ( both p and q are false) | | Exportation | [(p ∧ q) → r] ⊢ [p → (q → r)] | from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true) | | Importation | [p → (q → r)] ⊢ [(p ∧ q) → r] | | | Tautology | p ⊢ (p ∨ p) | p is true is equiv. to p is true or p is true | Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. ...
In Logic, 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: P → Q P ⊢ Q where ⊢ represents the logical assertion. ...
Modus tollens (Latin for mode that denies) is the formal name for indirect proof or proof by contrapositive (contrapositive inference), often abbreviated to MT. It can also be referred to as denying the consequent, and is a valid form of argument (unlike similarly-named but invalid arguments such as affirming...
In logic, a hypothetical syllogism has two uses. ...
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. ...
A logical constructive dilemma is a formal logical argument that takes the form: 1a) P → Q. b) R → S. 2) Either P or Q is true. ...
A logical destructive dilemma is a formal logical argument that takes the form: 1a) P → Q. b) R → S. 2) Either not-Q or not-S is true. ...
In logic, De Morgans laws (or De Morgans theorem) are rules in formal logic relating pairs of dual logical operators in a systematic manner expressed in terms of negation. ...
Commutation may mean: In Mathematics, commutation refers to a Commutative operation, where a x b = b x a In Law, commutation refers to a reduction in sentence for a criminal act. ...
Tautology refers to a use of redundant language in speech or writing, or, put simply, saying the same thing twice. Within the study of logic, a tautology is a statement that is true by its own definition. ...
Axiomatization In formal terms, a deduction is a sequence of statements such that each statement can be derived from the preceeding one. This leaves open the question of how to prove the first sentence (since it has no predecessor). Axiomatic propositional logic solves this by requiring the following conditions for a proof:
A proof of α from an ensemble Σ of well-formed formulas (wffs) is a finite sequence of wffs: In logic, WFF is an abbreviation for well-formed formula. ...
- β1,...,βi,...,βn
where - βn = α
and for each βi (1 ≤ i ≤ n), either -
or -
or -
- βi is the output of Modus Ponens for two previous wffs, βi-g and βi-h.
Different versions of axiomatic propositional logics contain a few axioms, usually three or more, in addition to one or more inference rules. For instance, Gottlob Frege's axiomatization of propositional logic, which is also the first instance of such an attempt, has six propositional axioms and two rules. Bertrand Russell and Alfred North Whitehead also suggested a system with five axioms. In Logic, 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: P → Q P ⊢ Q where ⊢ represents the logical assertion. ...
For the algebra software named Axiom, see Axiom computer algebra system. ...
Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ...
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, and mathematician, working mostly in the 20th century. ...
Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England– December 30, 1947 Cambridge, Massachusetts, USA) was an English mathematician who became an American philosopher. ...
For instance a version of axiomatic propositional logic due to Jan Lukasiewicz (1878-1956) has a set A of axioms adopted as follows: The title given to this article is incorrect due to technical limitations. ...
-
- [PL1] p → (q → p)
- [PL2] (p → (q → r)) → ((p → q) → (p → r))
- [PL3] (¬p → ¬q) → (q → p)
and it has the set R of Rules of inference with one rule in it that is Modu Ponendo Ponens as follows: -
- [MP] from α and α → β, infer β.
The inference rule(s) allows us to derive the statements following the axioms or given wffs of the ensemble Σ.
Natural deductive logic One version of natural deductive logic has no axioms. System L, developed by E.J. Lemmon, has only nine primitive rules that govern the syntax of a proof.
The nine primitive rules of system L are:
- The Rule of Assumption (A)
- Modus Ponendo Ponens (MPP)
- The Rule of Double Negation (DN)
- The Rule of Conditional Proof (CP)
- The Rule of ∧-introduction (∧I)
- The Rule of ∧-elimination (∧E)
- The Rule of ∨-introduction (∨I)
- The Rule of ∨-elimination (∨E)
- Reductio Ad Absurdum (RAA)
In system L, a proof has a definition with the following conditions: - has a finite sequence of wffs (well-formed formula)
- each line of it is justified by a rule of the system L
- the last line of the proof is what is intended (Q.E.D, quod erat demonstrandum, is a Latin expression that means: which was the thing to be proved), and this last line of the proof uses the only premise(s) that is given; or no premise if nothing is given.
Then if no premise is given, the sequent is called theorem. Therefore, the definitions of a theorem in system L is: - a theorem is a sequent that can be proved in system L, using an empty set of assumption.
or in other words: - a theorem is a sequent that can be proved from an empty set of assumptions in system L
An example of the proof of a sequent (Modus Tollendo Tollens in this case): | p → q, ¬q ⊢ ¬p [Modus Tollendo Tollens (MTT)] | | Assumption number | Line number | Formula (wff) | Lines in-use and Justification | | 1 | (1) | (p → q) | A | | 2 | (2) | ¬q | A | | 3 | (3) | p | A (for RAA) | | 1,3 | (4) | q | 1,3,MPP | | 1,2,3 | (5) | q ∧ ¬q | 2,4,∧I | | 1,2 | (6) | ¬p | 3,5,RAA | | Q.E.D | An example of the proof of a sequent (a theorem in this case): | ⊢p ∨ ¬p | | Assumption number | Line number | Formula (wff) | Lines in-use and Justification | | 1 | (1) | ¬(p ∨ ¬p) | A (for RAA) | | 2 | (2) | ¬p | A (for RAA) | | 2 | (3) | (p ∨ ¬p) | 2, ∨I | | 1, 2 | (4) | (p ∨ ¬p) ∧ ¬(p ∨ ¬p) | 1, 2, ∧I | | 1 | (5) | ¬¬p | 2, 4, RAA | | 1 | (6) | p | 5, DN | | 1 | (7) | (p ∨ ¬p) | 6, ∨I | | 1 | (8) | (p ∨ ¬p) ∧ ¬(p ∨ ¬p) | 1, 7, ∧I | | (9) | ¬¬(p ∨ ¬p) | 1, 8, RAA | | (10) | (p ∨ ¬p) | 9, DN | | Q.E.D | Each rule of system L has its own requirements for the type of input(s) or entry(es) that it can accept and has its own way of treating and calculating the assumptions used by its inputs.
References - Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, 2005, ISBN 87-991013-7-8
- Jennings, R. E., Continuing Logic, the course book of 'Axiomatic Logic' in Simon Fraser University, Vancouver, Canada
- Zarefsky, David, Argumentation: The Study of Effective Reasoning Parts I and II, The Teaching Company 2002
Vincent F. Hendricks is Professor of Epistemology, Logic and Methodology and member of IIP - Institut Internationale de Philosophie. ...
See also |
Feeds |
Contact