Deductive reasoning

In traditional Aristotelian logic, Deductive reasoning is reasoning in which the conclusion is necessitated by previously known facts - the premises: if the premises are true, the conclusion must be true. This is as opposed to abductive and inductive reasoning, where the premises may predict a high probability of the conclusion, but do not ensure that the conclusion is true.

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.

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", 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 all men are mortal,

and since Socrates is a man,

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 all birds have wings,

and since a cardinal is a bird,

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.

Forms of logical argumentation in math form

Name	Sequent	Description
Basic argument forms of the calculus
Modus Ponens	→ q) ∧ p ⊢ q	if p then q; p; therefore q
Modus Tollens	→ q) ∧ ¬q ⊢ ¬p	if p then q; not q; therefore not p
Hypothetical Syllogism	→ q) ∧ (q → r) ⊢ (p → ''r)	if p then q; if q then r; therefore, if p then r
Disjunctive Syllogism	∨ q) ∧ ¬p ⊢ q	Either p or q; not p; therefore, q
Constructive Dilemma	→ 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	→ 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	→ q) ∧ (p → r) ⊢ → (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)	∨ (q ∨ r) ⊢ ∨ q) ∨ r	p or (q or r) is equiv. to (p or q) or r
Association (2)	∧ (q ∧ r) ⊢ ∧ q) ∧ r	p and (q and r) is equiv. to (p and q) and r
Distribution (1)	∧ (q ∨ r) ⊢ ∧ q) ∨ (p ∧ r)	p and (q or r) is equiv. to (p and q) or (p and r)
Distribution (2)	∨ (q ∧ r) ⊢ ∨ 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) ⊢ → 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) ⊢ ∧ q) ∨ (¬q ∧ ¬p)	(p is equiv. to q) means, either (p and q are true) or ( both p and q are false)
Exportation	∧ q) → r ⊢ → (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	→ (q → r) ⊢ ∧ q) → r
Tautology	p ⊢ (p ∨ p)	p is true is equiv. to p is true or p is true

Axiomatization

In more formal terms, a deduction is a sequence of statements such that every statement can be derived from those before it. It is understandable, then, that this leaves open the question of how we prove the first sentence (since it cannot follow from anything). Axiomatic propositional logic solves this by requiring the following conditions for a proof to be met:

A proof of α from an ensemble Σ of well-formed-formulas (wffs) is a finite sequence of wffs:

β1,...,βi,...,βn

where

βn = α

and for each βi (1 ≤ i ≤ n), either

βi ∈ Σ

βi is an axiom,

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 than three, 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.

For instance a version of axiomatic propositional logic due to Jan Lukasiewicz (1878-1956) has a set A of axioms adopted as follows:

^* p → (q → p)
^* (p → (q → r)) → ((p → q) → (p → r))
^* (¬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:

^* 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

In one version of natural deductive logic presented by E.J. Lemmon that we should refer to it as system L, we do not have any axiom to begin with. We only have 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):

Assumption number	Line number	Formula (wff)	Lines in-use and Justification
p → q, ¬q ⊢ ¬p Tollendo Tollens (MTT)
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):

Assumption number	Line number	Formula (wff)	Lines in-use and Justification
⊢p ∨ ¬p
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

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

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How it works

Forms of logical argumentation in math form

Axiomatization

Natural deductive logic

References

See also