allzermalmer

Truth suffers from too much analysis

Posts Tagged ‘Inference’

Proof of Modus Ponens

Posted by allzermalmer on August 29, 2013

There is a rule of Inference known as Modus Ponens. It is also sometimes known as rule of Detachment.

When it comes to rules of inference, some rules are taking as primitive and some are taken as derived. A primitive rule is one in which no proof is given for the rule, and a derived rule is one that is derived from the primitive rule.

The proof of Modus Ponens, which is to be given here, takes Modus Ponens as a derived rule and will use Disjunctive Syllogism as a primitive rule. So from Two premises and the primitive rule of Disjunctive Syllogism, Modus Ponens as a rule of inference can be derived.

Before we go through the proof, Modus Ponens form of argumentation shall be shown.

(1.) P–>Q [Premise]
(2.) P         [Premise]
(3.) Q         [Conclusion]

Here is an example of Disjunctive Syllogism.

(1.) ~PvQ [Premise]
(2.) P        [Premise]
(3.) Q       [Conclusion]

So Disjunctive Syllogism shall be used as the primitive rule, and from this primitive rule will be able to derive the inference rule of Modus Ponens.

Now there is one point that needs to be gone over first, which is that of Modus Ponens works with conditional statements, or if…then statements. Disjunctive Syllogism works with disjunctive statements, or ‘or’ statements. Conditional statements have equivalent forms of disjunctive statements. This is how conditional statements are known as material implications statements.

So, P–>Q is a material implication, and thus can be switched into it’s disjunctive form. So P–>Q, as a material implication, states that ~PvQ. This equivalence shall be used in the proof of modus ponens.

(1.) P–>Q [Premise]
(2.) P         [Premise]
(3.) ~PvQ [Definition of (1)]
(4.) Q       [(2)-(3) by Disjunctive Syllogism] [Conclusion]

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Proof of Modus Tollens

Posted by allzermalmer on July 28, 2013

Language

(I) Symbols: Ø = contradiction, → = conditional, and [] = Modal Operator
(II) Variables: p, q, r, p’, q’, r’. (Variables lower case)

Well Formed Formula for Language

(i) Ø and any variable is a modal sentence.
(ii) If A is a modal sentence, then []A is a modal sentence.
(iii) If A is a modal sentence and B is a modal sentence, then A implies B (A→B) is a modal sentence.

* A, B, and C are modal sentences, i.e. upper case letters are modal sentences. These upper case letters are “variables as well”. They represent the lower case variables in conjunction with contradiction, conditional, or modal operator.

So A may possibly stand for p, or q, or r. It may also possibly stand for a compound of variables and symbols. So A may stand for q, or A may stand for p→Ø, and etc.

Negation (~) = A→Ø
Conjunction (&) = ~(A→B)
Disjunction (v) = ~A→B
Biconditional (↔) = (A→B) & (B→A)

Because Ø indicates contradiction, Ø is always false. But by the truth table of material implication, A → Ø is true if and only if either A is false or Ø is true. But Ø can’t be true. So A → Ø is true if and only if A is false.

This symbol ∞ will stand for something being proved.

(1) Hypothesis (HY) : A new hypothesis may be added to a proof anytime, but the hypothesis begins a new sub-proof.

(2) Modus Ponens (MP) : If A implies B and A, then B must lie in exactly the same sub-proof.

(3) Conditional Proof (CP): When proof of B is derived from the hypothesis A, it follows that A implies B, where A implies B lies outside hypothesis A.

(4) Double Negation (DN): Removal of double negation ~~A & A lie in the same same sub-proof.

(5) Reiteration (R): Sentence A may be copied into a new sub-proof.

Proof of Modus Tollens: Given the conditional claim that the consequent is true if the antecedent is true, and given that the consequent is false, we can infer that the antecedent is also false.

(If p implies q & ~q, then necessarily true that ~p)

Premise (1) p implies q (Hypothesis)
Premise (2) ~q (Hypothesis)
(3) q implies Ø ((2) and of Definition ~)
(4) p (Hypothesis)
(5) p implies q (Reiteration of (1))
(6) q (Modus Ponens by (4) and (5))
(7) q implies Ø (Reiteration of (3))
(8) Ø (Modus Ponens by (6) and (7))
(9) p implies Ø ( Conditional Proof by  (5) through (8))
Conclusion (10) ~p ((9) and Definition of ~)

Shortened version, with some steps omitted, would go as follows.

P (1) p implies q
P (2) ~q
(3) q implies Ø ((2) and Definition of ~)
(4) p (Hypothesis)
(5) q (Modus Ponens by (1) and (4))
(6) Ø (Modus Ponens by (3) and (5))
(7) p implies Ø (Conditional Proof by (3) through (6))
C (8)  ~p ((7) and Definition ~)

Here is an even shorter proof of Modus Tollens, and it only requires the rule of inference of Hypothetical Syllogism:

(1) p implies q (Hypothesis)
(2) q implies Ø (Hypothesis)
(3) p implies Ø (Hypothetical Syllogism by (1) and (2))
(4) ~p (Reiteration of (3) by Definition of ~)

So we have proved that If p implies q and ~q, then ~p is necessarily true.

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Logically Valid Arguments

Posted by allzermalmer on April 8, 2013

Categorically Valid Syllogisms

M stands for Middle Term; P stands for Major Term; S stands for Minor Term

Figure 1

(1) Barabara:If all M are P and all S are M, then all S are P
P. All M are P
P. All S are M
C. All S are P

(2) Celarent: If no M are P and all S are M, then no S are P
P. No M are P
P. All S are M
C. No S are P

(3) Darii: If all M are P and some S are M, then some S are P
P. All M are P
P. Some S are M
C. Some S are P

(4) Ferio: If no M are P and some S are M, then some S are not P
P. No M are P
P. Some S are M
C. Some S are not P

Figure 2

(1) Camestres: If all P are M and no S are M, then no S are P
P. All P are M
P. No S are M
C. No S are P

(2) Cesare: If no P are M and all S are M then no S are P
P. No P are M
P. All S are M
C. No S are P

(3) Baroko: If all P are M and some S are not M, then some S are not P
P. All P are M
P. Some S are not M
C. Some S are not P

(4) Festino: If no P are M and some S are M, then some S are not P
P. No P are M
P. Some S are M
C. Some S are not P

Figure 3

(1) Datisi: If all M are P and some M are S, then some S are P
P. All M are P
P. Some M are S
C. Some S are P

(2) Disamis: If some M are P and all M are S, then some S are P
P. Some M are P
P. Some M are S
C. Some S are P

(3) Ferison: if no M are P and some M are S, then some S are not P
P. No M are P
P. Some M are S
C. Some S are not P

(4) Bokardo: If some M are not P and all M are S, then some S are not P
P. Some M are not P
P. All M are S
C. Some S are not P

Figure 4

(1) Camenes: If all P are M and no M are S, then no S are P
P. All P are M
P. No M are S
C. No S are P

(2) Dimaris: If some P are M and all M are S, then Some S are P
P. Some P are M
P. All M are S
C. Some S are P

(3) Fresison: If no P are M and some M are S, then some S are not P
P. No P are M
P. Some M are S
C. Some S are not P

Propositional Logic

Modus Ponens: Given the conditional claim that the consequent is true if the antecedent is true, and given that the antecedent is true, we can infer the consequent.
P. If S then P
P. S
C. Q

Modus Tollens: Given the conditional claim that the consequent is true if the antecedent is true, and given that the consequent is false, we can infer that the antecedent is also false.
P. If S then P
P. Not P
C. Not S

Hypothetical Syllogism: Given two conditional such that the antecedent of the second is the consequent of the first, we can infer a conditional such that its antecedent of the first premise and its consequent is the same as the consequent of the second premise.
P. If S then M
P. If M then P
C. If S then P

Constructive Dilemma: Given two conditionals, and given the disjunction of their antecedents, we can infer the disjunction of their consequents.
P. If S then P                 P. If S then P
P. If M then N               P. If M then P
P. S or M                        P. S or M
C. P or N                        C. P or P

Destructive Dilemma: Given two conditionals, and given the disjunction of the negation of their consequents, we can infer the disjunction of the negation of their antecedents.
P. If S then P                P. If S then P
P. If M then N              P. If S then N
P. Not P or Not N        P. Not P or Not N
C. No S or Not M        C. Not S or Not S

Biconditional Argument: Given a biconditional and given the truth value of one side is known, we can infer that the other side has exactly the same truth value.
P. S<–>P    P. S<–>P   P. S<–>P   P. S<–>P
P. S               P. P              P. Not S      P. Not P
C. P              C. S               C. Not P     C. Not S

Disjunctive Addition: Given that a statement is true, we can infer that a disjunct comprising it and any other statement is true, because only one disjunct needs to be true for the disjunctive compound to be true.
P. S
C. S or P

Disjunctive Syllogism: Because at least one disjunct must be true, by knowing one is false we can infer tat the other is true.
P. S or P   P. S or P
P. Not P   P. Not S
C. S          C. P

Simplification: Because both components of a conjunctive argument are true, it is permissible to infer that either of its conjuncts is true.
P. S & P   P. S & P
C. S          C. P

Adjunction: Because both premises are presumed true, we can infer their conjunction.
P. S
P. P
C. S & P

Conjunctive Argument: Because the first premise says that at least one of the conjuncts is false and the second premise identifies a true conjunct, we can infer that the other conjunct is false.
P. ~(S & P)   P. ~(S & P)
P. S                P. P
C. Not P        C. Not S

 

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