allzermalmer

Truth suffers from too much analysis

Posts Tagged ‘Rules of 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]

Advertisements

Posted in Philosophy | Tagged: , , , , , , , , | Leave a Comment »

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

 

Posted in Philosophy | Tagged: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a Comment »