# Posts Tagged ‘Simplification’

## Desiring, Believing, Knowing, Obligation, and Fitch’s Paradox

Posted by allzermalmer on April 26, 2013

Assume that Striving, Doing, Believing, & Knowing (SDBK) have some fairly simple properties.
Assume that (SDBK) Striving, Doing, Believing, & Knowing are two-termed relations between an Agent and a Possible State of Affairs.

It shall be a convention to treat Possible State of Affairs as Propositions. So, Φ is assumed to be an agent and  p is assumed to be a proposition.
(i) “Φ strives for p” is equivalent to saying “Φ strives to bring about or realize the (possible) state of affairs expressed by the proposition p.”
(ii) “Φ does p” is equivalent to saying “Φ brings about the (possible) state of affairs expressed by the proposition p.”
(iii) “Φ believes p” is equivalent to saying “Φ believes about or realize the (possible) state of affairs expressed by the proposition p.”
(iv) “Φ knows p” is equivalent to saying “Φ knows about or realize the (possible) state of affairs expressed by the proposition p.”

It shall be a convention to ignore the Agent and treat concepts to be considered, i.e. Striving, Doing, Believing, & Knowing (SDBK), as a “Class of Propositions” instead of Two-Termed relation.
(i) ‘Striving’ means the Class of Propositions striven for (that is striven to be realized).
(ii) ‘Believing’ means the Class of Propositions believed.
(iii) ‘Doing’ means the Class of Propositions doing (that is doing to be realized).
(iv) ‘Knows’ means the Class of Propositions known.

Here are two basic rules of Logical Inference in propositional logic. They are known, respectively, as Simplification and Adjunction.

Simplification: Because both components of a conjunctive argument are true, it is permissible to infer that either of it conjuncts is true.
(Premise): Germany Lost World War One & Germany Lost World War Two
(Conclusion): Germany Lost World War One

(Premise): Germany Lost World War One & Germany Lost World War Two
(Conclusion): Germany Lost World War Two

Adjunction: Because both premises are presumed true, we can infer their conjunction.
(Premise): Germany Lost World War One
(Premise): Germany Lost World War Two
(Conclusion): Germany Lost World War One & Germany Lost World War Two

Assume p and q are variables for two different propositions. Assume Ω stand for Class of Propositions, which can be either Striving, Believing, Obligation, and Knowing, or etc. Assume → stands for Strict Implication, Logical Entailment, Entailment, Logical Consequence.

Closed with Respect to Conjunction Elimination

Class of Propositions are Closed with Respect to Conjunction Elimination whenever a conjunctive proposition is in the Class, then those two propositions themselves are in that Class. Closed with Respect to Conjunction Elimination follows the logical inference of Simplification, but it uses one the relation of (SDBK).

Class of Propositions Closed with Respect to Conjunction Elimination:
(p)(q)[(Ω[p & q]) → [(Ωp) & (Ωq)]]

Assume that Ω stands for the Class of Propositions “know”. So the Class of Propositions (know) Closed with Respect to Conjunction Elimination means that “If (know both p & q) then logically entails (know p) & (know q)”. We can replace Ω with “Believe”, “Striving”, “Doing”, or the others listed.

Class of Propositions covered by Closed with Respect to Conjunction Elimination are: Striving (for), Doing, Believing, Knowing, Proving, Truth, Causal Necessity (in the sense of Burks), Causal Possibility ( in the sense of Burks), (Logical) Necessity, (Logical) Possibility, Obligation (Deontic Necessity), Permission (Deontic Possibility), Desire (for),

Closed with Respect to Conjunction Elimination

Class of Propositions Closed with Respect to Conjunction Introduction whenever two propositions are in the class, then so is the conjunction of the two propositions. Closed with Respect to Conjunction Introduction follows the logical inference of Adjunction, but it uses one of the relations of (SDBK).

Class of Propositions Closed with Respect to Conjunction Introduction:
(p)(q)[[(Ωp) & (Ωq)] → (Ω[p & q])]

Assume Ω stands for the Class of Propositions “know”. So the Class of Propositions (know) Closed with Respect to Conjunction Introduction means that “If (know p) and (know q), then logically entails (Know both p & q).” We can replace Ω with “Believe”, “Strive”, “Doing”, or others listed.

Class of Propositions covered by Closed with Respect to Conjunction Introduction are:
Truth, Causal Necessity (in the sense of Burks), Logical Necessity, Obligation (Deontic Necessity).

Class of Propositions possibly covered by Closed with Respect to Conjunction Introduction are: Striving (for), Doing, Believing, Knowing, Proving, Desire (for).

Class of Propositions not covered by Closed with Respect to Conjunction Introduction are: Causal possibility (in the sense of Burks), Logical Possibility, and Permission (Deontic Possibility).

Truth Class

Class of Propositions are a Truth Class if every member of it is true.

Class of Propositions Truth Class:
(p)[(Ωp) → p]

Assume Ω stands for Class of Propositions Truth Class “knows”. So the Class of Propositions Truth Class (knows) means “If (knows p) then logical entails p.”

Class of Propositions Truth Class are: Truth, Causal Necessity (in the sense of Burks), Logical Necessity, Knowing, Done, and Proving.

Theorems about Truth Classes

Theorem 1: If (Class of Propositions) is a Truth Class which is Closed with Respect to Conjunction Elimination, then the proposition, [p & ~(Ωp)], which asserts that p is true but not a member of (Class of Propositions) (where p is any proposition), is itself necessarily not a member of (Class of Propositions).

Proof: Suppose the contrary, [p & ~(Ωp)], is a member of (Class of Propositions), i.e. suppose that (Ω[p & ~(Ωp)]) is a member of (Class of Propositions). Since (Class of Propositions) are Closed with Respect to Conjunction Elimination, the propositions p and ~(Ωp) must both be members of (Class of Propositions), so that the propositions (Ωp) and (Ω(~(Ωp))) must both be true. But the fact that (Class of Propositions) is a truth class and has ~(Ωp) is true, and this contradicts the result that (Ωp) is true. Thus from the assumption that [p & ~(Ωp)] is a member of (Class of Propositions) we have derived contradictory results. Hence, that assumption is necessarily false.

Theorem 2: If (Class of Propositions) is a Truth Class which is Closed with Respect to Conjunction Elimination, and if p is a true proposition which is not a member of (Class of Proposition), then the proposition, [p & ~(Ωp)], is a true proposition which is necessarily not a member of (Class of Propositions).

Proof: The proposition [p & ~(Ωp)] is clearly true, and by Theorem 1 it is necessarily not a member of (Class of Propositions).

Omnipotent and Fallibility

Theorem 3: If an Agent is all-powerful in the sense that for each situation that is the case, it is logically possible that that situation was brought about by that Agent, then whatever is the case was brought about (done) by that Agent.

Proof: Suppose that p is the case but was not brought about by the agent in question. Then, since (doing) is a Truth Class Closed with Respect to Conjunction Elimination, we conclude from Theorem 2 that there is some actual situation which could not have been brought about by that Agent, and hence that Agent is not all-powerful in the sense described. Hence, that assumption is necessarily false.

Theorem 4: For each Agent who is not omniscient, there is a true proposition which that Agent cannot know.

Proof: Suppose that p is true but not known by the Agent. Then, since (knowing) is a Truth Class Closed with Respect to Conjunction Elimination, we conclude from Theorem 2 that there is some true proposition which cannot be known by the Agent.

Theorem 5: If there is some true proposition which nobody knows (or has known or will know) to be true, then there is a true proposition which nobody can know to be true.

Proof: Similar to proof in Theorem 4.

Proved True Never Proved

Theorem 6: If there is some true proposition about proving that nobody has ever proved or ever will prove, then there is some true proposition about proving that nobody can prove.

Proof: Similar to the proof in Theorem 4, using the fact that if p is a proposition about proving, so is [p & ~(Ωp)].

## 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