# Posts Tagged ‘Fitch’

## 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.

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)].

Posted by allzermalmer on April 12, 2013

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

“There are truths that cannot be known. For suppose that all truths can be known. Then all truths actually are known. Otherwise, we may suppose for some p that p but it is not known that p. Then it can be known that p but it is not known that p. But when it is known that thus and such, it is known that thus and it is known that such. So it could be known that p and known that it is not known that p. But what is known is true. So it could be known that p and not known that p. But that is a contradiction, and no contradiction can be true. So all truths are actually known.” W.D. Hart

(1) Assume that if X is true then possible to know that X is true. (2) Then, if X is true & do not know that X is true, then possible to know that both X is true & do not know X is true. (3) But, not possible to know that both X is true & do not know X is true. (4) Not both X is true & do not know X is true. (5)  If X is true then do not not know that X is true. (6) If X is true then know that X is true.

What if the World is non-omniscient? This would mean that nobody knows all truths, and nobody ever will. Therefore, there are unknowable truths. If some truth is unknown, then that it is unknown is itself unknowable; Because the world is non-omniscient, there is some unknowable truth. If there at exists at least one Truth, such that Truth is true and Truth is unknown, then there exists at least one Truth, such that Truth is unknown and Truth is unknowable. If there does not exist at least one Truth, such that Truth is unknown and Truth is unknowable, then there does not exist at least one Truth, such that Truth is true and Truth is unknown.

It is possible that it is known by someone at some time that both X is true & It is not known by someone at some time that X is true. It is possible that both It is known by someone at some time that X is true & It is not known by someone at some time that X is true (reduction ad absurdum)

Non-Omniscience: X is true & It is not known by someone at some time that X is true.

Verdicality (KV): If it is known by someone at some time that X is true, then X is true.

Distribution (KC): If it is known by someone at some time that both X is true & Y is true, then both it is known by someone at some time that X is true & It is known by someone at some time that Y is true.

Non-Contradiction (LNC): It is not possible that both X is true & X is not true.

Clousure (CP): If X is true implies Y is true & it is possible that X is true, then it is possible that Y is true.

Knowability (KP): If X is true then it is possible that it is known by someone at some time that X is true.

(1) Assume that X is true & It is not known by someone at some time that X is true

(2) It is possible that it is known by someone at some time that both X is true & It is not known by someone at some time that X is true. (By KP & (1).

(3) It is known by someone at some time that both X is true & It is not known by someone at some time that X is true. It is known by someone at some time that X is true & It is known by someone at some time that it is not known by someone at some time that X is true.

(4) It is known by someone at some time that both X is true & it is not known by someone at some time that X is true. It is known by someone at some time that X is true & It is not known by someone at some time that X is true. (By Simp, VK, and Adjunction (and Transitivity implication))

(5) It is possible that both It is known by someone at some time that X is true & It is not known by someone at some time that X is true. (by CP)

(6) It is not possible that both It is known by someone at some time that X is true & It is not known by someone at some time that X is true. (by LNC)

(7) It is necessary that not both X is true & X is not true.

*(8) X is true & It is known by someone at some time that X is true. (by Reduction Ad Absurdim)

Thus, If X is true, then it is known by someone at some time that X is true:: If it is not known by someone at some time that X is true, then X is not true.