Fundamental Tautologies

First I shall list all the truth tables for basic logical operators. They shall each be given their own symbol as an operator. I will give both two different symbols for them, one for symbolic notation and one in polish notation.

Φ and Ψ will be used as meta-variables, which may be replaced by propositions at any time.

Meta-Variable for proposition Φ:
Given that Φ=True then Φ=True.
Given that Φ=False then Φ=False.

Symbolic (~) and Polish (N): Not..
Given that Φ=True then NΦ=False or (~Φ=False).
Given that Φ=False then NΦ=True or (~Φ=True).

Symbolic(&) and Polish (K): Both…and…
Given that Φ=True and Ψ=True, then KΦΨ=True or (Φ&Ψ)=True.
Given that Φ=True and Ψ=False, then KΦΨ=False or (Φ&Ψ)=False.
Given that Φ=False and Ψ=True, then KΦΨ=False or (Φ&Ψ)=False.
Given that Φ=False and Ψ=False, then KΦΨ=False or (Φ&Ψ)=False.

Symbolic (↓) and Polish (X): Neither…nor…
Given that Φ=True and Ψ=True, then XΦΨ=False or (Φ↓Ψ)=False.
Given that Φ=True and Ψ=False, then XΦΨ=False or (Φ↓Ψ)=False.
Given that Φ=False and Ψ=True, then XΦΨ=False or (Φ↓Ψ)=False.
Given that Φ=False and Ψ=False, then XΦΨ=True or (Φ↓Ψ)=True.

Symbolic (<->) and Polish (E): …if and only if…
Given that Φ=True and Ψ=True, then EΦΨ=True or (Φ<->Ψ)=True.
Given that Φ=True and Ψ=False, then EΦΨ=False or (Φ<->Ψ)=False.
Given that Φ=False and Ψ=False, then EΦΨ=False or (Φ<->Ψ)=False.
Given that Φ=False and Ψ=False, then EΦΨ=True or (Φ<->Ψ)=True.

Symbolic (v) and Polish (A): Either…or…both
Given that Φ=True and Ψ=True, then AΦΨ=True or (ΦvΨ)=True.
Given that Φ=True and Ψ=False, then AΦΨ=True or (ΦvΨ)=True.
Given that Φ=False and Ψ=True, then AΦΨ=True or (ΦvΨ)=True.
Given that Φ=False and Ψ=False, then AΦΨ=False or (ΦvΨ)=False.

Symbolic (↑) and Polish (D): Not both…and…
Given that Φ=True and Ψ=True, then DΦΨor (Φ↑Ψ)=False.
Given that Φ=True and Ψ=False, then DΦΨ or (Φ↑Ψ)=True.
Given that Φ=False and Ψ=True, then DΦΨ or (Φ↑Ψ)=True.
Given that Φ=False and Ψ=False, then DΦΨ or (Φ↑Ψ)=True.

Symbolic (->) and Polish (C): If…then…
Given that Φ=True and Ψ=True, then CΦΨ or (Φ->Ψ)=True.
Given that Φ=True and Ψ=False, then CΦΨ or (Φ->Ψ)=False.
Given that Φ=False and Ψ=True, then CΦΨ or (Φ->Ψ)=True.
Given that Φ=False and Ψ=False, then CΦΨor (Φ->Ψ)=True.

Tautologies:

Symbolic (&) and Polish (K): Both…and…
~(Φ&~Φ)=NKΦNΦ
~(~Φ&Φ)=NKNΦΦ

Symbolic (↓) and Polish (X):Neither…nor…
~(~Φ↓Φ)=NXNΦΦ
~(Φ↓~Φ)=NXΦNΦ

Symbolic (<->) and Polish (E):…if and only if…
(Φ<->Φ)=EΦΦ
(~Φ<->~Φ)=ENΦNΦ

Symbolic (v) and Polish (A):Either…or…both
(Φv~Φ)=AΦNΦ
(~ΦvΦ)=ANΦΦ

Symbolic (↑) and Polish (D):Not both…and…
(~Φ↑Φ)=DNΦΦ
(Φ↑~Φ)=DΦNΦ

Symbolic (->) and Polish (C): If…then…
(Φ->Φ)=CΦΦ
(~Φ->~Φ)=CNΦNΦ

Equivalence:

The order of these equivalence follow those above: (&), (↓), (<->), (v), (->), (↑)

(K) (Φ&Ψ): (Φ&Ψ), (~Φ&~Ψ), ~(Φ&~Ψ)&~(~Φ&Ψ), ~(~Φ&~Ψ), ~(Φ&~Ψ), ~(Φ&Ψ)

(X) (Φ↓Ψ): (~Φ↓~Ψ), (Φ↓Ψ), ~((~Φ↓~Ψ)↓(Φ↓Ψ)), ~(Φ↓Ψ), ~(~Φ↓Ψ), ~(~Φ↓~Ψ)

(A) (ΦvΨ): ~(~Φv~Ψ), ~(ΦvΨ), ~(Φv~Ψ)v~(ΦvΨ), (ΦvΨ), (~ΦvΨ), (~Φv~Ψ)

(D) (Φ↑Ψ): ~(Φ↑Ψ), ~(~Φ↑~Ψ), ~(Φ↑Ψ)↑(Ψ↑~Φ), (~Φ↑~Ψ), (Φ↑~Ψ), (Φ↑Ψ)

(C) (Φ->Ψ): ~(Φ->~Ψ), ~(~Φ->Ψ), ~((Φ->Ψ)->~(Ψ->Φ)), (~Φ->Ψ), (Φ->Ψ), (Φ->~Ψ)

Source of Knowledge and Epistemology

It will be assumed that there are only two sources for knowledge. These sources are both Cognition and Senses.

It will be assumed that each source of knowledge has three possible truth values to be attached to it. The truth values are always true, sometimes true, and never true.

From these two assumptions we are able to derive different epistemological systems that can take on any of these sources or truth values.

1. For all sources of knowledge, if knowledge source is Cognition then knowledge source is always true & if knowledge source is Senses then knowledge source is always true.

2. For all sources of knowledge, if knowledge source is Cognition then knowledge source is always true & if knowledge source is Senses then knowledge source is sometimes true.

3. For all sources of knowledge, if knowledge source is Cognition then knowledge source is always true & if knowledge source is Senses then knowledge source is never true.

4. For all sources of knowledge, if knowledge source is Cognition then knowledge source is sometimes true & if knowledge source is Senses then knowledge source is always true.

5. For all sources of knowledge, if knowledge source is Cognition then knowledge source is sometimes true & if knowledge source is Senses then knowledge source is sometimes true.

6. For all sources of knowledge, if knowledge source is Cognition then knowledge source is sometimes true & if knowledge source is Senses then knowledge source is never true.

7. For all sources of knowledge, if knowledge source is Cognition then knowledge source is never true & if knowledge source is Senses then knowledge source is always true.

8. For all sources of knowledge, if knowledge source is Cognition then knowledge source is never true & if knowledge source is Senses then knowledge source is sometimes true.

9. For all sources of knowledge, if knowledge source is Cognition then knowledge source is never true & if knowledge source is Senses then knowledge source is never true.

It should be made immediately clear that (2) and (4), and (3) and (7), and (6) and (8), are the converse of one another.

It should be made immediately clear that (1) and (9) are contrary to one another, since both may be false but both can’t be true.

It should be made clear that (1) and (5), and (5) and (9) are both contradictory, since both can’t be false but one can be true.  Either (1) or (5) or (9) are true.

Those epistemological hypothesis, like those of (2)-(8), all are fallible. They are sometimes true. It can be supposed that there are different degrees contained within those epistemological hypothesis.

For example, it may be supposed that since some truth comes one of the two sources and that 99% is true or that 1% is true from one of the two sources.

Here is an example of fallibility, i.e. sometimes true, and different degrees of truth contained within it.  Suppose that X stands for Cognition or Senses but not both together.

For all knowledge, if knowledge source is X then knowledge source is 99% true
For all knowledge, if knowledge source is X then knowledge source is 89% true
For all knowledge, if knowledge source is X then knowledge source is 79% true
For all knowledge, if knowledge source is X then knowledge source is 69% true
For all knowledge, if knowledge source is X then knowledge source is 59% true
For all knowledge, If knowledge source is X then knowledge source is 49% true
For all knowledge, if knowledge source is X then knowledge source is 39% true
For all knowledge, if knowledge source is X then knowledge source is 29% true
For all knowledge, if knowledge source is X then knowledge source is 19% true
For all knowledge, if knowledge source is X then knowledge source is 9% true
For all knowledge, if knowledge source is X then knowledge source is 1% true.

Categorical Propositions aren’t Same as Conditional Propositions

It is sometimes held that categorical propositions are equivalent to conditional proposition. However, at least with propositional logic, this isn’t necessarily true.

Categorical proposition: All X are Y.
Conditional proposition: If X then Y.

In other words, it is sometimes held that, All X are Y if and only if X implies Y.

This can be shown false by a very simple method that Categorical propositions aren’t equivalent to Conditional propositions. All we have to do is replace the variables X or Y with Truth Values, and see what the Truth of the proposition as a whole will come out to.

Suppose that X is True and Y is True. Now we replace those variables with their Truth Values in the Statements.

Categorical Proposition: All True are True= True.
Conditional Proposition: If True then True= True.

Suppose that X is False and Y is False.

Categorical Proposition: All False are False=True.
Conditional Proposition: If False then False=True.

Suppose that X is True and Y is False.

Categorical Proposition: All True are False= False.
Conditional Proposition: If True then False= False.

Here is where the Fallacy comes in of thinking Categorical Propositions are equivalent to Conditional Propositions.Suppose that X is False and Y is True.

Categorical Proposition: All False are True= False.
Conditional Proposition: If False then True=True.

We immediately notice that their truth value’s are not equivalent when each variable has the same truth value. This shows that categorical propositions necessarily say something different from conditional propositions.

The only way that Categorical Propositions will say the same thing as Conditional Propositions is if the Subject of the Categorical Proposition isn’t False & the Predicate isn’t True. In other words, the Subject of the Categorical Proposition must Exist.

All Mermaids are creatures that swim in the Ocean if and only if Mermaids implies creatures swim in the Ocean. Mermaids can’t not exist for this equivalency to hold with the Conditional, while the Conditional doesn’t need that Mermaids exist.

Different Models of both Knowledge & Epistemology

One of the differences in epistemology are different theories of knowledge. Renee Descartes helped to present one theory of knowledge, which followed a general form of Rationalism. There is another general form known as Empiricism.

We shall have two Categories and three Truth-values.

Category 1: Cognitive or Cognition
Truth Value: Either Cognition is always true or Cognition is sometimes true & sometimes false or Cognition is always false.

Category 2: Sensory or Senses
Truth Value: Either Senses are always true or Senses are sometimes true & sometimes false or Senses are always false.

Now we can combine both of these categories together to form Both Cognition & Senses, apply the Truth Values, and derive 9 different Models of Epistemology or Knowledge.

Hypothesis *1:
Both cognition is always true & senses are always true.
Both senses are always true & cognition is always true.

Hypothesis *2:
Both cognition is always true & senses are sometimes true and sometimes false.
Both senses are sometimes true and sometimes false & cognition is always true.

Hypothesis *3:
Both cognition is always true & senses are always false.
Both senses are always false & cognition is always true.

Hypothesis 1*:
Both cognition is sometimes true and sometimes false & senses are always true.
Both senses are always true & cognition is sometimes true and sometimes false.

Hypothesis 2*:
Both cognition is sometimes true and sometimes false & senses are sometimes true and sometimes false.
Both senses are sometimes true and sometimes false & cognition is sometimes true and sometimes false.

Hypothesis 3*:
Both cognition is sometimes true and sometimes false & senses are always false.
Both senses are always false & cognition is sometimes true and sometimes false.

Hypothesis *1*:
Both cognition is always false & senses are always true.
Both senses are always true & cognition is always false.

Hypothesis *2*:
Both cognition is always false & senses are sometimes true and sometimes false.
Both senses are sometimes true and sometimes false & cognition is always false.

Hypothesis *3*:
Both cognition is always false & senses are always false.
Both senses are always false & cognition is always false.

These models of knowledge, or epistemology, exhaust all logically possible positions given only these two categories and these three truth values. Some possible subdivisions could be made, especially when either categories, or both, take on the truth value of sometimes true and sometimes false.

One basic idea is that cognition, under Rationalism, would always be true & senses, under empiricism, would always be true.