Infinite-valued Logics

Equation form expr-01208e159d2aec8c

\land

Read as: conjunction

Means: conjunction

2 occurrences in this chapter

Equation form expr-043b130790fae68b

\lnot !A \lor !B

Read as: the disjunction of not A with B

Means: the disjunction of not A with B

1 occurrence in this chapter

Occurrence projected-formula-0024133

Equation form expr-05337f415c1099c6

\tf{\lfalse} & = 0\\ \tf{\lnot}[\LogGod](x) & = \begin{cases} $1$ & \text{if } x =0\\ $0$ & \text{otherwise} \end{cases}\\ \tf{\land}[\LogGod](x,y) & = \min(x,y)\\ \tf{\lor}[\LogGod](x,y) & = \max(x,y)\\ \tf{\lif}[\LogGod](x,y) & = \begin{cases} 1 & \text{if } x \le y\\ y & \text{otherwise.} \end{cases}

Read as: Goedel truth functions. Falsity has value zero. Negation of x has value one if x equals zero, and zero otherwise. Conjunction of x and y has the minimum of x and y as its value. Disjunction has their maximum as its value. The conditional with antecedent value x and consequent value y has value one if x is less than or equal to y, and value y otherwise. End of truth function definitions.

Means: Goedel truth functions. Falsity has value zero. Negation of x has value one if x equals zero, and zero otherwise. Conjunction of x and y has the minimum of x and y as its value. Disjunction has their maximum as its value. The conditional with antecedent value x and consequent value y has value one if x is less than or equal to y, and value y otherwise. End of truth function definitions.

1 occurrence in this chapter

Occurrence projected-formula-0024146

Equation form expr-069ac2aed6bc166f

\tf{\lif}[\LogLuk[3]]

Read as: the three valued Lukasiewicz conditional truth function

Means: the three valued Lukasiewicz conditional truth function

1 occurrence in this chapter

Occurrence projected-formula-0024114

Equation form expr-0fc07e559bd4cc67

m \ge 2

Read as: m greater than or equal to two

Means: m greater than or equal to two

2 occurrences in this chapter

Equation form expr-16f38629e1f81ade

\LogLuk[3]

Read as: three valued Lukasiewicz logic

Means: three valued Lukasiewicz logic

2 occurrences in this chapter

Equation form expr-1a47830e263c0c3b

(p \lif q) \lor (q \lif r) \lor (r \lif s)

Read as: the disjunction of the three conditionals: if p then q; if q then r; and if r then s

Means: the disjunction of the three conditionals: if p then q; if q then r; and if r then s

1 occurrence in this chapter

Occurrence projected-formula-0024222

Equation form expr-1c09ff1165c1a636

V_{[0,1]} = [0,1]

Read as: V subscript the closed interval from zero to one equals that closed interval

Means: V subscript the closed interval from zero to one equals that closed interval

1 occurrence in this chapter

Occurrence projected-formula-0024057

Equation form expr-1d06c3cd92e108cd

\LogGod[\infty]

Read as: G subscript infinity

Means: G subscript infinity

8 occurrences in this chapter

Equation form expr-20ddfc618734c4ad

\tf{\lor}[\LogGod]

Read as: the Goedel disjunction truth function

Means: the Goedel disjunction truth function

1 occurrence in this chapter

Occurrence projected-formula-0024174

Equation form expr-284c47be48fb0c66

\lnot

Read as: negation

Means: negation

2 occurrences in this chapter

Equation form expr-2f31c694e82f1471

V_\infty & = \Setabs{\frac{n}{m}}{n,m \in \Nat \text{ and } n\le m}. \intertext{When considering this infinite truth value set, it is often useful to also consider the subsets} V_m & = \Setabs{\frac{n}{m-1}}{n \in \Nat \text{ and } n\le m} \intertext{For instance, $V_5$ is the set with $5$ evenly spaced truth values,} V_5 & = \{0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\}.

Read as: V subscript infinity equals the set of fractions n divided by m such that n and m are natural numbers and n is less than or equal to m. When considering this infinite truth value set, it is often useful to also consider the subsets V subscript m, which the source writes as the set of n divided by the quantity m minus one, such that n is a natural number and n is less than or equal to m. For instance, V subscript five is the set with five evenly spaced truth values: V subscript five equals the set containing zero, one quarter, one half, three quarters, and one. End of displayed value sets.

Means: V subscript infinity equals the set of fractions n divided by m such that n and m are natural numbers and n is less than or equal to m. When considering this infinite truth value set, it is often useful to also consider the subsets V subscript m, which the source writes as the set of n divided by the quantity m minus one, such that n is a natural number and n is less than or equal to m. For instance, V subscript five is the set with five evenly spaced truth values: V subscript five equals the set containing zero, one quarter, one half, three quarters, and one. End of displayed value sets.

1 occurrence in this chapter

Occurrence projected-formula-0024051

Equation form expr-3163c74b28295918

\tf{\lor}[\LogLuk[3]]

Read as: the three valued Lukasiewicz disjunction truth function

Means: the three valued Lukasiewicz disjunction truth function

1 occurrence in this chapter

Occurrence projected-formula-0024098

Equation form expr-463f2998327eb3a6

[0,1]

Read as: the closed interval from zero to one

Means: the closed interval from zero to one

1 occurrence in this chapter

Occurrence projected-formula-0024060

Equation form expr-529ad2daacd7efcb

\lor

Read as: disjunction

Means: disjunction

2 occurrences in this chapter

Equation form expr-58197ff8ca27696b

\tf{\lnot}

Read as: the negation truth function

Means: the negation truth function

1 occurrence in this chapter

Occurrence projected-formula-0024075

Equation form expr-59762fad561b366c

\Log{LC}

Read as: L C

Means: L C

1 occurrence in this chapter

Occurrence projected-formula-0024219

Equation form expr-5d94be88a66c614f

\tf{\lif}[\LogGod]

Read as: the Goedel conditional truth function

Means: the Goedel conditional truth function

1 occurrence in this chapter

Occurrence projected-formula-0024190

Equation form expr-5feceb66ffc86f38

0

Read as: zero

Means: zero

35 occurrences in this chapter

Equation form expr-62c66a7a5dd70c31

m

Read as: m

Means: m

2 occurrences in this chapter

Equation form expr-6b4edfc5fe38d079

\LogGod[3]

Read as: three valued Goedel logic

Means: three valued Goedel logic

5 occurrences in this chapter

Equation form expr-6b86b273ff34fce1

1

Read as: one

Means: one

46 occurrences in this chapter

Equation form expr-79b01b537984bd57

V_\infty = [0,1] \cap \Rat

Read as: V subscript infinity equals the intersection of the closed interval from zero to one with the rational numbers

Means: V subscript infinity equals the intersection of the closed interval from zero to one with the rational numbers

1 occurrence in this chapter

Occurrence projected-formula-0024050

Equation form expr-8124eb9a8372e942

\tf{\land}[\LogGod]

Read as: the Goedel conjunction truth function

Means: the Goedel conjunction truth function

1 occurrence in this chapter

Occurrence projected-formula-0024158

Equation form expr-82623f345da734fd

\tf{\land}[\LogLuk[3]]

Read as: the three valued Lukasiewicz conjunction truth function

Means: the three valued Lukasiewicz conjunction truth function

1 occurrence in this chapter

Occurrence projected-formula-0024082

Equation form expr-87b3f0a70b64ebaf

\tf{\lnot}[\LogGod[3]]

Read as: the three valued Goedel negation truth function

Means: the three valued Goedel negation truth function

1 occurrence in this chapter

Occurrence projected-formula-0024151

Equation form expr-8fb9f63317f1a904

\tf{\lnot}[\LogLuk](x) & = 1 - x\\ \tf{\land}[\LogLuk](x,y) & = \min(x,y)\\ \tf{\lor}[\LogLuk](x,y) & = \max(x,y)\\ \tf{\lif}[\LogLuk](x,y) & = \min(1,1-(x-y)) = \begin{cases} 1 & \text{if } x \le y\\ 1-(x-y) & \text{otherwise.} \end{cases}

Read as: Lukasiewicz truth functions. Negation of x has value one minus x. Conjunction of x and y has the minimum of x and y as its value. Disjunction has their maximum as its value. The conditional with antecedent value x and consequent value y has value the minimum of one and the quantity one minus the difference x minus y. Equivalently, its value is one if x is less than or equal to y, and one minus the difference x minus y otherwise. End of truth function definitions.

Means: Lukasiewicz truth functions. Negation of x has value one minus x. Conjunction of x and y has the minimum of x and y as its value. Disjunction has their maximum as its value. The conditional with antecedent value x and consequent value y has value the minimum of one and the quantity one minus the difference x minus y. Equivalently, its value is one if x is less than or equal to y, and one minus the difference x minus y otherwise. End of truth function definitions.

1 occurrence in this chapter

Occurrence projected-formula-0024070

Equation form expr-9244c801c50f58fd

(p \lif q) \lor (q \lif p)

Read as: the disjunction of if p then q with if q then p

Means: the disjunction of if p then q with if q then p

3 occurrences in this chapter

Equation form expr-9ef1dd9e99e03eb0

\Gamma \Entails[\LogLuk[m]] !B

Read as: Gamma entails B in m valued Lukasiewicz logic

Means: Gamma entails B in m valued Lukasiewicz logic

1 occurrence in this chapter

Occurrence projected-formula-0024131

Equation form expr-ab2c0c588ded3e79

V = V_m

Read as: V equals V subscript m

Means: V equals V subscript m

2 occurrences in this chapter

Equation form expr-ad9293b54c2b636f

(!A \lif !B) \lor (!B \lif !A)

Read as: the disjunction of if A then B with if B then A

Means: the disjunction of if A then B with if B then A

1 occurrence in this chapter

Occurrence projected-formula-0024217

Equation form expr-b1c615d83e93e6d0

\Gamma \Entails[\LogGod[\infty]] !B

Read as: Gamma entails B in infinite valued Goedel logic

Means: Gamma entails B in infinite valued Goedel logic

1 occurrence in this chapter

Occurrence projected-formula-0024206

Equation form expr-b85f62f6e470cee8

\Gamma \Entails[\LogLuk[\infty]] !B

Read as: Gamma entails B in infinite valued Lukasiewicz logic

Means: Gamma entails B in infinite valued Lukasiewicz logic

1 occurrence in this chapter

Occurrence projected-formula-0024130

Equation form expr-bd91dd8f754d52fc

V_\infty

Read as: V subscript infinity

Means: V subscript infinity

2 occurrences in this chapter

Equation form expr-be94b5694f7121c8

V^+ = \{1\}

Read as: the designated value set V plus is the singleton set containing one

Means: the designated value set V plus is the singleton set containing one

3 occurrences in this chapter

Equation form expr-cdc2ed7d3b3d72c2

\lfalse

Read as: the falsity constant

Means: the falsity constant

1 occurrence in this chapter

Occurrence projected-formula-0024138

Equation form expr-d04ff80d9f6dc462

\lif

Read as: conditional

Means: conditional

2 occurrences in this chapter

Equation form expr-d0d17125f775c2da

\Gamma \Entails[\LogGod[m]] !B

Read as: Gamma entails B in m valued Goedel logic

Means: Gamma entails B in m valued Goedel logic

1 occurrence in this chapter

Occurrence projected-formula-0024207

Equation form expr-d4e220a9305a8728

\LogLuk[\infty]

Read as: L subscript infinity

Means: L subscript infinity

2 occurrences in this chapter

Equation form expr-d72c2cedf15e5fa4

\Lang L_0

Read as: L subscript zero

Means: L subscript zero

2 occurrences in this chapter

Equation form expr-d939926f05444b0f

1/2

Read as: one half

Means: one half

30 occurrences in this chapter

Equation form expr-de5a6f78116eca62

V

Read as: V

Means: V

1 occurrence in this chapter

Occurrence projected-formula-0024056

Equation form expr-e54931a36854f35f

& p \lor \lnot p && (p \lif q) \lif (\lnot p \lor q) \\ & \lnot\lnot p \lif p && \lnot(\lnot p \land \lnot q) \lif (p \lor q) \\ & ((p \lif q) \lif p) \lif p && \lnot(p \lif q) \lif (p \land \lnot q)

Read as: Six formulas, read across each source row from left to right. Row one, left: p or not p. Row one, right: if the conditional if p then q holds, then not p or q. Row two, left: if not not p then p. Row two, right: if it is not the case that both not p and not q, then p or q. Row three, left: if the conditional from if p then q to p holds, then p. Row three, right: if it is not the case that if p then q, then both p and not q. End of six formulas.

Means: Six formulas, read across each source row from left to right. Row one, left: p or not p. Row one, right: if the conditional if p then q holds, then not p or q. Row two, left: if not not p then p. Row two, right: if it is not the case that both not p and not q, then p or q. Row three, left: if the conditional from if p then q to p holds, then p. Row three, right: if it is not the case that if p then q, then both p and not q. End of six formulas.

1 occurrence in this chapter

Occurrence projected-formula-0024212

Infinite and finite sets of truth values

The display gives a rational value set, a finite evenly spaced value set, and the five element example zero, one quarter, one half, three quarters and one. The source finite-set upper bound and omitted denominator restriction are retained with a caveat.

Source

Infinite and finite valued Lukasiewicz matrices

Only one is designated. Negation is one minus the input, conjunction is minimum, disjunction is maximum, and the conditional is the minimum of one and one minus the difference of antecedent and consequent values. The finite version restricts the value set. The source identifies this section as a stub.

Source

Lukasiewicz truth function equations

The displayed conditional has two branches: one when the antecedent value is at most the consequent value, and one minus their difference otherwise. The subtraction groups the antecedent minus consequent difference as a whole.

Source

Agreement of the two three valued Lukasiewicz definitions

The proposition identifies the earlier symbolic true, undefined and false truth tables with the numeric values one, one half and zero, respectively. Four tables demonstrate the agreement.

Source

Numeric Lukasiewicz negation table

In source input order one, one half, zero, negation gives zero, one half, one respectively.

Source

Numeric Lukasiewicz conjunction table

Both input axes are ordered one, one half, zero. Each result is the lesser input. The full source row sequence is retained in the separate table reading.

Source

Numeric Lukasiewicz disjunction table

Both input axes are ordered one, one half, zero. Each result is the greater input. The full source row sequence is retained in the separate table reading.

Source

Numeric Lukasiewicz conditional table

Rows are antecedent values and columns consequent values, each ordered one, one half, zero. The middle row is one, one, one half, distinguishing this conditional from the Goedel conditional.

Source

From infinite to finite Lukasiewicz consequence

Consequence in infinite valued Lukasiewicz logic implies consequence in each m valued version for every m at least two. The proof is left as an exercise; the subsequent source sentence also states the converse without proving it.

Source

Exercise on restricting Lukasiewicz truth values

Prove that infinite valued consequence implies consequence for every finite number of values at least two. No solution is supplied.

Source

Exercise on Lukasiewicz prelinearity

Show that the disjunction of if p then q and if q then p is a tautology of infinite valued Lukasiewicz logic. No solution is supplied.

Source

Infinite and finite valued Goedel matrices

Only one is designated. Falsity is zero; negation is one only at zero and otherwise zero; conjunction is minimum; disjunction is maximum. The conditional is one when antecedent is at most consequent and otherwise equals the consequent. The source identifies this section as a stub.

Source

Goedel truth function equations

The equations include a zero-valued falsity constant and explicit two-branch definitions of negation and conditional. Negation at a positive intermediate value is zero, not its complement.

Source

Agreement of the two three valued Goedel definitions

The proposition identifies the earlier symbolic truth values with one, one half and zero. The following four numeric tables show the same truth functions.

Source

Numeric Goedel negation table

In source input order one, one half, zero, negation gives zero, zero, one respectively.

Source

Numeric Goedel conjunction table

Both axes are ordered one, one half, zero, and each cell is their minimum. All rows and cells are explicitly linearized.

Source

Numeric Goedel disjunction table

Both axes are ordered one, one half, zero, and each cell is their maximum. All rows and cells are explicitly linearized.

Source

Numeric Goedel conditional table

Rows are antecedent values and columns consequent values, each ordered one, one half, zero. The middle row is one, one, zero. That final zero differs from the one half in Lukasiewicz logic.

Source

From infinite to finite Goedel consequence

Consequence in infinite valued Goedel logic implies consequence in each m valued version for every m at least two. The source leaves the proof to an exercise and subsequently states the converse without proof.

Source

Exercise on restricting Goedel truth values

Prove the preceding implication from infinite valued Goedel consequence to every finite valued version with at least two values. No solution is supplied.

Source

Six non-tautologies of Goedel logic

The source arranges six formulas in three rows and two columns. They are read across each row, left then right, retaining scope. The list includes excluded middle, double negation elimination and Peirce law among other formulas.

Source

Exercise on Goedel prelinearity

Show that the disjunction of if p then q with if q then p is a tautology of infinite valued Goedel logic. No solution is supplied.

Source

Exercise separating finite and infinite Goedel tautologies

Show that the disjunction of the three conditionals from p to q, q to r, and r to s, which is a three valued Goedel tautology, is not an infinite valued Goedel tautology. No solution is supplied.

Source