Classic-like Analytic Tableaux for Non-Classical Logics Joo Marcos - - PowerPoint PPT Presentation

Classic-like Analytic Tableaux for Non-Classical Logics

João Marcos

UFRN, BR & RUB, DE

PhDs in Logic IX Bochum

2–4 May 2017

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Tableaux are a kind of magic

dedicated to Raymond Smullyan

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Tableaux are super-easy

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Tableaux are super-easy

A logical matrix for Classical Logic: (where D “ t1u)

• 1

1 Ñ 1 1 1 1 1

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Tableaux are super-easy

A logical matrix for Classical Logic: (where D “ t1u)

• 1

1 Ñ 1 1 1 1 1

Tableaux:

pαq α α Ñ β α β pα Ñ βq α β α α ¸

Exercises:

Is pp Ñ qq Ñ r $ p Ñ pq Ñ rq provable? What about its converse? Check if $ pp Ñ pq Ñ p is provable.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Tableaux are super-easy

A logical matrix for Classical Logic: (where D “ t1u)

• 1

1 Ñ 1 1 1 1 1

Signed Tableaux:

F:α T:α T:α F:α T:α Ñ β F:α T:β F:α Ñ β T:α F:β T:α F:α ¸

Exercises:

Is pp Ñ qq Ñ r $ p Ñ pq Ñ rq provable? What about its converse? Check if $ pp Ñ pq Ñ p is provable.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Tableaux are super-easy

A logical matrix for Classical Logic: (where D “ t1u)

• 1

1 Ñ 1 1 1 1 1

Signed Tableaux:

F:α T:α T:α F:α T:α Ñ β F:α T:β F:α Ñ β T:α F:β T:α F:α ¸

Feature: Subformula Property

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

An infinite-valued fragment of CL

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

An infinite-valued fragment of CL

A non-deterministic matrix for K/2: (where D “ t1u)

• t0, 1u

1 t0u Ñ 1 t1u t1u 1 t0u t1u

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

An infinite-valued fragment of CL

A non-deterministic matrix for K/2: (where D “ t1u)

• t0, 1u

1 t0u Ñ 1 t1u t1u 1 t0u t1u

Signed Tableaux:

T:α F:α T:α Ñ β F:α T:β F:α Ñ β T:α F:β T:α F:α ¸

Exercises:

Check if $ pp Ñ pq Ñ p is still provable. Check if $ ppp Ñ pq Ñ pq Ñ p is provable.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

Consider now the following matrix: (where D “ t1

2, 1u)

• ‚

1

1 2 1 2

1 1

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

Consider now the following matrix: (where D “ t1

2, 1u)

• ‚

1

1 2 1 2

1 1

Contrast Tableaux #1. . .

F:‚α F:α F:α T:‚α T:α T:α F:α T:α T:α F:α T:‚α F:α F:α T:α T:α T:‚‚α ¸ T:α F:α ¸

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

Contrast Tableaux #1. . .

F:‚α F:α F:α T:‚α T:α T:α F:α T:α T:α F:α T:‚α F:α F:α T:α T:α T:‚‚α ¸ T:α F:α ¸

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

Contrast Tableaux #1. . .

F:‚α F:α F:α T:‚α T:α T:α F:α T:α T:α F:α T:‚α F:α F:α T:α T:α T:‚‚α ¸ T:α F:α ¸

. . . to Tableaux #2

T:‚α T:α F:α T:α F:‚α T:α F:α T:‚α T:‚α T:‚α F:‚α F:α F:α T:‚‚α ¸ T:α F:α ¸

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

Contrast Tableaux #1. . .

F:‚α F:α F:α T:‚α T:α T:α F:α T:α T:α F:α T:‚α F:α F:α T:α T:α T:‚‚α ¸ T:α F:α ¸

. . . to Tableaux #2

T:‚α T:α F:α T:α F:‚α T:α F:α T:‚α T:‚α T:‚α F:‚α F:α F:α T:‚‚α ¸ T:α F:α ¸

Exercises:

Show that the rules of #1 and #2 are interderivable. Check that #1 allows for loops.

(check what happens with T:‚α) João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

. . . to Tableaux #2

T:‚α T:α F:α T:α F:‚α T:α F:α T:‚α T:‚α T:‚α F:‚α F:α F:α T:‚‚α ¸ T:α F:α ¸

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

. . . to Tableaux #2

T:‚α T:α F:α T:α F:‚α T:α F:α T:‚α T:‚α T:‚α F:‚α F:α F:α T:‚‚α ¸ T:α F:α ¸

A more interesting exercise:

Check that #2 does not allow for loops.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

. . . to Tableaux #2

T:‚α T:α F:α T:α F:‚α T:α F:α T:‚α T:‚α T:‚α F:‚α F:α F:α T:‚‚α ¸ T:α F:α ¸

A more interesting exercise:

Check that #2 does not allow for loops.

Solution: Consider the non-canonical complexity measure ℓpϕq “ # ℓpψq ` 1, if ϕ “ ‚ψ ℓpψq ` 2, if ϕ “ ψ

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Please make it stop!

. . . to Tableaux #2

T:‚α T:α F:α T:α F:‚α T:α F:α T:‚α T:‚α T:‚α F:‚α F:α F:α T:‚‚α ¸ T:α F:α ¸

A more interesting exercise:

Check that #2 does not allow for loops.

Solution: Consider the non-canonical complexity measure ℓpϕq “ # ℓpψq ` 1, if ϕ “ ‚ψ ℓpψq ` 2, if ϕ “ ψ Note: This hints to a generalization of the Subformula Property!

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Should proof systems include a proof strategy?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Should proof systems include a proof strategy? Does the rule application order matter?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

Should proof systems include a proof strategy? Does the rule application order matter? More about this at a later example!

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

As linear as can be

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

As linear as can be

Signed Tableaux for Classical Logic:

F:α T:α T:α F:α F:α Ñ β T:α F:β T:α Ñ β F:α T:β T:α F:α ¸

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

As linear as can be

Cut-based Signed Tableaux for Classical Logic:

F:α T:α F:α T:α T:α F:α F:α Ñ β T:α F:β T:α Ñ β T:α T:β T:α Ñ β F:β F:α T:α F:α ¸

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

As linear as can be

Cut-based Signed Tableaux for Classical Logic:

F:α T:α F:α T:α T:α F:α F:α Ñ β T:α F:β T:α Ñ β T:α T:β T:α Ñ β F:β F:α T:α F:α ¸

Exercises:

Check again the provability of pp Ñ qq Ñ r $ p Ñ pq Ñ rq and of its converse. Check again the provability of $ pp Ñ pq Ñ p.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Classical Logic A Non-classical example Analyticity generalized Rule application strategies Cut-based tableaux

As linear as can be

Cut-based Signed Tableaux for Classical Logic:

F:α T:α F:α T:α T:α F:α F:α Ñ β T:α F:β T:α Ñ β T:α T:β T:α Ñ β F:β F:α T:α F:α ¸

Exercises:

Check again the provability of pp Ñ qq Ñ r $ p Ñ pq Ñ rq and of its converse. Check again the provability of $ pp Ñ pq Ñ p.

Are tableaux advantageous over truth-tables? The ‘average case’ is better, the ‘worst case’ much worse! Cut-based tableaux can polynomially simulate truth-tables.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

TWO is not enough!

“The philosophical logic simply had

no appreciation for the finer conceptual distinctions because it did not operate with sharply delineated concepts and unambiguously determined symbols; rather it sank into the swamp of the fluid and vague speech used in everyday.”

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

A 3-valued logic: Ł3

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

A 3-valued logic: Ł3

The logical matrices of Ł3: (where D “ t1u)

• 1

1 2 1 2

1 Ñ

1 2

1 1 1 1

1 2 1 2

1 1 1

1 2

1

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

A 3-valued logic: Ł3

The logical matrices of Ł3: (where D “ t1u)

• 1

1 2 1 2

1 Ñ

1 2

1 1 1 1

1 2 1 2

1 1 1

1 2

1

Tableaux through brute force: Examples of logical rules:

1 2 :α 1 2 :α 1 2 :α Ñ β 1 2 :α

0 :β 1 :α

1 2 :β

Examples of closure rules: 0 :α

1 2 :α

¸

1 2 :α

1 :α ¸

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

What about classic-like tableaux for Ł3?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

What about classic-like tableaux for Ł3?

Finding an adequate bivalent semantics S V tT, Fu

h ¶ bh “ ¶ ˝ h

where V “ D Y U and D X U “ ∅, ¶pxq “ T if x P D, ¶pxq “ F if x P U. Then: Γ | ùbh α iff Γ | ùh α

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

What about classic-like tableaux for Ł3?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

What about classic-like tableaux for Ł3?

The logical matrices of Ł3: (where D “ t1u)

• 1

1 2 1 2

1 Ñ

1 2

1 1 1 1

1 2 1 2

1 1 1

1 2

1

How the logical rules may now look like: F:α Ñ β T:α F:β F:α T:β T:pα Ñ βq T:α T:β New closure rules: T:α T:α ¸ T:α F:α ¸

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

First step:

(algebraic ˆ logical values)

Pairwise distinguishing the truth-values in terms of ‘binary prints’.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

First step:

(algebraic ˆ logical values)

Pairwise distinguishing the truth-values in terms of ‘binary prints’.

x ¶pxq F

1 3

F

2 3

F 1 T João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

Example: Consider the separating formulas: θ0pϕq “ ϕ θ1pϕq “ ϕ θ2pϕq “ pϕ Ñ ϕq

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

Example: Consider the separating formulas: θ0pϕq “ ϕ θ1pϕq “ ϕ θ2pϕq “ pϕ Ñ ϕq

x ¶pxq θ1pxq ¶pĂ θ1pxqq θ2pxq ¶pĂ θ2pxqq F 1 T 1 T

1 3

F

2 3

F 1 T

2 3

F

1 3

F

1 3

F 1 T F F João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

Second step: Provide a bivalent description of the truth-tables using the binary prints of the truth-values.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

Examples of logical rules:

F:α F:α F:θ1pαq T:θ2pαq F:α F:θ1pαq F:θ2pαq T:α F:θ1pαq F:θ2pαq T:α F:α T:θ1pαq T:θ2pαq T:θ1pαq T:α F:θ1pαq F:θ2pαq João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

Examples of logical rules:

T:θ2pα Ñ βq F:α F:θ1pαq F:θ2pαq F:β T:θ1pβq T:θ2pβq T:α F:θ1pαq F:θ2pαq F:β T:θ1pβq T:θ2pβq T:α F:θ1pαq F:θ2pαq F:β F:θ1pβq T:θ2pβq João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

Third step: Take into account the unobtainable semantic scenarios.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

Some binary prints do not correspond to any algebraic truth-value:

x ¶pĂ θ0pxqq ¶pĂ θ1pxqq ¶pĂ θ2pxqq ¸ T T T ¸ T T F ¸ T F T 1 T F F F T T ¸ F T F

1 3

F F T

2 3

F F F João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

Minimize the corresponding information:

x ¶pĂ θ0pxqq ¶pĂ θ1pxqq ¶pĂ θ2pxqq ¸ T T T ¸ T T F ¸ T F T ¸ F T F becomes x ¶pĂ θ0pxqq ¶pĂ θ1pxqq ¶pĂ θ2pxqq ¸ T T ´ ¸ T ´ T ¸ F T F João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

The latter, in turn, originate the following closure rules:

F:α T:θ1pαq F:θ2pαq ¸ T:α T:θ2pαq ¸ T:α T:θ1pαq ¸ T:α F:α ¸

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The logical matrices of Ł4: (where D “ t1u)

• 1

1 3 2 3 2 3 1 3

1 Ñ

1 3 2 3

1 1 1 1 1

1 3 2 3

1 1 1

2 3 1 3 2 3

1 1 1

1 3 2 3

1

The latter, in turn, originate the following closure rules:

F:α T:θ1pαq F:θ2pαq ¸ T:α T:θ2pαq ¸ T:α T:θ1pαq ¸ T:α F:α ¸

For reflection: What if we had used 0 ,

1 3 , 2 3 and 1 as labels?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

Fourth step: Consider a strategy for rule application, in order to guarantee analyticity.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

Fourth step: Consider a strategy for rule application, in order to guarantee analyticity. Example: Consider a signed formula of the form T:ϕ, where ϕ is ppα Ñ βq Ñ pα Ñ βqq.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

Example: Consider a signed formula of the form T:ϕ, where ϕ is ppα Ñ βq Ñ pα Ñ βqq.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The corresponding labelled node can match the head

• f some Ł4-rule in 3 different ways, namely:

Rule rT:θ0s is applied: Rule rT:θ1s is applied: T:ppα Ñ βq Ñ pα Ñ βqq F:ppα Ñ βq Ñ pα Ñ βqq T:θ1pppα Ñ βq Ñ pα Ñ βqqq T:θ2pppα Ñ βq Ñ pα Ñ βqqq T:θ1pppα Ñ βq Ñ pα Ñ βqqq T:ppα Ñ βq Ñ pα Ñ βqq F:θ1pppα Ñ βq Ñ pα Ñ βqqq F:θ2pppα Ñ βq Ñ pα Ñ βqqq Rule rT:θ2 Ñs is applied: T:θ2pα Ñ βq F:α F:θ1pαq F:θ2pαq F:β T:θ1pβq T:θ2pβq T:α F:θ1pαq F:θ2pαq F:β T:θ1pβq T:θ2pβq T:α F:θ1pαq F:θ2pαq F:β F:θ1pβq T:θ2pβq João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Adding more values

The following rule application strategy, according to which: (1) θk-rules, for k ą 0, are to be preferred over θ0-rules (2) if there are θi and θj such that θipϕiq “ ϕ “ θjpϕjq, for some appropriate ϕi and ϕj, one should give preference to the θ-rule whose head is more ‘concrete’ (m.g.u.) guarantees the decrease of the following non-canonical complexity measure: (ℓ0) ℓpθkpϕqq “ ℓpϕq, where k ą 0 and k is ‘minimal’ (see (2)) (ℓ1) ℓppq “ 0, where p is an atom (ℓ2) ℓpϕ1q “ ℓpϕ1q ` 1 (ℓ3) ℓpϕ2 Ñ ϕ3q “ ℓpϕ2q ` ℓpϕ3q ` 1

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Deriving rules

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Beyond two truth-values A bivalent approach Further detail on the associated procedure

Deriving rules On comparing logics How could one use the latter proof systems to compare the logics Ł3 and Ł4?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Is many-valuedness but a farse?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Is many-valuedness but a farse?

‘‘`a˜fˇt´eˇrffl 50 ”y´e´a˚r¯s ”wfle ¯sfi˚tˇi˜l¨l ˜f´a`c´e `a‹nffl ˚i˜l¨l´oˆgˇi`c´a˜l ¯p`a˚r`a`d˚i¯sfi`e `o˝f ”m`a‹n‹y ˚tˇr˚u˚t‚h¯s `a‹n`dffl ˜f´a˜l˙ sfi`e‚h`oˆoˆd¯s’’

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Is many-valuedness but a farse?

‘‘˛h`o“w ”wˆa¯s ˚i˚t ¯p`o¸ sfi¯sfi˚i˜b˝l´e ˚t‚h`a˚t ˚t‚h`e ˛h˚u‹m˜b˘u`g `o˝f ”m`a‹n‹y ˜l´oˆgˇi`c´a˜l ”vˆa˜lˇu`e˙ s ¯p`eˇr¯sfi˚i¯sfi˚t´e´dffl `o“vfleˇrffl ˚t‚h`e ˜l´a¯sfi˚t ˜fˇi˜fˇt›y ”y´e´a˚r¯s?’’

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Is many-valuedness but a farse?

‘‘Łu˛k`a¯sfi˚i`e›w˘i`cˇz ˚i¯s ˚t‚h`e `c‚h˚i`e¨f ¯p`eˇr¯p`eˇtˇr`a˚t´o˘rffl `o˝f `affl ”m`a`g›n˚i˜fˇi`c´e›n˚t `c´o“n`c´e˙ p˚tˇu`a˜l `d`e´c´eˇi¯p˚t ˜l´a¯sfi˚tˇi‹n`g `o˘u˚t ˚i‹nffl ”m`a˚t‚h`e›m`a˚tˇi`c´a˜l ˜l´oˆgˇi`c ˚t´o ˚t‚h`e ¯p˚r`e˙ sfi`e›n˚t `d`a‹y.’’

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Is many-valuedness but a farse?

‘‘`o˝b“v˘i`o˘u¯sfi˜l›y, `a‹n‹y ”m˚u˜lˇtˇi¯p˜lˇi`c´a˚tˇi`o“nffl `o˝f ˜l´oˆgˇi`c´a˜l ”vˆa˜lˇu`e˙ s ˚i¯s `affl ”m`a`dffl ˚i`d`e´affl’’

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to find separators

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to find separators

The previously illustrated procedure may be fully automated.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to find separators

Algorithm #1. Starting from a logical matrix over a given signature, find out whether it is sufficiently expressive to separate the truth-values.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to find separators

Algorithm #1. Starting from a logical matrix over a given signature, find out whether it is sufficiently expressive to separate the truth-values. In case it is not, produce a minimal conservative extension that does the job.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to produce a bivalent characterization

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to produce a bivalent characterization

Algorithm #2. Starting from a sufficiently expressive logical matrix, extract an axiomatization (using a classical metalanguage) that describes a bivalent characterization of it (as previously explained in Steps 1 and 2).

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to produce a bivalent characterization

Algorithm #2. Starting from a sufficiently expressive logical matrix, extract an axiomatization (using a classical metalanguage) that describes a bivalent characterization of it (as previously explained in Steps 1 and 2). Make sure this characterization takes into account an appropriate well-founded notion of complexity that allows for a generalization of the compositionality principle.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to produce analytic tableaux

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to produce analytic tableaux

Algorithm #3. Given an appropriate bivalent characterization of a given logic, calculate the minimal closuring sequences and set up a rule application strategy (as previously explained in Steps 3 and 4).

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

Algorithm to produce analytic tableaux

Algorithm #3. Given an appropriate bivalent characterization of a given logic, calculate the minimal closuring sequences and set up a rule application strategy (as previously explained in Steps 3 and 4). Describe analytic tableaux based on the data obtained so far.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

More values, or just different?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! The role of separators Bivalence is the key Analytic tableaux

More values, or just different? Again on the comparison of logics We have already discussed cases of logics L1 and L2 s.t. #pV1q ‰ #pV2q. What if #pV1q “ #pV2q, yet #pD1q ‰ #pD2q?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Is FOUR more than enough?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Variations around FDE

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Variations around FDE

b

b b b b

f J t K info (ď2) truth (ď1)

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Variations around FDE

b

b b b b

f J t K info (ď2) truth (ď1) Syntax & Semantics: V “ tf , K, J, tu L1 “ pV, ^1, _1, 1q L2 “ pV, ^2, _2, 2q B “ pL1♥L2, t, J, K, f q

FOUR BiLat “ CL BA

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Variations around FDE

b

b b b b

f J t K info (ď2) truth (ď1)

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Variations around FDE

• -entailment:

Γ | ùo

i ∆ iff Ű irΓu ďi

Ů

ir∆u

p-entailment: Vj “ Uj Y Dj, and Γ | ùp

j ∆ iff rΓu Uj or r∆uDj

b

b b b b

f J t K info (ď2) truth (ď1)

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Variations around FDE

• -entailment:

Γ | ùo

i ∆ iff Ű irΓu ďi

Ů

ir∆u

p-entailment: Vj “ Uj Y Dj, and Γ | ùp

j ∆ iff rΓu Uj or r∆uDj

Contrast now: Ueℓ“tf , Ku / Deℓ“tJ, tu

b

b b b b

f J t K info (ď2) truth (ď1) eℓ

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Variations around FDE

• -entailment:

Γ | ùo

i ∆ iff Ű irΓu ďi

Ů

ir∆u

p-entailment: Vj “ Uj Y Dj, and Γ | ùp

j ∆ iff rΓu Uj or r∆uDj

Contrast now: Ueℓ“tf , Ku / Deℓ“tJ, tu Un“tf , K, Ju / Dn“ttu

b

b b b b

f J t K info (ď2) truth (ď1) eℓ n

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Variations around FDE

• -entailment:

Γ | ùo

i ∆ iff Ű irΓu ďi

Ů

ir∆u

p-entailment: Vj “ Uj Y Dj, and Γ | ùp

j ∆ iff rΓu Uj or r∆uDj

Contrast now: Ueℓ“tf , Ku / Deℓ“tJ, tu Un“tf , K, Ju / Dn“ttu Ub“tf u / Db“tK, J, tu

b

b b b b

f J t K info (ď2) truth (ď1) eℓ n b

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu “ T ñ rαu “ T and rβu “ T [and2] rα&βu “ T ð rαu “ T and rβu “ T

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu “ T ñ rαu “ T and rβu “ T [and2] rα&βu “ T ð rαu “ T and rβu “ T [or1] rα || βu “ T ñ rαu “ T

• r

rβu “ T [or2] rα || βu “ T ð rαu “ T

• r

rβu “ T

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu “ T ñ rαu “ T and rβu “ T [and2] rα&βu “ T ð rαu “ T and rβu “ T [or1] rα || βu “ T ñ rαu “ T

• r

rβu “ T [or2] rα || βu “ T ð rαu “ T

• r

rβu “ T

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu “ T ñ rαu “ T and rβu “ T [and2] rα&βu “ F ñ rαu “ F

• r

rβu “ F [or1] rα || βu “ T ñ rαu “ T

• r

rβu “ T [or2] rα || βu “ F ñ rαu “ F and rβu “ F

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu “ T ñ rαu “ T and rβu “ T [and2] rα&βu “ F ñ rαu “ F

• r

rβu “ F [or1] rα || βu “ T ñ rαu “ T

• r

rβu “ T [or2] rα || βu “ F ñ rαu “ F and rβu “ F

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu P D ñ rαu P D and rβu P D [and2] rα&βu P U ñ rαu P U

• r

rβu P U [or1] rα || βu P D ñ rαu P D

• r

rβu P D [or2] rα || βu P U ñ rαu P U and rβu P U

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu P D ñ rαu P D and rβu P D [and2] rα&βu P U ñ rαu P U

• r

rβu P U [or1] rα || βu P D ñ rαu P D

• r

rβu P D [or2] rα || βu P U ñ rαu P U and rβu P U

CRs Operators Properties

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu P D ñ rαu P D and rβu P D [and2] rα&βu P U ñ rαu P U

• r

rβu P U [or1] rα || βu P D ñ rαu P D

• r

rβu P D [or2] rα || βu P U ñ rαu P U and rβu P U

CRs Operators Properties | ùo

1, |

ùo

2, |

ùp

eℓ

& P t^1, ^2u [and1], [and2] | ùo

1, |

ùo

2, |

ùp

eℓ

|| P t_1, _2u [or1], [or2]

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu P D ñ rαu P D and rβu P D [and2] rα&βu P U ñ rαu P U

• r

rβu P U [or1] rα || βu P D ñ rαu P D

• r

rβu P D [or2] rα || βu P U ñ rαu P U and rβu P U

CRs Operators Properties | ùo

1, |

ùo

2, |

ùp

eℓ

& P t^1, ^2u [and1], [and2] | ùo

1, |

ùo

2, |

ùp

eℓ

|| P t_1, _2u [or1], [or2] | ùp

b, |

ùp

n

& “ ^1 [and1], [and2] | ùp

b, |

ùp

n

|| “ _1 [or1], [or2]

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

A closer look at ‘ands’ and ‘ors’

[and1] rα&βu P D ñ rαu P D and rβu P D [and2] rα&βu P U ñ rαu P U

• r

rβu P U [or1] rα || βu P D ñ rαu P D

• r

rβu P D [or2] rα || βu P U ñ rαu P U and rβu P U

CRs Operators Properties | ùo

1, |

ùo

2, |

ùp

eℓ

& P t^1, ^2u [and1], [and2] | ùo

1, |

ùo

2, |

ùp

eℓ

|| P t_1, _2u [or1], [or2] | ùp

b, |

ùp

n

& “ ^1 [and1], [and2] | ùp

b, |

ùp

n

|| “ _1 [or1], [or2] | ùp

b, |

ùp

n

& “ ^2 [and2] | ùp

b, |

ùp

n

|| “ _2 [or1]

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

How to find dissimilarities between these logics?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

How to find dissimilarities between these logics?

Examples 2pα _2 βq | ùp

n 2α ^2 2β

2pα _2 βq ­| ùp

b 2α ^2 2β

α ^2 pβ _2 γq ­| ùp

n pα ^1 βq _1 pα ^2 γq

α ^2 pβ _2 γq | ùp

b pα ^1 βq _1 pα ^2 γq

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

How to find dissimilarities between these logics?

Examples 2pα _2 βq | ùp

n 2α ^2 2β

2pα _2 βq ­| ùp

b 2α ^2 2β

α ^2 pβ _2 γq ­| ùp

n pα ^1 βq _1 pα ^2 γq

α ^2 pβ _2 γq | ùp

b pα ^1 βq _1 pα ^2 γq

• Note. Ingenuity is often required!

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

How to find dissimilarities between these logics?

Examples 2pα _2 βq | ùp

n 2α ^2 2β

2pα _2 βq ­| ùp

b 2α ^2 2β

α ^2 pβ _2 γq ­| ùp

n pα ^1 βq _1 pα ^2 γq

α ^2 pβ _2 γq | ùp

b pα ^1 βq _1 pα ^2 γq

• Note. Ingenuity is often required!

Can this task also be automated?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Matrices in, tableaux out

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Matrices in, tableaux out

Recall that:

x ¶eℓpxq ¶bpxq ¶npxq f F F F K F T F J T T F t T T T

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Matrices in, tableaux out

Recall that:

x ¶eℓpxq ¶bpxq ¶npxq f F F F K F T F J T T F t T T T

Consider the (definable) separator:

x r θpxq f f K t J f t t

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Matrices in, tableaux out

Recall that:

x ¶eℓpxq ¶bpxq ¶npxq f F F F K F T F J T T F t T T T

Consider the (definable) separator:

x r θpxq f f K t J f t t

Run the algorithms. . .

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Matrices in, tableaux out

Recall that:

x ¶eℓpxq ¶bpxq ¶npxq f F F F K F T F J T T F t T T T

Consider the (definable) separator:

x r θpxq f f K t J f t t

Run the algorithms. . . and note that the outputs are different, yet the metalanguage is uniform!

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more! Another line of experimentation Same number of values, but other meanings for them

Matrices in, tableaux out

Recall that:

x ¶eℓpxq ¶bpxq ¶npxq f F F F K F T F J T T F t T T T

Consider the (definable) separator:

x r θpxq f f K t J f t t

Run the algorithms. . . and note that the outputs are different, yet the metalanguage is uniform!

Check if the tableau rules of L1 are derivable in L2.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more!

Kein Ende in Sicht

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more!

Some questions & some lessons to be learned

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more!

Some questions & some lessons to be learned

What are the goals in searching for uniform frameworks?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more!

Some questions & some lessons to be learned

What are the goals in searching for uniform frameworks? Analyticity is more than just a word!

(a well-founded complexity measure is welcome / cuts are not always malign)

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more!

Some questions & some lessons to be learned

What are the goals in searching for uniform frameworks? Analyticity is more than just a word!

(a well-founded complexity measure is welcome / cuts are not always malign)

How much further can we automate the construction

• f proof systems?

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more!

Some questions & some lessons to be learned

What are the goals in searching for uniform frameworks? Analyticity is more than just a word!

(a well-founded complexity measure is welcome / cuts are not always malign)

How much further can we automate the construction

• f proof systems?

Note the important role of proof strategies.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more!

Some questions & some lessons to be learned

What are the goals in searching for uniform frameworks? Analyticity is more than just a word!

(a well-founded complexity measure is welcome / cuts are not always malign)

How much further can we automate the construction

• f proof systems?

Note the important role of proof strategies. Other proof formalisms may benefit from the same lessons.

João Marcos Classic-like Analytic Tableaux for Non-Classical Logics

Tableaux More on finite-valued logics Through Suszko Reduction Comparing different logics . . . and so much more!

Some questions & some lessons to be learned

What are the goals in searching for uniform frameworks? Analyticity is more than just a word!

(a well-founded complexity measure is welcome / cuts are not always malign)

How much further can we automate the construction

• f proof systems?

Note the important role of proof strategies. Other proof formalisms may benefit from the same lessons. Some online references may be found here.

[J.M. acknowledges support from the Humboldt Foundation.] João Marcos Classic-like Analytic Tableaux for Non-Classical Logics