Tableaux Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! Cut-based tableaux Please make it stop! . . . to Tableaux #2 T : ‚ α F : � α T : � α T : ‚� α F : ‚� α T : ‚‚ α T : α F : α T : α T : α F : α T : ‚ α T : ‚ α F : � α F : α ¸ F : ‚ α ¸ A more interesting exercise: Check that #2 does not allow for loops. Solution: Consider the non-canonical complexity measure # ℓ p ψ q ` 1 , if ϕ “ ‚ ψ ℓ p ϕ q “ ℓ p ψ q ` 2 , if ϕ “ � ψ João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! Cut-based tableaux Please make it stop! . . . to Tableaux #2 T : ‚ α F : � α T : � α T : ‚� α F : ‚� α T : ‚‚ α T : α F : α T : α T : α F : α T : ‚ α T : ‚ α F : � α F : α ¸ F : ‚ α ¸ A more interesting exercise: Check that #2 does not allow for loops. Solution: Consider the non-canonical complexity measure # ℓ p ψ q ` 1 , if ϕ “ ‚ ψ ℓ p ϕ q “ ℓ p ψ q ` 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 Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! Cut-based tableaux Should proof systems include a proof strategy? João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! 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 Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! 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 Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! Cut-based tableaux As linear as can be João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! Cut-based tableaux As linear as can be Signed Tableaux for Classical Logic: F : � α T : � α F : α Ñ β T : α Ñ β T : α F : α F : α T : β T : α F : α T : α ¸ F : β João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! Cut-based tableaux As linear as can be Cut-based Signed Tableaux for Classical Logic: F : � α T : � α F : α Ñ β T : α Ñ β T : α Ñ β T : α T : α F : β F : α F : α T : α T : α F : α T : α ¸ F : β T : β F : α João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! Cut-based tableaux As linear as can be Cut-based Signed Tableaux for Classical Logic: F : � α T : � α F : α Ñ β T : α Ñ β T : α Ñ β T : α T : α F : β F : α F : α T : α T : α F : α T : α ¸ F : β T : β F : α Exercises: Check again the provability of p p Ñ q q Ñ r $ p Ñ p q Ñ r q and of its converse. Check again the provability of $ p� p Ñ p q Ñ p . João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux Classical Logic More on finite-valued logics A Non-classical example Through Suszko Reduction Analyticity generalized Comparing different logics Rule application strategies . . . and so much more! Cut-based tableaux As linear as can be Cut-based Signed Tableaux for Classical Logic: F : � α T : � α F : α Ñ β T : α Ñ β T : α Ñ β T : α T : α F : β F : α F : α T : α T : α F : α T : α ¸ F : β T : β F : α Exercises: Check again the provability of p p Ñ q q Ñ r $ p Ñ p q Ñ r q and of its converse. Check again the provability of $ p� p Ñ p q 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 Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! 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 Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! A 3-valued logic: Ł3 João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! A 3-valued logic: Ł3 The logical matrices of Ł3: (where D “ t 1 u ) 1 � Ñ 0 1 2 0 1 0 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 0 1 0 1 2 João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! A 3-valued logic: Ł3 The logical matrices of Ł3: (where D “ t 1 u ) 1 � Ñ 0 1 2 0 1 0 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 0 1 0 1 2 Tableaux through brute force : Examples of logical rules : Examples of closure rules : 1 1 2 : � α 2 : α Ñ β 0 : α 1 2 : α 1 2 : α 1 : α 1 1 : α 1 2 : α 2 : α 1 2 : β ¸ ¸ 0 : β João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! What about classic-like tableaux for Ł3? João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! What about classic-like tableaux for Ł3? Finding an adequate bivalent semantics where V “ D Y U and D X U “ ∅ , h S V ¶ p x q “ T if x P D , b h “ ¶ ˝ h ¶ p x q “ F if x P U . ¶ Then: t T , F u Γ | ù b h α iff Γ | ù h α João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! What about classic-like tableaux for Ł3? João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! What about classic-like tableaux for Ł3? The logical matrices of Ł3: (where D “ t 1 u ) 1 � Ñ 0 1 2 0 1 0 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 0 1 0 1 2 How the logical rules may now look like: New closure rules : F : α Ñ β T : �p α Ñ β q T : α T : α T : � α F : α F : � α T : α T : α F : β T : � β T : � β ¸ ¸ João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 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 Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 x ¶ p x q First step : (algebraic ˆ logical values) 0 F Pairwise distinguishing the truth-values 1 F 3 2 in terms of ‘ binary prints ’. F 3 1 T João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 Example: Consider the separating formulas : θ 0 p ϕ q “ ϕ θ 1 p ϕ q “ � ϕ θ 2 p ϕ q “ ��p ϕ Ñ � ϕ q João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 Example: Consider the ¶ p Ă ¶ p Ă x ¶ p x q θ 1 p x q θ 1 p x qq θ 2 p x q θ 2 p x qq separating formulas : 0 1 1 F T T θ 0 p ϕ q “ ϕ 1 2 F F 1 T 3 3 2 1 1 θ 1 p ϕ q “ � ϕ F F F 3 3 3 1 0 0 T F F θ 2 p ϕ q “ ��p ϕ Ñ � ϕ q João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 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 Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 Examples of logical rules : F : � α T : � α T : θ 1 p� α q F : α T : α T : θ 1 p α q F : θ 1 p α q F : α F : α T : α T : θ 2 p α q F : θ 2 p α q F : θ 1 p α q F : θ 1 p α q F : θ 1 p α q T : θ 2 p α q F : θ 2 p α q F : θ 2 p α q João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 Examples of T : θ 2 p α Ñ β q logical rules : F : α T : α T : α F : θ 1 p α q F : θ 1 p α q F : θ 1 p α q F : θ 2 p α q F : θ 2 p α q F : θ 2 p α q F : β F : β F : β T : θ 1 p β q T : θ 1 p β q F : θ 1 p β q T : θ 2 p β q T : θ 2 p β q T : θ 2 p β q João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 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 Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 Some binary prints do not correspond to any algebraic truth-value : ¶ p Ă ¶ p Ă ¶ p Ă x θ 0 p x qq θ 1 p x qq θ 2 p x qq ¸ T T T ¸ T T F ¸ T F T 1 T F F 0 F T T ¸ F T F 1 F F T 3 2 F F F 3 João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 Minimize the corresponding information: ¶ p Ă ¶ p Ă ¶ p Ă x θ 0 p x qq θ 1 p x qq θ 2 p x qq ¶ p Ă ¶ p Ă ¶ p Ă x θ 0 p x qq θ 1 p x qq θ 2 p x qq ¸ T T T ¸ T T ´ becomes ¸ T T F ¸ T ´ T ¸ T F T ¸ F T F ¸ F T F João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 The latter, in turn, originate the following closure rules : F : α T : α T : α T : α T : θ 1 p α q T : θ 2 p α q T : θ 1 p α q F : α F : θ 2 p α q ¸ ¸ ¸ ¸ João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The logical matrices of Ł4: (where D “ t 1 u ) 1 2 � Ñ 0 1 3 3 0 1 0 1 1 1 1 1 2 1 2 1 1 1 3 3 3 3 2 1 2 1 2 1 1 3 3 3 3 3 1 2 1 0 1 0 1 3 3 The latter, in turn, originate the following closure rules : F : α T : α T : α T : α T : θ 1 p α q T : θ 2 p α q T : θ 1 p α q F : α F : θ 2 p α q ¸ ¸ ¸ ¸ 1 2 For reflection: What if we had used 0 , 3 , 3 and 1 as labels? João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! 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 Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! 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 ��pp α Ñ β q Ñ �p α Ñ β qq . João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values Example: Consider a signed formula of the form T : ϕ , where ϕ is ��pp α Ñ β q Ñ �p α Ñ β qq . João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The corresponding labelled node can match the head of some Ł 4 -rule in 3 different ways, namely: Rule r T : θ 0 s is applied: Rule r T : θ 1 �s is applied: T : ��pp α Ñ β q Ñ �p α Ñ β qq T : θ 1 p�pp α Ñ β q Ñ �p α Ñ β qqq F : �pp α Ñ β q Ñ �p α Ñ β qq T : pp α Ñ β q Ñ �p α Ñ β qq T : θ 1 p�pp α Ñ β q Ñ �p α Ñ β qqq F : θ 1 ppp α Ñ β q Ñ �p α Ñ β qqq T : θ 2 p�pp α Ñ β q Ñ �p α Ñ β qqq F : θ 2 ppp α Ñ β q Ñ �p α Ñ β qqq Rule r T : θ 2 Ñs is applied: T : θ 2 p α Ñ β q F : α T : α T : α F : θ 1 p α q F : θ 1 p α q F : θ 1 p α q F : θ 2 p α q F : θ 2 p α q F : θ 2 p α q F : β F : β F : β T : θ 1 p β q T : θ 1 p β q F : θ 1 p β q T : θ 2 p β q T : θ 2 p β q T : θ 2 p β q João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Adding more values The following rule application strategy , according to which: (1) θ k -rules, for k ą 0, are to be preferred over θ 0 -rules (2) if there are θ i and θ j such that θ i p ϕ i q “ ϕ “ θ j p ϕ j q , for some appropriate ϕ i and ϕ j , 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 : ( ℓ 0) ℓ p θ k p ϕ qq “ ℓ p ϕ q , where k ą 0 and k is ‘minimal’ (see (2)) ( ℓ 1) ℓ p p q “ 0, where p is an atom ( ℓ 2) ℓ p� ϕ 1 q “ ℓ p ϕ 1 q ` 1 ( ℓ 3) ℓ p ϕ 2 Ñ ϕ 3 q “ ℓ p ϕ 2 q ` ℓ p ϕ 3 q ` 1 João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! Deriving rules João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Beyond two truth-values Through Suszko Reduction A bivalent approach Comparing different logics Further detail on the associated procedure . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! Is many-valuedness but a farse? João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! Algorithm to find separators João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! Algorithm to produce a bivalent characterization João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! Algorithm to produce analytic tableaux João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! 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 The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! More values, or just different? João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics The role of separators Through Suszko Reduction Bivalence is the key Comparing different logics Analytic tableaux . . . and so much more! More values, or just different? Again on the comparison of logics We have already discussed cases of logics L 1 and L 2 s.t. # p V 1 q ‰ # p V 2 q . What if # p V 1 q “ # p V 2 q , yet # p D 1 q ‰ # p D 2 q ? João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Is FOUR more than enough? João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Variations around FDE João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
b b b b b Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Variations around FDE info ( ď 2 ) J t f K truth ( ď 1 ) João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
b b b b b Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Variations around FDE info ( ď 2 ) J Syntax & Semantics : V “ t f , K , J , t u L 1 “ p V , ^ 1 , _ 1 , � 1 q t f L 2 “ p V , ^ 2 , _ 2 , � 2 q B “ p L 1 ♥L 2 , t , J , K , f q K FOUR BiLat “ CL BA truth ( ď 1 ) João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
b b b b b Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Variations around FDE info ( ď 2 ) J t f K truth ( ď 1 ) João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
b b b b b Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Variations around FDE info ( ď 2 ) J o -entailment: i ∆ iff Ű Ů ù o Γ | i r Γ u ď i i r ∆ u p -entailment: V j “ U j Y D j , and ù p Γ | j ∆ iff r Γ u� U j or r ∆ u� D j t f K truth ( ď 1 ) João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
b b b b b Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Variations around FDE info ( ď 2 ) J o -entailment: i ∆ iff Ű Ů ù o Γ | i r Γ u ď i i r ∆ u p -entailment: V j “ U j Y D j , and ù p Γ | j ∆ iff r Γ u� U j or r ∆ u� D j t f Contrast now: e ℓ U e ℓ “t f , Ku / D e ℓ “tJ , t u K truth ( ď 1 ) João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
b b b b b Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Variations around FDE info ( ď 2 ) J o -entailment: i ∆ iff Ű Ů ù o Γ | i r Γ u ď i i r ∆ u p -entailment: V j “ U j Y D j , and ù p Γ | j ∆ iff r Γ u� U j or r ∆ u� D j t f Contrast now: e ℓ U e ℓ “t f , Ku / D e ℓ “tJ , t u U n “t f , K , Ju / D n “t t u n K truth ( ď 1 ) João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
b b b b b Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Variations around FDE info ( ď 2 ) J o -entailment: i ∆ iff Ű Ů ù o Γ | i r Γ u ď i i r ∆ u p -entailment: V j “ U j Y D j , and ù p Γ | j ∆ iff r Γ u� U j or r ∆ u� D j t f Contrast now: e ℓ U e ℓ “t f , Ku / D e ℓ “tJ , t u U n “t f , K , Ju / D n “t t u n b K U b “t f u / D b “tK , J , t u truth ( ď 1 ) João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u “ T ñ r α u “ T and r β u “ T [and 2 ] r α & β u “ T ð r α u “ T and r β u “ T João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u “ T ñ r α u “ T and r β u “ T [and 2 ] r α & β u “ T ð r α u “ T and r β u “ T [or 1 ] r α || β u “ T ñ r α u “ T or r β u “ T [or 2 ] r α || β u “ T ð r α u “ T or r β u “ T João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u “ T ñ r α u “ T and r β u “ T [and 2 ] r α & β u “ T ð r α u “ T and r β u “ T [or 1 ] r α || β u “ T ñ r α u “ T or r β u “ T [or 2 ] r α || β u “ T ð r α u “ T or r β u “ T João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u “ T ñ r α u “ T and r β u “ T [and 2 ] r α & β u “ F ñ r α u “ F or r β u “ F [or 1 ] r α || β u “ T ñ r α u “ T or r β u “ T [or 2 ] r α || β u “ F ñ r α u “ F and r β u “ F João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u “ T ñ r α u “ T and r β u “ T [and 2 ] r α & β u “ F ñ r α u “ F or r β u “ F [or 1 ] r α || β u “ T ñ r α u “ T or r β u “ T [or 2 ] r α || β u “ F ñ r α u “ F and r β u “ F João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u P D ñ r α u P D and r β u P D [and 2 ] r α & β u P U ñ r α u P U or r β u P U [or 1 ] r α || β u P D ñ r α u P D or r β u P D [or 2 ] r α || β u P U ñ r α u P U and r β u P U João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u P D ñ r α u P D and r β u P D [and 2 ] r α & β u P U ñ r α u P U or r β u P U [or 1 ] r α || β u P D ñ r α u P D r β u P D or [or 2 ] r α || β u P U ñ r α u P U and r β u P U CRs Operators Properties João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u P D ñ r α u P D and r β u P D [and 2 ] r α & β u P U ñ r α u P U or r β u P U [or 1 ] r α || β u P D ñ r α u P D r β u P D or [or 2 ] r α || β u P U ñ r α u P U and r β u P U CRs Operators Properties ù p ù o ù o [and 1 ], [and 2 ] | 1 , | 2 , | & P t^ 1 , ^ 2 u e ℓ ù p ù o ù o [or 1 ], [or 2 ] | 1 , | 2 , | || P t_ 1 , _ 2 u e ℓ João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u P D ñ r α u P D and r β u P D [and 2 ] r α & β u P U ñ r α u P U or r β u P U [or 1 ] r α || β u P D ñ r α u P D r β u P D or [or 2 ] r α || β u P U ñ r α u P U and r β u P U CRs Operators Properties ù p ù o ù o [and 1 ], [and 2 ] | 1 , | 2 , | & P t^ 1 , ^ 2 u e ℓ ù p ù o ù o [or 1 ], [or 2 ] | 1 , | 2 , | || P t_ 1 , _ 2 u e ℓ ù p ù p [and 1 ], [and 2 ] | b , | & “ ^ 1 n ù p ù p [or 1 ], [or 2 ] | b , | || “ _ 1 n João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! A closer look at ‘ands’ and ‘ors’ [and 1 ] r α & β u P D ñ r α u P D and r β u P D [and 2 ] r α & β u P U ñ r α u P U or r β u P U [or 1 ] r α || β u P D ñ r α u P D r β u P D or [or 2 ] r α || β u P U ñ r α u P U and r β u P U CRs Operators Properties ù p ù o ù o [and 1 ], [and 2 ] | 1 , | 2 , | & P t^ 1 , ^ 2 u e ℓ ù p ù o ù o [or 1 ], [or 2 ] | 1 , | 2 , | || P t_ 1 , _ 2 u e ℓ ù p ù p [and 1 ], [and 2 ] | b , | & “ ^ 1 n ù p ù p [or 1 ], [or 2 ] | b , | || “ _ 1 n ù p ù p [and 2 ] | b , | & “ ^ 2 n ù p ù p [or 1 ] | b , | || “ _ 2 n João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! How to find dissimilarities between these logics? João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! How to find dissimilarities between these logics? Examples ù p � 2 p α _ 2 β q | n � 2 α ^ 2 � 2 β ù p � 2 p α _ 2 β q | b � 2 α ^ 2 � 2 β ù p α ^ 2 p β _ 2 γ q | n p α ^ 1 β q _ 1 p α ^ 2 γ q ù p α ^ 2 p β _ 2 γ q | b p α ^ 1 β q _ 1 p α ^ 2 γ q João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! How to find dissimilarities between these logics? Examples ù p � 2 p α _ 2 β q | n � 2 α ^ 2 � 2 β ù p � 2 p α _ 2 β q | b � 2 α ^ 2 � 2 β ù p α ^ 2 p β _ 2 γ q | n p α ^ 1 β q _ 1 p α ^ 2 γ q ù p α ^ 2 p β _ 2 γ q | b p α ^ 1 β q _ 1 p α ^ 2 γ q Note . Ingenuity is often required! João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! How to find dissimilarities between these logics? Examples ù p � 2 p α _ 2 β q | n � 2 α ^ 2 � 2 β ù p � 2 p α _ 2 β q | b � 2 α ^ 2 � 2 β ù p α ^ 2 p β _ 2 γ q | n p α ^ 1 β q _ 1 p α ^ 2 γ q ù p α ^ 2 p β _ 2 γ q | b p α ^ 1 β q _ 1 p α ^ 2 γ q 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 Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Matrices in, tableaux out João Marcos Classic-like Analytic Tableaux for Non-Classical Logics
Tableaux More on finite-valued logics Another line of experimentation Through Suszko Reduction Same number of values, but other meanings for them Comparing different logics . . . and so much more! Matrices in, tableaux out Recall that: ¶ e ℓ p x q ¶ b p x q ¶ n p x q x 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
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