Truth, Truth-values,
and the like
Fabien Schang
National Research University
Higher School of Economics
Moscow (Russia)
[email protected]
Content
1. Truth
2. Truth-values
3. The like
1.
Truth
A basic difference, in philosophy, between ontology and epistemology
Ontology is about what is: being
Epistemology is about how to access to what is: knowing
Logic is about everything meaningful
A set of sets of sentences, related to each other in a formal language
One common concern in ontology, logic, and epistemology: truth
All the three sections deal with truth, but from different perspectives
Ontology and epistemology: the material truth of atoms (p, q, …)
Logic: the formal truth of molecules (p  q, p  q, …)
At the crossroad of logic and epistemology: epistemic logic
A couple of different combinations between these two concepts
A basic difference, in philosophy, between ontology and epistemology
Ontology is about what is: being
Epistemology is about how to access to what is: knowing
Logic is about everything meaningful
A set of sets of sentences, related to each other in a formal language
One common concern in ontology, logic, and epistemology: truth
All the three sections deal with truth, but from different perspectives
Ontology and epistemology: the material truth of atoms (p, q, …)
Logic: the formal truth of molecules (p  q, p  q, …)
At the crossroad of logic and epistemology: epistemic logic
A couple of different combinations between these two concepts
A basic difference, in philosophy, between ontology and epistemology
Ontology is about what is: being
Epistemology is about how to access to what is: knowing
Logic is about everything meaningful
A set of sets of sentences, related to each other in a formal language
One common concern in ontology, logic, and epistemology: truth
All the three sections deal with truth, but from different perspectives
Ontology and epistemology: the material truth of atoms (p, q, …)
Logic: the formal truth of molecules (p  q, p  q, …)
At the crossroad of logic and epistemology: epistemic logic
A couple of different combinations between these two concepts
A basic difference, in philosophy, between ontology and epistemology
Ontology is about what is: being
Epistemology is about how to access to what is: knowing
Logic is about everything meaningful
A set of sets of sentences, related to each other in a formal language
One common concern in ontology, logic, and epistemology: truth
All the three sections deal with truth, but from different perspectives
Ontology and epistemology: the material truth of atoms (p, q, …)
Logic: the formal truth of molecules (p  q, p  q, …)
At the crossroad of logic and epistemology: epistemic logic
A couple of different combinations between these two concepts
A basic difference, in philosophy, between ontology and epistemology
Ontology is about what is: being
Epistemology is about how to access to what is: knowing
Logic is about everything meaningful
A set of sets of sentences, related to each other in a formal language
One common concern in ontology, logic, and epistemology: truth
All the three sections deal with truth, but from different perspectives
Ontology and epistemology: the material truth of atoms (p, q, …)
Logic: the formal truth of molecules (p  q, p  q, …)
At the crossroad of logic and epistemology: epistemic logic
A couple of different combinations between these two concepts
“Logic of epistemology”: about the foundations of scientific theories
“Epistemic logic”: about the sentences with epistemic concepts
“Epistemology of logic”: about the foundations of the theory of logic
Two sorts of logic for epistemology:
A logical analysis of epistemic concepts (Erkennntnislehre)
(knowledge, belief, doubt, justification): epistemic logic
A logical analysis of the scientific methods (Wissenschaftslehre)
(Bayesianism, causation, induction): formal epistemology
Epistemic logic: What is it?
A set of formal truths about sentences including modal operators
Strong operators : K for knowledge, B for belief
Weak operators : P for possible knowledge, C for possible belief
A minimal criterion for logical relations: consistency
“Logic of epistemology”: about the foundations of scientific theories
“Epistemic logic”: about the sentences with epistemic concepts
“Epistemology of logic”: about the foundations of the theory of logic
Two sorts of logic for epistemology:
A logical analysis of epistemic concepts (Erkennntnislehre)
(knowledge, belief, doubt, justification): epistemic logic
A logical analysis of the scientific methods (Wissenschaftslehre)
(Bayesianism, causation, induction): formal epistemology
Epistemic logic: What is it?
A set of formal truths about sentences including modal operators
Strong operators : K for knowledge, B for belief
Weak operators : P for possible knowledge, C for possible belief
A minimal criterion for logical relations: consistency
“Logic of epistemology”: about the foundations of scientific theories
“Epistemic logic”: about the sentences with epistemic concepts
“Epistemology of logic”: about the foundations of the theory of logic
Two sorts of logic for epistemology:
A logical analysis of epistemic concepts (Erkennntnislehre)
(knowledge, belief, doubt, justification): epistemic logic
A logical analysis of the scientific methods (Wissenschaftslehre)
(Bayesianism, causation, induction): formal epistemology
Epistemic logic: What is it?
A set of formal truths about sentences including modal operators
Strong operators : K for knowledge, B for belief
Weak operators : P for possible knowledge, C for possible belief
A minimal criterion for logical relations: consistency
Epistemic logic: What is it for?
A logical analysis of concepts through a set of relative axioms
K-structure: p, p  q ├K q
D-structure: ├D p  p
T-structure: ├T p  p
4-structure: ├4 p  p
5-structure: ├5 p  p
Epistemic paradoxes: unaccepted conclusions from accepted premises
Examples: Fitch’s Paradox of Knowability, Moore’s Paradox
How to solve a logical paradox?
Resolution: reject one inference rule between axioms and theorems
Dissolution: reject a premise as ill-formed
Epistemic logic: What is it for?
A logical analysis of concepts through a set of relative axioms
K-structure: p, p  q ├K q
D-structure: ├D p  p
T-structure: ├T p  p
4-structure: ├4 p  p
5-structure: ├5 p  p
Epistemic paradoxes: unaccepted conclusions from accepted premises
Examples: Fitch’s Paradox of Knowability, Moore’s Paradox
How to solve a logical paradox?
Resolution: reject one inference rule between axioms and theorems
Dissolution: reject a premise as ill-formed
Epistemic logic: What is it for?
A logical analysis of concepts through a set of relative axioms
K-structure: p, p  q ├K q
D-structure: ├D p  p
T-structure: ├T p  p
4-structure: ├4 p  p
5-structure: ├5 p  p
Epistemic paradoxes: unaccepted conclusions from accepted premises
Examples: Fitch’s Paradox of Knowability, Moore’s Paradox
How to solve a logical paradox?
Resolution: reject one inference rule between axioms and theorems
Dissolution: reject a premise as ill-formed
Epistemic logic: What is it for?
A logical analysis of concepts through a set of relative axioms
K-structure: p, p  q ├K q
D-structure: ├D p  p
T-structure: ├T p  p
4-structure: ├4 p  p
5-structure: ├5 p  p
Epistemic paradoxes: unaccepted conclusions from accepted premises
Examples: Fitch’s Paradox of Knowability, Moore’s Paradox
How to solve a logical paradox?
Resolution: reject one inference rule between axioms and theorems
Dissolution: reject a premise as ill-formed
A trade-off between material and formal truth: informal validity
(material truth of molecular sentences, i.e. logical relations)
A discussion about the extra-validity of axioms, outside logical systems
How can the axioms of a logical system be justified themselves?
Examples: the truth-clause
Kp  p: Every sentence p that is known is thereby true
If p is known, therefore p is true (in every T-model)
Formal truths: logical relations true in every model
In every K-model, the truth of Kp entails the truth of p
A relative sense of truth: truth-in-a-model (set of true sentences)
Does it make sense to talk about extra-validity (cf. matter vs form)?
Axioms and obviousness (axiom of parallels, LEM, truth-clause, etc.)
Axioms are assumed to be obviously true, naturally accepted
A trade-off between material and formal truth: informal validity
(material truth of molecular sentences, i.e. logical relations)
A discussion about the extra-validity of axioms, outside logical systems
How can the axioms of a logical system be justified themselves?
Examples: the truth-clause
Kp  p: Every sentence p that is known is thereby true
If p is known, therefore p is true (in every T-model)
Formal truths: logical relations true in every model
In every K-model, the truth of Kp entails the truth of p
A relative sense of truth: truth-in-a-model (set of true sentences)
Does it make sense to talk about extra-validity (cf. matter vs form)?
Axioms and obviousness (axiom of parallels, LEM, truth-clause, etc.)
Axioms are assumed to be obviously true, naturally accepted
A trade-off between material and formal truth: informal validity
(material truth of molecular sentences, i.e. logical relations)
A discussion about the extra-validity of axioms, outside logical systems
How can the axioms of a logical system be justified themselves?
Examples: the truth-clause
Kp  p: Every sentence p that is known is thereby true
If p is known, therefore p is true (in every T-model)
Formal truths: logical relations true in every model
In every K-model, the truth of Kp entails the truth of p
A relative sense of truth: truth-in-a-model (set of true sentences)
Does it make sense to talk about extra-validity (cf. matter vs form)?
Axioms and obviousness (axiom of parallels, LEM, truth-clause, etc.)
Axioms are assumed to be obviously true, naturally accepted
A trade-off between material and formal truth: informal validity
(material truth of molecular sentences, i.e. logical relations)
A discussion about the extra-validity of axioms, outside logical systems
How can the axioms of a logical system be justified themselves?
Examples: the truth-clause
Kp  p: Every sentence p that is known is thereby true
If p is known, therefore p is true (in every T-model)
Formal truths: logical relations true in every model
In every K-model, the truth of Kp entails the truth of p
A relative sense of truth: truth-in-a-model (set of true sentences)
Does it make sense to talk about extra-validity (cf. matter vs form)?
Axioms and obviousness (axiom of parallels, LEM, truth-clause, etc.)
Axioms are assumed to be obviously true, naturally accepted
What if given axioms happen to be false?
The whole argument is made irrelevant …
… but neither materially, not formally false
Extra-validity has to do not with truth, but relevance
Can an axiom be said to be “relevant” and “false” at once?
Relevant for what? For whom? False of what? For whom?
Is “relevance” another name for pragmatic truth?
Back to the truth-clause
Kp  p
p: “I have a hand”
The skeptic accepts this axiom, but denies the premise Kp
Is such an axiom relevant for a skeptic, especially a Pyrrhonian?
What if given axioms happen to be false?
The whole argument is made irrelevant …
… but neither materially, not formally false
Extra-validity has to do not with truth, but relevance
Can an axiom be said to be “relevant” and “false” at once?
Relevant for what? For whom? False of what? For whom?
Is “relevance” another name for pragmatic truth?
Back to the truth-clause
Kp  p
p: “I have a hand”
The skeptic accepts this axiom, but denies the premise Kp
Is such an axiom relevant for a skeptic, especially a Pyrrhonian?
What if given axioms happen to be false?
The whole argument is made irrelevant …
… but neither materially, not formally false
Extra-validity has to do not with truth, but relevance
Can an axiom be said to be “relevant” and “false” at once?
Relevant for what? For whom? False of what? For whom?
Is “relevance” another name for pragmatic truth?
Back to the truth-clause
Kp  p
p: “I have a hand”
The skeptic accepts this axiom, but denies the premise Kp
Is such an axiom relevant for a skeptic, especially a Pyrrhonian?
Pragmatism, as understood here, means a way
of doing philosophy that takes seriously the
practical human life as a starting point for all
philosophic contemplation.
(Martela 2010: 2)
Entailment thesis
K,    ├ K ( and  are metavariables)
: “I have a hand”, : “I am not a brain in a vat”
A case of deaf dialogue: logical agreement, material disagreement
G. E. Moore, Pyrrho: both accept the entailment thesis
G. E. Moore: accepts K, accepts   , accepts K
Pyrrho: denies K, accepts   , denies K
accepts K, accepts   , accepts K
G. E. Moore: reasons by Modus Ponens
Pyrrho: reasons by Modus Tollens
Dialogue needs a minimal agreement about the premises to be relevant
(cf. Socratic dialogues: from formal to material agreement, through
consistency of the whole)
Entailment thesis
K,    ├ K ( and  are metavariables)
: “I have a hand”, : “I am not a brain in a vat”
A case of deaf dialogue: logical agreement, material disagreement
G. E. Moore, Pyrrho: both accept the entailment thesis
G. E. Moore: accepts K, accepts   , accepts K
Pyrrho: denies K, accepts   , denies K
(= accepts K, accepts   , accepts K ?)
G. E. Moore: reasons by Modus Ponens
Pyrrho: reasons by Modus Tollens
Dialogue needs a minimal agreement about the premises to be relevant
(cf. Socratic dialogues: from formal to material agreement, through
consistency of the whole)
Entailment thesis
K,    ├ K ( and  are metavariables)
: “I have a hand”, : “I am not a brain in a vat”
A case of deaf dialogue: logical agreement, material disagreement
G. E. Moore, Pyrrho: both accept the entailment thesis
G. E. Moore: accepts K, accepts   , accepts K
Pyrrho: denies K, accepts   , denies K
(= accepts K, accepts   , accepts K ?)
G. E. Moore: reasons by Modus Ponens
Pyrrho: reasons by Modus Tollens
Dialogue needs a minimal agreement about the premises to be relevant
(cf. Socratic dialogues: from formal to material agreement, through
consistency of the whole)
Logical truth needs material truth to be relevant (cf. truth-preservation)
How to obtain material agreement?
Any relevant logical truth needs a reflection about truth simpliciter
How to warrant the truth of a sentence (material truth)?
Is truth absolute or relative?
A number of competing theories of truth:
Correspondence (truth is related to corresponding facts)
Coherence (truth is consistency between sentences/beliefs)
Pragmatic (truth is an epistemic agreement between agents)
Only 3 theories? No overlapping about the nature of truth?
2 opposite: objective-subjective, ontic-epistemic views of truth
Logical truth needs material truth to be relevant (cf. truth-preservation)
How to obtain material agreement?
Any relevant logical truth needs a reflection about truth simpliciter
How to warrant the truth of a sentence (material truth)?
Is truth absolute or relative?
A number of competing theories of truth:
Correspondence (truth is related to corresponding facts)
Coherence (truth is consistency between sentences/beliefs)
Pragmatic (truth is an epistemic agreement between agents)
Only 3 theories? No overlapping about the nature of truth?
2 opposite: objective-subjective, ontic-epistemic views of truth
Logical truth needs material truth to be relevant (cf. truth-preservation)
How to obtain material agreement?
Any relevant logical truth needs a reflection about truth simpliciter
How to warrant the truth of a sentence (material truth)?
Is truth absolute or relative?
A number of competing theories of truth:
Correspondence (truth is related to corresponding facts)
Coherence (truth is consistency between sentences/beliefs)
Pragmatic (truth is an epistemic agreement between agents)
Only 3 theories? No overlapping about the nature of truth?
2 opposite: objective-subjective, ontic-epistemic views of truth
Logical truth needs material truth to be relevant (cf. truth-preservation)
How to obtain material agreement?
Any relevant logical truth needs a reflection about truth simpliciter
How to warrant the truth of a sentence (material truth)?
Is truth absolute or relative?
A number of competing theories of truth:
Correspondence (truth is related to corresponding facts)
Coherence (truth is consistency between sentences/beliefs)
Pragmatic (truth is an epistemic agreement between agents)
Only 3 theories? No overlap about the nature of truth?
2 opposite: objective-subjective, ontic-epistemic views of truth
Niiniluoto (2013): “Is truth absolute or relative?”
A list of overlapping theories from the aforementioned pairs
-
Fallibilism (strong, weak)
Pragmatism
Critical realism
Probabilism
Verisimilitude
Cultural relativism
Perspectivism
Provability
Niiniluoto (2013): “Is truth absolute or relative?”
A list of overlapping theories from the aforementioned pairs:
-
Fallibilism (strong, weak)
Pragmatism
Critical realism
Probabilism
Verisimilitude
Cultural relativism
Perspectivism
Provability
Subjective truth:
Subjective relativism
(Protagoras: “Man is the measurement of everything”)
Bp  p
Plato against Protagoras’ relativism: reduction ad absurdum (logical vs
material truth)
The agent a believes p: “This wine is sweet”, therefore p is true for a
Bap  p
The agent b disbelieves p, therefore p is false for b
Bbp  p
Subjective truth:
Subjective relativism
(Protagoras: “Man is the measurement of everything”)
Bp  p
Plato against Protagoras’ relativism: reduction ad absurdum (logical vs
material truth)
The agent a believes p: “This wine is sweet”, therefore p is true for a
Bap  p
The agent b disbelieves p, therefore p is false for b
Bbp  p
Plato’s reasoning by contraposition:   ,  ├ 
1. If both a and b are right, then it is right to state both p and p
├ (Bap  Bbp)  (p  p)
2. Now every contradiction is logically false, i.e. its negation is true
├ (p  p)
3. Therefore a and b cannot be right together, i.e. one of them is wrong
├ (Bap  Bbp)
Niiniluoto (2013): according to Twardoswki, Protagorean personal truth
predicate would violate classical principles of logic
Plato’s reasoning by contraposition:   ,  ├ 
1. If both a and b are right, then it is right to state both p and p
├ (Bap  Bbp)  (p  p)
2. Now every contradiction is logically false, i.e. its negation is true
├ (p  p)
3. Therefore a and b cannot be right together, i.e. one of them is wrong
├ (Bap  Bbp)
Niiniluoto (2013): according to Twardoswki, Protagorean personal truth
predicate would violate classical principles of logic
Plato’s reasoning by contraposition:   ,  ├ 
1. If both a and b are right, then it is right to state both p and p
├ (Bap  Bbp)  (p  p)
2. Now every contradiction is logically false, i.e. its negation is true
├ (p  p)
3. Therefore a and b cannot be right together, i.e. one of them is wrong
├ (Bap  Bbp)
Niiniluoto (2013): according to Twardoswki, Protagorean personal truth
predicate would violate classical principles of logic
Plato’s reasoning by contraposition:   ,  ├ 
1. If both a and b are right, then it is right to state both p and p
├ (Bap  Bbp)  (p  p)
2. Now every contradiction is logically false, i.e. its negation is true
├ (p  p)
3. Therefore a and b cannot be right together, i.e. one of them is wrong
├ (Bap  Bbp)
Niiniluoto (2013): according to Twardoswki, Protagorean personal truth
predicate would violate classical principles of logic
Plato’s reasoning by contraposition:   ,  ├ 
1. If both a and b are right, then it is right to state both p and p
├ (Bap  Bbp)  (p  p)
2. Now every contradiction is logically false, i.e. its negation is true
├ (p  p)
3. Therefore a and b cannot be right together, i.e. one of them is wrong
├ (Bap  Bbp)
Niiniluoto (2013): according to Twardoswki, Protagorean personal truth
predicate would violate classical principles of logic
Plato assumes objective truth in the first premise: what is true for an
agent is made true simpliciter (beyond anyone’s beliefs)
Any agreement between a and b about p requires a justification of their
beliefs
Tp  (Bp  Jp)
A reversal of Plato’s classical definition of knowledge: epistemic truth
├ Kp  (Bp  Tp  Jp)
├ Kp  Bp
├ Kp  Tp We assume Tarski’s T-scheme: Tp  p (in L)
├ Kp  Jp
Gettier’s Problem: justification may be insufficient to ground truth
├\ Jp  Kp
Plato assumes objective truth in the first premise: what is true for an
agent is made true simpliciter (beyond anyone’s beliefs)
Any agreement between a and b about p requires a justification of their
beliefs
Tp  (Bp  Jp)
A reversal of Plato’s classical definition of knowledge: epistemic truth
├ Kp  (Bp  Tp  Jp)
├ Kp  Bp
├ Kp  Tp We assume Tarski’s T-scheme: Tp  p (in L)
├ Kp  Jp
Gettier’s Problem: justification may be insufficient to ground truth
├\ Jp  Kp
Plato assumes objective truth in the first premise: what is true for an
agent is made true simpliciter (beyond anyone’s beliefs)
Any agreement between a and b about p requires a justification of their
beliefs
Tp  (Bp  Jp)
A reversal of Plato’s classical definition of knowledge: epistemic truth
├ Kp  (Bp  Tp  Jp)
├ Kp  Bp
├ Kp  Tp We assume Tarski’s T-scheme: Tp  p (in L)
├ Kp  Jp
Gettier’s Problem: justification may be insufficient to ground truth
├\ Jp  Kp
Plato assumes objective truth in the first premise: what is true for an
agent is made true simpliciter (beyond anyone’s beliefs)
Any agreement between a and b about p requires a justification of their
beliefs
Tp  (Bp  Jp)
A reversal of Plato’s classical definition of knowledge: epistemic truth
├ Kp  (Bp  Tp  Jp)
├ Kp  Bp
├ Kp  Tp We assume Tarski’s T-scheme: Tp  p (in L)
├ Kp  Jp
Gettier’s Problem: justification may be insufficient to ground truth
├* Jp  Kp
Do a and b discuss within the same model, M?
If they disagree about p then, by consistency (logical truth):
p is true-in-Ma
p is true-in-Mb (or, equivalently: p is false-in-Mb)
“Deaf dialogue”: more than one language in the dialogue
Logical truth is the sole basic criterion for material truth, thus far
Plato’s argument assumes uniqueness/universality of truth
How to obtain common agreement about p, accordingly?
Intersubjective truth (fallibilism)
Which theory of truth gives the best explanation of the relation
between knowledge, truth, belief, and justification? (cf. relevance)
My answer: epistemic truth (truth as assertion: Tap, “p is true-for-a”)
Do a and b discuss within the same model, M?
If they disagree about p then, by consistency (logical truth):
p is true-in-Ma
p is true-in-Mb (or, equivalently: p is false-in-Mb)
“Deaf dialogue”: more than one language in the dialogue
Logical truth is the sole basic criterion for material truth, thus far
Plato’s argument assumes uniqueness/universality of truth
How to obtain common agreement about p, accordingly?
Intersubjective truth (fallibilism)
Which theory of truth gives the best explanation of the relation
between knowledge, truth, belief, and justification? (cf. relevance)
My answer: epistemic truth (truth as assertion: Tap, “p is true-for-a”)
Do a and b discuss within the same model, M?
If they disagree about p then, by consistency (logical truth):
p is true-in-Ma
p is true-in-Mb (or, equivalently: p is false-in-Mb)
“Deaf dialogue”: more than one language in the dialogue
Logical truth is the sole basic criterion for material truth, thus far
Plato’s argument assumes uniqueness/universality of truth
How to obtain common agreement about p, accordingly?
Intersubjective truth (fallibilism)
Which theory of truth gives the best explanation of the relation
between knowledge, truth, belief, and justification? (cf. relevance)
My answer: epistemic truth (truth as assertion: Tap, “p is true-for-a”)
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(1) Relative truth Tap fails to satisfy Von Wright’s truth-logic
├ Ta(p  q)  Tap  Taq
├ Tap  Ta(p  q)
├ Tap  Tap
├ Tap  TaTap
├* Ta(p  q)  Tap  Taq
├* Tap  Tap
├* TaTap  Tap
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(1) Relative truth Tap fails to satisfy Von Wright’s truth-logic
├ Ta(p  q)  Tap  Taq
├ Tap  Ta(p  q)
├ Tap  Tap
├ Tap  TaTap
├* Ta(p  q)  Tap  Taq
├* Tap  Tap
├* TaTap  Tap
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(1) Relative truth Tap fails to satisfy Von Wright’s truth-logic
├ Ta(p  q)  Tap  Taq
├ Tap  Ta(p  q)
├ Tap  Tap
├ Tap  TaTap
├* Ta(p  q)  Tap  Taq
├* Tap  Tap
├* TaTap  Tap
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(2) It introduces omniscience into the concept of truth:
Tap, p  q, Taq
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(3) There are no external constraints for truth and falsity, accordingly
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(4) Tarski’s T-equivalence cannot be sustained, because
Tap  p does not make sense
not valid: Bap  p and p  Bap are not accepted in doxastic logic
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(4) Tarski’s T-equivalence cannot be sustained, because
Tap  p does not make sense
not valid: Bap  p and p  Bap are not accepted in doxastic logic
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(4) Tarski’s T-equivalence cannot be sustained, because
Tap  p does not make sense
not valid: Bap  p and p  Bap are not accepted in doxastic logic
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(5) Either relative truth has absolute truth-conditions: self-refuting
Or it doesn’t, and it results in endless iterations: Bap, BaBap,
BaBaBaBaBap, …
Niiniluoto (2013): epistemic definitions of truth are not relevant, for:
(5) Either relative truth has absolute truth-conditions: self-refuting
Or it doesn’t, and it results in endless iterations: Bap, BaBap,
BaBaBaBaBap, …
Reply to (1)
“Relative truth Tap fails to satisfy Von Wright’s truth-logic”
- Why should all of von Wright’s axioms be maintained?
- Why is relative truth reduced to the doxastic operator B?
- Isn’t relative truth a more complex operator like (Tap  Bap & Jap)?
- What if relative truth behaves like a weak modality, ?
Reply to (1)
“Relative truth Tap fails to satisfy Von Wright’s truth-logic”
- Why should all of von Wright’s axioms be maintained?
- Why is relative truth reduced to the doxastic operator B?
- Isn’t relative truth a more complex operator like (Tap  Bap & Jap)?
- What if relative truth behaves like a weak modality, ?
Reply to (1)
“Relative truth Tap fails to satisfy Von Wright’s truth-logic”
- Why should all of von Wright’s axioms be maintained?
- Why is relative truth reduced to the doxastic operator B?
- Isn’t relative truth a more complex operator like (Tap  Bap & Jap)?
- What if relative truth behaves like a weak modality, ?
Reply to (1)
“Relative truth Tap fails to satisfy Von Wright’s truth-logic”
- Why should all of von Wright’s axioms be maintained?
- Why is relative truth reduced to the doxastic operator B?
- Isn’t relative truth a more complex operator like (Tap  Bap & Jap)?
- What if relative truth behaves like a weak modality, ?
Reply to (1)
“Relative truth Tap fails to satisfy Von Wright’s truth-logic”
- Why should all of von Wright’s axioms be maintained?
- Why is relative truth reduced to the doxastic operator B?
- Isn’t relative truth a more complex operator like (Tap  Bap & Jap)?
- What if relative truth behaves like a weak modality, ?
Reply to (2)
“It introduces the case of omniscience into the concept of truth”
What is p  q, if not Ta(p  q)?
Unless objective truth is restored, omniscience is just Modus Ponens
Reply to (2)
“It introduces the case of omniscience into the concept of truth”
- What is p  q, if not Ta(p  q)?
- Unless objective truth is restored, omniscience is just Modus Ponens
Reply to (2)
“It introduces the case of omniscience into the concept of truth”
- What is p  q, if not Ta(p  q)?
- Unless objective truth is restored, omniscience is just Modus Ponens
Reply to (3):
“There are no external constraints for truth and falsity, accordingly”
- Tap can be enriched beyond merely personal belief (see (1))
- “Personal” needn’t mean “individual” (cf. intersubjective agreement)
Reply to (3):
“There are no external constraints for truth and falsity, accordingly”
- Tap can be enriched beyond merely personal belief (see (1))
- “Personal” needn’t mean “single” (cf. intersubjective agreement)
Reply to (3):
“There are no external constraints for truth and falsity, accordingly”
- Tap can be enriched beyond merely personal belief (see (1))
- “Personal” needn’t mean “single” (cf. intersubjective agreement)
Reply to (4):
“Tarski’s T-equivalence cannot be sustained”
- Objective truth is restored again through the formula “p”
- What if “p” means p-in-L, or “p for a”?
Reply to (4):
“Tarski’s T-equivalence cannot be sustained”
- Objective truth is restored again through the formula “p”
- What if “p” means p-in-L, or “p for a”?
Reply to (4):
“Tarski’s T-equivalence cannot be sustained”
- Objective truth is restored again through the formula “p”
- What if “p” means p-in-L, or “p for a”?
Reply to (5):
“Either relative truth has absolute truth-conditions: self-refuting
Or it doesn’t, and it results in endless iterations”
- Relativity needn’t be universal, or self-referential
- What does iterated (relative) truth mean?
Endless iteration relies upon the failure of Axiom 4 (Hintikka (1962))
- An argument against the modal interpretation of truth, at the best
What if truth is rendered as assertion, or truth-claim?
Reply to (5):
“Either relative truth has absolute truth-conditions: self-refuting
Or it doesn’t, and it results in endless iterations”
- Relativity needn’t be universal, or self-referential
- What does iterated (relative) truth mean?
Endless iteration relies upon the failure of Axiom 4 (Hintikka (1962))
- An argument against the modal interpretation of truth, at the best
What if truth is rendered as assertion, or truth-claim?
Reply to (5):
“Either relative truth has absolute truth-conditions: self-refuting
Or it doesn’t, and it results in endless iterations”
- Relativity needn’t be universal, or self-referential
- What does iterated (relative) truth mean?
Endless iteration relies upon the failure of Axiom 4 (Hintikka (1962))
- An argument against the modal interpretation of truth, at the best
What if truth is rendered as assertion, or truth-claim?
Reply to (5):
“Either relative truth has absolute truth-conditions: self-refuting
Or it doesn’t, and it results in endless iterations”
- Relativity needn’t be universal, or self-referential
- What does iterated (relative) truth mean?
Endless iteration relies upon the failure of Axiom 4 (Hintikka (1962))
- An argument against the modal interpretation of truth, at the best
What if truth is rendered as assertion, or truth-claim?
(1)-(5) reduce epistemic truth to mere relativism: Tap = Bap
assume objective truth in the definition of Ta (cf. Plato)
Two biases in the objections to epistemic truth:
- Uniqueness of truth is taken to be granted
- Truth is presented as a value (can there be more than one?)
An alternative solution: deflationism?
“p is true”: the sentence S is (true-)in-M
You can escape your shadow, by turning the light off
You can avoid the debate about the nature of truth, by begging it out
How to think of the nature of truth, if not as some agreement?
With what: reality, system, community, …, ?
(1)-(5) reduce epistemic truth to mere relativism: Tap = Bap
assume objective truth in the definition of Ta (cf. Plato)
Two biases in the objections to epistemic truth:
- Uniqueness of truth is taken to be granted
- Truth is presented as a value (can there be more than one?)
An alternative solution: deflationism?
“p is true”: the sentence S is (true-)in-M
You can escape your shadow, by turning the light off
You can avoid the debate about the nature of truth, by begging it out
How to think of the nature of truth, if not as some agreement?
With what: reality, system, community, …, ?
(1)-(5) reduce epistemic truth to mere relativism: Tap = Bap
assume objective truth in the definition of Ta (cf. Plato)
Two biases in the objections to epistemic truth:
- Uniqueness of truth is taken to be granted
- Truth is presented as a value (can there be more than one?)
An alternative solution: deflationism?
“p is true”: the sentence S is (true-)in-M
You can escape your shadow, by turning the light off
You can avoid the debate about the nature of truth, by begging it out
How to think of the nature of truth, if not as some agreement?
With what: reality, system, community, …, ?
(1)-(5) reduce epistemic truth to mere relativism: Tap = Bap
assume objective truth in the definition of Ta (cf. Plato)
Two biases in the objections to epistemic truth:
- Uniqueness of truth is taken to be granted
- Truth is presented as a value (can there be more than one?)
An alternative solution: deflationism?
“p is true”: the sentence S is (true-)in-M
You can escape your shadow, by turning the light off
You can avoid the debate about the nature of truth, by begging it out
How to think of the nature of truth, if not as some agreement?
With what: reality, system, community, …, ?
(1)-(5) reduce epistemic truth to mere relativism: Tap = Bap
assume objective truth in the definition of Ta (cf. Plato)
Two biases in the objections to epistemic truth:
- Uniqueness of truth is taken to be granted
- Truth is presented as a value (can there be more than one?)
An alternative solution: deflationism?
“p is true”: the sentence S is (true-)in-M
You can escape your shadow, by turning the light off
You can avoid the debate about the nature of truth, by begging it out
How to think of the nature of truth, if not as some agreement?
With what: reality, system, community, …, ?
Russell (1923): truth as correspondence with facts
A proposition is therefore a class of facts, psychological or linguistic,
defined as standing into a certain relation (it can be either assertion or
denial, according to the cases) to a certain fact.
Beliefs/sentences: truth-bearers
“Psychological facts”: assertions
Russell (1923): truth as correspondence with facts
A proposition is therefore a class of facts, psychological or linguistic,
defined as standing into a certain relation (it can be either assertion or
denial, according to the cases) to a certain fact.
Fact: truth-maker
Beliefs/sentences: truth-bearers
“Psychological facts”: assertions, denials
“Linguistic facts”: sentences (affirmative, negative)
Beliefs/Sentences are individuated by a fact making these true
A proposition is the class of such beliefs/sentences
How to warrant the occurrence of such facts, in practice?
Russell (1923): truth as correspondence with facts
A proposition is therefore a class of facts, psychological or linguistic,
defined as standing into a certain relation (it can be either assertion or
denial, according to the cases) to a certain fact.
Fact: truth-maker
Beliefs/sentences: truth-bearers
“Psychological facts”: assertions, denials
“Linguistic facts”: sentences (affirmative, negative)
Beliefs/Sentences are individuated by a fact making these true
A proposition is the class of such beliefs/sentences
How to warrant the occurrence of such facts, in practice?
Russell (1923): truth as correspondence with facts
A proposition is therefore a class of facts, psychological or linguistic,
defined as standing into a certain relation (it can be either assertion or
denial, according to the cases) to a certain fact.
Fact: truth-maker
Beliefs/sentences: truth-bearers
“Psychological facts”: assertions, denials
“Linguistic facts”: sentences (affirmative, negative)
Beliefs/Sentences are individuated by a fact making these true
A proposition is the class of such beliefs/sentences
How to warrant the occurrence of such facts, in practice?
Russell (1923): truth as correspondence with facts
A proposition is therefore a class of facts, psychological or linguistic,
defined as standing into a certain relation (it can be either assertion or
denial, according to the cases) to a certain fact.
Fact: truth-maker
Beliefs/sentences: truth-bearers
“Psychological facts”: assertions, denials
“Linguistic facts”: sentences (affirmative, negative)
Beliefs/Sentences are individuated by a fact making these true
A proposition is the class of such beliefs/sentences
How to warrant the occurrence of such facts, in practice?
Peirce (1877: 7): truth as ideal convergence of opinions
The question therefore is, how is true belief (or belief in the real)
distinguished from false belief (or belief in fiction). Now, as we have
seen (…) the ideas of truth and falsehood, in their full development,
appertain exclusively to the experiential method of settling opinion.
Truth: agreement between speakers in an ideal community
any sentence that ought to be believed by every agent
the result of an inquiry process related to agreed beliefs
What is the rationale (proto-logic) of such an inquiry process?
How to come from simple sentences expressing beliefs to true
propositions warranting knowledge?
Peirce (1877: 7): truth as ideal convergence of opinions
The question therefore is, how is true belief (or belief in the real)
distinguished from false belief (or belief in fiction). Now, as we have
seen (…) the ideas of truth and falsehood, in their full development,
appertain exclusively to the experiential method of settling opinion.
Truth: agreement between speakers in an ideal community
any sentence that ought to be believed by every agent
the result of an inquiry process related to agreed beliefs
What is the rationale (proto-logic) of such an inquiry process?
How to come from simple sentences expressing beliefs to true
propositions warranting knowledge?
Peirce (1877: 7): truth as ideal convergence of opinions
Peirce (1877: 7): truth as ideal convergence of opinions
The question therefore is, how is true belief (or belief in the real)
distinguished from false belief (or belief in fiction). Now, as we have
seen (…) the ideas of truth and falsehood, in their full development,
appertain exclusively to the experiential method of settling opinion.
Truth: agreement between speakers in an ideal community
any sentence that ought to be believed by every agent
the result of an inquiry process related to agreed beliefs
What is the rationale (proto-logic) of such an inquiry process?
How to come from simple sentences expressing beliefs to true
propositions warranting knowledge?
2.
Truth-Values
Frege: truth is the value of a proposition (its logical content)
The word “true” indicates the aim of logic as does “beautiful” that of
aesthetics or “good” that of ethics.
(Frege 1956: 289)
Cannot truth be relative as a multi-faceted value (cf. cultural relativism)?
Proposition: the “thought” (Gedanke) expressed by a sentence
“A sentence proper is a proper name, and its Bedeutung, if it has one, is
a truth-value: the True or the False.
Truth is an objective value: only one value for every proposition
(true/false), but different propositions (senses) for the same value
Frege: truth is the value of a proposition (its logical content)
The word “true” indicates the aim of logic as does “beautiful” that of
aesthetics or “good” that of ethics.
(Frege 1956: 289)
Cannot truth be relative as a multi-faceted value (cf. cultural relativism)?
Proposition: the “thought” (Gedanke) expressed by a sentence
“A sentence proper is a proper name, and its Bedeutung, if it has one, is
a truth-value: the True or the False.
Truth is an objective value: only one value for every proposition
(true/false), but different propositions (senses) for the same value
Frege: truth is the value of a proposition (its logical content)
The word “true” indicates the aim of logic as does “beautiful” that of
aesthetics or “good” that of ethics.
(Frege 1956: 289)
Cannot truth be relative as a multi-faceted value (cf. cultural relativism)?
Proposition: the “thought” (Gedanke) expressed by a sentence
A sentence proper is a proper name, and its reference, if it has one, is a
truth-value: the True or the False.
Truth is an objective value: only one value for every proposition
(true/false), but different propositions (senses) for the same value
Frege: truth is the value of a proposition (its logical content)
The word “true” indicates the aim of logic as does “beautiful” that of
aesthetics or “good” that of ethics.
(Frege 1956: 289)
Cannot truth be relative as a multi-faceted value (cf. cultural relativism)?
Proposition: the “thought” (Gedanke) expressed by a sentence
A sentence proper is a proper name, and its reference, if it has one, is a
truth-value: the True or the False.
Truth is an objective value: only one value for every proposition
(true/false), but different propositions (senses) for the same value
“Frege’s Axiom” (Suszko): a unique referent for declarative sentences
We are therefore driven into accepting the truth-value of a sentence as
constituting its reference. By the truth value of a sentence I understand
the circumstance that it is true or false. There are no further truthvalues. For brevity I call the one the True, the other the False. Every
declarative sentence concerned with the reference of its words is
therefore to be regarded as a proper name, and its reference, if it has
one, is either the true or the false.
(Frege 1960: 63)
One, or two truth-values? The meaning of bivalence
“Falsity”: whatever rejected by the speaker in the inquiry process
Only one expected value, two possible outcomes (success vs failure)
“Frege’s Axiom” (Suszko): a unique referent for declarative sentences
We are therefore driven into accepting the truth-value of a sentence as
constituting its reference. By the truth value of a sentence I understand
the circumstance that it is true or false. There are no further truthvalues. For brevity I call the one the True, the other the False. Every
declarative sentence concerned with the reference of its words is
therefore to be regarded as a proper name, and its reference, if it has
one, is either the true or the false.
(Frege 1960: 63)
One, or two truth-values? The meaning of bivalence
“Falsity”: whatever rejected by the speaker in the inquiry process
Only one expected value, two possible outcomes (success vs failure)
2 preconditions for sentences to express a “thought” (Frege 1960: 127)
- A common object of investigation:
The being of a thought may also be taken to lie in the possibility of
different thinkers’ grasping the thought as one and the same thought.
- An object prior to any investigation:
But even the act of grasping a thought is not a production of the thought,
is not an act of setting its parts in order; for the thought was already
true, and so was already there with its parts in order, before it was
grasped. A traveler who crosses a mountain-range does not thereby
make the mountain-range; no more does the judging subject make a
thought by acknowledging its truth.
Thought is prior to judgment; what is prior to thought itself?
2 preconditions for sentences to express a “thought” (Frege 1960: 127)
- A common object of investigation:
The being of a thought may also be taken to lie in the possibility of
different thinkers’ grasping the thought as one and the same thought.
- An object prior to any investigation:
But even the act of grasping a thought is not a production of the thought,
is not an act of setting its parts in order; for the thought was already
true, and so was already there with its parts in order, before it was
grasped. A traveler who crosses a mountain-range does not thereby
make the mountain-range; no more does the judging subject make a
thought by acknowledging its truth.
Thought is prior to judgment; what is prior to thought itself?
2 preconditions for sentences to express a “thought” (Frege 1960: 127)
- A common object of investigation:
The being of a thought may also be taken to lie in the possibility of
different thinkers’ grasping the thought as one and the same thought.
- An object prior to any investigation:
But even the act of grasping a thought is not a production of the thought,
is not an act of setting its parts in order; for the thought was already
true, and so was already there with its parts in order, before it was
grasped. A traveler who crosses a mountain-range does not thereby
make the mountain-range; no more does the judging subject make a
thought by acknowledging its truth.
Thought is prior to judgment; what is prior to thought itself?
2 preconditions for sentences to express a “thought” (Frege 1960: 127)
- A common object of investigation:
The being of a thought may also be taken to lie in the possibility of
different thinkers’ grasping the thought as one and the same thought.
- An object prior to any investigation:
But even the act of grasping a thought is not a production of the thought,
is not an act of setting its parts in order; for the thought was already
true, and so was already there with its parts in order, before it was
grasped. A traveler who crosses a mountain-range does not thereby
make the mountain-range; no more does the judging subject make a
thought by acknowledging its truth.
Thought is prior to judgment; what is prior to thought itself?
2 problems:
1. The reference: a “truth-value”, not a fact? (cf. Slingshot’s Argument)
The theory with which Frege’s name is especially associated is one
which is apt to strike one at first rather fantastic, being usually expressed
as a theory that sentences are names of truth-values.
(Prior 1953: 55)
2. What is a Fregean truth-maker, accordingly?
The value is not an ideal object, but an ideal activity of agreement
It is the striving for truth that drives us always to advance from the
sense to the reference.
(Frege 1960: 63)
2 problems:
1. The reference: a “truth-value”, not a fact? (cf. Slingshot’s Argument)
The theory with which Frege’s name is especially associated is one
which is apt to strike one at first rather fantastic, being usually expressed
as a theory that sentences are names of truth-values.
(Prior 1953: 55)
2. What is a Fregean truth-maker, accordingly?
The value is not an ideal object, but an ideal activity of agreement
It is the striving for truth that drives us always to advance from the
sense to the reference.
(Frege 1960: 63)
2 problems:
1. The reference: a “truth-value”, not a fact? (cf. Slingshot’s Argument)
The theory with which Frege’s name is especially associated is one
which is apt to strike one at first rather fantastic, being usually expressed
as a theory that sentences are names of truth-values.
(Prior 1953: 55)
2. What is a Fregean truth-maker, accordingly?
The value is not an ideal object, but an ideal activity of agreement
It is the striving for truth that drives us always to advance from the
sense to the reference.
(Frege 1960: 63)
Frege on skeptics: scientific models assume existence (no justification)
Assumption (judgeable content) vs Assertion (judgment) of propositions
Idealists or sceptics will perhaps long since have objected: ‘You talk,
without further ado, of the Moon as an object; but how do you know that
the name ‘the Moon’ has any reference? How do you know anything
whatsoever has a reference?’ I reply that when we say ‘the Moon’, we
do not intend to speak of our idea of the Moon, nor are we satisfied with
the sense alone, but we presuppose a reference.”
(Frege 1960: 61)
Bivalence holds for sentences whose referents are assumed
- some sentences are neither-true-nor-false
- these do not express propositions (out of the scientific inquiry)
Frege on skeptics: scientific models assume existence (no justification)
Assumption (judgeable content) vs Assertion (judgment) of propositions
Idealists or sceptics will perhaps long since have objected: ‘You talk,
without further ado, of the Moon as an object; but how do you know that
the name ‘the Moon’ has any reference? How do you know anything
whatsoever has a reference?’ I reply that when we say ‘the Moon’, we
do not intend to speak of our idea of the Moon, nor are we satisfied with
the sense alone, but we presuppose a reference.”
(Frege 1960: 61)
Bivalence holds for sentences whose referents are assumed
- some sentences are neither-true-nor-false
- these do not express propositions (out of the scientific inquiry)
The scientific inquiry: a question-answer game
A propositional question (Satzfrage) contains a demand that we should
either acknowledge the truth of a thought, or reject it as false. (…) The
answer to a question is an assertion based upon a judgment; this is so
equally whether the answer is affirmative or negative.
(Frege 1960: 117)
A three-fold distinction: sentence, proposition, statement (judgment)
Logic: a science related to the laws of truth-preservation
A sentence is the expression of a proposition
The statement “p” is a truth-claim: acknowledging the truth of p
Assertion: a truth-claim, the statement that p (“p is the case”)
The scientific inquiry: a question-answer game
A propositional question (Satzfrage) contains a demand that we should
either acknowledge the truth of a thought, or reject it as false. (…) The
answer to a question is an assertion based upon a judgment; this is so
equally whether the answer is affirmative or negative.
(Frege 1960: 117)
A three-fold distinction: sentence, proposition, statement (judgment)
Logic: a science related to the laws of truth-preservation
A sentence is the expression of a proposition
The statement “p” is a truth-claim: acknowledging the truth of p
Assertion: a truth-claim, the statement that p (“p is the case”)
The scientific inquiry: a question-answer game
A propositional question (Satzfrage) contains a demand that we should
either acknowledge the truth of a thought, or reject it as false. (…) The
answer to a question is an assertion based upon a judgment; this is so
equally whether the answer is affirmative or negative.
(Frege 1960: 117)
A three-fold distinction: sentence, proposition, statement (judgment)
Logic: a science related to the laws of truth-preservation
A sentence is the expression of a proposition
The statement “p” is a truth-claim: acknowledging the truth of p
Assertion: a truth-claim, the statement that p (“p is the case”)
A description of the inquiry process: Begriffschrift (ideography)
Frege’s turnstile ├: symbol of a truth-claim
├p: “the proposition (that) p is the case” (stated by a speaker)
“p is a logical truth (axiom, or theorem)”
Difference between relative (assertion) and absolute (logical truth)
Three grades of epistemic truth
├p may mean “I take p to be true”
“p is true for everyone-in-the-model”
“p is true in every model (everyone-in-every-model)”
There is no difference of nature between these three grades of
epistemic truth, but a difference of degree (of acceptance)
A description of the inquiry process: Begriffschrift (ideography)
Frege’s turnstile ├: symbol of a truth-claim
├p: “the proposition (that) p is the case” (stated by a speaker)
“p is a logical truth (axiom, or theorem)”
Difference between relative (assertion) and absolute (logical truth)
Three grades of epistemic truth
├p may mean “I take p to be true”
“p is true for everyone-in-the-model”
“p is true in every model (everyone-in-every-model)”
There is no difference of nature between these three grades of
epistemic truth, but a difference of degree (of acceptance)
A description of the inquiry process: Begriffschrift (ideography)
Frege’s turnstile ├: symbol of a truth-claim
├p: “the proposition (that) p is the case” (stated by a speaker)
“p is a logical truth (axiom, or theorem)”
Difference between relative (assertion) and absolute (logical truth)
Three grades of epistemic truth
├p may mean “I take p to be true”
“p is true for everyone-in-the-model”
“p is true in every model (everyone-in-every-model)”
There is no difference of nature between these three grades of
epistemic truth, but a difference of degree (of acceptance)
A description of the inquiry process: Begriffschrift (ideography)
Frege’s turnstile ├: symbol of a truth-claim
├p: “the proposition (that) p is the case” (stated by a speaker)
“p is a logical truth (axiom, or theorem)”
Difference between relative (assertion) and absolute (logical truth)
Three grades of epistemic truth
├p may mean “I take p to be true”
“p is true for everyone-in-the-model”
“p is true in every model (everyone-in-every-model)”
There is no difference of nature between these three grades of
epistemic truth, but a difference of degree (of acceptance)
An example of reasoning : by Modus Ponens
If the accused was not in Berlin at the time of the deed, he did not
commit the murder; now the accused was not in Berlin at the time of the
murder; therefore he did not commit the murder.
(Frege 1960: 125)
p: “the accused was not in Berlin at the time of the deed”
q: “he (the accused) did not commit the murder”
├ (p  q)
├p
├ q
├ (  )
├
├
├ (  ), , 
Compare with the modal K-structure, replacing “” by “├”
An example of reasoning : by Modus Ponens
If the accused was not in Berlin at the time of the deed, he did not
commit the murder; now the accused was not in Berlin at the time of the
murder; therefore he did not commit the murder.
(Frege 1960: 125)
p: “the accused was not in Berlin at the time of the deed”
q: “he (the accused) did not commit the murder”
├ (p  q)
├p
├ q
├ (  )
├
├
├ (  ), , 
Compare with the modal K-structure, replacing “” by “├”
An example of reasoning : by Modus Ponens
If the accused was not in Berlin at the time of the deed, he did not
commit the murder; now the accused was not in Berlin at the time of the
murder; therefore he did not commit the murder.
(Frege 1960: 125)
p: “the accused was not in Berlin at the time of the deed”
q: “he (the accused) did not commit the murder”
├ (p  q)
├ p
├ q
├ (  )
├
├
├ (  ), , 
Compare with the modal K-structure, replacing “” by “├”
An example of reasoning : by Modus Ponens
If the accused was not in Berlin at the time of the deed, he did not
commit the murder; now the accused was not in Berlin at the time of the
murder; therefore he did not commit the murder.
(Frege 1960: 125)
p: “the accused was not in Berlin at the time of the deed”
q: “he (the accused) did not commit the murder”
├ (p  q)
├ p
├ q
├ (  )
├
├
├ (  ), , 
Compare with the modal K-structure, replacing “” by “├”
An example of reasoning : by Modus Ponens
If the accused was not in Berlin at the time of the deed, he did not
commit the murder; now the accused was not in Berlin at the time of the
murder; therefore he did not commit the murder.
(Frege 1960: 125)
p: “the accused was not in Berlin at the time of the deed”
q: “he (the accused) did not commit the murder”
├ (p  q)
├ p
├ q
├ (  )
├
├
├ (  ), , 
Compare with the modal K-structure, replacing “” by “├”
An example of reasoning : by Modus Ponens
If the accused was not in Berlin at the time of the deed, he did not
commit the murder; now the accused was not in Berlin at the time of the
murder; therefore he did not commit the murder.
(Frege 1960: 125)
p: “the accused was not in Berlin at the time of the deed”
q: “he (the accused) did not commit the murder”
├ (p  q)
├ p
├ q
├ (  )
├
├
├ (  ), , 
Compare with the modal K-structure, replacing “” by “├”
Back to Niiniluoto (2003), objection #1 to epistemic truth:
How to account for the failure of:
├* Ta(p  q)  Tap  Taq
├* Tap  Tap
├* TaTap  Tap
Reading Ta as a truth-claim, or assertion ├:
- asserting a disjunction needn’t entail any of the disjuncts (cf. -E)
- failure of conclusive evidence may prevent from asserting either (Ta)
- what should asserting an assertion mean?
Assertion is not an operator but, rather, an operand: the logical value at
hand in any scientific inquiry
Back to Niiniluoto (2003), objection #1 to epistemic truth:
How to account for the failure of:
├* Ta(p  q)  Tap  Taq
├* Tap  Tap
├* TaTap  Tap
Reading Ta as a truth-claim, or assertion ├:
- asserting a disjunction needn’t entail any of the disjuncts (cf. -E)
- failure of conclusive evidence may prevent from asserting either (Ta)
- what should asserting an assertion mean?
Assertion is not an operator but, rather, an operand: the logical value at
hand in any scientific inquiry
Back to Niiniluoto (2003), objection #1 to epistemic truth:
How to account for the failure of:
├* Ta(p  q)  Tap  Taq
├* Tap  Tap
├* TaTap  Tap
Reading Ta as a truth-claim, or assertion ├:
- asserting a disjunction needn’t entail any of the disjuncts (cf. -E)
- failure of conclusive evidence may prevent from asserting either (Ta)
- what should asserting an assertion mean?
Assertion is not an operator but, rather, an operand: the logical value at
hand in any scientific inquiry
Back to Niiniluoto (2003), objection #1 to epistemic truth:
How to account for the failure of:
├* Ta(p  q)  Tap  Taq
├* Tap  Tap
├* TaTap  Tap
Reading Ta as a truth-claim, or assertion ├:
- asserting a disjunction needn’t entail any of the disjuncts (cf. -E)
- failure of conclusive evidence may prevent from asserting either (Ta)
- what should asserting an assertion mean?
Assertion is not an operator but, rather, an operand: the logical value at
hand in any scientific inquiry
3.
The like
A Question-Answer Semantics for a logic of epistemic truth
Any sentence p of a language has:
- a sense: a set of n ordered questions about p
Q(p) = q1(p), …, qn(p)
- a reference: a corresponding set of ordered answers
A(p) = a1(p), …, an(p)
n: the number of relevant questions expressing the value of p
Frege: n = 1, i.e. “Is p true?”
There is only one sort of judgment: assertive judgment
- either a1(p) = 1, therefore p is (claimed to be) true
- or a1(p) = 0, therefore p is (claimed to be) false
Frege assumes that every sentence can be assessed about its value
For every p: a(p) = 1 or a(p) = 0, i.e. a(p) = 1
A Question-Answer Semantics for a logic of epistemic truth
Any sentence p of a language has:
- a sense: a set of n ordered questions about p
Q(p) = q1(p), …, qn(p)
- a reference: a corresponding set of ordered answers
A(p) = a1(p), …, an(p)
n: the number of relevant questions expressing the value of p
Frege: n = 1, i.e. “Is p true?”
There is only one sort of judgment: assertive judgment
- either a1(p) = 1, therefore p is (claimed to be) true
- or a1(p) = 0, therefore p is (claimed to be) false
Frege assumes that every sentence can be assessed about its value
For every p: a(p) = 1 or a(p) = 0, i.e. a(p) = 1
A Question-Answer Semantics for a logic of epistemic truth
Any sentence p of a language has:
- a sense: a set of n ordered questions about p
Q(p) = q1(p), …, qn(p)
- a reference: a corresponding set of ordered answers
A(p) = a1(p), …, an(p)
n: the number of relevant questions expressing the value of p
Frege: n = 1, i.e. “Is p accepted?”
There is only one sort of judgment: assertive judgment
- either a1(p) = 1, therefore p is (claimed to be) true
- or a1(p) = 0, therefore p is (claimed to be) false
Frege assumes that every sentence can be assessed about its value
For every p: a(p) = 1 or a(p) = 0, i.e. a(p) = 1
A Question-Answer Semantics for a logic of epistemic truth
Any sentence p of a language has:
- a sense: a set of n ordered questions about p
Q(p) = q1(p), …, qn(p)
- a reference: a corresponding set of ordered answers
A(p) = a1(p), …, an(p)
n: the number of relevant questions expressing the value of p
Frege: n = 1, i.e. “Is p accepted?”
There is only one sort of judgment: assertive judgment
- either a1(p) = 1, therefore p is (claimed to be) true
- or a1(p) = 0, therefore p is (claimed to be) false
Frege assumes that every sentence can be assessed about its value
For every p: a(p) = 1 or a(p) = 0, i.e. a(p) = 1
A Question-Answer Semantics for a logic of epistemic truth
Any sentence p of a language has:
- a sense: a set of n ordered questions about p
Q(p) = q1(p), …, qn(p)
- a reference: a corresponding set of ordered answers
A(p) = a1(p), …, an(p)
n: the number of relevant questions expressing the value of p
Frege: n = 1, i.e. “Is p accepted?”
There is only one sort of judgment: assertive judgment
- either a(p) = 1, therefore p is (claimed to be) true
- or a(p) = 0, therefore p is (claimed to be) false
Frege assumes that every sentence can be assessed about its value
For every p: a(p) = 1 or a(p) = 0, i.e. a(p) = 1
A Question-Answer Semantics for a logic of epistemic truth
Any sentence p of a language has:
- a sense: a set of n ordered questions about p
Q(p) = q1(p), …, qn(p)
- a reference: a corresponding set of ordered answers
A(p) = a1(p), …, an(p)
n: the number of relevant questions expressing the value of p
Frege: n = 1, i.e. “Is p accepted?”
There is only one sort of judgment: assertive judgment
- either a(p) = 1, therefore p is (claimed to be) true
- or a(p) = 0, therefore p is (claimed to be) false
Frege assumes that every sentence can be assessed about its value
For every p: a(p) = 1 or a(p) = 0, i.e. a(p) = 1
On negative judgments: sentential vs statemental negation
- Not every sentence can be asserted or denied:
Positive assertion: “p is the case” (assertion)
Negative assertion: “p is not the case”, or “p is the case” (strong denial)
- Two independent logical values (n = 2): acceptance, and denial
aj(p) = 1 iff p is accepted
aj(p) = 0 iff p is denied (not accepted)
A broader logic of acceptance and refusal:
Q(p) = q1(p),q2(p)
q1(p): “Is p accepted?”
q2(p): “Is p denied”?
On negative judgments: sentential vs statemental negation
- Not every sentence can be asserted or denied:
Positive assertion: “p is the case” (assertion)
Negative assertion: “p is not the case”, or “p is the case” (strong denial)
- Two independent logical values (n = 2): acceptance, and denial
aj(p) = 1 iff p is accepted
aj(p) = 0 iff p is denied (not accepted)
A broader logic of acceptance and refusal:
Q(p) = q1(p),q2(p)
q1(p): “Is p accepted?”
q2(p): “Is p denied”?
On negative judgments: sentential vs statemental negation
- Not every sentence can be asserted or denied:
Positive assertion: “p is the case” (assertion)
Negative assertion: “p is not the case”, or “p is the case” (strong denial)
- Two independent logical values (n = 2): acceptance, and denial
aj(p) = 1 iff p is accepted
aj(p) = 0 iff p is denied (not accepted)
A broader logic of acceptance and refusal:
Q(p) = q1(p),q2(p)
q1(p): “Is p accepted?”
q2(p): “Is p denied”?
On negative judgments: sentential vs statemental negation
- Not every sentence can be asserted or denied:
Positive assertion: “p is the case” (assertion)
Negative assertion: “p is not the case”, or “p is the case” (strong denial)
- Two independent logical values (n = 2): acceptance, and denial
aj(p) = 1 iff p is accepted
aj(p) = 0 iff p is denied (not accepted)
A broader logic of acceptance and refusal:
Q(p) = q1(p),q2(p)
q1(p): “Is p accepted?”
q2(p): “Is p denied”?
The logical value of a sentence p is the resulting pair of answers to it:
1,1, 1,0, 0,1, 0,0
Frege’s “truth-values” T and F correspond to 1,0 and 0,1
1,1 and 0,0 correspond to the “glutty” (B) and “gappy” (N) values
Many-valuedness: there is more than 2 sorts of statements of p
Pragmatic bivalence: there are only 2 sorts of answers to each question
consistency: no sentence can be both accepted and denied
There can be various constraints on a truth-claim
1,1: weak affirmation (plausible evidence) for paraconsistent logics
0,0: strong affirmation (conclusive evidence) for paracomplete logics
The logical value of a sentence p is the resulting pair of answers to it:
1,1, 1,0, 0,1, 0,0
Frege’s “truth-values” T and F correspond to 1,0 and 0,1
1,1 and 0,0 correspond to the “glutty” (B) and “gappy” (N) values
Many-valuedness: there is more than 2 sorts of statements of p
Pragmatic bivalence: there are only 2 sorts of answers to each question
consistency: no sentence can be both accepted and denied
There can be various constraints on a truth-claim
1,1: weak affirmation (plausible evidence) for paraconsistent logics
0,0: strong affirmation (conclusive evidence) for paracomplete logics
The logical value of a sentence p is the resulting pair of answers to it:
1,1, 1,0, 0,1, 0,0
Frege’s “truth-values” T and F correspond to 1,0 and 0,1
1,1 and 0,0 correspond to the “glutty” (B) and “gappy” (N) values
Many-valuedness: there is more than 2 sorts of statements of p
Pragmatic bivalence: there are only 2 sorts of answers to each question
consistency: no sentence can be both accepted and denied
There can be various constraints on a truth-claim
1,1: weak affirmation (plausible evidence) for paraconsistent logics
0,0: strong affirmation (conclusive evidence) for paracomplete logics
The logical value of a sentence p is the resulting pair of answers to it:
1,1, 1,0, 0,1, 0,0
Frege’s “truth-values” T and F correspond to 1,0 and 0,1
1,1 and 0,0 correspond to the “glutty” (B) and “gappy” (N) values
Many-valuedness: there is more than 2 sorts of statements of p
Pragmatic bivalence: there are only 2 sorts of answers to each question
consistency: no sentence can be both accepted and denied
There can be various constraints on a truth-claim
1,1: weak affirmation (plausible evidence) for paraconsistent logics
0,0: strong affirmation (conclusive evidence) for paracomplete logics
The logical value of a sentence p is the resulting pair of answers to it:
1,1, 1,0, 0,1, 0,0
Frege’s “truth-values” T and F correspond to 1,0 and 0,1
1,1 and 0,0 correspond to the “glutty” (B) and “gappy” (N) values
Many-valuedness: there is more than 2 sorts of statements of p
Pragmatic bivalence: there are only 2 sorts of answers to each question
consistency: no sentence can be both accepted and denied
There can be various constraints on a truth-claim
1,1: weak affirmation (plausible evidence) for paraconsistent logics
0,0: strong affirmation (conclusive evidence) for paracomplete logics
Frege’s Begriffschrift: an ideography about propositions and judgments
─
content stroke: symbol for propositions
─p
the proposition (that) p
┬p
the proposition that not-p
│
judgment-stroke
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
question about a proposition: q1(…) or q2(…)
─p
the proposition (that) p
┬p
the proposition that not-p
│
judgment-stroke
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
content stroke: symbol for propositions
─p
the proposition (that) p
┬p
the proposition that not-p
│
judgment-stroke
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
content stroke: symbol for propositions
─p
the proposition (that) p: q1(p)
┬p
the proposition that not-p
│
judgment-stroke
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
content stroke: symbol for propositions
─p
the proposition (that) p
┬p
the proposition that not-p
│
judgment-stroke
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
content stroke: symbol for propositions
─p
the proposition (that) p
┬p
the proposition that not-p: q2(p)
│
judgment-stroke
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
content stroke: symbol for propositions
─p
the proposition (that) p
┬p
the proposition that not-p
│
judgment-stroke
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
content stroke: symbol for propositions
─p
the proposition (that) p
┬p
the proposition that not-p
│
the assertion of a proposition: a1(…) or a2(…)
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
content stroke: symbol for propositions
─p
the proposition (that) p
┬p
the proposition that not-p
│
judgment-stroke
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
─p
┬p
content stroke: symbol for propositions
the proposition (that) p
the proposition that not-p
│
judgment-stroke
┼p
assertion or denial of p (p is the case, or p is not the case)
├
the assertion of p (p is the case): a1(p) = 1
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
─p
┬p
content stroke: symbol for propositions
the proposition (that) p
the proposition that not-p
│
judgment-stroke
┼p
assertion or denial of p (p is the case, or p is not the case)
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
─p
┬p
content stroke: symbol for propositions
the proposition (that) p
the proposition that not-p
│
judgment-stroke
┼p
assertion or denial of p (p is the case, or p is not the case)
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case): a2(p) = 1
┤
the denial of p (not: p is the case)
Frege’s Begriffschrift: an ideography about propositions and judgments
─
─p
┬p
content stroke: symbol for propositions
the proposition (that) p
the proposition that not-p
│
judgment-stroke
┼p
assertion or denial of p (p is the case, or p is not the case)
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
+ one additional judgment: denial
the denial of p (not: p is the case)
┤
Frege’s Begriffschrift: an ideography about propositions and judgments
─
─p
┬p
content stroke: symbol for propositions
the proposition (that) p
the proposition that not-p
│
judgment-stroke
┼p
assertion or denial of p (p is the case, or p is not the case)
├
the assertion of p (p is the case)
├┬
the assertion of not-p (p is not the case)
+ one additional judgment: denial
the denial of p (not: p is the case): a1(p) = 0
┤
From epistemic logic to epistemology of logic:
- axioms of epistemic logic are made relevant by equating K and ├
- assertion, truth-claim, belief are on a par
- no conflation of knowledge and belief: degrees of epistemic truth
From to sentential to statemental logic:
- truth-claims as the pragmatic value of sentences (what is done with)
science is an inquiry games where answers are given to questions
- Frege’s anti-psychologism made reference independent of the inquiry
statements afford values within a logic of acceptance-preservation
From classical logic to alternative language-games
- logic is an activity with ruled purposes: a language-game
- truth-preservation is only but one of these language-games
- opposition beyond consequence: about agreement and disagreement
From epistemic logic to epistemology of logic:
- axioms of epistemic logic are made relevant by equating K and ├
- assertion, truth-claim, belief are on a par
- no conflation of knowledge and belief: degrees of epistemic truth
From to sentential to statemental logic:
- truth-claims as the pragmatic value of sentences (what is done with)
science is an inquiry games where answers are given to questions
- Frege’s anti-psychologism made reference independent of the inquiry
statements afford values within a logic of acceptance-preservation
From classical logic to alternative language-games
- logic is an activity with ruled purposes: a language-game
- truth-preservation is only but one of these language-games
- opposition beyond consequence: about agreement and disagreement
From epistemic logic to epistemology of logic:
- axioms of epistemic logic are made relevant by equating K and ├
- assertion, truth-claim, belief are on a par
- no conflation of knowledge and belief: degrees of epistemic truth
From to sentential to statemental logic:
- truth-claims as the pragmatic value of sentences (what is done with)
science is an inquiry games where answers are given to questions
- Frege’s anti-psychologism made reference independent of the inquiry
statements afford values within a logic of acceptance-preservation
From classical logic to alternative language-games:
- logic is an activity with ruled purposes: a language-game
- truth-preservation is only but one of these language-games
- opposition beyond consequence: about agreement and disagreement
L,Cn
L,Op
The language L: formal ontology (arbitrary entities)
- A finite set of meaningful objects, given by questions-answers
Every object: a set of properties given by sentences
- A reversal in formal ontology: objects are made by sentences
The number of relevant sentences/questions is context-dependent
- Example: A(a) = 1011001, in a bitstring of 7 ordered answers
Each single answer is normative: what ought be accepted/denied
The abstract relation Op: opposition (as difference)
- A set of relations between objects in L
The relation between logical values yields an algebraic semantics: QAS
The language L: formal ontology (arbitrary entities)
- A finite set of meaningful objects, given by questions-answers
Every object: a set of properties given by sentences
- A reversal in formal ontology: objects are made by sentences
The number of relevant sentences/questions is context-dependent
- Example: A(a) = 1011001, in a bitstring of 7 ordered answers
Each single answer is normative: what ought be accepted/denied
The abstract relation Op: opposition (as difference)
- A set of relations between objects in L
The relation between logical values yields an algebraic semantics: QAS
The language L: formal ontology (arbitrary entities)
- A finite set of meaningful objects, given by questions-answers
Every object: a set of properties given by sentences
- A reversal in formal ontology: objects are made by sentences
The number of relevant sentences/questions is context-dependent
- Example: A(a) = 1011001, in a bitstring of 7 ordered answers
Each single answer is normative: what ought be accepted/denied
The abstract relation Op: opposition (as difference)
- A set of relations between objects in L
The relation between logical values yields an algebraic semantics: QAS
The language L: formal ontology (arbitrary entities)
- A finite set of meaningful objects, given by questions-answers
Every object: a set of properties given by sentences
- A reversal in formal ontology: objects are made by sentences
The number of relevant sentences/questions is context-dependent
- Example: A(a) = 1011001, in a bitstring of 7 ordered answers
Each single answer is normative: what ought be accepted/denied
The abstract relation Op: opposition (as difference)
- A set of relations between objects in L
The relation between logical values yields an algebraic semantics: QAS
The language L: formal ontology (arbitrary entities)
- A finite set of meaningful objects, given by questions-answers
Every object: a set of properties given by sentences
- A reversal in formal ontology: objects are made by sentences
The number of relevant sentences/questions is context-dependent
- Example: A(a) = 1011001, in a bitstring of 7 ordered answers
Each single answer is normative: what ought be accepted/denied
The abstract relation Op: opposition (as difference)
- A set of relations between objects in L
The relation between logical values yields an algebraic semantics: QAS
The language L: formal ontology (arbitrary entities)
- A finite set of meaningful objects, given by questions-answers
Every object: a set of properties given by sentences
- A reversal in formal ontology: objects are made by sentences
The number of relevant sentences/questions is context-dependent
- Example: A(a) = 1011001, in a bitstring of 7 ordered answers
Each single answer is normative: what ought be accepted/denied
The abstract relation Op: opposition (as difference)
- A set of relations between objects in L
The relation between logical values yields an algebraic semantics: QAS
References
G. Frege. “The Thought. A Logical Inquiry”, Mind, Vol. 65(1956): 289-311
G. Frege. Transcription for the Philosophical Writings of Gottlob Frege, P. Geach & Max Black
(eds.), Basil Blackwell, Oxford (1960)
J. Hintikka. Knowledge and Belief, Ithaca Press (1962)
F. Martela. “Truth as intersubjective epistemological commitment – a pragmatic account of
truth”, draft (2010)
C. S. Peirce. “The Fixation of Belief”, Popular Science Monthly, Vol. 12(1877): 1-15
I. Niiniluoto. “Is truth relative or absolute?”, talk presented at the conference Logic and
Philosophy, University of Kiev (23-25 May 2013)
B. Russell. “Truth-functions and meaning-functions”, in The Collected Pa-pers of Bertrand Russell.
Vol. 9 : “Language, Mind and Matter : 1919-26”. London, Boston: Unwin Hyman, 1988: 158
Merci!
Thank you!
Cпасибо!
Kiitos!