Formal systems
| University of Edinburgh | PHIL08004 |
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January 16, 2020 Puzzle 3
A man was looking at a portrait. Someone asked him, “Whose picture are you looking at?” He replied:
“Brothers and sisters have I none, but this man’s father is my father’s son.”
Whose picture was the man looking at?
2 / 32 3 / 32 I Is the man in the picture the man himself?
I Self-portrait?
4 / 32 I Is the man in the picture the man himself?
I Self-portrait?
4 / 32 I Is the man in the picture the man himself?
I Self-portrait?
4 / 32 I No.
I If the man in the picture is himself, then this father would be his own son!
I “this man is my father’s son” vs.
I “this man’s father is my father’s son”
5 / 32 I No.
I If the man in the picture is himself, then this father would be his own son!
I “this man is my father’s son” vs.
I “this man’s father is my father’s son”
5 / 32 I No.
I If the man in the picture is himself, then this father would be his own son!
I “this man is my father’s son” vs.
I “this man’s father is my father’s son”
5 / 32 I No.
I If the man in the picture is himself, then this father would be his own son!
I “this man is my father’s son” vs.
I “this man’s father is my father’s son”
5 / 32 I No.
I If the man in the picture is himself, then this father would be his own son!
I “this man is my father’s son” vs.
I “this man’s father is my father’s son”
5 / 32 I No.
I If the man in the picture is himself, then this father would be his own son!
I “this man is my father’s son” vs.
I “this man’s father is my father’s son”
5 / 32 6 / 32 This man’s father is my father’s son
I (1) The father of the man in the portrait is a son of the father of the speaker.
I (2) A son of the father of the speaker must have the same father as the speaker.
I (3) So, the father of the man in the portrait and the speaker have the same father.
I (4) But the speaker has no brothers (“brothers and sisters I have none”)
I (5) So, the speaker is the father of the man in the portrait.
I (6) So, the man in the portrait is the son of the speaker. 7 / 32 This man’s father is my father’s son
I (1) The father of the man in the portrait is a son of the father of the speaker.
I (2) A son of the father of the speaker must have the same father as the speaker.
I (3) So, the father of the man in the portrait and the speaker have the same father.
I (4) But the speaker has no brothers (“brothers and sisters I have none”)
I (5) So, the speaker is the father of the man in the portrait.
I (6) So, the man in the portrait is the son of the speaker. 7 / 32 This man’s father is my father’s son
I (1) The father of the man in the portrait is a son of the father of the speaker.
I (2) A son of the father of the speaker must have the same father as the speaker.
I (3) So, the father of the man in the portrait and the speaker have the same father.
I (4) But the speaker has no brothers (“brothers and sisters I have none”)
I (5) So, the speaker is the father of the man in the portrait.
I (6) So, the man in the portrait is the son of the speaker. 7 / 32 This man’s father is my father’s son
I (1) The father of the man in the portrait is a son of the father of the speaker.
I (2) A son of the father of the speaker must have the same father as the speaker.
I (3) So, the father of the man in the portrait and the speaker have the same father.
I (4) But the speaker has no brothers (“brothers and sisters I have none”)
I (5) So, the speaker is the father of the man in the portrait.
I (6) So, the man in the portrait is the son of the speaker. 7 / 32 This man’s father is my father’s son
I (1) The father of the man in the portrait is a son of the father of the speaker.
I (2) A son of the father of the speaker must have the same father as the speaker.
I (3) So, the father of the man in the portrait and the speaker have the same father.
I (4) But the speaker has no brothers (“brothers and sisters I have none”)
I (5) So, the speaker is the father of the man in the portrait.
I (6) So, the man in the portrait is the son of the speaker. 7 / 32 This man’s father is my father’s son
I (1) The father of the man in the portrait is a son of the father of the speaker.
I (2) A son of the father of the speaker must have the same father as the speaker.
I (3) So, the father of the man in the portrait and the speaker have the same father.
I (4) But the speaker has no brothers (“brothers and sisters I have none”)
I (5) So, the speaker is the father of the man in the portrait.
I (6) So, the man in the portrait is the son of the speaker. 7 / 32 This man’s father is my father’s son
I (1) The father of the man in the portrait is a son of the father of the speaker.
I (2) A son of the father of the speaker must have the same father as the speaker.
I (3) So, the father of the man in the portrait and the speaker have the same father.
I (4) But the speaker has no brothers (“brothers and sisters I have none”)
I (5) So, the speaker is the father of the man in the portrait.
I (6) So, the man in the portrait is the son of the speaker. 7 / 32 Formal languages
I Symbols: the atomic expressions of the language
I Formation rules: say what strings of symbols are formulas (what strings are grammatical or “well-formed”)
8 / 32 Formal languages
I Symbols: the atomic expressions of the language
I Formation rules: say what strings of symbols are formulas (what strings are grammatical or “well-formed”)
8 / 32 Formal languages
I Symbols: the atomic expressions of the language
I Formation rules: say what strings of symbols are formulas (what strings are grammatical or “well-formed”)
8 / 32 The language BANG
I Symbols: , , !, β I Formulation rules: I and are formulas, I If X is a formula, !X is a formula, I Nothing else is a formula.
9 / 32 The language BANG
I Symbols: , , !, β I Formulation rules: I and are formulas, I If X is a formula, !X is a formula, I Nothing else is a formula.
9 / 32 The language BANG
I Symbols: , , !, β I Formulation rules: I and are formulas, I If X is a formula, !X is a formula, I Nothing else is a formula.
9 / 32 The language BANG
I Symbols: , , !, β I Formulation rules: I and are formulas, I If X is a formula, !X is a formula, I Nothing else is a formula.
9 / 32 The language BANG
I Symbols: , , !, β I Formulation rules: I and are formulas, I If X is a formula, !X is a formula, I Nothing else is a formula.
9 / 32 The language BANG
I Symbols: , , !, β I Formulation rules: I and are formulas, I If X is a formula, !X is a formula, I Nothing else is a formula.
9 / 32 Formal systems
I Formal languages: Symbols + Formation rules
I Formal systems: Formal language + Deductive apparatus
I This generates a special subset of the well-formed formulas that are the “theorems” of the system.
I Theorem: A formula producible by the deductive apparatus.
10 / 32 Formal systems
I Formal languages: Symbols + Formation rules
I Formal systems: Formal language + Deductive apparatus
I This generates a special subset of the well-formed formulas that are the “theorems” of the system.
I Theorem: A formula producible by the deductive apparatus.
10 / 32 Formal systems
I Formal languages: Symbols + Formation rules
I Formal systems: Formal language + Deductive apparatus
I This generates a special subset of the well-formed formulas that are the “theorems” of the system.
I Theorem: A formula producible by the deductive apparatus.
10 / 32 Formal systems
I Formal languages: Symbols + Formation rules
I Formal systems: Formal language + Deductive apparatus
I This generates a special subset of the well-formed formulas that are the “theorems” of the system.
I Theorem: A formula producible by the deductive apparatus.
10 / 32 Formal systems
I Formal languages: Symbols + Formation rules
I Formal systems: Formal language + Deductive apparatus
I This generates a special subset of the well-formed formulas that are the “theorems” of the system.
I Theorem: A formula producible by the deductive apparatus.
10 / 32 11 / 32 Deductive apparatus
Axioms are formulas that are “starting points, or “free theorems” I
Inference rules specify what transitions between formu- las are allowed I
I (Notice that formal systems can have only axioms, only inference rules, or some of each)
12 / 32 Deductive apparatus
Axioms are formulas that are “starting points, or “free theorems” I
Inference rules specify what transitions between formu- las are allowed I
I (Notice that formal systems can have only axioms, only inference rules, or some of each)
12 / 32 Deductive apparatus
Axioms are formulas that are “starting points, or “free theorems” I
Inference rules specify what transitions between formu- las are allowed I
I (Notice that formal systems can have only axioms, only inference rules, or some of each)
12 / 32 Deductive apparatus
Axioms are formulas that are “starting points, or “free theorems” I
Inference rules specify what transitions between formu- las are allowed I
I (Notice that formal systems can have only axioms, only inference rules, or some of each)
12 / 32 Deductive apparatus
Axioms are formulas that are “starting points, or “free theorems” I
Inference rules specify what transitions between formu- las are allowed I
I (Notice that formal systems can have only axioms, only inference rules, or some of each)
12 / 32 Derivation
A derivation is an explicit line by line demonstration of how to produce a theorem according to the rules and axioms of the formal system.
13 / 32 Hofstatder’s MIU-system
14 / 32 MIU-system
I Language: I Symbols: M, I, U I Formation rule: Any finite string (of MIU symbols) is well-formed.
15 / 32 MIU-system
I Language: I Symbols: M, I, U I Formation rule: Any finite string (of MIU symbols) is well-formed.
15 / 32 MIU-system
I Language: I Symbols: M, I, U I Formation rule: Any finite string (of MIU symbols) is well-formed.
15 / 32 MIU-system
I Language: I Symbols: M, I, U I Formation rule: Any finite string (of MIU symbols) is well-formed.
15 / 32 MIU’s deductive apparatus
I Axiom: MI
I Rule 1: Add a U to the end of any string ending in I. For example: MI to MIU.
I Rule 2: Double any string after the M (that is, change Mx, to Mxx). For example: MIU to MIUIU.
I Rule 3: Replace any III with a U. For example: MUIIIU to MUUU.
I Rule 4: Remove any UU. For example: MUUU to MU.
16 / 32 MIU’s deductive apparatus
I Axiom: MI
I Rule 1: Add a U to the end of any string ending in I. For example: MI to MIU.
I Rule 2: Double any string after the M (that is, change Mx, to Mxx). For example: MIU to MIUIU.
I Rule 3: Replace any III with a U. For example: MUIIIU to MUUU.
I Rule 4: Remove any UU. For example: MUUU to MU.
16 / 32 MIU’s deductive apparatus
I Axiom: MI
I Rule 1: Add a U to the end of any string ending in I. For example: MI to MIU.
I Rule 2: Double any string after the M (that is, change Mx, to Mxx). For example: MIU to MIUIU.
I Rule 3: Replace any III with a U. For example: MUIIIU to MUUU.
I Rule 4: Remove any UU. For example: MUUU to MU.
16 / 32 MIU’s deductive apparatus
I Axiom: MI
I Rule 1: Add a U to the end of any string ending in I. For example: MI to MIU.
I Rule 2: Double any string after the M (that is, change Mx, to Mxx). For example: MIU to MIUIU.
I Rule 3: Replace any III with a U. For example: MUIIIU to MUUU.
I Rule 4: Remove any UU. For example: MUUU to MU.
16 / 32 MIU’s deductive apparatus
I Axiom: MI
I Rule 1: Add a U to the end of any string ending in I. For example: MI to MIU.
I Rule 2: Double any string after the M (that is, change Mx, to Mxx). For example: MIU to MIUIU.
I Rule 3: Replace any III with a U. For example: MUIIIU to MUUU.
I Rule 4: Remove any UU. For example: MUUU to MU.
16 / 32 MIU’s deductive apparatus (sleek) I Axiom: MI I Rule 1: xI xIU I Rule 2: Mx Mxx I Rule 3: xIIIy xUy I Rule 4: xUUy xy
17 / 32 MIU’s deductive apparatus (sleek) I Axiom: MI I Rule 1: xI xIU I Rule 2: Mx Mxx I Rule 3: xIIIy xUy I Rule 4: xUUy xy
17 / 32 MIU’s deductive apparatus (sleek) I Axiom: MI I Rule 1: xI xIU I Rule 2: Mx Mxx I Rule 3: xIIIy xUy I Rule 4: xUUy xy
17 / 32 MIU’s deductive apparatus (sleek) I Axiom: MI I Rule 1: xI xIU I Rule 2: Mx Mxx I Rule 3: xIIIy xUy I Rule 4: xUUy xy
17 / 32 MIU’s deductive apparatus (sleek) I Axiom: MI I Rule 1: xI xIU I Rule 2: Mx Mxx I Rule 3: xIIIy xUy I Rule 4: xUUy xy
17 / 32 Derive MUI
18 / 32 I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MUI [rule 3]
Derive MUI
19 / 32 I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MUI [rule 3]
Derive MUI
I 1. MI [axiom]
19 / 32 I 3. MIIII [rule 2] I 4. MUI [rule 3]
Derive MUI
I 1. MI [axiom] I 2. MII [rule 2]
19 / 32 I 4. MUI [rule 3]
Derive MUI
I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2]
19 / 32 Derive MUI
I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MUI [rule 3]
19 / 32 20 / 32 Derive MIUI
21 / 32 Derive MUUII
22 / 32 Derive MUUII
I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MIIIIIIII [rule 2] I 5. MUIIIII [rule 3] I 6. MUUII [rule 3]
23 / 32 Derive MUUII
I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MIIIIIIII [rule 2] I 5. MUIIIII [rule 3] I 6. MUUII [rule 3]
23 / 32 Derive MUUII
I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MIIIIIIII [rule 2] I 5. MUIIIII [rule 3] I 6. MUUII [rule 3]
23 / 32 Derive MUUII
I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MIIIIIIII [rule 2] I 5. MUIIIII [rule 3] I 6. MUUII [rule 3]
23 / 32 Derive MUUII
I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MIIIIIIII [rule 2] I 5. MUIIIII [rule 3] I 6. MUUII [rule 3]
23 / 32 Derive MUUII
I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MIIIIIIII [rule 2] I 5. MUIIIII [rule 3] I 6. MUUII [rule 3]
23 / 32 Derive MUUII
I 1. MI [axiom] I 2. MII [rule 2] I 3. MIIII [rule 2] I 4. MIIIIIIII [rule 2] I 5. MUIIIII [rule 3] I 6. MUUII [rule 3]
23 / 32 Mechanical method for deriving MIU-theorems
I Step 1. Apply every applicable rule to the axiom.
I Step 2. Apply every applicable rule to the theorems resulting from step 1.
I Step 3. Apply every applicable rule to the theorems resulting from step 2.
I ...
24 / 32 Mechanical method for deriving MIU-theorems
I Step 1. Apply every applicable rule to the axiom.
I Step 2. Apply every applicable rule to the theorems resulting from step 1.
I Step 3. Apply every applicable rule to the theorems resulting from step 2.
I ...
24 / 32 Mechanical method for deriving MIU-theorems
I Step 1. Apply every applicable rule to the axiom.
I Step 2. Apply every applicable rule to the theorems resulting from step 1.
I Step 3. Apply every applicable rule to the theorems resulting from step 2.
I ...
24 / 32 Mechanical method for deriving MIU-theorems
I Step 1. Apply every applicable rule to the axiom.
I Step 2. Apply every applicable rule to the theorems resulting from step 1.
I Step 3. Apply every applicable rule to the theorems resulting from step 2.
I ...
24 / 32 Mechanical method for deriving MIU-theorems
I Step 1. Apply every applicable rule to the axiom.
I Step 2. Apply every applicable rule to the theorems resulting from step 1.
I Step 3. Apply every applicable rule to the theorems resulting from step 2.
I ...
24 / 32 25 / 32 26 / 32 27 / 32 Derive MUIIU
28 / 32 Derive MUUIU
29 / 32 Can you derive MU?
30 / 32 MU playground homework
31 / 32 online interface 32 / 32