Defjnition 2.1. The basic symbols of SL are of three kinds: 1. - - PDF document

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Defjnition 2.1. The basic symbols of SL are of three kinds: 1. - - PDF document

Jan 21, 2023 213 likes •484 views

Defjnition 2.1. The basic symbols of SL are of three kinds: 1. Logical Connectives: , , , , 2. Punctuation Symbols: (, ) 3. Sentence Letters: , , , , , , 1 , 1 , 1 , , 1 , 1 ,

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Defjnition 2.1. The basic symbols of SL are of three kinds:

  • 1. Logical Connectives: ∼, ∧, ∨, →, ↔
  • 2. Punctuation Symbols: (, )
  • 3. Sentence Letters: 𝐵, 𝐶, 𝐷, … , 𝑇, 𝑈, 𝐵1, 𝐶1, 𝐷1, … , 𝑇1, 𝑈1, 𝐵2, …

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Defjnition 2.1. The basic symbols of SL are of three kinds:

  • 1. Logical Connectives: ∼, ∧, ∨, →, ↔
  • 2. Punctuation Symbols: (, )
  • 3. Sentence Letters: 𝐵, 𝐶, 𝐷, … , 𝑇, 𝑈, 𝐵1, 𝐶1, 𝐷1, … , 𝑇1, 𝑈1, 𝐵2, …

Defjnition 2.2. The sentences of SL are given by the following recursive defjnition:

Base Clause: Every sentence letter is a sentence. Generating Clauses:

  • 1. If 𝜚 is a sentence, then so is ∼𝜚.
  • 2. If 𝜚 and 𝜄 are sentences, then so are both (𝜚→𝜄) and (𝜚↔𝜄).
  • 3. If all of 𝜚1, 𝜚2, 𝜚3, 𝜚4, … , 𝜚𝑜 are sentences (the list must include at

least two sentences and be fjnite), then so are (𝜚1 ∧ 𝜚2 ∧ 𝜚3 ∧ 𝜚4 ∧ … ∧ 𝜚𝑜) and (𝜚1 ∨ 𝜚2 ∨ 𝜚3 ∨ 𝜚4 ∨ … ∨ 𝜚𝑜).

Closure Clause: A sequence of symbols is an SL sentence ifg its being a sen-

tence follows from the previous two clauses.

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Defjnition 2.1. The basic symbols of SL are of three kinds:

  • 1. Logical Connectives: ∼, ∧, ∨, →, ↔
  • 2. Punctuation Symbols: (, )
  • 3. Sentence Letters: 𝐵, 𝐶, 𝐷, … , 𝑇, 𝑈, 𝐵1, 𝐶1, 𝐷1, … , 𝑇1, 𝑈1, 𝐵2, …

Defjnition 2.2. The sentences of SL are given by the following recursive defjnition:

Base Clause: Every sentence letter is a sentence. Generating Clauses:

  • 1. If 𝜚 is a sentence, then so is ∼𝜚.
  • 2. If 𝜚 and 𝜄 are sentences, then so are both (𝜚→𝜄) and (𝜚↔𝜄).
  • 3. If all of 𝜚1, 𝜚2, 𝜚3, 𝜚4, … , 𝜚𝑜 are sentences (the list must include at

least two sentences and be fjnite), then so are (𝜚1 ∧ 𝜚2 ∧ 𝜚3 ∧ 𝜚4 ∧ … ∧ 𝜚𝑜) and (𝜚1 ∨ 𝜚2 ∨ 𝜚3 ∨ 𝜚4 ∨ … ∨ 𝜚𝑜).

Closure Clause: A sequence of symbols is an SL sentence ifg its being a sen-

tence follows from the previous two clauses. Defjnition 2.2 defjnes the offjcial sentences of SL. Defjnition 2.4 A string of symbols is an unoffjcial sentence of SL ifg we can obtain it from an offjcial sentence by

  • 1. deleting outer parentheses, or
  • 2. replacing one or more pairs of offjcial round parentheses ( ) with square

brackets [ ] or curly brackets { }.

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Defjnition 2.5. The following clauses defjne when one sentence is a sub- sentence of another:

  • 1. Every sentence is a subsentence of itself.
  • 2. 𝜚 is a subsentence of ∼𝜚.
  • 3. 𝜚 and 𝜄 are subsentences of (𝜚→𝜄) and (𝜚↔𝜄).
  • 4. All of 𝜚1, 𝜚2, 𝜚3, 𝜚4, … , 𝜚𝑜 are subsentences of (𝜚1 ∧ 𝜚2 ∧ 𝜚3 ∧ 𝜚4 ∧ … ∧ 𝜚𝑜)

and (𝜚1 ∨ 𝜚2 ∨ 𝜚3 ∨ 𝜚4 ∨ … ∨ 𝜚𝑜).

  • 5. (Transitivity) If 𝜚 is a subsentence of 𝜄 and 𝜄 is a subsentence of 𝜔,

then 𝜚 is a subsentence of 𝜔.

  • 6. That’s all.

A sentence 𝜚 is a proper subsentence of 𝜔 ifg 𝜚 is a subsentence of, but isn’t identical to 𝜔. So, while each sentence is a subsentence of itself, no sentence is a proper subsentence of itself.

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Defjnition 2.8 The following clauses defjne the order of every SL sen-

  • tence. Let ORD𝜚 be the order of 𝜚. Then:
  • 1. If 𝜚 is an atomic sentence (a sentence letter), then ORD𝜚 = 1.
  • 2. For any sentence 𝜚, ORD∼𝜚 = ORD𝜚 + 1.
  • 3. For any sentences 𝜚 and 𝜄, ORD(𝜚→𝜄) is one greater than the max of

ORD𝜚 and ORD𝜄. Likewise, ORD(𝜚↔𝜄) is one greater than the max

  • f ORD𝜚 and ORD𝜄.
  • 4. For any sentences 𝜚1, … , 𝜚𝑜, ORD(𝜚1 ∧ … ∧ 𝜚𝑜) is one greater than

the max of ORD𝜚1, …, ORD𝜚𝑜.

  • 5. For any sentences 𝜚1, … , 𝜚𝑜, ORD(𝜚1 ∨ … ∨ 𝜚𝑜) is one greater than

the max of ORD𝜚1, …, ORD𝜚𝑜.

  • 6. That’s all.

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Defjnition 2.10. The main connective is the connective token (or tokens) that occur(s) in the sentence but in no proper subsentence.

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Defjnition 2.12 The construction tree for a sentence is a diagram of how the sentence is generated through the recursive clauses of the defjnition

  • f SL sentences. We put atomic sentences as leaves at the top, and the gen-

erating clauses specify how we can join nodes of the tree together (starting with the leaves at the top) into new nodes. The complete sentence is the node at the base of the tree.

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Defjnition 2.12 The construction tree for a sentence is a diagram of how the sentence is generated through the recursive clauses of the defjnition

  • f SL sentences. We put atomic sentences as leaves at the top, and the gen-

erating clauses specify how we can join nodes of the tree together (starting with the leaves at the top) into new nodes. The complete sentence is the node at the base of the tree. Example: {𝐵 ∨ (𝐸→𝐶)} ∧ ∼𝐻. {𝐵 ∨ (𝐸→𝐶)} ∧ ∼𝐻 𝐵 ∨ (𝐸→𝐶) 𝐵 𝐸→𝐶 𝐸 𝐶 𝐻 ∼𝐻

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Defjnition 2.12 The construction tree for a sentence is a diagram of how the sentence is generated through the recursive clauses of the defjnition

  • f SL sentences. We put atomic sentences as leaves at the top, and the gen-

erating clauses specify how we can join nodes of the tree together (starting with the leaves at the top) into new nodes. The complete sentence is the node at the base of the tree. Example: {𝐵 ∨ (𝐸→𝐶)} ∧ ∼𝐻. {𝐵 ∨ (𝐸→𝐶)} ∧ ∼𝐻 𝐵 ∨ (𝐸→𝐶) 𝐵 𝐸→𝐶 𝐸 𝐶 𝐻 ∼𝐻 NB:

  • The subsentences of a sentence are the nodes in the sentence’s construc-

tion tree.

  • The order of a sentence is the number of nodes of its longest branch.
  • The main connective of a sentence is the connective added last (at the

bottom) of the construction tree.

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Defjnition 2.12 The construction tree for a sentence is a diagram of how the sentence is generated through the recursive clauses of the defjnition

  • f SL sentences. We put atomic sentences as leaves at the top, and the gen-

erating clauses specify how we can join nodes of the tree together (starting with the leaves at the top) into new nodes. The complete sentence is the node at the base of the tree. Example: ((𝐷 ∧ 𝐸) → 𝐵) ↔ (𝐸 ∨ 𝐼). ((𝐷 ∧ 𝐸) → 𝐵) ↔ (𝐸 ∨ 𝐼) (𝐷 ∧ 𝐸) → 𝐵 𝐷 ∧ 𝐸 𝐷 𝐸 𝐵 𝐸 ∨ 𝐼 𝐸 𝐼 NB:

  • The subsentences of a sentence are the nodes in the sentence’s construc-

tion tree.

  • The order of a sentence is the number of nodes of its longest branch.
  • The main connective of a sentence is the connective added last (at the

bottom) of the construction tree. f

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Truth in a Model Model of single sentence: Defjnition 2.17 Given that 𝜚 is a sentence of SL, a model for 𝜚 is an assignment of a truth value, either true or false, to each sentence letter in 𝜚. notation: if a model 𝔫 assigns a sentence letter 𝜔 the value true, we will write 𝔫(𝜔) = T If 𝔫 assigns a sentence letter 𝜔 the value false, we will write 𝔫(𝜔) = F

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Truth in a Model Model of single sentence: Defjnition 2.17 Given that 𝜚 is a sentence of SL, a model for 𝜚 is an assignment of a truth value, either true or false, to each sentence letter in 𝜚. notation:

  • if a model 𝔫 assigns a sentence letter 𝜔 the value true, we will write

𝔫(𝜔) = T

  • If 𝔫 assigns a sentence letter 𝜔 the value false, we will write 𝔫(𝜔) =

F.

Model of a set of sentences: Defjnition 2.18 Given that Δ is a set of SL sentences, 𝔫 is a model for Δ ifg 𝔫 is a model for each sentence in Δ [i.e. ifg 𝔫 is a model for each sentence letter in each sentence in Δ (by Def 2.17)]

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Truth in a Model Model of single sentence: Defjnition 2.17 Given that 𝜚 is a sentence of SL, a model for 𝜚 is an assignment of a truth value, either true or false, to each sentence letter in 𝜚. notation:

  • if a model 𝔫 assigns a sentence letter 𝜔 the value true, we will write

𝔫(𝜔) = T

  • If 𝔫 assigns a sentence letter 𝜔 the value false, we will write 𝔫(𝜔) =

F.

Model of a set of sentences: Defjnition 2.18 Given that Δ is a set of SL sentences, 𝔫 is a model for Δ ifg 𝔫 is a model for each sentence in Δ [i.e. ifg 𝔫 is a model for each sentence letter in each sentence in Δ (by Def 2.17)] Model for SL Defjnition 2.19 𝔫 is a model for SL ifg 𝔫 is a model for every sentence

  • f SL.

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Sentences more complex than single sentence letters are not directly assigned truth values by models:

Model: 𝔫(𝐵) = T, 𝔫(𝐶) = F, 𝔫(𝐸) = F, 𝔫(𝐻) = T sentence: 𝐵 ∨ (𝐸 → 𝐶)) ∧ ¬𝐻

𝔫[(𝐵 ∨ (𝐸 → 𝐶)) ∧ ¬𝐻] = ???

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Truth Functions Defjnition 2.20 The following clauses defjne when an SL sentence 𝜄 is true (or false) on a model 𝔫 for 𝜄:

  • 1. A sentence letter 𝜚 is true on 𝔫 ifg 𝔫 assigns true to it, i.e. ifg 𝔫(𝜚) = T.
  • 2. A negation ∼𝜚 is true on 𝔫 ifg the sentence 𝜚 is false on 𝔫.
  • 3. A conjunction 𝜚1 ∧ 𝜚2 ∧ 𝜚3 ∧ 𝜚4 ∧ … ∧ 𝜚𝑜 is true on 𝔫 ifg all of the

conjuncts are true on 𝔫.

  • 4. A disjunction 𝜚1 ∨ 𝜚2 ∨ 𝜚3 ∨ 𝜚4 ∨ … ∨ 𝜚𝑜 is true on 𝔫 ifg at least one
  • f the disjuncts is true on 𝔫.
  • 5. A conditional 𝜚→𝜔 is true on 𝔫 ifg the LHS is false or the RHS is true
  • n 𝔫.
  • 6. A biconditional 𝜚 ↔ 𝜔 is true on 𝔫 ifg both 𝜚 and 𝜔 have the same

truth value on 𝔫.

  • 7. If a sentence is false on 𝔫 ifg it’s not true on 𝔫.

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Truth Functions Defjnition 2.20 The following clauses defjne when an SL sentence 𝜄 is true (or false) on a model 𝔫 for 𝜄:

  • 1. A sentence letter 𝜚 is true on 𝔫 ifg 𝔫 assigns true to it, i.e. ifg 𝔫(𝜚) = T.
  • 2. A negation ∼𝜚 is true on 𝔫 ifg the sentence 𝜚 is false on 𝔫.
  • 3. A conjunction 𝜚1 ∧ 𝜚2 ∧ 𝜚3 ∧ 𝜚4 ∧ … ∧ 𝜚𝑜 is true on 𝔫 ifg all of the

conjuncts are true on 𝔫.

  • 4. A disjunction 𝜚1 ∨ 𝜚2 ∨ 𝜚3 ∨ 𝜚4 ∨ … ∨ 𝜚𝑜 is true on 𝔫 ifg at least one
  • f the disjuncts is true on 𝔫.
  • 5. A conditional 𝜚→𝜔 is true on 𝔫 ifg the LHS is false or the RHS is true
  • n 𝔫.
  • 6. A biconditional 𝜚 ↔ 𝜔 is true on 𝔫 ifg both 𝜚 and 𝜔 have the same

truth value on 𝔫.

  • 7. If a sentence is false on 𝔫 ifg it’s not true on 𝔫.

Important: truth value assignments vary from model to model, truth-funcitonal defjnitions of connectives do not.

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Computing truth values of complex sentences Method 1: compute 𝔫(𝜚) by computing truth in 𝔫 for each of 𝜚’s subsen- tences, using Defjnition 2.20

Model: 𝔫(𝐵) = T, 𝔫(𝐶) = F, 𝔫(𝐸) = F, 𝔫(𝐻) = T sentence: (𝐵 ∨ (𝐸 → 𝐶)) ∧ ¬𝐻)

𝔫[(𝐵 ∨ (𝐸 → 𝐶)) ∧ ¬𝐻] =

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Computing truth values of complex sentences Method 2: construction tree method

  • 1. construct the tree 𝜚
  • 2. 𝔫 and Clause 1 of Def. 2.20 establish the truth values of the sentence

letters at the top of the tree

  • 3. work you way down, using the relevant clause from Def 2.20 to determine

the truth value of more complex subsentences of 𝜚

  • 4. the relevant clause of Def 2.20 for each subsentence is determined by its

main connective

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Truth Tables and Truth Functional Connectives ∼, ∧, ∨, → and ↔ are truth functions: functions from truth values to truth values, often given in truth tables. Example: 𝜚 𝜔 𝜚→𝜔

T T T T F F F T T F F T

Recall: Defjnition 2.20(5) The following clauses defjne when an SL sentence 𝜄 is true (or false) on a model 𝔫 for 𝜄:

  • 5. A conditional 𝜚→𝜔 is true on 𝔫 ifg the LHS is false or the RHS is true
  • n 𝔫.

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Truth Tables and Truth Functional Connectives ∼, ∧, ∨, → and ↔ are truth functions: functions from truth values to truth values, often given in truth tables. Example: 𝜚 𝜔 𝜚→𝜔

T T T T F F F T T F F T

Recall: Defjnition 2.20(5) The following clauses defjne when an SL sentence 𝜄 is true (or false) on a model 𝔫 for 𝜄:

  • 5. A conditional 𝜚→𝜔 is true on 𝔫 ifg the LHS is false or the RHS is true
  • n 𝔫.

What are the truth tables for the other truth functional connectives?

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Logical Truth Distinguish three types of sentences of SL: Defjnition 2.26: A sentence 𝜚 of SL is truth functionally true (TFT) ifg it is true on all models for 𝜚. Examples of TFT’s: 𝐵 ∨ ∼𝐵, 𝐵 ↔ 𝐵 Defjnition 2.27: A sentence 𝜚 of SL is truth functionally false (TFF) ifg it is false on all models for 𝜚. Examples of TFF’s: 𝐵 ∧ ∼𝐵, 𝐵 ↔ ∼𝐵 Defjnition 2.28: A sentence 𝜚 of SL is truth functionally contingent (TFC) ifg it is true on some models for 𝜚 and false on others. Examples of TFC’s: 𝐵, 𝐵 ∨ 𝐶

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Checking whether it’s a TFT, TFF, or TFC with a truth table 𝐵 B (𝐵 ∨ 𝐶) → 𝐵

T T T T F T F T F F F T

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Checking whether it’s a TFT, TFF, or TFC with a truth table 𝐵 B (𝐵 ∧ 𝐶) → 𝐵

T T T T F T F T T F F T

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Checking whether it’s a TFT, TFF, or TFC with a truth table 𝐵 𝐶 𝐸 𝐻 ({𝐵 ∨ (𝐸→𝐶)} ∧ ∼𝐻)

T T T T F T T T F T T T F T F T T F F T T F T T F T F T F T T F F T F T F F F T F T T T F F T T F T F T F T F F T F F T F F T T F F F T F F F F F T F F F F F T

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Comments on truth tables

  • 1. they get big, fast
  • with 2 truth values and 𝑜 sentence letters, there are 2𝑜 lines on a

truth table

  • 2. though helpful in studying simple languages such as SL, they’re less

helpful in studying more sophisticated languages

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