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, 𝐵 1 , 𝐶 1 , 𝐷 1 , … , 𝑇 1 , 𝑈 1 , 𝐵 2 , … 1
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, 𝐵 1 , 𝐶 1 , 𝐷 1 , … , 𝑇 1 , 𝑈 1 , 𝐵 2 , … 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) . Closure Clause: A sequence of symbols is an SL sentence ifg its being a sen- tence follows from the previous two clauses. 2 3. If all of 𝜚 1 , 𝜚 2 , 𝜚 3 , 𝜚 4 , … , 𝜚 𝑜 are sentences (the list must include at least two sentences and be fjnite), then so are (𝜚 1 ∧ 𝜚 2 ∧ 𝜚 3 ∧ 𝜚 4 ∧ … ∧ 𝜚 𝑜 ) and (𝜚 1 ∨ 𝜚 2 ∨ 𝜚 3 ∨ 𝜚 4 ∨ … ∨ 𝜚 𝑜 ) .
Defjnition 2.1. The basic symbols of SL are of three kinds: 1. Logical Connectives: ∼ , ∧ , ∨ , → , ↔ brackets [ ] or curly brackets { }. 2. replacing one or more pairs of offjcial round parentheses ( ) with square 1. deleting outer parentheses, or we can obtain it from an offjcial sentence by Defjnition 2.4 A string of symbols is an unoffjcial sentence of SL ifg Defjnition 2.2 defjnes the offjcial sentences of SL. tence follows from the previous two clauses. Closure Clause: A sequence of symbols is an SL sentence ifg its being a sen- 3 2. If 𝜚 and 𝜄 are sentences, then so are both (𝜚→𝜄) and (𝜚↔𝜄) . 1. If 𝜚 is a sentence, then so is ∼𝜚 . Generating Clauses: Base Clause: Every sentence letter is a sentence. defjnition: Defjnition 2.2. The sentences of SL are given by the following recursive 3. Sentence Letters: 𝐵, 𝐶, 𝐷, … , 𝑇, 𝑈, 𝐵 1 , 𝐶 1 , 𝐷 1 , … , 𝑇 1 , 𝑈 1 , 𝐵 2 , … 2. Punctuation Symbols: (, ) 3. If all of 𝜚 1 , 𝜚 2 , 𝜚 3 , 𝜚 4 , … , 𝜚 𝑜 are sentences (the list must include at least two sentences and be fjnite), then so are (𝜚 1 ∧ 𝜚 2 ∧ 𝜚 3 ∧ 𝜚 4 ∧ … ∧ 𝜚 𝑜 ) and (𝜚 1 ∨ 𝜚 2 ∨ 𝜚 3 ∨ 𝜚 4 ∨ … ∨ 𝜚 𝑜 ) .
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 . 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. 4 3. 𝜚 and 𝜄 are subsentences of (𝜚→𝜄) and (𝜚↔𝜄) . 4. All of 𝜚 1 , 𝜚 2 , 𝜚 3 , 𝜚 4 , … , 𝜚 𝑜 are subsentences of (𝜚 1 ∧ 𝜚 2 ∧ 𝜚 3 ∧ 𝜚 4 ∧ … ∧ 𝜚 𝑜 ) and (𝜚 1 ∨ 𝜚 2 ∨ 𝜚 3 ∨ 𝜚 4 ∨ … ∨ 𝜚 𝑜 ) .
Defjnition 2.8 The following clauses defjne the order of every SL sen- tence. Let ORD 𝜚 be the order of 𝜚 . Then: 1. If 𝜚 is an atomic sentence (a sentence letter), then ORD 𝜚 = 1 . 2. For any sentence 𝜚 , ORD ∼𝜚 = ORD 𝜚 + 1 . ORD 𝜚 and ORD 𝜄 . Likewise, ORD (𝜚↔𝜄) is one greater than the max of ORD 𝜚 and ORD 𝜄 . the max of ORD 𝜚 1 , … , ORD 𝜚 𝑜 . the max of ORD 𝜚 1 , … , ORD 𝜚 𝑜 . 6. That’s all. 5 3. For any sentences 𝜚 and 𝜄 , ORD (𝜚→𝜄) is one greater than the max of 4. For any sentences 𝜚 1 , … , 𝜚 𝑜 , ORD (𝜚 1 ∧ … ∧ 𝜚 𝑜 ) is one greater than 5. For any sentences 𝜚 1 , … , 𝜚 𝑜 , ORD (𝜚 1 ∨ … ∨ 𝜚 𝑜 ) is one greater than
Defjnition 2.10. The main connective is the connective token (or tokens) that occur(s) in the sentence but in no proper subsentence. 6
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 of 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. 7
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 of 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. {𝐵 ∨ (𝐸→𝐶)} ∧ ∼𝐻 𝐵 ∨ (𝐸→𝐶) 𝐵 𝐸→𝐶 𝐸 𝐶 𝐻 ∼𝐻 8 Example: {𝐵 ∨ (𝐸→𝐶)} ∧ ∼𝐻 .
Defjnition 2.12 The construction tree for a sentence is a diagram of 𝐶 bottom) of the construction tree. • The main connective of a sentence is the connective added last (at the • The order of a sentence is the number of nodes of its longest branch. tion tree. • The subsentences of a sentence are the nodes in the sentence’s construc- NB: ∼𝐻 𝐻 𝐸 how the sentence is generated through the recursive clauses of the defjnition 𝐸→𝐶 𝐵 𝐵 ∨ (𝐸→𝐶) {𝐵 ∨ (𝐸→𝐶)} ∧ ∼𝐻 node at the base of the tree. with the leaves at the top) into new nodes. The complete sentence is the erating clauses specify how we can join nodes of the tree together (starting of SL sentences. We put atomic sentences as leaves at the top, and the gen- 9 Example: {𝐵 ∨ (𝐸→𝐶)} ∧ ∼𝐻 .
Defjnition 2.12 The construction tree for a sentence is a diagram of 𝐵 bottom) of the construction tree. f • The main connective of a sentence is the connective added last (at the • The order of a sentence is the number of nodes of its longest branch. tion tree. • The subsentences of a sentence are the nodes in the sentence’s construc- NB: 𝐼 𝐸 𝐸 ∨ 𝐼 𝐸 how the sentence is generated through the recursive clauses of the defjnition 𝐷 𝐷 ∧ 𝐸 (𝐷 ∧ 𝐸) → 𝐵 ((𝐷 ∧ 𝐸) → 𝐵) ↔ (𝐸 ∨ 𝐼) Example: ((𝐷 ∧ 𝐸) → 𝐵) ↔ (𝐸 ∨ 𝐼) . node at the base of the tree. with the leaves at the top) into new nodes. The complete sentence is the erating clauses specify how we can join nodes of the tree together (starting of SL sentences. We put atomic sentences as leaves at the top, and the gen- 10
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 11
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)] 12
Truth in a Model F . of SL. Defjnition 2.19 𝔫 is a model for SL ifg 𝔫 is a model for every sentence Model for SL sentence letter in each sentence in Δ (by Def 2.17)] Δ ifg 𝔫 is a model for each sentence in Δ [i.e. ifg 𝔫 is a model for each Defjnition 2.18 Given that Δ is a set of SL sentences, 𝔫 is a model for Model of a set of sentences: If 𝔫 assigns a sentence letter 𝜔 the value false, we will write 𝔫(𝜔) = Model of single sentence: • 𝔫(𝜔) = T if a model 𝔫 assigns a sentence letter 𝜔 the value true, we will write • notation: assignment of a truth value, either true or false, to each sentence letter in 𝜚 . Defjnition 2.17 Given that 𝜚 is a sentence of SL, a model for 𝜚 is an 13
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] = ??? 14
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