Syntax and Semantics of Propositional Logic

INTRODUCTIONTOLOGIC Lecture 2 Syntax and Semantics of Propositional Logic.

Dr. James Studd

Logic is the beginning of wisdom. Thomas Aquinas Outline

1 Syntax vs Semantics.

2 Syntax of L1.

3 Semantics of L1. 4 Truth-table methods. Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence.

Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence.

Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences.

Examples of syntactic claims ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence.

Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences.

Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence.

Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences.

Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. Combining a proper noun and a verb phrase in this way makes a sentence.

Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences.

Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences.

Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence. Examples of semantic claims ‘Bertrand Russell’ refers to a British philosopher. ‘Bertrand Russell’ refers to Bertrand Russell. ‘likes logic’ expresses a property Russell has. ‘Bertrand Russell likes logic’ is true.

Syntax vs. Semantics Semantics Semantics is all about meanings of expressions. ‘Bertrand Russell’ refers to a British philosopher. ‘Bertrand Russell’ refers to Bertrand Russell. ‘likes logic’ expresses a property Russell has. ‘Bertrand Russell likes logic’ is true.

Syntax vs. Semantics Semantics Semantics is all about meanings of expressions. Examples of semantic claims ‘Bertrand Russell’ refers to Bertrand Russell. ‘likes logic’ expresses a property Russell has. ‘Bertrand Russell likes logic’ is true.

Syntax vs. Semantics Semantics Semantics is all about meanings of expressions. Examples of semantic claims ‘Bertrand Russell’ refers to a British philosopher. ‘likes logic’ expresses a property Russell has. ‘Bertrand Russell likes logic’ is true.

Syntax vs. Semantics Semantics Semantics is all about meanings of expressions. Examples of semantic claims ‘Bertrand Russell’ refers to a British philosopher. ‘Bertrand Russell’ refers to Bertrand Russell. ‘likes logic’ expresses a property Russell has. Syntax vs. Semantics Semantics Semantics is all about meanings of expressions. Examples of semantic claims ‘Bertrand Russell’ refers to a British philosopher. ‘Bertrand Russell’ refers to Bertrand Russell. ‘likes logic’ expresses a property Russell has. ‘Bertrand Russell likes logic’ is true. This occurrence (with quotes) refers to an expression.

This occurrence (without quotes) refers to a man.

‘Bertrand Russell’ refers to Bertrand Russell.

Mention The ﬁrst occurrence of ‘Bertrand Russell’ is an example of mention.

Use The second occurrence of ‘Bertrand Russell’ is an example of use.

Syntax vs. Semantics Use vs Mention Note our use of quotes to talk about expressions. This occurrence (with quotes) refers to an expression.

This occurrence (without quotes) refers to a man.

Mention The ﬁrst occurrence of ‘Bertrand Russell’ is an example of mention.

Use The second occurrence of ‘Bertrand Russell’ is an example of use.

Syntax vs. Semantics Use vs Mention Note our use of quotes to talk about expressions.

‘Bertrand Russell’ refers to Bertrand Russell. This occurrence (without quotes) refers to a man.

This occurrence (with quotes) refers to an expression.

Use The second occurrence of ‘Bertrand Russell’ is an example of use.

Syntax vs. Semantics Use vs Mention Note our use of quotes to talk about expressions.

‘Bertrand Russell’ refers to Bertrand Russell.

Mention The ﬁrst occurrence of ‘Bertrand Russell’ is an example of mention. This occurrence (without quotes) refers to a man.

Use The second occurrence of ‘Bertrand Russell’ is an example of use.

Syntax vs. Semantics Use vs Mention Note our use of quotes to talk about expressions.

‘Bertrand Russell’ refers to Bertrand Russell.

Mention The ﬁrst occurrence of ‘Bertrand Russell’ is an example of mention. This occurrence (with quotes) refers to an expression. This occurrence (without quotes) refers to a man.

Syntax vs. Semantics Use vs Mention Note our use of quotes to talk about expressions.

‘Bertrand Russell’ refers to Bertrand Russell.

Mention The ﬁrst occurrence of ‘Bertrand Russell’ is an example of mention. This occurrence (with quotes) refers to an expression.

Use The second occurrence of ‘Bertrand Russell’ is an example of use. Syntax vs. Semantics Use vs Mention Note our use of quotes to talk about expressions.

‘Bertrand Russell’ refers to Bertrand Russell.

Mention The ﬁrst occurrence of ‘Bertrand Russell’ is an example of mention. This occurrence (with quotes) refers to an expression.

Use The second occurrence of ‘Bertrand Russell’ is an example of use. This occurrence (without quotes) refers to a man. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc..

L1 has just two sorts of basic expression.

Some basic expressions of L1 (1) Sentence letters: e.g. ‘P’, ‘Q’. (2) Connectives: e.g. ‘¬’, ‘∧’.

2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc..

L1 has just two sorts of basic expression.

Some basic expressions of L1 (1) Sentence letters: e.g. ‘P’, ‘Q’. (2) Connectives: e.g. ‘¬’, ‘∧’.

2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. Some expressions of English (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc..

L1 has just two sorts of basic expression.

Some basic expressions of L1 (1) Sentence letters: e.g. ‘P’, ‘Q’. (2) Connectives: e.g. ‘¬’, ‘∧’.

2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc..

L1 has just two sorts of basic expression.

Some basic expressions of L1 (1) Sentence letters: e.g. ‘P’, ‘Q’. (2) Connectives: e.g. ‘¬’, ‘∧’.

2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc..

L1 has just two sorts of basic expression.

Some basic expressions of L1 (1) Sentence letters: e.g. ‘P’, ‘Q’. (2) Connectives: e.g. ‘¬’, ‘∧’.

2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc..

L1 has just two sorts of basic expression.

Some basic expressions of L1 (1) Sentence letters: e.g. ‘P’, ‘Q’. (2) Connectives: e.g. ‘¬’, ‘∧’.

2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. L1 has just two sorts of basic expression.

Some basic expressions of L1 (1) Sentence letters: e.g. ‘P’, ‘Q’. (2) Connectives: e.g. ‘¬’, ‘∧’.

2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc.. (1) Sentence letters: e.g. ‘P’, ‘Q’. (2) Connectives: e.g. ‘¬’, ‘∧’.

2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc..

L1 has just two sorts of basic expression.

Some basic expressions of L1 (2) Connectives: e.g. ‘¬’, ‘∧’.

2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc..

L1 has just two sorts of basic expression.

Some basic expressions of L1 (1) Sentence letters: e.g. ‘P’, ‘Q’. 2.2 The Syntax of the Language of Propositional

Syntax: English vs. L1. English has many diﬀerent sorts of expression. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like conceptual analysis’, etc.. (2) Connectives: ‘it is not the case that’, ‘and’, etc.. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. (5) Also: nouns, verbs, pronouns, etc., etc., etc..

L1 has just two sorts of basic expression.

Some basic expressions of L1 (1) Sentence letters: e.g. ‘P’, ‘Q’. (2) Connectives: e.g. ‘¬’, ‘∧’. Some complex sentences ‘It is not the case that’ and ‘Bertrand Russell likes logic’ make: ‘It is not the case that Bertrand Russell likes logic’. ‘¬’ and ‘P’ make: ‘¬P ’. ‘Bertrand Russell likes logic’ and ‘and’ and ‘Philosophers like conceptual analysis’ make: ‘Bertrand Russell likes logic and philosophers like conceptual analysis’. ‘P ’, ‘∧’ and ‘Q’ make: ‘(P ∧ Q)’.

Logic convention: no quotes around L1-expressions. P , ∧ and Q make: (P ∧ Q).

2.2 The Syntax of the Language of Propositional

Combining sentences and connectives makes new sentences. ‘It is not the case that’ and ‘Bertrand Russell likes logic’ make: ‘It is not the case that Bertrand Russell likes logic’. ‘¬’ and ‘P’ make: ‘¬P ’. ‘Bertrand Russell likes logic’ and ‘and’ and ‘Philosophers like conceptual analysis’ make: ‘Bertrand Russell likes logic and philosophers like conceptual analysis’. ‘P ’, ‘∧’ and ‘Q’ make: ‘(P ∧ Q)’.

Logic convention: no quotes around L1-expressions. P , ∧ and Q make: (P ∧ Q).

2.2 The Syntax of the Language of Propositional

Combining sentences and connectives makes new sentences. Some complex sentences ‘¬’ and ‘P’ make: ‘¬P ’. ‘Bertrand Russell likes logic’ and ‘and’ and ‘Philosophers like conceptual analysis’ make: ‘Bertrand Russell likes logic and philosophers like conceptual analysis’. ‘P ’, ‘∧’ and ‘Q’ make: ‘(P ∧ Q)’.

Logic convention: no quotes around L1-expressions. P , ∧ and Q make: (P ∧ Q).

2.2 The Syntax of the Language of Propositional

Combining sentences and connectives makes new sentences. Some complex sentences ‘It is not the case that’ and ‘Bertrand Russell likes logic’ make: ‘It is not the case that Bertrand Russell likes logic’. ‘Bertrand Russell likes logic’ and ‘and’ and ‘Philosophers like conceptual analysis’ make: ‘Bertrand Russell likes logic and philosophers like conceptual analysis’. ‘P ’, ‘∧’ and ‘Q’ make: ‘(P ∧ Q)’.

Logic convention: no quotes around L1-expressions. P , ∧ and Q make: (P ∧ Q).

2.2 The Syntax of the Language of Propositional

Logic convention: no quotes around L1-expressions. P , ∧ and Q make: (P ∧ Q).

2.2 The Syntax of the Language of Propositional

Combining sentences and connectives makes new sentences. Some complex sentences ‘It is not the case that’ and ‘Bertrand Russell likes logic’ make: ‘It is not the case that Bertrand Russell likes logic’. ‘¬’ and ‘P’ make: ‘¬P ’. ‘Bertrand Russell likes logic’ and ‘and’ and ‘Philosophers like conceptual analysis’ make: ‘Bertrand Russell likes logic and philosophers like conceptual analysis’. Logic convention: no quotes around L1-expressions. P , ∧ and Q make: (P ∧ Q).

2.2 The Syntax of the Language of Propositional

Combining sentences and connectives makes new sentences. Some complex sentences ‘It is not the case that’ and ‘Bertrand Russell likes logic’ make: ‘It is not the case that Bertrand Russell likes logic’. ‘¬’ and ‘P’ make: ‘¬P ’. ‘Bertrand Russell likes logic’ and ‘and’ and ‘Philosophers like conceptual analysis’ make: ‘Bertrand Russell likes logic and philosophers like conceptual analysis’. ‘P ’, ‘∧’ and ‘Q’ make: ‘(P ∧ Q)’. P , ∧ and Q make: (P ∧ Q).

2.2 The Syntax of the Language of Propositional

Combining sentences and connectives makes new sentences. Some complex sentences ‘It is not the case that’ and ‘Bertrand Russell likes logic’ make: ‘It is not the case that Bertrand Russell likes logic’. ‘¬’ and ‘P’ make: ‘¬P ’. ‘Bertrand Russell likes logic’ and ‘and’ and ‘Philosophers like conceptual analysis’ make: ‘Bertrand Russell likes logic and philosophers like conceptual analysis’. ‘P ’, ‘∧’ and ‘Q’ make: ‘(P ∧ Q)’.

Logic convention: no quotes around L1-expressions. 2.2 The Syntax of the Language of Propositional

Combining sentences and connectives makes new sentences. Some complex sentences ‘It is not the case that’ and ‘Bertrand Russell likes logic’ make: ‘It is not the case that Bertrand Russell likes logic’. ‘¬’ and ‘P’ make: ‘¬P ’. ‘Bertrand Russell likes logic’ and ‘and’ and ‘Philosophers like conceptual analysis’ make: ‘Bertrand Russell likes logic and philosophers like conceptual analysis’. ‘P ’, ‘∧’ and ‘Q’ make: ‘(P ∧ Q)’.

Logic convention: no quotes around L1-expressions. P , ∧ and Q make: (P ∧ Q). 2.2 The Syntax of the Language of Propositional Connectives

Here’s the full list of L1-connectives.

name in English symbol conjunction and ∧ disjunction or ∨ negation it is not the ¬ case that arrow if . . . then → double arrow if and only if ↔ 2.2 The Syntax of the Language of Propositional Connectives

Here’s the full list of L1-connectives.

name in English symbol conjunction and ∧ disjunction or ∨ negation it is not the ¬ case that arrow if . . . then → double arrow if and only if ↔ Deﬁnition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the oﬃcial deﬁnition of L1-sentence. (i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: (ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,... ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∨ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (φ ↔ ψ) Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1.

2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1. 2.2 The Syntax of the Language of Propositional

The syntax of L1

Here’s the official definition of L1-sentence. Definition

(i) All sentence letters are sentences of L1: P, Q, R, P1,Q1,R1,P2,Q2,R2,P3,...

(ii) If φ and ψ are sentences of L1, then so are: ¬φ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) (φ ↔ ψ)

(iii) Nothing else is a sentence of L1.

Greek letters: φ (‘PHI’) and ψ (‘PSI’): not part of L1. ¬¬(((P ∧ Q) → (P ∨ ¬R45)) ↔ ¬((P3 ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬((( ∧ Q) → (P ∨ ¬R45)) ↔ ¬((P3 ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬((( ∧ ) → (P ∨ ¬R45)) ↔ ¬((P3 ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

P Q

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬(( → (P ∨ ¬R45)) ↔ ¬((P3 ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

(P ∧ Q)

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬(( → (P ∨ ¬ )) ↔ ¬((P3 ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

(P ∧ Q) R45

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬(( → (P ∨ )) ↔ ¬((P3 ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

(P ∧ Q) ¬R45

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬(( → ( ∨ )) ↔ ¬((P3 ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

(P ∧ Q) P ¬R45

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬(( → ) ↔ ¬((P3 ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

(P ∧ Q) (P ∨ ¬R45)

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬( ↔ ¬((P3 ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

((P ∧ Q) → (P ∨ ¬R45))

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬( ↔ ¬(( ∨ R) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

((P ∧ Q) → (P ∨ ¬R45)) P3

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬( ↔ ¬(( ∨ ) ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

((P ∧ Q) → (P ∨ ¬R45)) P3 R

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬( ↔ ¬( ∨ R))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

((P ∧ Q) → (P ∨ ¬R45)) (P3 ∨ R)

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬( ↔ ¬( ∨ ))

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

((P ∧ Q) → (P ∨ ¬R45)) (P3 ∨ R) R

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬( ↔ ¬ )

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

((P ∧ Q) → (P ∨ ¬R45)) ((P3 ∨ R) ∨ R)

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬( ↔ )

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

((P ∧ Q) → (P ∨ ¬R45)) ¬((P3 ∨ R) ∨ R)

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬¬

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

(((P ∧ Q) → (P ∨ ¬R45)) ↔ ¬((P3 ∨ R) ∨ R))

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. ¬

2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

¬(((P ∧ Q) → (P ∨ ¬R45)) ↔ ¬((P3 ∨ R) ∨ R))

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. 2.2 The Syntax of the Language of Propositional

How to build a sentence of L1

Example

The following is a sentence of L1:

¬¬(((P ∧ Q) → (P ∨ ¬R45)) ↔ ¬((P3 ∨ R) ∨ R))

Deﬁnition of L1-sentences (repeated from previous page)

(i) All sentence letters are sentences of L1.

(ii) If φ and ψ are sentences of L1, then ¬φ, (φ ∧ ψ), (φ ∨ ψ), (φ → ψ) and (φ ↔ ψ) are sentences of L1.

(iii) Nothing else is a sentence of L1. (but is not one itself). e.g. ¬P , ¬(Q ∨ R), ¬(P ↔ (Q ∨ R)), ...

¬φ describes many L1-sentences

The object language is L1.

The metalanguage is (augmented) English.

¬P is a L1-sentence.

φ and ψ are part of the metalanguage, not the object one. Object language The object language is the one we’re theorising about.

Metalanguage The metalanguage is the one we’re theorising in.

φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1. (but is not one itself). e.g. ¬P , ¬(Q ∨ R), ¬(P ↔ (Q ∨ R)), ...

The object language is L1.

The metalanguage is (augmented) English.

¬φ describes many L1-sentences

φ and ψ are part of the metalanguage, not the object one. Object language The object language is the one we’re theorising about.

Metalanguage The metalanguage is the one we’re theorising in.

φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence. The object language is L1.

The metalanguage is (augmented) English.

(but is not one itself). e.g. ¬P , ¬(Q ∨ R), ¬(P ↔ (Q ∨ R)), ... φ and ψ are part of the metalanguage, not the object one. Object language The object language is the one we’re theorising about.

Metalanguage The metalanguage is the one we’re theorising in.

φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence.

¬φ describes many L1-sentences The object language is L1.

The metalanguage is (augmented) English.

(but is not one itself).

φ and ψ are part of the metalanguage, not the object one. Object language The object language is the one we’re theorising about.

Metalanguage The metalanguage is the one we’re theorising in.

φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence.

¬φ describes many L1-sentences e.g. ¬P , ¬(Q ∨ R), ¬(P ↔ (Q ∨ R)), ... The object language is L1.

The metalanguage is (augmented) English.

φ and ψ are part of the metalanguage, not the object one. Object language The object language is the one we’re theorising about.

Metalanguage The metalanguage is the one we’re theorising in.

φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence.

¬φ describes many L1-sentences (but is not one itself). e.g. ¬P , ¬(Q ∨ R), ¬(P ↔ (Q ∨ R)), ... The object language is L1.

The metalanguage is (augmented) English.

Object language The object language is the one we’re theorising about.

Metalanguage The metalanguage is the one we’re theorising in.

φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence.

¬φ describes many L1-sentences (but is not one itself). e.g. ¬P , ¬(Q ∨ R), ¬(P ↔ (Q ∨ R)), ... φ and ψ are part of the metalanguage, not the object one. The metalanguage is (augmented) English.

The object language is L1.

Metalanguage The metalanguage is the one we’re theorising in.

φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence.

¬φ describes many L1-sentences (but is not one itself). e.g. ¬P , ¬(Q ∨ R), ¬(P ↔ (Q ∨ R)), ... φ and ψ are part of the metalanguage, not the object one. Object language The object language is the one we’re theorising about. The metalanguage is (augmented) English.

Metalanguage The metalanguage is the one we’re theorising in.

φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence.

The object language is L1. The metalanguage is (augmented) English. φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence.

The object language is L1.

Metalanguage The metalanguage is the one we’re theorising in. φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language.

2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence.

The object language is L1.

Metalanguage The metalanguage is the one we’re theorising in. The metalanguage is (augmented) English. 2.2 The Syntax of the Language of Propositional Object vs. Metalanguage I mentioned that φ and ψ are not part of L1.

¬P is a L1-sentence.

The object language is L1.

Metalanguage The metalanguage is the one we’re theorising in. The metalanguage is (augmented) English. φ and ψ are used as variables in the metalanguage: in order to generalise about sentences of the object language. ( )

does not abbreviate (4 + 5) × 3.

4 + 5 × 3 abbreviates 4 +(5 × 3).

(P → Q ∧ R) abbreviates (P → (Q ∧ R)).

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

4 + 5 × 3 × ‘binds more strongly’ than +.

∧ and ∨ bind more strongly than → and ↔.

One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

Some are similar to rules used for + and × in arithmetic. Example in arithmetic

Examples in L1

2.3 Rules for Dropping Brackets Bracketing conventions

There are conventions for dropping brackets in L1. ( )

does not abbreviate (4 + 5) × 3.

4 + 5 × 3 abbreviates 4 +(5 × 3).

(P → Q ∧ R) abbreviates (P → (Q ∧ R)).

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

4 + 5 × 3 × ‘binds more strongly’ than +.

∧ and ∨ bind more strongly than → and ↔.

One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

Example in arithmetic

Examples in L1

2.3 Rules for Dropping Brackets Bracketing conventions

There are conventions for dropping brackets in L1. Some are similar to rules used for + and × in arithmetic. ( )

(P → Q ∧ R) abbreviates (P → (Q ∧ R)).

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

does not abbreviate (4 + 5) × 3.

4 + 5 × 3 abbreviates 4 +(5 × 3).

∧ and ∨ bind more strongly than → and ↔.

One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

4 + 5 × 3 × ‘binds more strongly’ than +.

Examples in L1

2.3 Rules for Dropping Brackets Bracketing conventions

There are conventions for dropping brackets in L1. Some are similar to rules used for + and × in arithmetic. Example in arithmetic ( )

(P → Q ∧ R) abbreviates (P → (Q ∧ R)).

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

4 + 5 × 3 abbreviates 4 +(5 × 3).

∧ and ∨ bind more strongly than → and ↔.

One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

does not abbreviate (4 + 5) × 3. × ‘binds more strongly’ than +.

Examples in L1

2.3 Rules for Dropping Brackets Bracketing conventions

There are conventions for dropping brackets in L1. Some are similar to rules used for + and × in arithmetic. Example in arithmetic 4 + 5 × 3 ( )

(P → Q ∧ R) abbreviates (P → (Q ∧ R)).

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

4 + 5 × 3 abbreviates 4 +(5 × 3).

∧ and ∨ bind more strongly than → and ↔.

One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

× ‘binds more strongly’ than +.

Examples in L1

2.3 Rules for Dropping Brackets Bracketing conventions

There are conventions for dropping brackets in L1. Some are similar to rules used for + and × in arithmetic. Example in arithmetic 4 + 5 × 3 does not abbreviate (4 + 5) × 3. ( )

(P → Q ∧ R) abbreviates (P → (Q ∧ R)).

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

∧ and ∨ bind more strongly than → and ↔.

One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

4 + 5 × 3 abbreviates 4 +(5 × 3).

Examples in L1

2.3 Rules for Dropping Brackets Bracketing conventions

There are conventions for dropping brackets in L1. Some are similar to rules used for + and × in arithmetic. Example in arithmetic 4 + 5 × 3 does not abbreviate (4 + 5) × 3. × ‘binds more strongly’ than +. ( )

(P → Q ∧ R) abbreviates (P → (Q ∧ R)).

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

∧ and ∨ bind more strongly than → and ↔.

One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

Examples in L1

2.3 Rules for Dropping Brackets Bracketing conventions

(P → Q ∧ R) abbreviates (P → (Q ∧ R)).

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

∧ and ∨ bind more strongly than → and ↔.

One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

2.3 Rules for Dropping Brackets Bracketing conventions

Examples in L1 ( )

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

(P → Q ∧ R) abbreviates (P → (Q ∧ R)). One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

2.3 Rules for Dropping Brackets Bracketing conventions

Examples in L1 ∧ and ∨ bind more strongly than → and ↔. ( )

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)).

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

One may drop outer brackets.

One may drop brackets on strings of ∧s or ∨s.

2.3 Rules for Dropping Brackets Bracketing conventions

Examples in L1 ∧ and ∨ bind more strongly than → and ↔. (P → Q ∧ R) abbreviates (P → (Q ∧ R)). ( )

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)). One may drop brackets on strings of ∧s or ∨s.

2.3 Rules for Dropping Brackets Bracketing conventions

Examples in L1 ∧ and ∨ bind more strongly than → and ↔. (P → Q ∧ R) abbreviates (P → (Q ∧ R)). One may drop outer brackets. ( )

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

One may drop brackets on strings of ∧s or ∨s.

2.3 Rules for Dropping Brackets Bracketing conventions

Examples in L1 ∧ and ∨ bind more strongly than → and ↔. (P → Q ∧ R) abbreviates (P → (Q ∧ R)). One may drop outer brackets. P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)). ( )

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25

2.3 Rules for Dropping Brackets Bracketing conventions

Examples in L1 ∧ and ∨ bind more strongly than → and ↔. (P → Q ∧ R) abbreviates (P → (Q ∧ R)). One may drop outer brackets. P ∧ (Q → ¬P4) abbreviates (P ∧ (Q → ¬P4)). One may drop brackets on strings of ∧s or ∨s. 2.3 Rules for Dropping Brackets Bracketing conventions

(P ∧ Q ∧ R) abbreviates ((P ∧ Q) ∧ R). 25 2.3 Rules for Dropping Brackets Bracketing conventions

(P ∧ Q ∧ R) abbreviates ((P ∧ Q) ∧ R). 25 ( )

2.3 Rules for Dropping Brackets Bracketing conventions

P ∧ Q ∧ R abbreviates ((P ∧ Q) ∧ R). 25 Characterisation An argument is logically valid if and only if there is no interpretation of subject-speciﬁc expressions under which: (i) the premisses are all true, and (ii) the conclusion is false.

We’ll adapt this characterisation to L1. Logical expressions: ¬, ∧, ∨, → and ↔. Subject speciﬁc expressions: P, Q, R, ...

Interpretation: L1-structure.

2.4 The Semantics of Propositional Logic Semantics Recall the characterisation of validity from week 1. We’ll adapt this characterisation to L1. Logical expressions: ¬, ∧, ∨, → and ↔. Subject speciﬁc expressions: P, Q, R, ...

Interpretation: L1-structure.

2.4 The Semantics of Propositional Logic Semantics Recall the characterisation of validity from week 1. Characterisation An argument is logically valid if and only if there is no interpretation of subject-speciﬁc expressions under which: (i) the premisses are all true, and (ii) the conclusion is false. Logical expressions: ¬, ∧, ∨, → and ↔. Subject speciﬁc expressions: P, Q, R, ...

Interpretation: L1-structure.

We’ll adapt this characterisation to L1. Subject speciﬁc expressions: P, Q, R, ...

Interpretation: L1-structure.

We’ll adapt this characterisation to L1. Logical expressions: ¬, ∧, ∨, → and ↔. Interpretation: L1-structure.

We’ll adapt this characterisation to L1. Logical expressions: ¬, ∧, ∨, → and ↔. Subject speciﬁc expressions: P, Q, R, ... 2.4 The Semantics of Propositional Logic Semantics Recall the characterisation of validity from week 1. Characterisation An argument is logically valid if and only if there is no interpretation of subject-speciﬁc expressions under which: (i) the premisses are all true, and (ii) the conclusion is false.

We’ll adapt this characterisation to L1. Logical expressions: ¬, ∧, ∨, → and ↔. Subject speciﬁc expressions: P, Q, R, ...

Interpretation: L1-structure. PQRP1 Q1 R1 P2 Q2 R2 ··· A : TFFFTFTTF ··· B : FFFFFFFFF ···

We interpret sentence letters by assigning them truth-values: either T for True or F for False. Deﬁnition

An L1-structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1.

Examples

We can think of an L1-structure as an inﬁnite list that provides a value T or F for every sentence letter.

We use A, B, etc. to stand for L1-structures.

2.4 The Semantics of Propositional Logic

L1-structures PQRP1 Q1 R1 P2 Q2 R2 ··· A : TFFFTFTTF ··· B : FFFFFFFFF ···

either T for True or F for False. Deﬁnition

An L1-structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1.

Examples

We can think of an L1-structure as an inﬁnite list that provides a value T or F for every sentence letter.

We use A, B, etc. to stand for L1-structures.

2.4 The Semantics of Propositional Logic

L1-structures We interpret sentence letters by assigning them truth-values: PQRP1 Q1 R1 P2 Q2 R2 ··· A : TFFFTFTTF ··· B : FFFFFFFFF ···

Deﬁnition

An L1-structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1.

Examples

We can think of an L1-structure as an inﬁnite list that provides a value T or F for every sentence letter.

We use A, B, etc. to stand for L1-structures.

2.4 The Semantics of Propositional Logic

L1-structures We interpret sentence letters by assigning them truth-values: either T for True or F for False. PQRP1 Q1 R1 P2 Q2 R2 ··· A : TFFFTFTTF ··· B : FFFFFFFFF ···

Examples

We can think of an L1-structure as an inﬁnite list that provides a value T or F for every sentence letter.

We use A, B, etc. to stand for L1-structures.

2.4 The Semantics of Propositional Logic

L1-structures We interpret sentence letters by assigning them truth-values: either T for True or F for False. Deﬁnition

An L1-structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1. B : FFFFFFFFF ···

We use A, B, etc. to stand for L1-structures.

2.4 The Semantics of Propositional Logic

L1-structures We interpret sentence letters by assigning them truth-values: either T for True or F for False. Deﬁnition

An L1-structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1.

Examples

We can think of an L1-structure as an inﬁnite list that provides a value T or F for every sentence letter.

PQRP1 Q1 R1 P2 Q2 R2 ··· A : TFFFTFTTF ··· A : TFFFTFTTF ···

We use A, B, etc. to stand for L1-structures.

2.4 The Semantics of Propositional Logic

L1-structures We interpret sentence letters by assigning them truth-values: either T for True or F for False. Deﬁnition

An L1-structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1.

Examples

We can think of an L1-structure as an inﬁnite list that provides a value T or F for every sentence letter.

PQRP1 Q1 R1 P2 Q2 R2 ···

B : FFFFFFFFF ··· A : TFFFTFTTF ···

2.4 The Semantics of Propositional Logic

L1-structures We interpret sentence letters by assigning them truth-values: either T for True or F for False. Deﬁnition

An L1-structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1.

Examples

We can think of an L1-structure as an inﬁnite list that provides a value T or F for every sentence letter.

PQRP1 Q1 R1 P2 Q2 R2 ···

B : FFFFFFFFF ···

We use A, B, etc. to stand for L1-structures. The logical connectives have ﬁxed meanings. These determine the truth-values of complex sentences.

Notation: |φ|A is the truth-value of φ under A.

Truth-conditions for ¬ The meaning of ¬ is summarised in its truth table.

φ ¬φ T F F T

In words: |¬φ|A = T if and only if |φ|A = F.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 1/3

L1-structures only directly specify truth-values for P , Q, R,... These determine the truth-values of complex sentences.

Notation: |φ|A is the truth-value of φ under A.

Truth-conditions for ¬ The meaning of ¬ is summarised in its truth table.

φ ¬φ T F F T

In words: |¬φ|A = T if and only if |φ|A = F.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 1/3

L1-structures only directly specify truth-values for P , Q, R,... The logical connectives have ﬁxed meanings. Notation: |φ|A is the truth-value of φ under A.

Truth-conditions for ¬ The meaning of ¬ is summarised in its truth table.

φ ¬φ T F F T

In words: |¬φ|A = T if and only if |φ|A = F.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 1/3

L1-structures only directly specify truth-values for P , Q, R,... The logical connectives have ﬁxed meanings. These determine the truth-values of complex sentences. Truth-conditions for ¬ The meaning of ¬ is summarised in its truth table.

φ ¬φ T F F T

In words: |¬φ|A = T if and only if |φ|A = F.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 1/3

L1-structures only directly specify truth-values for P , Q, R,... The logical connectives have ﬁxed meanings. These determine the truth-values of complex sentences.

Notation: |φ|A is the truth-value of φ under A. In words: |¬φ|A = T if and only if |φ|A = F.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 1/3

L1-structures only directly specify truth-values for P , Q, R,... The logical connectives have ﬁxed meanings. These determine the truth-values of complex sentences.

Notation: |φ|A is the truth-value of φ under A.

Truth-conditions for ¬ The meaning of ¬ is summarised in its truth table.

φ ¬φ T F F T 2.4 The Semantics of Propositional Logic Truth-values of complex sentences 1/3

L1-structures only directly specify truth-values for P , Q, R,... The logical connectives have ﬁxed meanings. These determine the truth-values of complex sentences.

Notation: |φ|A is the truth-value of φ under A.

Truth-conditions for ¬ The meaning of ¬ is summarised in its truth table.

φ ¬φ T F F T

In words: |¬φ|A = T if and only if |φ|A = F. T F F F T T T F F

2.4 The Semantics of Propositional Logic Worked example 1 φ ¬φ |φ|A is the truth-value of φ under A. T F F T Compute the following truth-values. Let the structure A be partially speciﬁed as follows.

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = |Q|A = |R1|A =

|¬P |A = |¬Q|A = |¬R1|A =

|¬¬P |A = |¬¬Q|A = |¬¬R1|A = F F F T T T F F

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = T |Q|A = |R1|A =

|¬P |A = |¬Q|A = |¬R1|A =

|¬¬P |A = |¬¬Q|A = |¬¬R1|A = F F T T T F F

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = T |Q|A = F |R1|A =

|¬P |A = |¬Q|A = |¬R1|A =

|¬¬P |A = |¬¬Q|A = |¬¬R1|A = F T T T F F

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = T |Q|A = F |R1|A = F

|¬P |A = |¬Q|A = |¬R1|A =

|¬¬P |A = |¬¬Q|A = |¬¬R1|A = T T T F F

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = T |Q|A = F |R1|A = F

|¬P |A = F |¬Q|A = |¬R1|A =

|¬¬P |A = |¬¬Q|A = |¬¬R1|A = T T F F

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = T |Q|A = F |R1|A = F

|¬P |A = F |¬Q|A = T |¬R1|A =

|¬¬P |A = |¬¬Q|A = |¬¬R1|A = T F F

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = T |Q|A = F |R1|A = F

|¬P |A = F |¬Q|A = T |¬R1|A = T

|¬¬P |A = |¬¬Q|A = |¬¬R1|A = F F

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = T |Q|A = F |R1|A = F

|¬P |A = F |¬Q|A = T |¬R1|A = T

|¬¬P |A = T |¬¬Q|A = |¬¬R1|A = F

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = T |Q|A = F |R1|A = F

|¬P |A = F |¬Q|A = T |¬R1|A = T

|¬¬P |A = T |¬¬Q|A = F |¬¬R1|A = 2.4 The Semantics of Propositional Logic Worked example 1 φ ¬φ |φ|A is the truth-value of φ under A. T F F T Compute the following truth-values. Let the structure A be partially speciﬁed as follows.

PQRP 1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· Compute: |P |A = T |Q|A = F |R1|A = F

|¬P |A = F |¬Q|A = T |¬R1|A = T

|¬¬P |A = T |¬¬Q|A = F |¬¬R1|A = F |(φ ∨ ψ)|A = T if and only if |φ|A = T or |ψ|A = T (or both).

|(φ ∧ ψ)|A = T if and only if |φ|A = T and |ψ|A = T.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 2/3

Truth-conditions for ∧ and ∨ The meanings of ∧ and ∨ are given by the truth tables:

φ ψ (φ ∧ ψ) φ ψ (φ ∨ ψ) T T T T T T T F F T F T F T F F T T F F F F F F |(φ ∨ ψ)|A = T if and only if |φ|A = T or |ψ|A = T (or both).

|(φ ∧ ψ)|A = T if and only if |φ|A = T and |ψ|A = T.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 2/3

Truth-conditions for ∧ and ∨ The meanings of ∧ and ∨ are given by the truth tables:

φ ψ (φ ∧ ψ) φ ψ (φ ∨ ψ) T T T T T T T F F T F T F T F F T T F F F F F F |(φ ∨ ψ)|A = T if and only if |φ|A = T or |ψ|A = T (or both).

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 2/3

Truth-conditions for ∧ and ∨ The meanings of ∧ and ∨ are given by the truth tables:

φ ψ (φ ∧ ψ) φ ψ (φ ∨ ψ) T T T T T T T F F T F T F T F F T T F F F F F F

|(φ ∧ ψ)|A = T if and only if |φ|A = T and |ψ|A = T. |(φ ∨ ψ)|A = T if and only if |φ|A = T or |ψ|A = T (or both).

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 2/3

Truth-conditions for ∧ and ∨ The meanings of ∧ and ∨ are given by the truth tables:

φ ψ (φ ∧ ψ) φ ψ (φ ∨ ψ) T T T T T T T F F T F T F T F F T T F F F F F F

|(φ ∧ ψ)|A = T if and only if |φ|A = T and |ψ|A = T. 2.4 The Semantics of Propositional Logic Truth-values of complex sentences 2/3

Truth-conditions for ∧ and ∨ The meanings of ∧ and ∨ are given by the truth tables:

φ ψ (φ ∧ ψ) φ ψ (φ ∨ ψ) T T T T T T T F F T F T F T F F T T F F F F F F

|(φ ∧ ψ)|A = T if and only if |φ|A = T and |ψ|A = T. |(φ ∨ ψ)|A = T if and only if |φ|A = T or |ψ|A = T (or both). |(φ ↔ ψ)|A = T if and only if |φ|A = |ψ|A.

|(φ → ψ)|A = T if and only if |φ|A = F or |ψ|A = T.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 3/3

Truth-conditions for → and ↔ The meanings of → and ↔ are given by the truth tables:

φ ψ (φ → ψ) φ ψ (φ ↔ ψ) T T T T T T T F F T F F F T T F T F F F T F F T |(φ ↔ ψ)|A = T if and only if |φ|A = |ψ|A.

|(φ → ψ)|A = T if and only if |φ|A = F or |ψ|A = T.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 3/3

Truth-conditions for → and ↔ The meanings of → and ↔ are given by the truth tables:

φ ψ (φ → ψ) φ ψ (φ ↔ ψ) T T T T T T T F F T F F F T T F T F F F T F F T |(φ ↔ ψ)|A = T if and only if |φ|A = |ψ|A.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 3/3

Truth-conditions for → and ↔ The meanings of → and ↔ are given by the truth tables:

φ ψ (φ → ψ) φ ψ (φ ↔ ψ) T T T T T T T F F T F F F T T F T F F F T F F T

|(φ → ψ)|A = T if and only if |φ|A = F or |ψ|A = T.

35 |(φ ↔ ψ)|A = T if and only if |φ|A = |ψ|A.

2.4 The Semantics of Propositional Logic Truth-values of complex sentences 3/3

Truth-conditions for → and ↔ The meanings of → and ↔ are given by the truth tables:

φ ψ (φ → ψ) φ ψ (φ ↔ ψ) T T T T T T T F F T F F F T T F T F F F T F F T

|(φ → ψ)|A = T if and only if |φ|A = F or |ψ|A = T.

35 2.4 The Semantics of Propositional Logic Truth-values of complex sentences 3/3

Truth-conditions for → and ↔ The meanings of → and ↔ are given by the truth tables:

φ ψ (φ → ψ) φ ψ (φ ↔ ψ) T T T T T T T F F T F F F T T F T F F F T F F T

|(φ → ψ)|A = T if and only if |φ|A = F or |ψ|A = T. |(φ ↔ ψ)|A = T if and only if |φ|A = |ψ|A. 35 1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T 2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = T

3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T 3 |¬(P → Q) → (P ∧ Q)|B = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T = F

2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T 2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T 2.4 The Semantics of Propositional Logic Worked example 2

Let |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B What is the truth value of ¬(P → Q) → (P ∧ Q) under B?

1 |(P → Q)|B = F and |(P ∧ Q)|B = F

2 |¬(P → Q)|B = T

3 |¬(P → Q) → (P ∧ Q)|B = F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F T T F F F T F F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q)

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T T F F F T F F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F F F T F F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F F T F F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F F F F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T F T

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F F F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T F T F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F F F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T F T F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T F T F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T F T F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T F F T F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T T F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T F F T F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T T F F T F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T F

2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T T F F T F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T 2.4 The Semantics of Propositional Logic

For actual calculations it’s usually better to use tables.

Suppose |P |B = T and |Q|B = F.

Compute |¬(P → Q) → (P ∧ Q)|B P Q ¬ (P → Q) → (P ∧ Q) T F T T F F F T F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T F T T T T T T T T T F F F T F F F F T T T F F T F F T F T F F F

P Q ¬ (P → Q) → (P ∧ Q) T T T F F T F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

40 F T T T T T T T T T F F F T F F F F T T T F F T F F T F T F F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T F F T F F

40 F T T T T T T T T F F F T F F F F T T T F F T F F T F T F F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T F F T F F

40 F T T T T T T T F F F T F F F F T T T F F T F F T F T F F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T F T F T F F

40 F T T T T T T T F F F T F F F T T T F F T F F T F T F F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T F T F T F F F

40 F T T T T T T T F F F T F F F T T T F F T F T F T F F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T F T F T F F F F

40 F T T T T T T F F F T F F F T T T F F T F T F T F F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T F T F T F F F F

40 F T T T T T T F F F F F F T T T F F T F T F T F F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T F T T F T F F F F

40 F T T T T T T F F F F F F T T T F T F T F T F F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T F T T F T F F F F F

40 F T T T T T T F F F F F F T T T F T F T F T F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T F T T F T F F F F F F

40 F T T T T T F F F F F F T T T F T F T F T F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T F T T F T F F F F F F

40 F T T T T T F F F F F T T T F T F T F T F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T F T F T F T F F F F F F

40 F T T T T T F F F F F T T F T F T F T F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T F T F T F T F T F F F F F

40 F T T T T T F F F F F T T F T F T T F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T F T F T F T F T F F F F F F

40 F T T T T F F F F F T T F T F T T F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T F T F T F T F T F F F F F F

40 F T T T T F F F F T T F T F T T F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T F T F T F F T F T F F F F F F

40 F T T T T F F F F T T F F T T F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T F T F T F F T F T F T F F F F F

40 F T T T T F F F F T T F F T T F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T F T F T F F T F T F T F F F F F F

40 F T T T F F F F T T F F T T F

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T T F T F T F F T F T F T F F F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 F T T T F F F T T F F T T F

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T T F T F F T F F T F T F T F F F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 F T T T F F F T F F T T F

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T T F T F F T F F T F T T F T F F F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 F T T T F F F T F F T F

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T T F T F F T F F T F T T F T F F F T F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 F T T F F F T F F T F

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T T T F T F F T F F T F T T F T F F F T F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 F T T F F T F F T F

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T T T F T F F T F F F T F T T F T F F F T F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 F T T F F T F T F

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T T T F T F F T F F F T F T T F F T F F F T F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 F T T F F T F T

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T T T T T T T T F T F F T F F F T F T T F F T F F F T F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 T T F F T F T

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T F T T T T T T T F T F F T F F F T F T T F F T F F F T F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 T F F T F T

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T F T T T T T T T F T T F F T F F F T F T T F F T F F F T F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 T F T F T

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T F T T T T T T T F T T F F T F F F T F F T T F F T F F F T F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 T F T T

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T F T T T T T T T F T T F F T F F F T F F T T F F T F F F F T F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 F T T

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T F T T T T T T T T F T T F F T F F F T F F T T F F T F F F F T F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 T T

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T F T T T T T T T T F T T F F F T F F F T F F T T F F T F F F F T F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 T

The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T F T T T T T T T T F T T F F F T F F F T F F T T T F F T F F F F T F F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 The main column (underlined) gives the truth-value of the whole sentence.

2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T F T T T T T T T T F T T F F F T F F F T F F T T T F F T F F F F T F T F F F

φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 2.4 The Semantics of Propositional Logic

Using the same technique we can ﬁll out the full truth table for ¬(P → Q) → (P ∧ Q)

P Q ¬ (P → Q) → (P ∧ Q) T T F T T T T T T T T F T T F F F T F F F T F F T T T F F T F F F F T F T F F F

The main column (underlined) gives the truth-value of the whole sentence. φ ψ (φ ∧ ψ) (φ → ψ) φ ¬φ T T T T T F T F F F F T F T F T F F F T

40 Deﬁnition The argument with all sentences in Γ as premisses and φ as conclusion is valid if and only if there is no L1-structure under which: (i) all sentences in Γ are true; and (ii) φ is false.

Notation: when this argument is valid we write Γ φ.

{P → ¬Q, Q} |= ¬P means that the argument whose premises are P → ¬Q and Q, and whose conclusion is ¬P is valid. Also written: P → ¬Q, Q |= ¬P

2.4 The Semantics of Propositional Logic Validity

Let Γ be a set of sentences of L1 and φ a sentence of L1. Notation: when this argument is valid we write Γ φ.

{P → ¬Q, Q} |= ¬P means that the argument whose premises are P → ¬Q and Q, and whose conclusion is ¬P is valid. Also written: P → ¬Q, Q |= ¬P

2.4 The Semantics of Propositional Logic Validity

Let Γ be a set of sentences of L1 and φ a sentence of L1. Deﬁnition The argument with all sentences in Γ as premisses and φ as conclusion is valid if and only if there is no L1-structure under which: (i) all sentences in Γ are true; and (ii) φ is false. {P → ¬Q, Q} |= ¬P means that the argument whose premises are P → ¬Q and Q, and whose conclusion is ¬P is valid. Also written: P → ¬Q, Q |= ¬P

2.4 The Semantics of Propositional Logic Validity

Notation: when this argument is valid we write Γ φ. 2.4 The Semantics of Propositional Logic Validity

Notation: when this argument is valid we write Γ φ.

{P → ¬Q, Q} |= ¬P means that the argument whose premises are P → ¬Q and Q, and whose conclusion is ¬P is valid. Also written: P → ¬Q, Q |= ¬P

P Q P → ¬ Q Q ¬ P T T T F FT T F T T F T T TF F F T F T F T FT T T F F F F T TF F T F

Rows correspond to interpretations. One needs to check that there is no row in which all the premisses are assignedT and the conclusion is assignedF.

Example Show that {P → ¬Q, Q} |= ¬P .

2.4 The Semantics of Propositional Logic Worked example 3

We can use truth-tables to show that L1-arguments are valid.

P Q P → ¬ Q Q ¬ P T T T F FT T F T T F T T TF F F T F T F T FT T T F F F F T TF F T F

Rows correspond to interpretations. One needs to check that there is no row in which all the premisses are assignedT and the conclusion is assignedF.

2.4 The Semantics of Propositional Logic Worked example 3

We can use truth-tables to show that L1-arguments are valid. Example Show that {P → ¬Q, Q} |= ¬P . Rows correspond to interpretations. One needs to check that there is no row in which all the premisses are assignedT and the conclusion is assignedF.

2.4 The Semantics of Propositional Logic Worked example 3

We can use truth-tables to show that L1-arguments are valid. Example Show that {P → ¬Q, Q} |= ¬P .

P Q P → ¬ Q Q ¬ P T T T F FT T F T T F T T TF F F T F T F T FT T T F F F F T TF F T F One needs to check that there is no row in which all the premisses are assignedT and the conclusion is assignedF.

2.4 The Semantics of Propositional Logic Worked example 3

We can use truth-tables to show that L1-arguments are valid. Example Show that {P → ¬Q, Q} |= ¬P .

P Q P → ¬ Q Q ¬ P T T T F FT T F T T F T T TF F F T F T F T FT T T F F F F T TF F T F

Rows correspond to interpretations.

2.4 The Semantics of Propositional Logic Worked example 3

We can use truth-tables to show that L1-arguments are valid. Example Show that {P → ¬Q, Q} |= ¬P .

P Q P → ¬ Q Q ¬ P T T T F FT T F T T F T T TF F F T F T F T FT T T F F F F T TF F T F

Rows correspond to interpretations. One needs to check that there is no row in which all the premisses are assignedT and the conclusion is assignedF.

2.4 The Semantics of Propositional Logic Worked example 3

We can use truth-tables to show that L1-arguments are valid. Example Show that {P → ¬Q, Q} |= ¬P .

P Q P → ¬ Q Q ¬ P T T T F FT T F T T F T T TF F F T F T F T FT T T F F F F T TF F T F

Rows correspond to interpretations. One needs to check that there is no row in which all the premisses are assignedT and the conclusion is assignedF.

2.4 The Semantics of Propositional Logic Worked example 3

We can use truth-tables to show that L1-arguments are valid. Example Show that {P → ¬Q, Q} |= ¬P .

P Q P → ¬ Q Q ¬ P T T T F FT T F T T F T T TF F F T F T F T FT T T F F F F T TF F T F

Rows correspond to interpretations. One needs to check that there is no row in which all the premisses are assignedT and the conclusion is assignedF.

2.4 The Semantics of Propositional Logic Worked example 3

We can use truth-tables to show that L1-arguments are valid. Example Show that {P → ¬Q, Q} |= ¬P .

P Q P → ¬ Q Q ¬ P T T T F FT T F T T F T T TF F F T F T F T FT T T F F F F T TF F T F Rows correspond to interpretations. One needs to check that there is no row in which all the premisses are assignedT and the conclusion is assignedF.

2.4 The Semantics of Propositional Logic Worked example 3

We can use truth-tables to show that L1-arguments are valid. Example Show that {P → ¬Q, Q} |= ¬P .

Truth tables of tautologies Every row in the main column is a T.

P P ∨ ¬ P P → P T T T FT T T T F F T TF F T F

2.4 The Semantics of Propositional Logic Other logical notions

Deﬁnition

A sentence φ of L1 is logically true (a tautology) iﬀ:

φ is true under all L1-structures. Truth tables of tautologies Every row in the main column is a T.

P P ∨ ¬ P P → P T T T FT T T T F F T TF F T F

2.4 The Semantics of Propositional Logic Other logical notions

Deﬁnition

A sentence φ of L1 is logically true (a tautology) iﬀ:

φ is true under all L1-structures.

e.g. P ∨ ¬P , and P → P are tautologies. 2.4 The Semantics of Propositional Logic Other logical notions

Deﬁnition

A sentence φ of L1 is logically true (a tautology) iﬀ:

φ is true under all L1-structures.

e.g. P ∨ ¬P , and P → P are tautologies.

Truth tables of tautologies Every row in the main column is a T.

P P ∨ ¬ P P → P T T T FT T T T F F T TF F T F e.g. P ∧ ¬P , and ¬(P → P ) are contradictions.

Truth tables of contradictions Every row in the main column is an F.

P P ∧ ¬ P ¬ (P → P ) T T F FT F TTT F F F TF F FTF

2.4 The Semantics of Propositional Logic

Deﬁnition

A sentence φ of L1 is a contradiction iﬀ:

φ is not true under any L1-structure. Truth tables of contradictions Every row in the main column is an F.

P P ∧ ¬ P ¬ (P → P ) T T F FT F TTT F F F TF F FTF

2.4 The Semantics of Propositional Logic

Deﬁnition

A sentence φ of L1 is a contradiction iﬀ:

φ is not true under any L1-structure. e.g. P ∧ ¬P , and ¬(P → P ) are contradictions. 2.4 The Semantics of Propositional Logic

Deﬁnition

A sentence φ of L1 is a contradiction iﬀ:

φ is not true under any L1-structure. e.g. P ∧ ¬P , and ¬(P → P ) are contradictions.

Truth tables of contradictions Every row in the main column is an F.

P P ∧ ¬ P ¬ (P → P ) T T F FT F TTT F F F TF F FTF P and ¬¬P are logically equivalent. P ∧ Q and ¬(¬P ∨ ¬Q) are logically equivalent.

Truth tables of logical equivalents The truth-values in the main columns agree.

P Q P ∧ Q ¬ (¬ P ∨ ¬ Q) T T T T T T FTFFT T F T F F F FTTTF F T F F T F TFTFT F F F F F F TFTTF

2.4 The Semantics of Propositional Logic

Deﬁnition Sentences φ and ψ are logically equivalent iﬀ:

φ and ψ are true in exactly the same L1-structures. Truth tables of logical equivalents The truth-values in the main columns agree.

P Q P ∧ Q ¬ (¬ P ∨ ¬ Q) T T T T T T FTFFT T F T F F F FTTTF F T F F T F TFTFT F F F F F F TFTTF

2.4 The Semantics of Propositional Logic

Deﬁnition Sentences φ and ψ are logically equivalent iﬀ:

φ and ψ are true in exactly the same L1-structures.

P and ¬¬P are logically equivalent. P ∧ Q and ¬(¬P ∨ ¬Q) are logically equivalent. 2.4 The Semantics of Propositional Logic

Deﬁnition Sentences φ and ψ are logically equivalent iﬀ:

φ and ψ are true in exactly the same L1-structures.

P and ¬¬P are logically equivalent. P ∧ Q and ¬(¬P ∨ ¬Q) are logically equivalent.

Truth tables of logical equivalents The truth-values in the main columns agree.

P Q P ∧ Q ¬ (¬ P ∨ ¬ Q) T T T T T T FTFFT T F T F F F FTTTF F T F F T F TFTFT F F F F F F TFTTF PQR (P → (¬ Q ∧ R)) ∨ P TTT TFFTFT T T TTF TFFTFF T T TFT TTTFTT T T TFF TFTFFF T T FTT FTFTFT T F FTF FTFTFF T F FFT FTTFTT T F FFF FTTFFF T F

Method 1: Full truth table Write out the truth table for (P → (¬Q ∧ R)) ∨ P . Check there’s a T in every row of the main column.

2.4 The Semantics of Propositional Logic Worked example 4

Example Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology. PQR (P → (¬ Q ∧ R)) ∨ P TTT TFFTFT T T TTF TFFTFF T T TFT TTTFTT T T TFF TFTFFF T T FTT FTFTFT T F FTF FTFTFF T F FFT FTTFTT T F FFF FTTFFF T F

2.4 The Semantics of Propositional Logic Worked example 4

Example Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 1: Full truth table Write out the truth table for (P → (¬Q ∧ R)) ∨ P . Check there’s a T in every row of the main column. 2.4 The Semantics of Propositional Logic Worked example 4

Example Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 1: Full truth table Write out the truth table for (P → (¬Q ∧ R)) ∨ P . Check there’s a T in every row of the main column.

PQR (P → (¬ Q ∧ R)) ∨ P TTT TFFTFT T T TTF TFFTFF T T TFT TTTFTT T T TFF TFTFFF T T FTT FTFTFT T F FTF FTFTFF T F FFT FTTFTT T F FFF FTTFFF T F T3 F1 F F2

φ ψ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) φ ¬φ T T T T T T F T F F T F F T F T F T T F F F F T

2.4 The Semantics of Propositional Logic

Worked example 4 (cont.) Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 2: Backwards truth table. Put an F in the main column. Work backwards to show this leads to a contradiction.

P Q R (P → (¬ Q ∧ R)) ∨ P

50 T3 F1 F2

φ ψ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) φ ¬φ T T T T T T F T F F T F F T F T F T T F F F F T

2.4 The Semantics of Propositional Logic

Worked example 4 (cont.) Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 2: Backwards truth table. Put an F in the main column. Work backwards to show this leads to a contradiction.

P Q R (P → (¬ Q ∧ R)) ∨ P F

50 T3 F1 F2

2.4 The Semantics of Propositional Logic

Worked example 4 (cont.) Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 2: Backwards truth table. Put an F in the main column. Work backwards to show this leads to a contradiction.

P Q R (P → (¬ Q ∧ R)) ∨ P F

φ ψ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) φ ¬φ T T T T T T F T F F T F F T F T F T T F F F F T T3 F2

2.4 The Semantics of Propositional Logic

Worked example 4 (cont.) Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 2: Backwards truth table. Put an F in the main column. Work backwards to show this leads to a contradiction.

P Q R (P → (¬ Q ∧ R)) ∨ P F1 F

φ ψ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) φ ¬φ T T T T T T F T F F T F F T F T F T T F F F F T T3

2.4 The Semantics of Propositional Logic

Worked example 4 (cont.) Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 2: Backwards truth table. Put an F in the main column. Work backwards to show this leads to a contradiction.

P Q R (P → (¬ Q ∧ R)) ∨ P F1 F F2

φ ψ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) φ ¬φ T T T T T T F T F F T F F T F T F T T F F F F T T3

2.4 The Semantics of Propositional Logic

Worked example 4 (cont.) Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 2: Backwards truth table. Put an F in the main column. Work backwards to show this leads to a contradiction.

P Q R (P → (¬ Q ∧ R)) ∨ P F1 F F2

φ ψ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) φ ¬φ T T T T T T F T F F T F F T F T F T T F F F F T 2.4 The Semantics of Propositional Logic

Worked example 4 (cont.) Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 2: Backwards truth table. Put an F in the main column. Work backwards to show this leads to a contradiction.

P Q R (P → (¬ Q ∧ R)) ∨ P T3 F1 F F2

φ ψ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) φ ¬φ T T T T T T F T F F T F F T F T F T T F F F F T 2.4 The Semantics of Propositional Logic

Worked example 4 (cont.) Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 2: Backwards truth table. Put an F in the main column. Work backwards to show this leads to a contradiction.

P Q R (P → (¬ Q ∧ R)) ∨ P T3 F1 F F2

φ ψ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) φ ¬φ T T T T T T F T F F T F F T F T F T T F F F F T 2.4 The Semantics of Propositional Logic

Worked example 4 (cont.) Show that the sentence (P → (¬Q ∧ R)) ∨ P is a tautology.

Method 2: Backwards truth table. Put an F in the main column. Work backwards to show this leads to a contradiction.

P Q R (P → (¬ Q ∧ R)) ∨ P ? F1 F F2

φ ψ (φ ∧ ψ) (φ ∨ ψ) (φ → ψ) φ ¬φ T T T T T T F T F F T F F T F T F T T F F F F T Method 1: Full truth table Write out the full truth table. Check there’s no row in which the main column of the premiss is T and the main column of the conclusion is F.

2.4 The Semantics of Propositional Logic Worked example 5

Example Show that P ↔ ¬Q ¬(P ↔ Q) 2.4 The Semantics of Propositional Logic Worked example 5

Example Show that P ↔ ¬Q ¬(P ↔ Q)

Method 1: Full truth table Write out the full truth table. Check there’s no row in which the main column of the premiss is T and the main column of the conclusion is F. T4 T T5 ? F T2 T1 T3 F4 T F5 ? F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T

Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q)

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) T4 T T5 ? F T2 T1 T3 F4 T F5 ? F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T

P Q P ↔ ¬ Q ¬ (P ↔ Q)

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X T4 T T5 ? F T2 T1 T3 F4 T F5 ? F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T4 T T5 ? F T2 T1 T3 F4 T F5 ? F F2 T1 F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q)

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? F T2 T1 T3 F4 T F5 ? F F2 T1 F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? T2 T1 T3 F4 T F5 ? F F2 T1 F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T F

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? T2 T1 T3 F4 T F5 ? F F2 T1 F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T F

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? T2 T3 F4 T F5 ? F F2 T1 F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T F T1

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? T2 T3 F4 T F5 ? F F2 T1 F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T F T1

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? T2 T3 F4 F5 ? F2 F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T F T1 T F T1

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? T3 F4 F5 ? F2 F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T F T2 T1 T F T1

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? F4 F5 ? F2 F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T F T2 T1 T3 T F T1

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? F4 F5 ? F3

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T F T2 T1 T3 T F F2 T1

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T4 T5 ? F4 F5 ?

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T F T2 T1 T3 T F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T T5 ? F4 F5 ?

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T4 T F T2 T1 T3 T F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T ? F4 F5 ?

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T4 T T5 F T2 T1 T3 T F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T F4 F5 ?

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T4 T T5 ? F T2 T1 T3 T F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T F5 ?

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T4 T T5 ? F T2 T1 T3 F4 T F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T ?

2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T4 T T5 ? F T2 T1 T3 F4 T F5 F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T 2.4 The Semantics of Propositional Logic

Worked example 5 (cont.) Show that P ↔ ¬Q ¬(P ↔ Q) Method 2: Backwards truth table Put a T in the main column of the premiss and an F in the main column of the conclusion.

Work backwards to obtain a contradiction. X

P Q P ↔ ¬ Q ¬ (P ↔ Q) T4 T T5 ? F T2 T1 T3 F4 T F5 ? F F2 T1 F3

φ ψ (φ ↔ ψ) φ ¬φ T T T T F T F F F T F T F F F T http://logicmanual.philosophy.ox.ac.uk