Introduction to Logic Lecture 12 An introduction to Sentence Logic By David Kelsey

Sentence Logic Also called truth functional logic or propositional logic. Has as its basic unit the sentence. It gives us rules for making justified inferences about sentences. Thus, we might have 2 sentences, and there might be a rule which tells us that from these we can infer a third sentence. For example, given the sentences ‘if it rains, it pours’, and ‘it rains’ we can infer via the rule modus ponens the sentence ‘it pours’.

Simple and Compound Claims A compound claim links two or more simple claims together via a truth functional connector. Truth functional connectors include the words ‘and’, ‘or’, ‘if…then’, ‘not’, or any synonym of these words (for instance, but, however…) For every truth functional connector there is a truth functional symbol. For instance, we represent the truth functional connector ‘and’ with the symbol ‘&’. And we symbolize the truth functional connector ‘or’ with the symbol ‘v’.

An example of a compound claim ‘The moon is made of green cheese’ is a simple claim. ‘The moon isn’t made of green cheese’ is a compound claim because it makes use of the truth functional connector ‘not’. ‘If the moon is made of green cheese, then it orbits the Earth’ is a compound claim because it makes use of the ‘if…then’ connector. ‘The moon is made of green cheese and the moon orbits the Earth’ is a compound claim because it makes use of the ‘and’ connector.

Claim variables For any simple claim we assign it a claim variable. A claim variable is just an uppercase letter of the alphabet, which is used to stand for the simple claim. Thus, if we have a compound claim we must use more than one claim variable to symbolize it. To symbolize a compound claim we will use at least two claim variables as well as at least one truth functional symbol.

Claim variables: an example An example using claim variables: –We might assign the claim variable ‘G’ to the claim ‘The moon is made of green cheese.’ –And we might assign the variable ‘E’ to the claim ‘The moon orbits the Earth.’ –And we might use some truth functional connectors to relate these claims: –G and E. (I.e. The moon is made of green cheese and the moon orbits the Earth) –If G, then E. –G or E. –Not G.

Truth functional symbols We use truth functional symbols to stand for our truth functional connectors. –We use the truth functional symbol ‘~’ to stand for the truth functional connector ‘not’ (or any synonym of it). –And we use the symbol ‘v’ to stand for the connector ‘or’ (or any synonym of it.) –We use the symbol ‘&’ to stand for the connector ‘and’ (or any synonym of it.) –And we use the symbol ‘  ’ to stand for the connector ‘if…then’ (or any synonym of it.)

Truth Tables A truth table of some claim gives one a truth functional analysis of that claim. This analysis displays, pictorially, the truth value of the claim and the truth values of its simpler parts. Claims are truth functional: the truth value of any claim is dependent upon the truth values of its parts. Thus, a truth table displays the truth values for all of the parts of a claim. And a truth table displays the truth value of the claim itself. And the truth value of the claim is dependent upon the values the table gives for the claim’s simpler parts.

Claims are truth functional: an example Examples of claims that are truth functional: –The claim that Italy won the world cup and the Steelers won the Super Bowl is true. This is because it is true that Italy won the world cup. And it is true that the Steelers won the Super Bowl. Thus, the truth of the simpler claims entails the truth of the compound claim.

Truth tables A truth table displays all of the possible truth values for a claim and all of its parts. The function of a truth table is to act as a display of all of the possible circumstances that might occur. A claim might be true so we write this. And a claim might be false so we write this. Thus, when constructing a truth table for a claim we must display all of the possible truth values for the claim and all of the possible truth values for its parts. Take the claim ‘Parker went to the store.’ Let’s assign ‘P’ to this claim. The truth table for P looks like this: P T F

Negation Whatever truth value a claim P has, its negation will have the other. Thus, we get this: P ~P TF FT -We first write all of the possible truth values of P. -We write these in the left column. -It is to this column then that we apply the negation rule, as stated above. Be aware that the symbol ‘~’ stands for ‘it is not the case that’. Thus, if by P we mean ‘Parker is at home, Then by ~P we mean ‘It is not the case that Parker is at home’.

Conjunction A conjunction is a compound claim made from 2 simpler claims called ‘conjuncts’. Thus, the form of a conjunction is this: a claim, P, related to another claim, Q, by the word ‘and’ or any synonym of and, I.e. but, while, though… Susan is surfing and Kevin is playing ball is an example of a conjunction. –The 2 conjuncts are 1) Susan is surfing & 2) Kevin is playing ball. –We can now pick a claim variable to stand for each of the two conjuncts. Let us let ‘S’ stand for ‘Susan is surfing,’ and ‘K’ stand for ‘Kevin is playing ball’.

Conjunction: truth table We use the truth functional symbol ‘&’ to stand for the word ‘and’. The truth table for the conjunction looks like this: P Q P&Q T T T T F F F T F F F F

Disjunction A disjunction is a compound claim composed of two simpler claims called ‘disjuncts’. Thus, the form of a disjunction is this: a claim, P, related to another claim, Q, by the word ‘or’ or any of it’s synonyms. We use the truth functional symbol ‘v’ to stand for the word ‘or’. Stacy is studying or she is out on the town is an example. –The 2 disjuncts are 1) Stacy is studying & 2) She is out on the town. –We can now pick claim variables to stand for claims. We can let ‘S’ stand for ‘Stacy is studying’ and we can let ‘O’ stand for ‘She is out on the town’.

Disjunction: truth table The truth table for the disjunction looks like this: P Q PvQ T T T T F T F T T F F F -As you can see, a disjunction is true unless both disjuncts are false.

Conditionals A conditional is a compound claim made from two simpler claims. The form of a conditional is this: if…then… The claim that fills the first blank is the antecedent of the conditional and the claim that fills the second blank is the consequent of the conditional. We will use the truth functional symbol ‘  ’ to stand for if…then…

An example of a conditional claim An example of a conditional claim: –we might state that If Bolton won then Sunderland lost. –The antecedent of the conditional is Bolton won and the consequent is Sunderland lost. –We can let ‘B’ stand for ‘Bolton won’ and ‘S’ stand for ‘Sunderland lost’. –In this case we symbolize our claim as this: B  S.

Conditionals again The truth table for the conditional looks like this: P Q P  Q T T T T F F F T T F F T Why is the conditional claim true in the last 2 cases? –Consider the conditional ‘if I get paid today I will take Jenny to dinner’. –Let’s symbolize our claim as this: P  J –Let us consider this as a promise. I will take Jenny to dinner if I get paid today. –Then the claim is false just in case I break my promise. –But I break my promise if I get paid and yet don’t take Jenny to dinner. –Thus, the conditional is false when its antecedent P is true and yet its consequent J is false.