1. Sentences can be combined using connectives.
1.1. and, or, if, then
1.2. The basic connectives
2. The truth conditions of complex propositions
2.1. If we know the truth values of simple propositions, we can use these to get the truth-values of more complex propositions.
2.2. Truth
2.2.1. Truth conditions: What would the world have to be like for the sentence to be true?
2.2.2. Truth : Correspondence to facts
2.2.3. Truth value: Whether a sentence is true or false
2.2.3.1. If it is raining 'It's raining' has the truth value of true.
2.2.3.2. If it is NOT raining 'It's raining' has the truth value of false.
2.2.4. Truth Conditions: What the world would have to be like to make a sentence true or false.
2.3. Truth tables give you all the possible combinations of truth-values which can be assigned to a pair of propositions.
3. Capturing sentential relations using truth tables.
3.1. Entailment
3.1.1. A sentence expressing a proposition X entails a sentence expressing proposition Y is the truth of Y follows necessarily from the truth of X
3.1.2. A sentence p entails a sentence q when the truth of p guarantees the truth of q. The falsity of q guarantees the falsity of p
3.1.3. John saw and eagle (p) John saw a bird (q)
3.1.3.1. If john actually saw an eagle (p is true) then q must be true as eagles are birds. So p entails q
3.1.3.2. John saw something that wasn't an eagle (p is false) then q could be true or false. So p may entail q or p may not entail q.
3.1.3.3. John didn't see a bird (q is false) then john saw an eagle (p is false), if john didn't see a bird then he didn't see an eagle. So q doesn't entail p.
3.1.3.4. John did see a bird ( q is true) then john may have seen an eagle or he may not have, p is true or false. q may entail p or p may not entail q
3.2. Contradiction
3.2.1. Impossible for both propositions to be true.
3.2.1.1. James is married contradicts James is single
3.2.2. If a sentence entails the negation of another sentence, the sentence contradict each other.
3.3. Paraphrases
3.3.1. Paraphrase: Sentences with mutual/2-way entailments – express the same proposition
3.3.2. No one is irresponsible and Everyone is responsible , ENTAIL each other.
3.4. Presupposition
3.4.1. A presupposition of a sentence is information that is assumed to be true
3.4.2. Jane’s boyfriend is a journalist PRESUPPOSES Jane has a boyfriend
4. Logical connectives, truth tables
4.1. Inclusive disjunction
4.1.1. (∨)
4.1.2. (You can have coffee) or (you can have tea) p q
4.1.2.1. You can have either coffee or tea, or both.
4.1.3. p v q
4.1.4. The truth value of pvq depends on the value of the individual truth conditions for p and q.
4.1.4.1. You can actually have coffee (p is true) and you can actually have tea (q is true) THEN pvq is TRUE
4.1.4.2. You can actually have coffee (p is true) but there's no tea left (q is false) then pvq is TRUE
4.1.4.3. There is no coffee left (p is false) and there is no tea left (q is false) the pvq is FALSE`
4.1.4.4. There's no coffee left (p is false) but you can actually have tea (q is true) then pvq is TRUE
4.2. Conjunction
4.2.1. (∧)
4.2.2. (The library is open) AND (the Gulbenkian is closed). p q
4.2.3. p∧q
4.2.3.1. The truth value of p∧q depends on the value of the individual truth conditions for p and q.
4.2.3.1.1. If the library is actually open (p is true) and the Gulbenkian is actually closed (q is true) then p∧q is TRUE
4.2.3.1.2. If the library is actually closed (p is false) and the Gulbenkian is actually closed (q is true) then p∧q is FALSE
4.2.3.1.3. If the library is actually open (p is true) and the Gulbenkian is actually open (q is false) then p∧q is FALSE
4.2.3.1.4. If the library is actually closed (p is false) and the Gulbenkian is actually open (q is false) the p∧q is FALSE
4.3. Biconditional
4.3.1. (if and only if...then) (↔)
4.3.2. p↔q is equivalent to (p→q)∧(q→p)
4.3.3. (You will get a 1st) iff. (you achieve above 69%) p q
4.3.4. The truth value of p↔q depends on the value of the individual truth conditions for p and q.
4.3.4.1. You get a first ( p is true) and you get over 69% (q is true) then p↔q is TRUE
4.3.4.2. You get a first (p is true) but you didn't get above 69% (q is false) then p↔q is FALSE
4.3.4.3. You get a 2:1 (p is false) but you get over 69% (q s true) then p↔q is FALSE
4.3.4.4. You get a 2:1 (p is false) and you get lower than 69% (q is true) then p↔q is TRUE
4.4. Exclusive disjunction
4.4.1. Ve
4.4.2. (Sam's in his room)OR (he is in Starbucks ) p q
4.4.2.1. Sam's room is not in Starbucks, he is in one or the other, not both.
4.4.3. p Ve q
4.4.4. The truth value of pVeq depends on the value of the individual truth conditions for p and q.
4.4.4.1. Sam is actually in his room (p is true) and Sam is in Starbucks (q is true) the p Ve q is FALSE
4.4.4.2. Sam is in his room (p is true) and he is not in Starbucks (q is false) then pVeq is TRUE
4.4.4.3. Sam is not in his office (p is false) but Sam is in Starbucks (q is true) then pVeq is TRUE
4.4.4.4. Sam is not in his office (p is false) Sam is also not in Starbucks (q is false) then pVeq is FALSE.
4.5. Material implication
4.5.1. (if...then) (→)
4.5.2. If (the library is open) then (Maria is in the library) p q
4.5.2.1. Maria is always in the library when it is open
4.5.3. The truth value of p→q depends on the value of the individual truth conditions for p and q.
4.5.3.1. The library is actually open (p is true) and Maria is actually there (q is true) p→q is TRUE
4.5.3.2. The library is actually close (p is false) but Maria is there ( q is true) then p→q is TRUE
4.5.3.3. The library is actually open (p is true) but Maria isn't there (q is false) then p→q is FALSE
4.5.3.4. The library is actually closed (p is false) and Maria is at home (q is false) then p→q is TRUE
4.6. Negation
4.6.1. Negation (¬) reverses the truth value of a sentence:
4.6.2. The library is open = p
4.6.3. The library is not open = ¬p