Different systems of formal logic are based on which kinds of words we consider are topic neutral: eg. PL, ML etc.
However, there are some topic neutral words that are entirely non-controversially so, ie. The ones of PL.
A
-A
so B
→ deductively valid
→ logically valid
(because we cannot imagine a situation in which the premises are true at the same and the conclusion is false).
ie. inconsistent arguments.
”a fortiori” = further, or there is an even stronger argument given x argument
A
so B or -B
(the opposite kind of inconsistent argument that is still valid)
(Denying these as properly valid is a kind of logic known as relevance logic)
The kind of logic we are doing accepts these as simply limit-cases that do not have any actual application and so are in a sense vacuous.
A or B
- A
→ B
- (A and B)
A
→ -B
And connects two expressions of the same grammatical category
eg. nouns, verbs, adverbs, adjectives
It can also connect two sentences that do not need to have anything to do with one another.
”A and B” can be anything and it will make sense.
Thus, it is a propositional connective in logic. It is also a binary connective, it connects two propositions and no more.
The inputs are propositions and there can only be two: ie. A conjunction.
If A is true and B is true, then A and B is true as well
(if both conjuncts are true, then the whole conjunction is true, in all other cases it is false)
A B
T F - F
T T - T
F F - F
F T - F
A ^ B
this conjunction can also mean ”but”, but ”but” should mean ’and’ + contrast (we do not see contrast due to truth conditional meaning).
Lorenz trying to explain how a logical conjunct is not the same as saying ”and”
Temporal succession:
Eve married Adam and she became pregnant
or, Eve became pregnant and she married Adam
→ these sentences are the same, ie. A ^ B
Simultaneity:
Jean-Paul is reading the newspaper and smoking a cigarette.
→ This would just be A
Causality:
eg. The vase fell on the ground and it broke into pieces
→ this would be more like A>B
”Or” also connects sentences, propositions
AvB always makes sense
v is a binary propositional connective, however, it disjuncts the two parts, the disjuncts.
If at least one of A and B is true, then A or B is true as well
otherwise, when A and B are both false, then A or B is false. (Inclusive disjunction)
A B
T F - T
T T - T
F F - F
F T - T
The exclusive disjunction means A or B, but not both.
A B
T F - T
T T - F
F F - F
F T - T
We may use ”not” to deny or negate a sentence.
It is shown using the ”-” or ”~” or ”¬”
Be careful, how do we deny sentences like ”some students are smart”
the negation is still ”no students are smart” whilst ”some students are not smart” means something different.
(Some meaning ”at least 1”)
Negation is a unary connective. It applies to only one thing.
Not always flips all truth values.
If A is true, its negation is false; if A is false, its negation is true.
(We use double-negations in this logic)
A = ~~A