Lecture 2 The Language of S5

Propositional logic is concerned with the meaning of sentences of the form ‘It is not the case that ’, ‘ and ’, ‘ or ’ and ‘if  then ’, where  and  are sentences. Examples: ‘It is not the case that today is Thursday.’ ‘Today is Friday and grass is green.’ ‘Today is Thursday or today is Friday.’ ‘If today is Friday then yesterday was Thursday.’

x f(x) T F x y f(x, y) T F x y f(x, y) T F x y f(x, y) T F We take the meaning of a sentence to be a truth-value. And we take the meaning of the expressions ‘It is not the case that’, ‘and’, ‘or’ and ‘if … then …’ to be functions from truth-values to truth-values: ‘not’ ‘and’ ‘or’ ‘if’ x f(x) T F x y f(x, y) T F x y f(x, y) T F x y f(x, y) T F

The meaning of a complex sentence such as ‘Grass is green and snow is white’ is determined by the meanings of its constituents: ‘Grass is green’, ‘and’, and ‘snow is white’. How? The meaning of ‘Grass is green and snow is white’ is the meaning of ‘and’ (a function from truth-values to truth-values) applied to the meaning of ‘Grass is green’ (a truth-value) and the meaning of ‘snow is white’ (a truth-value). So ‘Grass is green and snow is white’ is true just in case ‘Grass is green’ is true and ‘snow is white’ is true.

Modal logic is concerned with the meaning of sentences of the form ‘It is necessary that ’, and ‘It is possible that ’, where  is a sentence. Examples: ‘It is necessary that 2 + 2 = 4.’ ‘It is possible that it will rain tomorrow.’ Also: ‘Necessarily, 2 + 2 = 4.’ ‘Possibly, it will rain tomorrow.’ ‘2 + 2 is necessarily 4.’ ‘It will possibly rain tomorrow.’ ‘2 + 2 must be 4.’ ‘It might rain tomorrow.’

It will no longer work to take the meaning of a sentence to be a truth-value, and the meanings of ‘It is necessary that’ and ‘It is possible that’ to be functions from truth-values to truth-values. Why not? Because the truth-values of ‘It is necessary that ’ and ‘It is possible that ’ depend on more than just the truth-value of : ‘It is necessary that 2 + 2 = 4.’ ‘It is necessary that grass is green.’

So what do we do? We take the meaning of a sentence to be not a truth-value, but a function from possible worlds to truth-values. Call such a function a proposition. So a sentence is not just true or false, it is true or false at a world, according to whether the proposition that it means is true or false at that world. If we single out one world and call it the actual world, then we can define a sentence to be true or false (simpliciter) according to whether it is true or false at the actual world.

What about the meaning of ‘It is necessary that’ and ‘It is possible that’? We take the meaning of each to be not a function from truth-values to truth-values, but a function from propositions to propositions. Which functions? The meaning of ‘It is necessary that’ is the function f such that if p is a proposition then f(p) is the proposition that is true at a world w just in case p is true at all possible worlds. The meaning of ‘It is possible that’ is the function f such that if p is a proposition then f(p) is the proposition that is true at a world w just in case p is true at some possible world.

The meaning of a complex sentence such as ‘It is necessary that grass is green’ is determined by the meanings of its constituents: ‘It is necessary that’ and ‘grass is green’. How? The meaning of ‘It is necessary that grass is green’ is the meaning of ‘It is necessary that’ (a function from propositions to propositions) applied to the meaning of ‘grass is green’ (a proposition). Which proposition do we get? The proposition that is true at a world w just in case ‘grass is green’ is true at all worlds.

In summary: ‘It is necessary that ’ is true at a world w just in case  is true at all worlds. ‘It is possible that ’ is true at a world w just in case  is true at some world. ‘It is necessary that ’ is true (simpliciter) just in case  is true at all worlds. ‘It is possible that ’ is true (simpliciter) just in case  is true at some world.

Since we are taking the meaning of a sentence to be a proposition, we have to take the meanings of ‘It is not the case that’, ‘and’, ‘or’, and ‘if … then …’ to be functions from propositions to propositions. Which functions? The meaning of ‘and’ is the function f such that if p and q are propositions then f(p, q) is the proposition that is true at a world w just in case p is true at w and q is true at w. This means that the sentence ‘ and ’ is true at a world w just in case  is true at w and  is true at w.

In summary: ‘It is not the case that ’ is true at a world w just in case  is false at w. ‘ and ’ is true at a world w just in case  is true at w and  is true at w. ‘ or ’ is true at a world w just in case  is true at w or  is true at w. ‘If  then ’ is true at a world w just in case  is false at w or  is true at w.

We extend the formal language PC to the formal language S5. Additional symbols: ‘□’ (for ‘It is necessary that’) ‘’ (for ‘It is possible that’) Additional wffs: If  is a wff then so is ‘□’ and ‘’

We interpret the language by specifying a set of possible worlds, and assigning a proposition to each sentence letter (that is, a truth-value at each world). The meaning of every complex wff is then giving by the following rules: ‘It is not the case that ’ is true at a world w just in case  is false at w. ‘ and ’ is true at a world w just in case  is true at w and  is true at w. ‘ or ’ is true at a world w just in case  is true at w or  is true at w. ‘If  then ’ is true at a world w just in case  is false at w or  is true at w. ‘It is necessary that ’ is true at a world w just in case  is true at every possible world. ‘It is possible that ’ is true at a world w just in case  is true at some possible world.

If  is a set of wffs (possibly empty) and  is wff, say that ‘╞ ’ is a correct semantic sequent just in case there is no interpretation on which every wff in  is true (simpliciter) but  is false (simpliciter). Examples of correct semantic sequents: A╞ A A╞ A A╞ A A╞ A A╞ A [A  B], A╞ B ╞ [A  A] ╞ [A  [A  B]] ╞ [A  A]

Examples of incorrect semantic sequents: A╞ A A╞ A [A & B]╞ [A & B] A, [A  B]╞ B

We also extend the deductive system of PC by adding new rules of deduction to S5: Def  From  deduce  (and vice-versa) From  deduce  (and vv) RN From [] deduce  ( not in the scope of an assumption in […]) RK From  and […] deduce […] ( not in the scope of an assumption) RT From  deduce  RB From  and […] deduce […] R4 From  and […] deduce […] R5 From  and […] deduce […]

If  is a set of wffs (possibly empty) and  is wff, say that ‘├ ’ is a correct syntactic sequent just in case  can be deduced from  using the rules of deduction of the system. Examples of correct syntactic sequents: A├ A A├ A A├ A A├ A A├ A [A  B], A├ B ├ [A  A] ├ [A  [A  B]] ├ [A  A]